Subgroups — C₆₇×#21507 #30700758

Order 17152.

1124 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#463×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#442×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#440×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#124×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1983	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1949	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1927	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1925	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1879	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1877	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#1804	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#1767	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#1763	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#1762	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₆₇	—	2	1	no	—
25	#463×C₆₇	C₄	1	8	yes	complemented
26	#442×C₆₇	—	2	4	no	—
27	#440×C₆₇	—	2	4	no	—
28	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
42	#391×C₆₇	C₂²	1	1	yes	characteristic
43	#390×C₆₇	C₂²	1	1	yes	characteristic
44	#388×C₆₇	—	2	1	no	—
45	#388×C₆₇	—	2	1	no	—
46	#382×C₆₇	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
49	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
51	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
52	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
53	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
54	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
55	C₄³×C₆₇	C₂²	1	1	yes	characteristic
56	#340×C₆₇	C₂²	1	1	yes	characteristic
57	#340×C₆₇	C₂²	1	1	yes	characteristic
58	#340×C₆₇	C₂²	1	1	yes	characteristic
59	#340×C₆₇	C₂²	1	1	yes	characteristic
60	#336×C₆₇	C₂²	1	1	yes	characteristic
61	#336×C₆₇	C₂²	1	1	yes	characteristic
62	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	—	2	1	no	—
66	C₂³×C₄×C₆₇	—	2	1	no	—
67	C₂³×C₄×C₆₇	—	2	1	no	—
68	C₂³×C₄×C₆₇	—	2	1	no	—
69	C₂×SD₁₆×C₆₇	—	2	2	no	—
70	C₂×SD₁₆×C₆₇	—	2	2	no	—
71	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
72	C₂×D₈×C₆₇	—	2	2	no	—
73	C₂×D₈×C₆₇	—	2	2	no	—
74	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
79	#124×C₆₇	—	2	2	no	—
80	#124×C₆₇	C₂×C₄	1	4	yes	complemented
81	C₄×Q₈×C₆₇	—	2	2	no	—
82	C₄×Q₈×C₆₇	—	2	2	no	—
83	C₄×D₄×C₆₇	—	2	2	no	—
84	C₄×D₄×C₆₇	—	2	2	no	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	—	2	2	no	—
87	C₄×D₄×C₆₇	—	2	2	no	—
88	C₄×D₄×C₆₇	—	2	2	no	—
89	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
90	C₄×D₄×C₆₇	D₄	1	4	yes	—
91	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
105	C₂×C₄²×C₆₇	—	2	1	no	—
106	C₂×C₄²×C₆₇	—	2	1	no	—
107	C₂×C₄²×C₆₇	—	2	1	no	—
108	C₂×C₄²×C₆₇	—	2	1	no	—
109	C₂×C₄²×C₆₇	—	4	4	no	—
110	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
111	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
117	#107×C₆₇	—	2	2	no	—
118	#106×C₆₇	—	2	2	no	—
119	C₄⋊C₈×C₆₇	—	2	2	no	—
120	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	complemented
121	#103×C₆₇	—	2	2	no	—
122	#103×C₆₇	—	2	2	no	—
123	#103×C₆₇	C₂×C₄	1	4	yes	complemented
124	#102×C₆₇	—	2	2	no	—
125	#102×C₆₇	—	2	2	no	—
126	#102×C₆₇	—	2	2	no	—
127	#102×C₆₇	—	2	2	no	—
128	#102×C₆₇	C₂×C₄	1	4	yes	complemented
129	#97×C₆₇	—	2	2	no	—
130	#97×C₆₇	—	2	2	no	—
131	#95×C₆₇	—	2	1	no	—
132	#95×C₆₇	—	2	1	no	—
133	#95×C₆₇	—	2	1	no	—
134	#95×C₆₇	—	2	1	no	—
135	#95×C₆₇	—	2	1	no	—
136	#95×C₆₇	—	2	1	no	—
137	#95×C₆₇	—	2	1	no	—
138	C₂⁴×C₆₇	—	2	1	no	—
139	C₂⁴×C₆₇	—	2	1	no	—
140	C₂×Q₈×C₆₇	—	2	1	no	—
141	C₂×Q₈×C₆₇	—	2	1	no	—
142	C₂×Q₈×C₆₇	—	2	1	no	—
143	C₂×Q₈×C₆₇	—	2	1	no	—
144	C₂×Q₈×C₆₇	—	2	1	no	—
145	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂×D₄×C₆₇	—	2	1	no	—
147	C₂×D₄×C₆₇	—	2	1	no	—
148	C₂×D₄×C₆₇	—	2	1	no	—
149	C₂×D₄×C₆₇	—	2	1	no	—
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	1	no	—
154	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
155	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
156	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
157	C₂×D₄×C₆₇	#41	1	2	yes	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	1	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	1	no	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	1	no	—
166	C₂²×C₄×C₆₇	—	2	1	no	—
167	C₂²×C₄×C₆₇	—	4	2	no	—
168	C₂²×C₄×C₆₇	—	4	2	no	—
169	C₂²×C₄×C₆₇	—	4	2	no	—
170	C₂²×C₄×C₆₇	—	4	2	no	—
171	C₂²×C₄×C₆₇	—	4	2	no	—
172	C₂²×C₄×C₆₇	—	4	2	no	—
173	C₂²×C₄×C₆₇	—	4	2	no	—
174	C₂²×C₄×C₆₇	—	4	2	no	—
175	C₂²×C₄×C₆₇	—	4	2	no	—
176	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	SD₁₆×C₆₇	—	2	8	no	—
184	D₈×C₆₇	—	2	8	no	—
185	C₂×C₈×C₆₇	—	2	1	no	—
186	C₂×C₈×C₆₇	—	2	1	no	—
187	C₂×C₈×C₆₇	—	2	1	no	—
188	C₂×C₈×C₆₇	—	2	1	no	—
189	C₂×C₈×C₆₇	—	2	1	no	—
190	C₂×C₈×C₆₇	—	2	1	no	—
191	C₂×C₈×C₆₇	—	2	1	no	—
192	C₂×C₈×C₆₇	—	2	1	no	—
193	C₂×C₈×C₆₇	—	2	1	no	—
194	C₂×C₈×C₆₇	—	2	1	no	—
195	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
196	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
197	C₄⋊C₄×C₆₇	—	2	1	no	—
198	C₄⋊C₄×C₆₇	—	2	1	no	—
199	C₄⋊C₄×C₆₇	—	2	1	no	—
200	C₄⋊C₄×C₆₇	—	2	1	no	—
201	C₄⋊C₄×C₆₇	—	2	1	no	—
202	C₄⋊C₄×C₆₇	—	2	1	no	—
203	C₄⋊C₄×C₆₇	—	2	1	no	—
204	C₄⋊C₄×C₆₇	—	4	2	no	—
205	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
206	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
207	C₄⋊C₄×C₆₇	#41	1	2	yes	—
208	C₂²⋊C₄×C₆₇	—	2	4	no	—
209	C₂²⋊C₄×C₆₇	—	2	4	no	—
210	C₂²⋊C₄×C₆₇	—	2	4	no	—
211	C₂²⋊C₄×C₆₇	—	2	4	no	—
212	C₂²⋊C₄×C₆₇	—	2	4	no	—
213	C₂²⋊C₄×C₆₇	—	4	2	no	—
214	C₂²⋊C₄×C₆₇	—	4	2	no	—
215	C₂²⋊C₄×C₆₇	—	4	2	no	—
216	C₄²×C₆₇	—	2	2	no	—
217	C₄²×C₆₇	—	2	1	no	—
218	C₄²×C₆₇	—	2	1	no	—
219	C₄²×C₆₇	—	4	1	no	—
220	C₄²×C₆₇	—	4	1	no	—
221	C₄²×C₆₇	—	4	1	no	—
222	C₄²×C₆₇	—	4	2	no	—
223	C₄²×C₆₇	—	4	2	no	—
224	C₄²×C₆₇	—	4	2	no	—
225	C₄²×C₆₇	—	4	2	no	—
226	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
227	C₄²×C₆₇	C₂×D₄	1	2	yes	—
228	C₄²×C₆₇	#41	1	2	yes	—
229	C₄²×C₆₇	#41	1	2	yes	—
230	C₂³×C₆₇	—	2	2	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	2	2	no	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	4	1	no	—
235	C₂³×C₆₇	—	4	1	no	—
236	C₂³×C₆₇	—	4	1	no	—
237	C₂³×C₆₇	—	4	1	no	—
238	C₂³×C₆₇	—	4	1	no	—
239	C₂³×C₆₇	—	4	1	no	—
240	C₂³×C₆₇	—	4	1	no	—
241	C₂³×C₆₇	—	4	1	no	—
242	C₂³×C₆₇	—	4	1	no	—
243	C₂³×C₆₇	—	4	1	no	—
244	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
245	Q₈×C₆₇	—	2	2	no	—
246	Q₈×C₆₇	—	2	2	no	—
247	Q₈×C₆₇	—	4	1	no	—
248	Q₈×C₆₇	—	4	1	no	—
249	D₄×C₆₇	—	2	2	no	—
250	D₄×C₆₇	—	2	2	no	—
251	D₄×C₆₇	—	2	2	no	—
252	D₄×C₆₇	—	2	2	no	—
253	D₄×C₆₇	—	2	2	no	—
254	D₄×C₆₇	—	4	1	no	—
255	D₄×C₆₇	—	4	1	no	—
256	D₄×C₆₇	C₄×D₄	1	4	yes	—
257	C₂×C₄×C₆₇	—	2	1	no	—
258	C₂×C₄×C₆₇	—	2	1	no	—
259	C₂×C₄×C₆₇	—	2	1	no	—
260	C₂×C₄×C₆₇	—	2	1	no	—
261	C₂×C₄×C₆₇	—	2	1	no	—
262	C₂×C₄×C₆₇	—	2	1	no	—
263	C₂×C₄×C₆₇	—	2	1	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	2	1	no	—
266	C₂×C₄×C₆₇	—	2	1	no	—
267	C₂×C₄×C₆₇	—	2	2	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	1	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	1	no	—
273	C₂×C₄×C₆₇	—	2	1	no	—
274	C₂×C₄×C₆₇	—	4	8	no	—
275	C₂×C₄×C₆₇	—	4	8	no	—
276	C₂×C₄×C₆₇	—	4	8	no	—
277	C₂×C₄×C₆₇	—	4	1	no	—
278	C₂×C₄×C₆₇	—	4	1	no	—
279	C₂×C₄×C₆₇	—	4	1	no	—
280	C₂×C₄×C₆₇	—	4	1	no	—
281	C₂×C₄×C₆₇	—	4	1	no	—
282	C₂×C₄×C₆₇	—	4	1	no	—
283	C₂×C₄×C₆₇	—	4	1	no	—
284	C₂×C₄×C₆₇	—	4	1	no	—
285	C₂×C₄×C₆₇	—	4	1	no	—
286	C₂×C₄×C₆₇	—	4	1	no	—
287	C₂×C₄×C₆₇	—	4	1	no	—
288	C₂×C₄×C₆₇	—	4	1	no	—
289	C₂×C₄×C₆₇	—	4	1	no	—
290	C₂×C₄×C₆₇	—	4	1	no	—
291	C₂×C₄×C₆₇	—	8	4	no	—
292	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
293	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
294	C₂×C₄×C₆₇	#121	1	2	yes	—
295	C₂×C₄×C₆₇	#121	1	2	yes	—
296	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
297	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
298	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
300	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
301	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
302	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
303	C₈×C₆₇	—	2	2	no	—
304	C₈×C₆₇	—	2	2	no	—
305	C₈×C₆₇	—	4	1	no	—
306	C₈×C₆₇	—	4	1	no	—
307	C₂²×C₆₇	—	2	4	no	—
308	C₂²×C₆₇	—	4	2	no	—
309	C₂²×C₆₇	—	4	2	no	—
310	C₂²×C₆₇	—	4	2	no	—
311	C₂²×C₆₇	—	4	2	no	—
312	C₂²×C₆₇	—	4	2	no	—
313	C₂²×C₆₇	—	4	2	no	—
314	C₂²×C₆₇	—	4	2	no	—
315	C₂²×C₆₇	—	4	2	no	—
316	C₂²×C₆₇	—	4	2	no	—
317	C₂²×C₆₇	—	8	1	no	—
318	C₂²×C₆₇	—	8	1	no	—
319	C₂²×C₆₇	—	8	1	no	—
320	C₂²×C₆₇	—	8	1	no	—
321	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
322	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
323	C₂²×C₆₇	#525	1	1	yes	characteristic
324	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
325	C₂²×C₆₇	#575	1	1	yes	characteristic
326	C₂²×C₆₇	#575	1	1	yes	characteristic
327	C₂²×C₆₇	#575	1	1	yes	characteristic
328	C₄×C₆₇	—	2	2	no	—
329	C₄×C₆₇	—	2	2	no	—
330	C₄×C₆₇	—	2	2	no	—
331	C₄×C₆₇	—	2	2	no	—
332	C₄×C₆₇	—	2	2	no	—
333	C₄×C₆₇	—	4	1	no	—
334	C₄×C₆₇	—	4	2	no	—
335	C₄×C₆₇	—	4	2	no	—
336	C₄×C₆₇	—	4	2	no	—
337	C₄×C₆₇	—	4	2	no	—
338	C₄×C₆₇	—	8	1	no	—
339	C₄×C₆₇	—	8	1	no	—
340	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
341	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
342	C₄×C₆₇	#529	1	1	yes	characteristic
343	C₄×C₆₇	#532	1	1	yes	characteristic
344	#21507	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
345	C₂×C₆₇	—	4	4	no	—
346	C₂×C₆₇	—	8	2	no	—
347	C₂×C₆₇	C₄×#121	1	1	yes	characteristic
348	C₂×C₆₇	#2943	1	1	yes	characteristic
349	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
350	C₂×C₆₇	#3036	1	1	yes	characteristic
351	C₂×C₆₇	#3040	1	1	yes	characteristic
352	C₂×C₆₇	#3111	1	1	yes	characteristic
353	C₂×C₆₇	#3114	1	1	yes	characteristic
354	C₂×#463	C₂×C₆₇	1	1	yes	characteristic
355	C₂×#442	C₂×C₆₇	1	1	yes	characteristic
356	C₂×#440	C₂×C₆₇	1	1	yes	characteristic, complemented
357	C₄×#124	C₂×C₆₇	1	1	yes	characteristic, complemented
358	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
359	#1983	C₂×C₆₇	1	1	yes	characteristic, complemented
360	#1949	C₂×C₆₇	1	1	yes	characteristic, complemented
361	#1927	C₂×C₆₇	1	1	yes	characteristic, complemented
362	#1925	C₂×C₆₇	1	1	yes	characteristic, complemented
363	#1879	C₂×C₆₇	1	1	yes	characteristic
364	#1877	C₂×C₆₇	1	1	yes	characteristic, complemented
365	#1804	C₂×C₆₇	1	1	yes	characteristic, complemented
366	#1767	C₂×C₆₇	1	1	yes	characteristic, complemented
367	#1763	C₂×C₆₇	1	1	yes	characteristic, complemented
368	#1762	C₂×C₆₇	1	1	yes	characteristic, complemented
369	C₆₇	#21507	1	1	yes	characteristic, direct product factor, Sylow
370	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
371	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
372	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
373	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
374	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
375	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
376	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
377	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
378	C₂²×C₄²	—	2	1	no	—
379	#463	C₄×C₆₇	1	8	yes	complemented
380	#442	—	2	4	no	—
381	#440	—	2	4	no	—
382	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
383	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
384	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
385	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
386	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
387	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
388	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
389	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
390	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
391	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
392	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
393	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
394	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
395	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
396	#391	C₂²×C₆₇	1	1	yes	characteristic
397	#390	C₂²×C₆₇	1	1	yes	characteristic
398	#388	—	2	1	no	—
399	#388	—	2	1	no	—
400	#382	C₂²×C₆₇	1	1	yes	characteristic
401	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
402	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
403	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
404	C₄×C₂²⋊C₄	—	2	1	no	—
405	C₄×C₂²⋊C₄	—	2	1	no	—
406	C₄×C₂²⋊C₄	—	2	1	no	—
407	C₄×C₂²⋊C₄	—	2	1	no	—
408	C₄×C₂²⋊C₄	—	2	1	no	—
409	C₄³	C₂²×C₆₇	1	1	yes	characteristic
410	#340	C₂²×C₆₇	1	1	yes	characteristic
411	#340	C₂²×C₆₇	1	1	yes	characteristic
412	#340	C₂²×C₆₇	1	1	yes	characteristic
413	#340	C₂²×C₆₇	1	1	yes	characteristic
414	#336	C₂²×C₆₇	1	1	yes	characteristic
415	#336	C₂²×C₆₇	1	1	yes	characteristic
416	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
417	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
418	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
419	C₂³×C₄	—	2	1	no	—
420	C₂³×C₄	—	2	1	no	—
421	C₂³×C₄	—	2	1	no	—
422	C₂³×C₄	—	2	1	no	—
423	C₂×SD₁₆	—	2	2	no	—
424	C₂×SD₁₆	—	2	2	no	—
425	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
426	C₂×D₈	—	2	2	no	—
427	C₂×D₈	—	2	2	no	—
428	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
429	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
430	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
432	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
433	#124	—	2	2	no	—
434	#124	C₂×C₄×C₆₇	1	4	yes	complemented
435	C₄×Q₈	—	2	2	no	—
436	C₄×Q₈	—	2	2	no	—
437	C₄×D₄	—	2	2	no	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	2	no	—
440	C₄×D₄	—	2	2	no	—
441	C₄×D₄	—	2	2	no	—
442	C₄×D₄	—	2	2	no	—
443	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
444	C₄×D₄	D₄×C₆₇	1	4	yes	—
445	C₂×C₄⋊C₄	—	2	1	no	—
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×C₂²⋊C₄	—	2	1	no	—
452	C₂×C₂²⋊C₄	—	2	1	no	—
453	C₂×C₂²⋊C₄	—	2	1	no	—
454	C₂×C₂²⋊C₄	—	2	1	no	—
455	C₂×C₂²⋊C₄	—	2	1	no	—
456	C₂×C₂²⋊C₄	—	2	1	no	—
457	C₂×C₂²⋊C₄	—	2	1	no	—
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₄²	—	2	1	no	—
460	C₂×C₄²	—	2	1	no	—
461	C₂×C₄²	—	2	1	no	—
462	C₂×C₄²	—	2	1	no	—
463	C₂×C₄²	—	4	4	no	—
464	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
465	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
466	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
467	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
468	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
469	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
470	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
471	#107	—	2	2	no	—
472	#106	—	2	2	no	—
473	C₄⋊C₈	—	2	2	no	—
474	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	complemented
475	#103	—	2	2	no	—
476	#103	—	2	2	no	—
477	#103	C₂×C₄×C₆₇	1	4	yes	complemented
478	#102	—	2	2	no	—
479	#102	—	2	2	no	—
480	#102	—	2	2	no	—
481	#102	—	2	2	no	—
482	#102	C₂×C₄×C₆₇	1	4	yes	complemented
483	#97	—	2	2	no	—
484	#97	—	2	2	no	—
485	#95	—	2	1	no	—
486	#95	—	2	1	no	—
487	#95	—	2	1	no	—
488	#95	—	2	1	no	—
489	#95	—	2	1	no	—
490	#95	—	2	1	no	—
491	#95	—	2	1	no	—
492	C₂⁴	—	2	1	no	—
493	C₂⁴	—	2	1	no	—
494	C₂×Q₈	—	2	1	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
509	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
510	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
511	C₂×D₄	#41×C₆₇	1	2	yes	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	4	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	4	2	no	—
522	C₂²×C₄	—	4	2	no	—
523	C₂²×C₄	—	4	2	no	—
524	C₂²×C₄	—	4	2	no	—
525	C₂²×C₄	—	4	2	no	—
526	C₂²×C₄	—	4	2	no	—
527	C₂²×C₄	—	4	2	no	—
528	C₂²×C₄	—	4	2	no	—
529	C₂²×C₄	—	4	2	no	—
530	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
531	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
532	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
533	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
534	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
535	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
536	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
537	SD₁₆	—	2	8	no	—
538	D₈	—	2	8	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
550	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	1	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	1	no	—
558	C₄⋊C₄	—	4	2	no	—
559	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
560	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
561	C₄⋊C₄	#41×C₆₇	1	2	yes	—
562	C₂²⋊C₄	—	2	4	no	—
563	C₂²⋊C₄	—	2	4	no	—
564	C₂²⋊C₄	—	2	4	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	—	2	4	no	—
567	C₂²⋊C₄	—	4	2	no	—
568	C₂²⋊C₄	—	4	2	no	—
569	C₂²⋊C₄	—	4	2	no	—
570	C₄²	—	2	2	no	—
571	C₄²	—	2	1	no	—
572	C₄²	—	2	1	no	—
573	C₄²	—	4	1	no	—
574	C₄²	—	4	1	no	—
575	C₄²	—	4	1	no	—
576	C₄²	—	4	2	no	—
577	C₄²	—	4	2	no	—
578	C₄²	—	4	2	no	—
579	C₄²	—	4	2	no	—
580	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
581	C₄²	C₂×D₄×C₆₇	1	2	yes	—
582	C₄²	#41×C₆₇	1	2	yes	—
583	C₄²	#41×C₆₇	1	2	yes	—
584	C₂³	—	2	2	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	2	no	—
587	C₂³	—	2	2	no	—
588	C₂³	—	4	1	no	—
589	C₂³	—	4	1	no	—
590	C₂³	—	4	1	no	—
591	C₂³	—	4	1	no	—
592	C₂³	—	4	1	no	—
593	C₂³	—	4	1	no	—
594	C₂³	—	4	1	no	—
595	C₂³	—	4	1	no	—
596	C₂³	—	4	1	no	—
597	C₂³	—	4	1	no	—
598	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
599	Q₈	—	2	2	no	—
600	Q₈	—	2	2	no	—
601	Q₈	—	4	1	no	—
602	Q₈	—	4	1	no	—
603	D₄	—	2	2	no	—
604	D₄	—	2	2	no	—
605	D₄	—	2	2	no	—
606	D₄	—	2	2	no	—
607	D₄	—	2	2	no	—
608	D₄	—	4	1	no	—
609	D₄	—	4	1	no	—
610	D₄	C₄×D₄×C₆₇	1	4	yes	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	4	8	no	—
629	C₂×C₄	—	4	8	no	—
630	C₂×C₄	—	4	8	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	8	4	no	—
646	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
647	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
648	C₂×C₄	#121×C₆₇	1	2	yes	—
649	C₂×C₄	#121×C₆₇	1	2	yes	—
650	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
651	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
652	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
653	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
654	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
655	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
656	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
657	C₈	—	2	2	no	—
658	C₈	—	2	2	no	—
659	C₈	—	4	1	no	—
660	C₈	—	4	1	no	—
661	C₂²	—	2	4	no	—
662	C₂²	—	4	2	no	—
663	C₂²	—	4	2	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	—	4	2	no	—
668	C₂²	—	4	2	no	—
669	C₂²	—	4	2	no	—
670	C₂²	—	4	2	no	—
671	C₂²	—	8	1	no	—
672	C₂²	—	8	1	no	—
673	C₂²	—	8	1	no	—
674	C₂²	—	8	1	no	—
675	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
676	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
677	C₂²	#525×C₆₇	1	1	yes	characteristic
678	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
679	C₂²	#575×C₆₇	1	1	yes	characteristic
680	C₂²	#575×C₆₇	1	1	yes	characteristic
681	C₂²	#575×C₆₇	1	1	yes	characteristic
682	C₄	—	2	2	no	—
683	C₄	—	2	2	no	—
684	C₄	—	2	2	no	—
685	C₄	—	2	2	no	—
686	C₄	—	2	2	no	—
687	C₄	—	4	2	no	—
688	C₄	—	4	2	no	—
689	C₄	—	4	1	no	—
690	C₄	—	4	2	no	—
691	C₄	—	4	2	no	—
692	C₄	—	8	1	no	—
693	C₄	—	8	1	no	—
694	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
695	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
696	C₄	#529×C₆₇	1	1	yes	characteristic
697	C₄	#532×C₆₇	1	1	yes	characteristic
698	C₂	—	4	4	no	—
699	C₂	—	8	2	no	—
700	C₂	C₄×#121×C₆₇	1	1	yes	characteristic
701	C₂	C₆₇×#2943	1	1	yes	characteristic
702	C₂	C₄×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
703	C₂	C₆₇×#3036	1	1	yes	characteristic
704	C₂	C₆₇×#3040	1	1	yes	characteristic
705	C₂	C₆₇×#3111	1	1	yes	characteristic
706	C₂	C₆₇×#3114	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×#440×C₆₇	C₂	semidirect
4. C₄×#124×C₆₇	C₂	semidirect
5. C₄²×D₄×C₆₇	C₂	semidirect
6. C₆₇×#1983	C₂	semidirect
7. C₆₇×#1949	C₂	semidirect
8. C₆₇×#1927	C₂	semidirect
9. C₆₇×#1925	C₂	semidirect
11. C₆₇×#1877	C₂	semidirect
12. C₆₇×#1804	C₂	semidirect
13. C₆₇×#1767	C₂	semidirect
14. C₆₇×#1763	C₂	semidirect
15. C₆₇×#1762	C₂	semidirect
25. #463×C₆₇	C₄	semidirect
71. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
80. #124×C₆₇	C₂×C₄	semidirect
89. C₄×D₄×C₆₇	C₂×C₄	semidirect
120. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
123. #103×C₆₇	C₂×C₄	semidirect
128. #102×C₆₇	C₂×C₄	semidirect
344. #21507	C₆₇	direct (N × H)
356. C₂×#440	C₂×C₆₇	semidirect
357. C₄×#124	C₂×C₆₇	semidirect
358. C₄²×D₄	C₂×C₆₇	semidirect
359. #1983	C₂×C₆₇	semidirect
360. #1949	C₂×C₆₇	semidirect
361. #1927	C₂×C₆₇	semidirect
362. #1925	C₂×C₆₇	semidirect
364. #1877	C₂×C₆₇	semidirect
365. #1804	C₂×C₆₇	semidirect
366. #1767	C₂×C₆₇	semidirect
367. #1763	C₂×C₆₇	semidirect
368. #1762	C₂×C₆₇	semidirect
369. C₆₇	#21507	direct (N × H)
379. #463	C₄×C₆₇	semidirect
425. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
434. #124	C₂×C₄×C₆₇	semidirect
443. C₄×D₄	C₂×C₄×C₆₇	semidirect
474. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
477. #103	C₂×C₄×C₆₇	semidirect
482. #102	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#463×C₆₇	2
C₂×#442×C₆₇	2
C₂×#440×C₆₇	2
C₄×#124×C₆₇	2
C₄²×D₄×C₆₇	2
C₆₇×#1983	2
C₆₇×#1949	2
C₆₇×#1927	2
C₆₇×#1925	2
C₆₇×#1879	2
C₆₇×#1877	2
C₆₇×#1804	2
C₆₇×#1767	2
C₆₇×#1763	2
C₆₇×#1762	2
#21507	67

Subgroups — C67×#21507 #30700758

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21507 #30700758