Subgroups — C₆₇×#23982 #30703233

Order 17152.

1424 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#552×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#450×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2408	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2025	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2000	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1894	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1880	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1874	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1870	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1825	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	#552×C₆₇	—	2	4	no	—
16	#552×C₆₇	—	2	4	no	—
17	C₂×#126×C₆₇	—	2	1	no	—
18	C₂×#125×C₆₇	—	2	1	no	—
19	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂×#123×C₆₇	—	2	1	no	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×#121×C₆₇	C₂²	1	2	yes	—
24	C₂×#120×C₆₇	—	2	1	no	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#117×C₆₇	—	2	1	no	—
28	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
30	#450×C₆₇	—	2	4	no	—
31	#418×C₆₇	—	2	4	no	—
32	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#102×C₆₇	C₂²	1	2	yes	—
34	C₂×#102×C₆₇	C₂²	1	2	yes	—
35	C₂×#102×C₆₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
38	#394×C₆₇	—	2	1	no	—
39	#392×C₆₇	—	2	1	no	—
40	#390×C₆₇	C₂²	1	2	yes	—
41	#380×C₆₇	C₂²	1	1	yes	characteristic
42	#379×C₆₇	—	2	1	no	—
43	#340×C₆₇	C₂²	1	2	yes	complemented
44	#340×C₆₇	C₂²	1	2	yes	complemented
45	#340×C₆₇	C₂²	1	2	yes	complemented
46	#336×C₆₇	C₂²	1	2	yes	complemented
47	C₂⁵×C₆₇	—	2	1	no	—
48	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	—	2	4	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	—	2	2	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	—	4	2	no	—
55	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
56	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×SD₁₆×C₆₇	—	2	4	no	—
61	C₂×SD₁₆×C₆₇	—	2	4	no	—
62	C₂×SD₁₆×C₆₇	—	4	2	no	—
63	C₂×M₁₆×C₆₇	—	2	2	no	—
64	C₂×M₁₆×C₆₇	—	4	2	no	—
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
67	#126×C₆₇	—	2	8	no	—
68	#125×C₆₇	—	2	8	no	—
69	#125×C₆₇	—	2	2	no	—
70	#125×C₆₇	—	2	2	no	—
71	#123×C₆₇	—	2	8	no	—
72	C₂²⋊Q₈×C₆₇	—	2	2	no	—
73	C₂²⋊Q₈×C₆₇	—	2	2	no	—
74	C₂²⋊Q₈×C₆₇	—	2	2	no	—
75	C₂²⋊Q₈×C₆₇	D₄	1	4	yes	complemented
76	#121×C₆₇	—	2	4	no	—
77	#121×C₆₇	—	4	2	no	—
78	#120×C₆₇	—	4	4	no	—
79	C₄×Q₈×C₆₇	—	2	2	no	—
80	C₄×Q₈×C₆₇	—	2	2	no	—
81	C₄×D₄×C₆₇	—	2	2	no	—
82	C₄×D₄×C₆₇	—	2	2	no	—
83	#117×C₆₇	—	2	8	no	—
84	#117×C₆₇	—	2	2	no	—
85	#117×C₆₇	D₄	1	4	yes	complemented
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
90	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
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	2	no	—
104	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
108	C₂×C₄²×C₆₇	—	2	1	no	—
109	C₂×C₄²×C₆₇	—	2	1	no	—
110	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
111	#103×C₆₇	—	4	2	no	—
112	#102×C₆₇	—	2	4	no	—
113	#102×C₆₇	—	2	4	no	—
114	#102×C₆₇	—	2	4	no	—
115	C₂²⋊C₈×C₆₇	—	2	2	no	—
116	C₂²⋊C₈×C₆₇	—	4	2	no	—
117	#95×C₆₇	—	2	1	no	—
118	C₂⁴×C₆₇	—	2	2	no	—
119	C₂⁴×C₆₇	—	4	4	no	—
120	C₂⁴×C₆₇	—	4	2	no	—
121	C₂⁴×C₆₇	—	4	2	no	—
122	C₂⁴×C₆₇	—	4	2	no	—
123	C₂⁴×C₆₇	—	8	2	no	—
124	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂×Q₈×C₆₇	—	2	1	no	—
127	C₂×Q₈×C₆₇	—	2	2	no	—
128	C₂×Q₈×C₆₇	—	2	2	no	—
129	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
130	C₂×D₄×C₆₇	—	2	8	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	2	2	no	—
134	C₂×D₄×C₆₇	—	2	2	no	—
135	C₂×D₄×C₆₇	—	2	2	no	—
136	C₂×D₄×C₆₇	—	2	2	no	—
137	C₂×D₄×C₆₇	—	2	2	no	—
138	C₂×D₄×C₆₇	—	4	8	no	—
139	C₂×D₄×C₆₇	—	4	1	no	—
140	C₂×D₄×C₆₇	—	4	4	no	—
141	C₂×D₄×C₆₇	—	4	4	no	—
142	C₂×D₄×C₆₇	—	4	4	no	—
143	C₂×D₄×C₆₇	—	4	4	no	—
144	C₂×D₄×C₆₇	—	4	2	no	—
145	C₂×D₄×C₆₇	—	8	2	no	—
146	C₂×D₄×C₆₇	#41	1	4	yes	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	1	no	—
151	C₂²×C₄×C₆₇	—	2	1	no	—
152	C₂²×C₄×C₆₇	—	2	1	no	—
153	C₂²×C₄×C₆₇	—	2	1	no	—
154	C₂²×C₄×C₆₇	—	2	1	no	—
155	C₂²×C₄×C₆₇	—	2	1	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	4	2	no	—
160	C₂²×C₄×C₆₇	—	4	2	no	—
161	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	SD₁₆×C₆₇	—	4	8	no	—
169	M₁₆×C₆₇	—	4	4	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	—	2	4	no	—
177	C₄⋊C₄×C₆₇	—	2	4	no	—
178	C₄⋊C₄×C₆₇	—	2	4	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	4	no	—
181	C₄⋊C₄×C₆₇	—	2	4	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	1	no	—
184	C₄⋊C₄×C₆₇	—	2	1	no	—
185	C₄⋊C₄×C₆₇	—	2	1	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
190	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
191	C₂²⋊C₄×C₆₇	—	2	4	no	—
192	C₂²⋊C₄×C₆₇	—	2	4	no	—
193	C₂²⋊C₄×C₆₇	—	2	4	no	—
194	C₂²⋊C₄×C₆₇	—	2	2	no	—
195	C₂²⋊C₄×C₆₇	—	2	2	no	—
196	C₂²⋊C₄×C₆₇	—	4	4	no	—
197	C₂²⋊C₄×C₆₇	—	4	4	no	—
198	C₂²⋊C₄×C₆₇	—	4	2	no	—
199	C₂²⋊C₄×C₆₇	—	8	2	no	—
200	C₄²×C₆₇	—	2	4	no	—
201	C₄²×C₆₇	—	2	4	no	—
202	C₄²×C₆₇	—	2	1	no	—
203	C₄²×C₆₇	—	2	1	no	—
204	C₂³×C₆₇	—	2	1	no	—
205	C₂³×C₆₇	—	2	2	no	—
206	C₂³×C₆₇	—	2	2	no	—
207	C₂³×C₆₇	—	2	1	no	—
208	C₂³×C₆₇	—	2	1	no	—
209	C₂³×C₆₇	—	2	1	no	—
210	C₂³×C₆₇	—	2	2	no	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	4	8	no	—
214	C₂³×C₆₇	—	4	2	no	—
215	C₂³×C₆₇	—	4	2	no	—
216	C₂³×C₆₇	—	4	2	no	—
217	C₂³×C₆₇	—	4	2	no	—
218	C₂³×C₆₇	—	4	2	no	—
219	C₂³×C₆₇	—	4	2	no	—
220	C₂³×C₆₇	—	8	4	no	—
221	C₂³×C₆₇	—	8	4	no	—
222	C₂³×C₆₇	—	8	8	no	—
223	C₂³×C₆₇	—	8	8	no	—
224	C₂³×C₆₇	#120	1	2	yes	—
225	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
226	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
227	Q₈×C₆₇	—	2	4	no	—
228	Q₈×C₆₇	—	4	1	no	—
229	Q₈×C₆₇	—	4	1	no	—
230	D₄×C₆₇	—	2	4	no	—
231	D₄×C₆₇	—	2	4	no	—
232	D₄×C₆₇	—	4	8	no	—
233	D₄×C₆₇	—	4	4	no	—
234	D₄×C₆₇	—	4	2	no	—
235	D₄×C₆₇	—	4	2	no	—
236	D₄×C₆₇	—	8	8	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	2	1	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	1	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	1	no	—
250	C₂×C₄×C₆₇	—	2	1	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	4	8	no	—
253	C₂×C₄×C₆₇	—	4	2	no	—
254	C₂×C₄×C₆₇	—	4	1	no	—
255	C₂×C₄×C₆₇	—	4	1	no	—
256	C₂×C₄×C₆₇	—	4	1	no	—
257	C₂×C₄×C₆₇	—	4	1	no	—
258	C₂×C₄×C₆₇	—	4	2	no	—
259	C₂×C₄×C₆₇	—	4	1	no	—
260	C₂×C₄×C₆₇	—	4	1	no	—
261	C₂×C₄×C₆₇	—	4	2	no	—
262	C₂×C₄×C₆₇	—	4	2	no	—
263	C₂×C₄×C₆₇	—	4	2	no	—
264	C₂×C₄×C₆₇	—	4	2	no	—
265	C₂×C₄×C₆₇	#120	1	2	yes	—
266	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
269	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
270	C₈×C₆₇	—	4	2	no	—
271	C₈×C₆₇	—	4	2	no	—
272	C₂²×C₆₇	—	2	2	no	—
273	C₂²×C₆₇	—	2	2	no	—
274	C₂²×C₆₇	—	2	2	no	—
275	C₂²×C₆₇	—	2	2	no	—
276	C₂²×C₆₇	—	2	4	no	—
277	C₂²×C₆₇	—	4	1	no	—
278	C₂²×C₆₇	—	4	1	no	—
279	C₂²×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	—	4	2	no	—
281	C₂²×C₆₇	—	4	2	no	—
282	C₂²×C₆₇	—	4	4	no	—
283	C₂²×C₆₇	—	4	4	no	—
284	C₂²×C₆₇	—	4	4	no	—
285	C₂²×C₆₇	—	8	16	no	—
286	C₂²×C₆₇	—	8	2	no	—
287	C₂²×C₆₇	—	8	2	no	—
288	C₂²×C₆₇	—	8	2	no	—
289	C₂²×C₆₇	—	8	2	no	—
290	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
291	C₂²×C₆₇	#546	1	2	yes	—
292	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₂²×C₆₇	#575	1	1	yes	characteristic
294	C₂²×C₆₇	#577	1	2	yes	—
295	C₄×C₆₇	—	2	2	no	—
296	C₄×C₆₇	—	4	1	no	—
297	C₄×C₆₇	—	4	2	no	—
298	C₄×C₆₇	—	4	1	no	—
299	C₄×C₆₇	—	4	2	no	—
300	C₄×C₆₇	—	4	2	no	—
301	C₄×C₆₇	—	4	1	no	—
302	C₄×C₆₇	—	4	2	no	—
303	C₄×C₆₇	—	8	2	no	—
304	C₄×C₆₇	#525	1	1	yes	characteristic
305	C₄×C₆₇	#535	1	1	yes	characteristic
306	C₄×C₆₇	#546	1	2	yes	—
307	#23982	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂×C₆₇	—	4	2	no	—
309	C₂×C₆₇	—	4	2	no	—
310	C₂×C₆₇	—	8	4	no	—
311	C₂×C₆₇	#2404	1	1	yes	characteristic
312	C₂×C₆₇	#3006	1	1	yes	characteristic
313	C₂×C₆₇	#3010	1	1	yes	characteristic
314	C₂×C₆₇	#3310	1	2	yes	—
315	C₂×C₆₇	#3323	1	2	yes	—
316	C₂×#552	C₂×C₆₇	1	1	yes	characteristic, complemented
317	C₂×#450	C₂×C₆₇	1	1	yes	characteristic, complemented
318	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
319	#2408	C₂×C₆₇	1	1	yes	characteristic
320	#2025	C₂×C₆₇	1	2	yes	complemented
321	#2000	C₂×C₆₇	1	2	yes	complemented
322	#1894	C₂×C₆₇	1	1	yes	characteristic, complemented
323	#1880	C₂×C₆₇	1	2	yes	complemented
324	#1874	C₂×C₆₇	1	2	yes	complemented
325	#1870	C₂×C₆₇	1	1	yes	characteristic, complemented
326	#1825	C₂×C₆₇	1	1	yes	characteristic, complemented
327	C₆₇	#23982	1	1	yes	characteristic, direct product factor, Sylow
328	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
329	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
330	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
331	#552	—	2	4	no	—
332	#552	—	2	4	no	—
333	C₂×#126	—	2	1	no	—
334	C₂×#125	—	2	1	no	—
335	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
336	C₂×#123	—	2	1	no	—
337	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
339	C₂×#121	C₂²×C₆₇	1	2	yes	—
340	C₂×#120	—	2	1	no	—
341	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
342	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
343	C₂×#117	—	2	1	no	—
344	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
345	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
346	#450	—	2	4	no	—
347	#418	—	2	4	no	—
348	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
349	C₂×#102	C₂²×C₆₇	1	2	yes	—
350	C₂×#102	C₂²×C₆₇	1	2	yes	—
351	C₂×#102	C₂²×C₆₇	1	2	yes	—
352	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
353	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
354	#394	—	2	1	no	—
355	#392	—	2	1	no	—
356	#390	C₂²×C₆₇	1	2	yes	—
357	#380	C₂²×C₆₇	1	1	yes	characteristic
358	#379	—	2	1	no	—
359	#340	C₂²×C₆₇	1	2	yes	complemented
360	#340	C₂²×C₆₇	1	2	yes	complemented
361	#340	C₂²×C₆₇	1	2	yes	complemented
362	#336	C₂²×C₆₇	1	2	yes	complemented
363	C₂⁵	—	2	1	no	—
364	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
365	C₂²×D₄	—	2	4	no	—
366	C₂²×D₄	—	2	2	no	—
367	C₂²×D₄	—	2	2	no	—
368	C₂²×D₄	—	2	2	no	—
369	C₂²×D₄	—	2	1	no	—
370	C₂²×D₄	—	4	2	no	—
371	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
372	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
373	C₂³×C₄	—	2	1	no	—
374	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
375	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
376	C₂×SD₁₆	—	2	4	no	—
377	C₂×SD₁₆	—	2	4	no	—
378	C₂×SD₁₆	—	4	2	no	—
379	C₂×M₁₆	—	2	2	no	—
380	C₂×M₁₆	—	4	2	no	—
381	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
382	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
383	#126	—	2	8	no	—
384	#125	—	2	8	no	—
385	#125	—	2	2	no	—
386	#125	—	2	2	no	—
387	#123	—	2	8	no	—
388	C₂²⋊Q₈	—	2	2	no	—
389	C₂²⋊Q₈	—	2	2	no	—
390	C₂²⋊Q₈	—	2	2	no	—
391	C₂²⋊Q₈	D₄×C₆₇	1	4	yes	complemented
392	#121	—	2	4	no	—
393	#121	—	4	2	no	—
394	#120	—	4	4	no	—
395	C₄×Q₈	—	2	2	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	#117	—	2	8	no	—
400	#117	—	2	2	no	—
401	#117	D₄×C₆₇	1	4	yes	complemented
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	—	2	1	no	—
408	C₂×C₄⋊C₄	—	2	2	no	—
409	C₂×C₄⋊C₄	—	2	2	no	—
410	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₂²⋊C₄	—	2	1	no	—
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₂²⋊C₄	—	2	2	no	—
420	C₂×C₂²⋊C₄	—	2	2	no	—
421	C₂×C₂²⋊C₄	—	2	2	no	—
422	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
423	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
424	C₂×C₄²	—	2	1	no	—
425	C₂×C₄²	—	2	1	no	—
426	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
427	#103	—	4	2	no	—
428	#102	—	2	4	no	—
429	#102	—	2	4	no	—
430	#102	—	2	4	no	—
431	C₂²⋊C₈	—	2	2	no	—
432	C₂²⋊C₈	—	4	2	no	—
433	#95	—	2	1	no	—
434	C₂⁴	—	2	2	no	—
435	C₂⁴	—	4	4	no	—
436	C₂⁴	—	4	2	no	—
437	C₂⁴	—	4	2	no	—
438	C₂⁴	—	4	2	no	—
439	C₂⁴	—	8	2	no	—
440	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
441	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	2	2	no	—
445	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	4	8	no	—
455	C₂×D₄	—	4	1	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	4	4	no	—
459	C₂×D₄	—	4	4	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	8	2	no	—
462	C₂×D₄	#41×C₆₇	1	4	yes	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	4	2	no	—
476	C₂²×C₄	—	4	2	no	—
477	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
482	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
484	SD₁₆	—	4	8	no	—
485	M₁₆	—	4	4	no	—
486	C₂×C₈	—	2	2	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	2	2	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	—	2	2	no	—
491	C₂×C₈	—	2	2	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
505	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
506	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	4	4	no	—
513	C₂²⋊C₄	—	4	4	no	—
514	C₂²⋊C₄	—	4	2	no	—
515	C₂²⋊C₄	—	8	2	no	—
516	C₄²	—	2	4	no	—
517	C₄²	—	2	4	no	—
518	C₄²	—	2	1	no	—
519	C₄²	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	—	2	2	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	1	no	—
524	C₂³	—	2	1	no	—
525	C₂³	—	2	1	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	4	8	no	—
530	C₂³	—	4	2	no	—
531	C₂³	—	4	2	no	—
532	C₂³	—	4	2	no	—
533	C₂³	—	4	2	no	—
534	C₂³	—	4	2	no	—
535	C₂³	—	4	2	no	—
536	C₂³	—	8	4	no	—
537	C₂³	—	8	4	no	—
538	C₂³	—	8	8	no	—
539	C₂³	—	8	8	no	—
540	C₂³	#120×C₆₇	1	2	yes	—
541	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
542	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
543	Q₈	—	2	4	no	—
544	Q₈	—	4	1	no	—
545	Q₈	—	4	1	no	—
546	D₄	—	2	4	no	—
547	D₄	—	2	4	no	—
548	D₄	—	4	8	no	—
549	D₄	—	4	4	no	—
550	D₄	—	4	2	no	—
551	D₄	—	4	2	no	—
552	D₄	—	8	8	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	4	8	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	#120×C₆₇	1	2	yes	—
582	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
584	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
585	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
586	C₈	—	4	2	no	—
587	C₈	—	4	2	no	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	4	no	—
590	C₂²	—	2	2	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	2	no	—
597	C₂²	—	4	1	no	—
598	C₂²	—	4	4	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	4	4	no	—
601	C₂²	—	8	16	no	—
602	C₂²	—	8	2	no	—
603	C₂²	—	8	2	no	—
604	C₂²	—	8	2	no	—
605	C₂²	—	8	2	no	—
606	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
607	C₂²	#546×C₆₇	1	2	yes	—
608	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
609	C₂²	#575×C₆₇	1	1	yes	characteristic
610	C₂²	#577×C₆₇	1	2	yes	—
611	C₄	—	2	2	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	1	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	8	2	no	—
620	C₄	#525×C₆₇	1	1	yes	characteristic
621	C₄	#535×C₆₇	1	1	yes	characteristic
622	C₄	#546×C₆₇	1	2	yes	—
623	C₂	—	4	2	no	—
624	C₂	—	4	2	no	—
625	C₂	—	8	4	no	—
626	C₂	C₆₇×#2404	1	1	yes	characteristic
627	C₂	C₆₇×#3006	1	1	yes	characteristic
628	C₂	C₆₇×#3010	1	1	yes	characteristic
629	C₂	C₆₇×#3310	1	2	yes	—
630	C₂	C₆₇×#3323	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₆₇×#2408	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#552×C₆₇	C₂	semidirect
2. C₂×#450×C₆₇	C₂	semidirect
3. C₂×#418×C₆₇	C₂	semidirect
5. C₆₇×#2025	C₂	semidirect
6. C₆₇×#2000	C₂	semidirect
7. C₆₇×#1894	C₂	semidirect
8. C₆₇×#1880	C₂	semidirect
9. C₆₇×#1874	C₂	semidirect
10. C₆₇×#1870	C₂	semidirect
11. C₆₇×#1825	C₂	semidirect
19. C₂×#125×C₆₇	C₂²	semidirect
22. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
25. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
29. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
32. C₂×#103×C₆₇	C₂²	semidirect
37. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
43. #340×C₆₇	C₂²	semidirect
44. #340×C₆₇	C₂²	semidirect
45. #340×C₆₇	C₂²	semidirect
46. #336×C₆₇	C₂²	semidirect
75. C₂²⋊Q₈×C₆₇	D₄	semidirect
85. #117×C₆₇	D₄	semidirect
307. #23982	C₆₇	direct (N × H)
316. C₂×#552	C₂×C₆₇	semidirect
317. C₂×#450	C₂×C₆₇	semidirect
318. C₂×#418	C₂×C₆₇	semidirect
320. #2025	C₂×C₆₇	semidirect
321. #2000	C₂×C₆₇	semidirect
322. #1894	C₂×C₆₇	semidirect
323. #1880	C₂×C₆₇	semidirect
324. #1874	C₂×C₆₇	semidirect
325. #1870	C₂×C₆₇	semidirect
326. #1825	C₂×C₆₇	semidirect
327. C₆₇	#23982	direct (N × H)
335. C₂×#125	C₂²×C₆₇	semidirect
338. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
341. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
345. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
348. C₂×#103	C₂²×C₆₇	semidirect
353. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
359. #340	C₂²×C₆₇	semidirect
360. #340	C₂²×C₆₇	semidirect
361. #340	C₂²×C₆₇	semidirect
362. #336	C₂²×C₆₇	semidirect
391. C₂²⋊Q₈	D₄×C₆₇	semidirect
401. #117	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#552×C₆₇	2
C₂×#450×C₆₇	2
C₂×#418×C₆₇	2
C₆₇×#2408	2
C₆₇×#2025	2
C₆₇×#2000	2
C₆₇×#1894	2
C₆₇×#1880	2
C₆₇×#1874	2
C₆₇×#1870	2
C₆₇×#1825	2
#23982	67

Subgroups — C67×#23982 #30703233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#23982 #30703233