Subgroups — C₆₇×#22437 #30701688

Order 17152.

1602 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#440×C₆₇	C₂	1	2	yes	maximal, complemented
3	Q₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1973	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1880	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1787	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1786	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₆₇	—	2	8	no	—
15	C₂²×SD₁₆×C₆₇	C₄	1	8	yes	complemented
16	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
24	#440×C₆₇	—	2	8	no	—
25	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
26	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
32	#391×C₆₇	—	2	2	no	—
33	#391×C₆₇	C₂²	1	2	yes	complemented
34	#390×C₆₇	C₂²	1	2	yes	—
35	#386×C₆₇	—	2	2	no	—
36	#380×C₆₇	—	2	1	no	—
37	#380×C₆₇	C₂²	1	1	yes	characteristic
38	#380×C₆₇	C₂²	1	1	yes	characteristic
39	#379×C₆₇	—	2	1	no	—
40	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
41	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	2	yes	complemented
43	#336×C₆₇	C₂²	1	2	yes	complemented
44	C₂⁵×C₆₇	—	2	1	no	—
45	C₂²×Q₈×C₆₇	—	2	1	no	—
46	C₂²×Q₈×C₆₇	—	2	2	no	—
47	C₂²×Q₈×C₆₇	—	2	4	no	—
48	C₂²×Q₈×C₆₇	—	4	2	no	—
49	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
50	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₆₇	—	2	4	no	—
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	—	2	2	no	—
55	C₂²×D₄×C₆₇	—	4	2	no	—
56	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	—	2	2	no	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×SD₁₆×C₆₇	—	2	4	no	—
63	C₂×SD₁₆×C₆₇	—	2	4	no	—
64	C₂×SD₁₆×C₆₇	—	2	32	no	—
65	C₂×SD₁₆×C₆₇	—	2	4	no	—
66	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
67	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	—
68	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
69	C₂×SD₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
70	C₂×SD₁₆×C₆₇	D₄	1	8	yes	—
71	C₂²×C₈×C₆₇	—	2	4	no	—
72	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
73	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₆₇	—	2	4	no	—
77	C₄×Q₈×C₆₇	—	4	2	no	—
78	C₄×D₄×C₆₇	—	2	4	no	—
79	C₄×D₄×C₆₇	—	4	2	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
89	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₆₇	—	2	2	no	—
92	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
93	#107×C₆₇	—	2	4	no	—
94	#103×C₆₇	—	2	4	no	—
95	#103×C₆₇	—	4	2	no	—
96	#102×C₆₇	—	2	4	no	—
97	#102×C₆₇	—	4	2	no	—
98	C₂²⋊C₈×C₆₇	—	2	2	no	—
99	#97×C₆₇	—	2	4	no	—
100	#95×C₆₇	—	2	2	no	—
101	#95×C₆₇	—	2	2	no	—
102	#95×C₆₇	—	2	1	no	—
103	#95×C₆₇	—	2	1	no	—
104	#95×C₆₇	D₄	1	2	yes	—
105	C₂⁴×C₆₇	—	2	1	no	—
106	C₂⁴×C₆₇	—	2	1	no	—
107	C₂⁴×C₆₇	—	2	4	no	—
108	C₂⁴×C₆₇	—	4	4	no	—
109	C₂⁴×C₆₇	—	4	2	no	—
110	C₂⁴×C₆₇	—	4	2	no	—
111	C₂⁴×C₆₇	—	8	2	no	—
112	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂×Q₈×C₆₇	—	2	4	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	8	no	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	4	4	no	—
121	C₂×Q₈×C₆₇	—	4	4	no	—
122	C₂×Q₈×C₆₇	—	4	8	no	—
123	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
124	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
125	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
126	C₂×D₄×C₆₇	—	2	8	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	1	no	—
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	4	4	no	—
134	C₂×D₄×C₆₇	—	4	8	no	—
135	C₂×D₄×C₆₇	—	4	4	no	—
136	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
138	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	1	no	—
144	C₂²×C₄×C₆₇	—	2	2	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	4	4	no	—
147	C₂²×C₄×C₆₇	—	4	4	no	—
148	C₂²×C₄×C₆₇	—	4	4	no	—
149	C₂²×C₄×C₆₇	—	8	2	no	—
150	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	#41	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
156	SD₁₆×C₆₇	—	2	8	no	—
157	SD₁₆×C₆₇	—	2	8	no	—
158	SD₁₆×C₆₇	—	2	8	no	—
159	SD₁₆×C₆₇	C₂²⋊C₄	1	16	yes	complemented
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	8	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	2	2	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
166	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
167	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	2	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	—	4	4	no	—
173	C₂²⋊C₄×C₆₇	—	2	2	no	—
174	C₂²⋊C₄×C₆₇	—	2	2	no	—
175	C₂²⋊C₄×C₆₇	—	4	4	no	—
176	C₂²⋊C₄×C₆₇	—	4	4	no	—
177	C₂²⋊C₄×C₆₇	—	4	2	no	—
178	C₂²⋊C₄×C₆₇	—	4	2	no	—
179	C₂²⋊C₄×C₆₇	—	8	2	no	—
180	C₄²×C₆₇	—	2	2	no	—
181	C₄²×C₆₇	—	2	2	no	—
182	C₄²×C₆₇	—	4	4	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	2	no	—
185	C₂³×C₆₇	—	2	2	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	2	no	—
188	C₂³×C₆₇	—	4	8	no	—
189	C₂³×C₆₇	—	4	1	no	—
190	C₂³×C₆₇	—	4	1	no	—
191	C₂³×C₆₇	—	4	2	no	—
192	C₂³×C₆₇	—	4	2	no	—
193	C₂³×C₆₇	—	4	1	no	—
194	C₂³×C₆₇	—	4	1	no	—
195	C₂³×C₆₇	—	4	2	no	—
196	C₂³×C₆₇	—	4	8	no	—
197	C₂³×C₆₇	—	8	4	no	—
198	C₂³×C₆₇	—	8	8	no	—
199	C₂³×C₆₇	—	8	8	no	—
200	C₂³×C₆₇	C₄×D₄	1	2	yes	—
201	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
202	C₂³×C₆₇	#137	1	2	yes	—
203	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₆₇	—	2	2	no	—
205	Q₈×C₆₇	—	2	4	no	—
206	Q₈×C₆₇	—	2	2	no	—
207	Q₈×C₆₇	—	2	2	no	—
208	Q₈×C₆₇	—	4	8	no	—
209	Q₈×C₆₇	—	4	2	no	—
210	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	complemented
211	D₄×C₆₇	—	2	2	no	—
212	D₄×C₆₇	—	2	2	no	—
213	D₄×C₆₇	—	2	2	no	—
214	D₄×C₆₇	—	2	4	no	—
215	D₄×C₆₇	—	4	8	no	—
216	D₄×C₆₇	—	4	2	no	—
217	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	4	8	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	—	4	2	no	—
234	C₂×C₄×C₆₇	—	4	2	no	—
235	C₂×C₄×C₆₇	—	4	2	no	—
236	C₂×C₄×C₆₇	—	4	2	no	—
237	C₂×C₄×C₆₇	—	4	2	no	—
238	C₂×C₄×C₆₇	—	4	2	no	—
239	C₂×C₄×C₆₇	—	8	8	no	—
240	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
241	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
242	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
244	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
245	C₈×C₆₇	—	2	2	no	—
246	C₈×C₆₇	—	4	2	no	—
247	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	complemented
248	C₂²×C₆₇	—	2	2	no	—
249	C₂²×C₆₇	—	2	2	no	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	2	4	no	—
252	C₂²×C₆₇	—	2	4	no	—
253	C₂²×C₆₇	—	4	2	no	—
254	C₂²×C₆₇	—	4	2	no	—
255	C₂²×C₆₇	—	4	4	no	—
256	C₂²×C₆₇	—	4	4	no	—
257	C₂²×C₆₇	—	4	2	no	—
258	C₂²×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	8	16	no	—
261	C₂²×C₆₇	—	8	2	no	—
262	C₂²×C₆₇	—	8	2	no	—
263	C₂²×C₆₇	#440	1	4	yes	complemented
264	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
265	C₂²×C₆₇	D₄²	1	2	yes	—
266	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
267	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
268	C₂²×C₆₇	#577	1	2	yes	—
269	C₄×C₆₇	—	2	2	no	—
270	C₄×C₆₇	—	2	4	no	—
271	C₄×C₆₇	—	4	2	no	—
272	C₄×C₆₇	—	4	2	no	—
273	C₄×C₆₇	—	4	2	no	—
274	C₄×C₆₇	—	8	2	no	—
275	C₄×C₆₇	—	8	2	no	—
276	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
277	C₄×C₆₇	#518	1	1	yes	characteristic
278	C₄×C₆₇	#546	1	2	yes	—
279	#22437	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂×C₆₇	—	2	4	no	—
281	C₂×C₆₇	—	4	4	no	—
282	C₂×C₆₇	—	8	2	no	—
283	C₂×C₆₇	D₄×C₂²⋊C₄	1	1	yes	characteristic
284	C₂×C₆₇	C₂×#440	1	1	yes	characteristic
285	C₂×C₆₇	#2949	1	1	yes	characteristic
286	C₂×C₆₇	#3265	1	2	yes	—
287	C₂×C₆₇	#3266	1	2	yes	—
288	C₂³×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
289	C₂×#440	C₂×C₆₇	1	2	yes	complemented
290	Q₈×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
291	D₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
292	#1973	C₂×C₆₇	1	2	yes	complemented
293	#1880	C₂×C₆₇	1	2	yes	complemented
294	#1877	C₂×C₆₇	1	2	yes	complemented
295	#1808	C₂×C₆₇	1	1	yes	characteristic, complemented
296	#1787	C₂×C₆₇	1	1	yes	characteristic, complemented
297	#1786	C₂×C₆₇	1	1	yes	characteristic
298	#1752	C₂×C₆₇	1	1	yes	characteristic, complemented
299	C₆₇	#22437	1	1	yes	characteristic, direct product factor, Sylow
300	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
301	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
302	C₂²×SD₁₆	—	2	8	no	—
303	C₂²×SD₁₆	C₄×C₆₇	1	8	yes	complemented
304	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
305	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
306	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
307	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
308	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
310	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
311	C₂²×C₂²⋊C₄	—	2	1	no	—
312	#440	—	2	8	no	—
313	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
314	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
315	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
316	C₂×#102	C₂²×C₆₇	1	2	yes	—
317	C₂×#102	C₂²×C₆₇	1	2	yes	—
318	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
320	#391	—	2	2	no	—
321	#391	C₂²×C₆₇	1	2	yes	complemented
322	#390	C₂²×C₆₇	1	2	yes	—
323	#386	—	2	2	no	—
324	#380	—	2	1	no	—
325	#380	C₂²×C₆₇	1	1	yes	characteristic
326	#380	C₂²×C₆₇	1	1	yes	characteristic
327	#379	—	2	1	no	—
328	C₄×C₂²⋊C₄	—	2	1	no	—
329	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
330	#340	C₂²×C₆₇	1	2	yes	complemented
331	#336	C₂²×C₆₇	1	2	yes	complemented
332	C₂⁵	—	2	1	no	—
333	C₂²×Q₈	—	2	1	no	—
334	C₂²×Q₈	—	2	2	no	—
335	C₂²×Q₈	—	2	4	no	—
336	C₂²×Q₈	—	4	2	no	—
337	C₂²×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
338	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
339	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
340	C₂²×D₄	—	2	4	no	—
341	C₂²×D₄	—	2	1	no	—
342	C₂²×D₄	—	2	2	no	—
343	C₂²×D₄	—	4	2	no	—
344	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
345	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
346	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	—	2	2	no	—
349	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
350	C₂×SD₁₆	—	2	4	no	—
351	C₂×SD₁₆	—	2	4	no	—
352	C₂×SD₁₆	—	2	32	no	—
353	C₂×SD₁₆	—	2	4	no	—
354	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
355	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	—
356	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
357	C₂×SD₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
358	C₂×SD₁₆	D₄×C₆₇	1	8	yes	—
359	C₂²×C₈	—	2	4	no	—
360	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
361	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
362	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	—	4	2	no	—
366	C₄×D₄	—	2	4	no	—
367	C₄×D₄	—	4	2	no	—
368	C₂×C₄⋊C₄	—	2	2	no	—
369	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
371	C₂×C₂²⋊C₄	—	2	1	no	—
372	C₂×C₂²⋊C₄	—	2	1	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	2	no	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	4	4	no	—
377	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	D₄×C₆₇	1	1	yes	characteristic
379	C₂×C₄²	—	2	2	no	—
380	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
381	#107	—	2	4	no	—
382	#103	—	2	4	no	—
383	#103	—	4	2	no	—
384	#102	—	2	4	no	—
385	#102	—	4	2	no	—
386	C₂²⋊C₈	—	2	2	no	—
387	#97	—	2	4	no	—
388	#95	—	2	2	no	—
389	#95	—	2	2	no	—
390	#95	—	2	1	no	—
391	#95	—	2	1	no	—
392	#95	D₄×C₆₇	1	2	yes	—
393	C₂⁴	—	2	1	no	—
394	C₂⁴	—	2	1	no	—
395	C₂⁴	—	2	4	no	—
396	C₂⁴	—	4	4	no	—
397	C₂⁴	—	4	2	no	—
398	C₂⁴	—	4	2	no	—
399	C₂⁴	—	8	2	no	—
400	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
401	C₂×Q₈	—	2	4	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	8	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	4	4	no	—
409	C₂×Q₈	—	4	4	no	—
410	C₂×Q₈	—	4	8	no	—
411	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
412	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
413	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
414	C₂×D₄	—	2	8	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	—	4	8	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
425	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
426	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	4	no	—
437	C₂²×C₄	—	8	2	no	—
438	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
439	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
440	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
442	C₂²×C₄	#41×C₆₇	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
444	SD₁₆	—	2	8	no	—
445	SD₁₆	—	2	8	no	—
446	SD₁₆	—	2	8	no	—
447	SD₁₆	C₂²⋊C₄×C₆₇	1	16	yes	complemented
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	8	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	2	2	no	—
453	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
454	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
455	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	4	4	no	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	4	4	no	—
464	C₂²⋊C₄	—	4	4	no	—
465	C₂²⋊C₄	—	4	2	no	—
466	C₂²⋊C₄	—	4	2	no	—
467	C₂²⋊C₄	—	8	2	no	—
468	C₄²	—	2	2	no	—
469	C₄²	—	2	2	no	—
470	C₄²	—	4	4	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	4	8	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	1	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	4	2	no	—
481	C₂³	—	4	1	no	—
482	C₂³	—	4	1	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	8	no	—
485	C₂³	—	8	4	no	—
486	C₂³	—	8	8	no	—
487	C₂³	—	8	8	no	—
488	C₂³	C₄×D₄×C₆₇	1	2	yes	—
489	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
490	C₂³	#137×C₆₇	1	2	yes	—
491	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
492	Q₈	—	2	2	no	—
493	Q₈	—	2	4	no	—
494	Q₈	—	2	2	no	—
495	Q₈	—	2	2	no	—
496	Q₈	—	4	8	no	—
497	Q₈	—	4	2	no	—
498	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	complemented
499	D₄	—	2	2	no	—
500	D₄	—	2	2	no	—
501	D₄	—	2	2	no	—
502	D₄	—	2	4	no	—
503	D₄	—	4	8	no	—
504	D₄	—	4	2	no	—
505	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	8	8	no	—
528	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
529	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
531	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
532	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
533	C₈	—	2	2	no	—
534	C₈	—	4	2	no	—
535	C₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	complemented
536	C₂²	—	2	4	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	2	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	2	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	8	16	no	—
549	C₂²	—	8	2	no	—
550	C₂²	—	8	2	no	—
551	C₂²	#440×C₆₇	1	4	yes	complemented
552	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
553	C₂²	D₄²×C₆₇	1	2	yes	—
554	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
555	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
556	C₂²	#577×C₆₇	1	2	yes	—
557	C₄	—	2	2	no	—
558	C₄	—	2	4	no	—
559	C₄	—	4	2	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	2	no	—
562	C₄	—	8	2	no	—
563	C₄	—	8	2	no	—
564	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
565	C₄	#518×C₆₇	1	1	yes	characteristic
566	C₄	#546×C₆₇	1	2	yes	—
567	C₂	—	2	4	no	—
568	C₂	—	4	4	no	—
569	C₂	—	8	2	no	—
570	C₂	D₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
571	C₂	C₂×#440×C₆₇	1	1	yes	characteristic
572	C₂	C₆₇×#2949	1	1	yes	characteristic
573	C₂	C₆₇×#3265	1	2	yes	—
574	C₂	C₆₇×#3266	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×#440×C₆₇	C₂	semidirect
3. Q₈×C₂²⋊C₄×C₆₇	C₂	semidirect
5. C₆₇×#1973	C₂	semidirect
6. C₆₇×#1880	C₂	semidirect
7. C₆₇×#1877	C₂	semidirect
8. C₆₇×#1808	C₂	semidirect
9. C₆₇×#1787	C₂	semidirect
11. C₆₇×#1752	C₂	semidirect
15. C₂²×SD₁₆×C₆₇	C₄	semidirect
21. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
25. C₂×#107×C₆₇	C₂²	semidirect
26. C₂×#103×C₆₇	C₂²	semidirect
27. C₂×#103×C₆₇	C₂²	semidirect
31. C₂×#97×C₆₇	C₂²	semidirect
33. #391×C₆₇	C₂²	semidirect
42. #340×C₆₇	C₂²	semidirect
43. #336×C₆₇	C₂²	semidirect
49. C₂²×Q₈×C₆₇	C₂×C₄	semidirect
68. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
69. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
72. C₂²×C₈×C₆₇	C₂×C₄	semidirect
159. SD₁₆×C₆₇	C₂²⋊C₄	semidirect
210. Q₈×C₆₇	C₂×C₂²⋊C₄	semidirect
247. C₈×C₆₇	C₂×C₂²⋊C₄	semidirect
263. C₂²×C₆₇	#440	semidirect
279. #22437	C₆₇	direct (N × H)
289. C₂×#440	C₂×C₆₇	semidirect
290. Q₈×C₂²⋊C₄	C₂×C₆₇	semidirect
292. #1973	C₂×C₆₇	semidirect
293. #1880	C₂×C₆₇	semidirect
294. #1877	C₂×C₆₇	semidirect
295. #1808	C₂×C₆₇	semidirect
296. #1787	C₂×C₆₇	semidirect
298. #1752	C₂×C₆₇	semidirect
299. C₆₇	#22437	direct (N × H)
303. C₂²×SD₁₆	C₄×C₆₇	semidirect
309. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
313. C₂×#107	C₂²×C₆₇	semidirect
314. C₂×#103	C₂²×C₆₇	semidirect
315. C₂×#103	C₂²×C₆₇	semidirect
319. C₂×#97	C₂²×C₆₇	semidirect
321. #391	C₂²×C₆₇	semidirect
330. #340	C₂²×C₆₇	semidirect
331. #336	C₂²×C₆₇	semidirect
337. C₂²×Q₈	C₂×C₄×C₆₇	semidirect
356. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
357. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
360. C₂²×C₈	C₂×C₄×C₆₇	semidirect
447. SD₁₆	C₂²⋊C₄×C₆₇	semidirect
498. Q₈	C₂×C₂²⋊C₄×C₆₇	semidirect
535. C₈	C₂×C₂²⋊C₄×C₆₇	semidirect
551. C₂²	#440×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×SD₁₆×C₆₇	2
C₂×#440×C₆₇	2
Q₈×C₂²⋊C₄×C₆₇	2
D₄×C₂²⋊C₄×C₆₇	2
C₆₇×#1973	2
C₆₇×#1880	2
C₆₇×#1877	2
C₆₇×#1808	2
C₆₇×#1787	2
C₆₇×#1786	2
C₆₇×#1752	2
#22437	67

Subgroups — C67×#22437 #30701688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22437 #30701688