Subgroups — C₆₇×#22480 #30701731

Order 17152.

1068 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	C₂×#552×C₆₇	C₂	1	1	yes	maximal, characteristic
2	D₄×M₁₆×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1998	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1901	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1901	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1822	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1821	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1801	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1742	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#343×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	#552×C₆₇	—	2	2	no	—
15	#552×C₆₇	—	2	2	no	—
16	#552×C₆₇	—	2	2	no	—
17	#552×C₆₇	—	2	2	no	—
18	C₂×#126×C₆₇	C₄	1	2	yes	—
19	C₂×#125×C₆₇	C₄	1	2	yes	—
20	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#123×C₆₇	C₄	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₆₇	C₄	1	2	yes	—
27	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#436×C₆₇	—	2	2	no	—
30	#435×C₆₇	—	2	2	no	—
31	#435×C₆₇	—	2	2	no	—
32	C₈×D₄×C₆₇	—	2	2	no	—
33	#423×C₆₇	C₂²	1	2	yes	—
34	C₂×#101×C₆₇	C₂²	1	2	yes	—
35	C₂×#101×C₆₇	C₂²	1	2	yes	—
36	#407×C₆₇	C₂²	1	2	yes	—
37	#407×C₆₇	C₂²	1	2	yes	—
38	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
39	#343×C₆₇	C₂²	1	1	yes	characteristic
40	#343×C₆₇	C₂²	1	2	yes	—
41	#343×C₆₇	C₂²	1	1	yes	characteristic
42	#343×C₆₇	C₂²	1	2	yes	—
43	#343×C₆₇	C₂²	1	1	yes	characteristic
44	#343×C₆₇	C₂²	1	2	yes	—
45	#343×C₆₇	C₂²	1	2	yes	—
46	#343×C₆₇	C₂²	1	1	yes	characteristic
47	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₆₇	—	2	2	no	—
53	C₂×M₁₆×C₆₇	—	2	2	no	—
54	C₂×M₁₆×C₆₇	—	2	4	no	—
55	C₂×M₁₆×C₆₇	—	2	4	no	—
56	C₂×M₁₆×C₆₇	—	4	2	no	—
57	C₂×M₁₆×C₆₇	—	4	2	no	—
58	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
59	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
60	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
61	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	—	2	2	no	—
63	C₂²×C₈×C₆₇	—	2	2	no	—
64	#126×C₆₇	—	2	4	no	—
65	#126×C₆₇	—	2	4	no	—
66	#125×C₆₇	—	2	4	no	—
67	#125×C₆₇	—	2	4	no	—
68	#125×C₆₇	—	2	2	no	—
69	#125×C₆₇	—	2	2	no	—
70	#123×C₆₇	—	2	4	no	—
71	#123×C₆₇	—	2	4	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	—	2	2	no	—
76	C₄×Q₈×C₆₇	—	2	2	no	—
77	C₄×Q₈×C₆₇	—	2	2	no	—
78	C₄×D₄×C₆₇	D₄	1	2	yes	—
79	C₄×D₄×C₆₇	D₄	1	2	yes	—
80	C₄×D₄×C₆₇	D₄	1	2	yes	—
81	C₄×D₄×C₆₇	D₄	1	2	yes	—
82	#117×C₆₇	—	2	2	no	—
83	#117×C₆₇	—	2	2	no	—
84	#117×C₆₇	—	2	2	no	—
85	#117×C₆₇	—	2	2	no	—
86	#117×C₆₇	—	2	2	no	—
87	#117×C₆₇	—	2	2	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂×C₄	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×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
106	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
107	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
108	C₄⋊C₈×C₆₇	—	2	2	no	—
109	C₄⋊C₈×C₆₇	—	2	2	no	—
110	#101×C₆₇	—	2	4	no	—
111	#101×C₆₇	—	4	2	no	—
112	C₂²⋊C₈×C₆₇	—	2	2	no	—
113	C₂²⋊C₈×C₆₇	—	2	2	no	—
114	C₂²⋊C₈×C₆₇	—	2	2	no	—
115	C₂²⋊C₈×C₆₇	—	2	2	no	—
116	#97×C₆₇	—	2	2	no	—
117	C₄×C₈×C₆₇	—	2	2	no	—
118	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×Q₈×C₆₇	—	2	2	no	—
121	C₂×Q₈×C₆₇	—	2	2	no	—
122	C₂×Q₈×C₆₇	—	2	1	no	—
123	C₂×Q₈×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	4	2	no	—
125	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
126	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	1	no	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	4	4	no	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
145	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
149	M₁₆×C₆₇	—	2	2	no	—
150	M₁₆×C₆₇	—	2	2	no	—
151	M₁₆×C₆₇	—	2	2	no	—
152	M₁₆×C₆₇	—	2	2	no	—
153	M₁₆×C₆₇	—	2	2	no	—
154	M₁₆×C₆₇	—	2	2	no	—
155	M₁₆×C₆₇	—	2	2	no	—
156	M₁₆×C₆₇	—	4	4	no	—
157	M₁₆×C₆₇	—	4	4	no	—
158	M₁₆×C₆₇	#41	1	4	yes	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	—	4	4	no	—
164	C₂×C₈×C₆₇	—	4	4	no	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	4	no	—
171	C₄⋊C₄×C₆₇	—	2	4	no	—
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	2	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
177	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
178	C₂²⋊C₄×C₆₇	—	2	2	no	—
179	C₂²⋊C₄×C₆₇	—	2	2	no	—
180	C₂²⋊C₄×C₆₇	—	2	4	no	—
181	C₂²⋊C₄×C₆₇	—	2	4	no	—
182	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
183	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
184	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
185	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
186	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
187	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
188	C₄²×C₆₇	—	2	2	no	—
189	C₄²×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	2	2	no	—
191	C₄²×C₆₇	—	2	2	no	—
192	C₄²×C₆₇	C₂×D₄	1	2	yes	—
193	C₄²×C₆₇	C₂×D₄	1	2	yes	—
194	C₂³×C₆₇	—	2	1	no	—
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	1	no	—
198	C₂³×C₆₇	—	2	1	no	—
199	C₂³×C₆₇	—	4	1	no	—
200	C₂³×C₆₇	—	4	1	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
203	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
206	Q₈×C₆₇	—	4	2	no	—
207	Q₈×C₆₇	—	4	2	no	—
208	D₄×C₆₇	—	2	4	no	—
209	D₄×C₆₇	—	4	2	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	2	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	2	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	2	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	—	4	1	no	—
229	C₂×C₄×C₆₇	—	4	1	no	—
230	C₂×C₄×C₆₇	—	4	4	no	—
231	C₂×C₄×C₆₇	—	4	4	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
240	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
241	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
243	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
244	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
245	C₈×C₆₇	—	2	4	no	—
246	C₈×C₆₇	—	4	2	no	—
247	C₈×C₆₇	—	4	2	no	—
248	C₈×C₆₇	—	4	2	no	—
249	C₂²×C₆₇	—	2	1	no	—
250	C₂²×C₆₇	—	2	1	no	—
251	C₂²×C₆₇	—	2	1	no	—
252	C₂²×C₆₇	—	2	1	no	—
253	C₂²×C₆₇	—	2	1	no	—
254	C₂²×C₆₇	—	2	2	no	—
255	C₂²×C₆₇	—	2	1	no	—
256	C₂²×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	2	1	no	—
258	C₂²×C₆₇	—	2	2	no	—
259	C₂²×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	4	2	no	—
261	C₂²×C₆₇	—	8	1	no	—
262	C₂²×C₆₇	—	8	1	no	—
263	C₂²×C₆₇	#413	1	2	yes	—
264	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
265	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
266	C₂²×C₆₇	#520	1	1	yes	characteristic
267	C₄×C₆₇	—	2	1	no	—
268	C₄×C₆₇	—	2	2	no	—
269	C₄×C₆₇	—	2	1	no	—
270	C₄×C₆₇	—	2	2	no	—
271	C₄×C₆₇	—	2	1	no	—
272	C₄×C₆₇	—	2	1	no	—
273	C₄×C₆₇	—	4	2	no	—
274	C₄×C₆₇	—	4	2	no	—
275	C₄×C₆₇	—	4	2	no	—
276	C₄×C₆₇	—	4	2	no	—
277	C₄×C₆₇	#546	1	2	yes	—
278	C₄×C₆₇	#547	1	2	yes	—
279	#22480	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂×C₆₇	—	2	1	no	—
281	C₂×C₆₇	—	2	1	no	—
282	C₂×C₆₇	—	2	2	no	—
283	C₂×C₆₇	—	4	1	no	—
284	C₂×C₆₇	—	4	2	no	—
285	C₂×C₆₇	#2339	1	1	yes	characteristic
286	C₂×C₆₇	C₂×#413	1	1	yes	characteristic
287	C₂×C₆₇	#2889	1	1	yes	characteristic
288	C₂×#552	C₂×C₆₇	1	1	yes	characteristic
289	D₄×M₁₆	C₂×C₆₇	1	2	yes	—
290	#1998	C₂×C₆₇	1	2	yes	complemented
291	#1901	C₂×C₆₇	1	2	yes	complemented
292	#1901	C₂×C₆₇	1	2	yes	complemented
293	#1822	C₂×C₆₇	1	1	yes	characteristic, complemented
294	#1821	C₂×C₆₇	1	1	yes	characteristic, complemented
295	#1801	C₂×C₆₇	1	1	yes	characteristic, complemented
296	C₄×#101	C₂×C₆₇	1	1	yes	characteristic, complemented
297	#1742	C₂×C₆₇	1	1	yes	characteristic, complemented
298	C₂×#343	C₂×C₆₇	1	1	yes	characteristic, complemented
299	C₆₇	#22480	1	1	yes	characteristic, direct product factor, Sylow
300	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
301	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
302	#552	—	2	2	no	—
303	#552	—	2	2	no	—
304	#552	—	2	2	no	—
305	#552	—	2	2	no	—
306	C₂×#126	C₄×C₆₇	1	2	yes	—
307	C₂×#125	C₄×C₆₇	1	2	yes	—
308	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
309	C₂×#123	C₄×C₆₇	1	2	yes	—
310	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
312	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
313	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
314	C₂×#117	C₄×C₆₇	1	2	yes	—
315	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
316	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
317	#436	—	2	2	no	—
318	#435	—	2	2	no	—
319	#435	—	2	2	no	—
320	C₈×D₄	—	2	2	no	—
321	#423	C₂²×C₆₇	1	2	yes	—
322	C₂×#101	C₂²×C₆₇	1	2	yes	—
323	C₂×#101	C₂²×C₆₇	1	2	yes	—
324	#407	C₂²×C₆₇	1	2	yes	—
325	#407	C₂²×C₆₇	1	2	yes	—
326	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
327	#343	C₂²×C₆₇	1	1	yes	characteristic
328	#343	C₂²×C₆₇	1	1	yes	characteristic
329	#343	C₂²×C₆₇	1	1	yes	characteristic
330	#343	C₂²×C₆₇	1	2	yes	—
331	#343	C₂²×C₆₇	1	2	yes	—
332	#343	C₂²×C₆₇	1	2	yes	—
333	#343	C₂²×C₆₇	1	2	yes	—
334	#343	C₂²×C₆₇	1	1	yes	characteristic
335	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
336	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
337	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
338	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂×M₁₆	—	2	2	no	—
341	C₂×M₁₆	—	2	2	no	—
342	C₂×M₁₆	—	2	4	no	—
343	C₂×M₁₆	—	2	4	no	—
344	C₂×M₁₆	—	4	2	no	—
345	C₂×M₁₆	—	4	2	no	—
346	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
347	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
348	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
349	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
350	C₂²×C₈	—	2	2	no	—
351	C₂²×C₈	—	2	2	no	—
352	#126	—	2	4	no	—
353	#126	—	2	4	no	—
354	#125	—	2	4	no	—
355	#125	—	2	4	no	—
356	#125	—	2	2	no	—
357	#125	—	2	2	no	—
358	#123	—	2	4	no	—
359	#123	—	2	4	no	—
360	C₂²⋊Q₈	—	2	2	no	—
361	C₂²⋊Q₈	—	2	2	no	—
362	C₂²⋊Q₈	—	2	2	no	—
363	C₂²⋊Q₈	—	2	2	no	—
364	C₄×Q₈	—	2	2	no	—
365	C₄×Q₈	—	2	2	no	—
366	C₄×D₄	D₄×C₆₇	1	2	yes	—
367	C₄×D₄	D₄×C₆₇	1	2	yes	—
368	C₄×D₄	D₄×C₆₇	1	2	yes	—
369	C₄×D₄	D₄×C₆₇	1	2	yes	—
370	#117	—	2	2	no	—
371	#117	—	2	2	no	—
372	#117	—	2	2	no	—
373	#117	—	2	2	no	—
374	#117	—	2	2	no	—
375	#117	—	2	2	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	2	no	—
378	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
379	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
380	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
381	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
382	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
383	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
384	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
388	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
389	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
390	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
391	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
392	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
393	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
394	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
395	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
396	C₄⋊C₈	—	2	2	no	—
397	C₄⋊C₈	—	2	2	no	—
398	#101	—	2	4	no	—
399	#101	—	4	2	no	—
400	C₂²⋊C₈	—	2	2	no	—
401	C₂²⋊C₈	—	2	2	no	—
402	C₂²⋊C₈	—	2	2	no	—
403	C₂²⋊C₈	—	2	2	no	—
404	#97	—	2	2	no	—
405	C₄×C₈	—	2	2	no	—
406	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
407	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×D₄	—	4	2	no	—
413	C₂×D₄	C₂²⋊C₄×C₆₇	1	2	yes	—
414	C₂×D₄	C₂²⋊C₄×C₆₇	1	2	yes	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	4	4	no	—
424	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
425	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
426	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
428	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
429	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
430	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
431	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
433	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
434	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
435	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
437	M₁₆	—	2	2	no	—
438	M₁₆	—	2	2	no	—
439	M₁₆	—	2	2	no	—
440	M₁₆	—	2	2	no	—
441	M₁₆	—	2	2	no	—
442	M₁₆	—	2	2	no	—
443	M₁₆	—	2	2	no	—
444	M₁₆	—	4	4	no	—
445	M₁₆	—	4	4	no	—
446	M₁₆	#41×C₆₇	1	4	yes	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	4	4	no	—
452	C₂×C₈	—	4	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
465	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
466	C₂²⋊C₄	—	2	2	no	—
467	C₂²⋊C₄	—	2	2	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
471	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
472	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
473	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
474	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
475	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
476	C₄²	—	2	2	no	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	C₂×D₄×C₆₇	1	2	yes	—
481	C₄²	C₂×D₄×C₆₇	1	2	yes	—
482	C₂³	—	2	1	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	1	no	—
487	C₂³	—	4	1	no	—
488	C₂³	—	4	1	no	—
489	C₂³	—	4	1	no	—
490	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
491	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
492	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
493	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
494	Q₈	—	4	2	no	—
495	Q₈	—	4	2	no	—
496	D₄	—	2	4	no	—
497	D₄	—	4	2	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	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	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
524	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
525	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
526	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
527	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
528	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
529	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
530	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
531	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
532	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
533	C₈	—	2	4	no	—
534	C₈	—	4	2	no	—
535	C₈	—	4	2	no	—
536	C₈	—	4	2	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	8	1	no	—
550	C₂²	—	8	1	no	—
551	C₂²	#413×C₆₇	1	2	yes	—
552	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
553	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
554	C₂²	#520×C₆₇	1	1	yes	characteristic
555	C₄	—	2	1	no	—
556	C₄	—	2	2	no	—
557	C₄	—	2	1	no	—
558	C₄	—	2	2	no	—
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	4	2	no	—
565	C₄	#546×C₆₇	1	2	yes	—
566	C₄	#547×C₆₇	1	2	yes	—
567	C₂	—	2	1	no	—
568	C₂	—	2	1	no	—
569	C₂	—	2	2	no	—
570	C₂	—	4	1	no	—
571	C₂	—	4	2	no	—
572	C₂	C₆₇×#2339	1	1	yes	characteristic
573	C₂	C₂×#413×C₆₇	1	1	yes	characteristic
574	C₂	C₆₇×#2889	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1998	C₂	semidirect
4. C₆₇×#1901	C₂	semidirect
5. C₆₇×#1901	C₂	semidirect
6. C₆₇×#1822	C₂	semidirect
7. C₆₇×#1821	C₂	semidirect
8. C₆₇×#1801	C₂	semidirect
9. C₄×#101×C₆₇	C₂	semidirect
10. C₆₇×#1742	C₂	semidirect
11. C₂×#343×C₆₇	C₂	semidirect
279. #22480	C₆₇	direct (N × H)
290. #1998	C₂×C₆₇	semidirect
291. #1901	C₂×C₆₇	semidirect
292. #1901	C₂×C₆₇	semidirect
293. #1822	C₂×C₆₇	semidirect
294. #1821	C₂×C₆₇	semidirect
295. #1801	C₂×C₆₇	semidirect
296. C₄×#101	C₂×C₆₇	semidirect
297. #1742	C₂×C₆₇	semidirect
298. C₂×#343	C₂×C₆₇	semidirect
299. C₆₇	#22480	direct (N × H)

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
D₄×M₁₆×C₆₇	2
C₆₇×#1998	2
C₆₇×#1901	2
C₆₇×#1822	2
C₆₇×#1821	2
C₆₇×#1801	2
C₄×#101×C₆₇	2
C₆₇×#1742	2
C₂×#343×C₆₇	2
#22480	67

Subgroups — C67×#22480 #30701731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22480 #30701731