Subgroups — C₆₇×#34356 #30713607

Order 17152.

4134 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#120×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2375	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2358	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2342	C₂	1	4	yes	maximal, complemented
5	D₄×C₂²⋊C₄×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2321	C₂	1	8	yes	maximal, complemented
7	C₆₇×#2312	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2309	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2308	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#379×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂×#120×C₆₇	C₄	1	16	yes	complemented
15	C₂×#120×C₆₇	C₂²	1	4	yes	—
16	C₂×#120×C₆₇	C₂²	1	2	yes	—
17	C₂×#120×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
19	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	complemented
26	#390×C₆₇	C₂²	1	4	yes	complemented
27	#390×C₆₇	C₂²	1	4	yes	complemented
28	#390×C₆₇	C₂²	1	8	yes	complemented
29	#388×C₆₇	C₂²	1	8	yes	complemented
30	#388×C₆₇	C₂²	1	8	yes	complemented
31	#388×C₆₇	C₂²	1	8	yes	complemented
32	#388×C₆₇	C₂²	1	8	yes	complemented
33	#386×C₆₇	C₂²	1	4	yes	complemented
34	#386×C₆₇	C₂²	1	4	yes	complemented
35	#386×C₆₇	C₂²	1	8	yes	complemented
36	#386×C₆₇	C₂²	1	8	yes	complemented
37	#385×C₆₇	C₂²	1	8	yes	complemented
38	#381×C₆₇	C₂²	1	2	yes	complemented
39	#380×C₆₇	C₂²	1	4	yes	complemented
40	#379×C₆₇	C₂²	1	2	yes	—
41	#379×C₆₇	C₂²	1	2	yes	—
42	#379×C₆₇	C₂²	1	4	yes	—
43	#379×C₆₇	C₂²	1	4	yes	complemented
44	#379×C₆₇	C₂²	1	4	yes	complemented
45	#379×C₆₇	C₂²	1	2	yes	complemented
46	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	complemented
47	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
48	C₂⁵×C₆₇	—	4	4	no	—
49	C₂⁵×C₆₇	—	4	8	no	—
50	C₂⁵×C₆₇	C₂×C₄	1	8	yes	complemented
51	C₂⁵×C₆₇	C₂³	1	2	yes	—
52	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
53	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
54	C₂⁵×C₆₇	C₂³	1	2	yes	—
55	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
56	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₆₇	—	4	4	no	—
58	C₂²×D₄×C₆₇	—	4	8	no	—
59	C₂²×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
60	C₂²×D₄×C₆₇	C₂×C₄	1	16	yes	complemented
61	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
62	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
64	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
68	C₂³×C₄×C₆₇	C₂³	1	4	yes	complemented
69	#120×C₆₇	C₂×C₄	1	16	yes	complemented
70	#120×C₆₇	C₂×C₄	1	32	yes	complemented
71	#120×C₆₇	C₂×C₄	1	16	yes	complemented
72	C₄×D₄×C₆₇	—	4	16	no	—
73	#117×C₆₇	—	4	4	no	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	complemented
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	complemented
76	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
78	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
80	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	16	yes	complemented
81	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
82	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
92	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	complemented
96	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	complemented
97	C₂×C₄²×C₆₇	C₂³	1	4	yes	complemented
98	C₂×C₄²×C₆₇	C₂³	1	8	yes	complemented
99	#95×C₆₇	C₂³	1	8	yes	complemented
100	#95×C₆₇	C₂³	1	8	yes	complemented
101	#95×C₆₇	C₂³	1	8	yes	complemented
102	C₂⁴×C₆₇	—	2	8	no	—
103	C₂⁴×C₆₇	—	2	8	no	—
104	C₂⁴×C₆₇	—	2	8	no	—
105	C₂⁴×C₆₇	—	2	8	no	—
106	C₂⁴×C₆₇	—	2	4	no	—
107	C₂⁴×C₆₇	—	2	4	no	—
108	C₂⁴×C₆₇	—	4	32	no	—
109	C₂⁴×C₆₇	—	4	16	no	—
110	C₂⁴×C₆₇	—	4	2	no	—
111	C₂⁴×C₆₇	—	4	4	no	—
112	C₂⁴×C₆₇	—	4	4	no	—
113	C₂⁴×C₆₇	—	4	4	no	—
114	C₂⁴×C₆₇	—	4	4	no	—
115	C₂⁴×C₆₇	—	4	2	no	—
116	C₂⁴×C₆₇	—	4	4	no	—
117	C₂⁴×C₆₇	—	4	4	no	—
118	C₂⁴×C₆₇	—	4	64	no	—
119	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
120	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
122	C₂⁴×C₆₇	C₂²×C₄	1	8	yes	—
123	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
125	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
126	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
127	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
129	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
131	C₂×D₄×C₆₇	—	2	16	no	—
132	C₂×D₄×C₆₇	—	4	16	no	—
133	C₂×D₄×C₆₇	—	4	2	no	—
134	C₂×D₄×C₆₇	—	4	2	no	—
135	C₂×D₄×C₆₇	—	4	8	no	—
136	C₂×D₄×C₆₇	—	4	2	no	—
137	C₂×D₄×C₆₇	—	4	2	no	—
138	C₂×D₄×C₆₇	—	4	32	no	—
139	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
140	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
141	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
142	C₂×D₄×C₆₇	C₂²×C₄	1	16	yes	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	2	16	no	—
145	C₂²×C₄×C₆₇	—	4	4	no	—
146	C₂²×C₄×C₆₇	—	4	8	no	—
147	C₂²×C₄×C₆₇	—	4	8	no	—
148	C₂²×C₄×C₆₇	—	4	4	no	—
149	C₂²×C₄×C₆₇	—	4	8	no	—
150	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
156	C₄⋊C₄×C₆₇	—	4	4	no	—
157	C₄⋊C₄×C₆₇	—	4	8	no	—
158	C₂²⋊C₄×C₆₇	—	2	8	no	—
159	C₂²⋊C₄×C₆₇	—	2	8	no	—
160	C₂²⋊C₄×C₆₇	—	2	16	no	—
161	C₂²⋊C₄×C₆₇	—	4	4	no	—
162	C₂²⋊C₄×C₆₇	—	4	8	no	—
163	C₂²⋊C₄×C₆₇	—	4	4	no	—
164	C₂²⋊C₄×C₆₇	—	4	8	no	—
165	C₂²⋊C₄×C₆₇	—	4	4	no	—
166	C₂²⋊C₄×C₆₇	—	4	4	no	—
167	C₂²⋊C₄×C₆₇	—	4	4	no	—
168	C₂²⋊C₄×C₆₇	—	4	8	no	—
169	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
170	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
171	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
172	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
173	C₄²×C₆₇	—	4	4	no	—
174	C₄²×C₆₇	—	4	8	no	—
175	C₂³×C₆₇	—	2	4	no	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	16	no	—
179	C₂³×C₆₇	—	2	8	no	—
180	C₂³×C₆₇	—	2	16	no	—
181	C₂³×C₆₇	—	2	16	no	—
182	C₂³×C₆₇	—	2	16	no	—
183	C₂³×C₆₇	—	2	2	no	—
184	C₂³×C₆₇	—	2	4	no	—
185	C₂³×C₆₇	—	2	4	no	—
186	C₂³×C₆₇	—	2	4	no	—
187	C₂³×C₆₇	—	2	4	no	—
188	C₂³×C₆₇	—	2	8	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	2	no	—
191	C₂³×C₆₇	—	2	4	no	—
192	C₂³×C₆₇	—	2	2	no	—
193	C₂³×C₆₇	—	2	8	no	—
194	C₂³×C₆₇	—	4	8	no	—
195	C₂³×C₆₇	—	4	8	no	—
196	C₂³×C₆₇	—	4	4	no	—
197	C₂³×C₆₇	—	4	8	no	—
198	C₂³×C₆₇	—	4	8	no	—
199	C₂³×C₆₇	—	4	4	no	—
200	C₂³×C₆₇	—	4	8	no	—
201	C₂³×C₆₇	—	4	8	no	—
202	C₂³×C₆₇	—	4	16	no	—
203	C₂³×C₆₇	—	4	32	no	—
204	C₂³×C₆₇	—	4	32	no	—
205	C₂³×C₆₇	—	4	16	no	—
206	C₂³×C₆₇	—	4	128	no	—
207	C₂³×C₆₇	—	4	16	no	—
208	C₂³×C₆₇	—	8	8	no	—
209	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
210	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
211	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
212	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
213	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
214	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
215	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
216	C₂³×C₆₇	H(5,𝔽₂)	1	8	yes	—
217	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
218	D₄×C₆₇	—	2	32	no	—
219	D₄×C₆₇	—	4	8	no	—
220	D₄×C₆₇	—	4	8	no	—
221	D₄×C₆₇	—	4	16	no	—
222	C₂×C₄×C₆₇	—	2	8	no	—
223	C₂×C₄×C₆₇	—	2	2	no	—
224	C₂×C₄×C₆₇	—	2	8	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	8	no	—
227	C₂×C₄×C₆₇	—	2	8	no	—
228	C₂×C₄×C₆₇	—	2	8	no	—
229	C₂×C₄×C₆₇	—	2	8	no	—
230	C₂×C₄×C₆₇	—	2	8	no	—
231	C₂×C₄×C₆₇	—	2	8	no	—
232	C₂×C₄×C₆₇	—	4	8	no	—
233	C₂×C₄×C₆₇	—	8	16	no	—
234	C₂×C₄×C₆₇	—	8	8	no	—
235	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
236	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
237	C₂²×C₆₇	—	2	4	no	—
238	C₂²×C₆₇	—	2	4	no	—
239	C₂²×C₆₇	—	2	4	no	—
240	C₂²×C₆₇	—	2	16	no	—
241	C₂²×C₆₇	—	2	8	no	—
242	C₂²×C₆₇	—	2	8	no	—
243	C₂²×C₆₇	—	2	2	no	—
244	C₂²×C₆₇	—	2	4	no	—
245	C₂²×C₆₇	—	2	2	no	—
246	C₂²×C₆₇	—	2	4	no	—
247	C₂²×C₆₇	—	2	4	no	—
248	C₂²×C₆₇	—	2	4	no	—
249	C₂²×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	4	32	no	—
252	C₂²×C₆₇	—	4	16	no	—
253	C₂²×C₆₇	—	4	16	no	—
254	C₂²×C₆₇	—	4	32	no	—
255	C₂²×C₆₇	—	4	16	no	—
256	C₂²×C₆₇	—	4	4	no	—
257	C₂²×C₆₇	—	4	4	no	—
258	C₂²×C₆₇	—	4	8	no	—
259	C₂²×C₆₇	—	4	4	no	—
260	C₂²×C₆₇	—	4	4	no	—
261	C₂²×C₆₇	—	8	8	no	—
262	C₂²×C₆₇	#518	1	4	yes	—
263	C₂²×C₆₇	#518	1	8	yes	—
264	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
265	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
266	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	4	yes	—
267	C₄×C₆₇	—	2	4	no	—
268	C₄×C₆₇	—	2	8	no	—
269	C₄×C₆₇	—	4	8	no	—
270	C₄×C₆₇	—	4	8	no	—
271	#34356	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
272	C₂×C₆₇	—	2	4	no	—
273	C₂×C₆₇	—	2	8	no	—
274	C₂×C₆₇	—	4	4	no	—
275	C₂×C₆₇	—	4	2	no	—
276	C₂×C₆₇	—	4	2	no	—
277	C₂×C₆₇	—	4	4	no	—
278	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
279	C₂×C₆₇	C₂×#518	1	2	yes	—
280	C₂×C₆₇	#3532	1	4	yes	—
281	C₂²×#120	C₂×C₆₇	1	1	yes	characteristic
282	#2375	C₂×C₆₇	1	4	yes	complemented
283	#2358	C₂×C₆₇	1	4	yes	complemented
284	#2342	C₂×C₆₇	1	4	yes	complemented
285	D₄×C₂²⋊C₄	C₂×C₆₇	1	4	yes	complemented
286	#2321	C₂×C₆₇	1	8	yes	complemented
287	#2312	C₂×C₆₇	1	2	yes	complemented
288	#2309	C₂×C₆₇	1	1	yes	characteristic, complemented
289	#2308	C₂×C₆₇	1	1	yes	characteristic, complemented
290	C₂×#379	C₂×C₆₇	1	2	yes	complemented
291	C₆₇	#34356	1	1	yes	characteristic, direct product factor, Sylow
292	C₂⁶	C₂²×C₆₇	1	1	yes	characteristic
293	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
294	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
295	C₂×#120	C₄×C₆₇	1	16	yes	complemented
296	C₂×#120	C₂²×C₆₇	1	4	yes	—
297	C₂×#120	C₂²×C₆₇	1	2	yes	—
298	C₂×#120	C₂²×C₆₇	1	2	yes	—
299	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
300	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
301	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
302	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
303	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
304	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
305	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
306	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	complemented
307	#390	C₂²×C₆₇	1	4	yes	complemented
308	#390	C₂²×C₆₇	1	4	yes	complemented
309	#390	C₂²×C₆₇	1	8	yes	complemented
310	#388	C₂²×C₆₇	1	8	yes	complemented
311	#388	C₂²×C₆₇	1	8	yes	complemented
312	#388	C₂²×C₆₇	1	8	yes	complemented
313	#388	C₂²×C₆₇	1	8	yes	complemented
314	#386	C₂²×C₆₇	1	4	yes	complemented
315	#386	C₂²×C₆₇	1	4	yes	complemented
316	#386	C₂²×C₆₇	1	8	yes	complemented
317	#386	C₂²×C₆₇	1	8	yes	complemented
318	#385	C₂²×C₆₇	1	8	yes	complemented
319	#381	C₂²×C₆₇	1	2	yes	complemented
320	#380	C₂²×C₆₇	1	4	yes	complemented
321	#379	C₂²×C₆₇	1	2	yes	—
322	#379	C₂²×C₆₇	1	2	yes	—
323	#379	C₂²×C₆₇	1	4	yes	—
324	#379	C₂²×C₆₇	1	4	yes	complemented
325	#379	C₂²×C₆₇	1	4	yes	complemented
326	#379	C₂²×C₆₇	1	2	yes	complemented
327	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	complemented
328	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
329	C₂⁵	—	4	4	no	—
330	C₂⁵	—	4	8	no	—
331	C₂⁵	C₂×C₄×C₆₇	1	8	yes	complemented
332	C₂⁵	C₂³×C₆₇	1	2	yes	—
333	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
334	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
335	C₂⁵	C₂³×C₆₇	1	2	yes	—
336	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
337	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
338	C₂²×D₄	—	4	4	no	—
339	C₂²×D₄	—	4	8	no	—
340	C₂²×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
341	C₂²×D₄	C₂×C₄×C₆₇	1	16	yes	complemented
342	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
343	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
344	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
345	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
346	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
348	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
349	C₂³×C₄	C₂³×C₆₇	1	4	yes	complemented
350	#120	C₂×C₄×C₆₇	1	16	yes	complemented
351	#120	C₂×C₄×C₆₇	1	32	yes	complemented
352	#120	C₂×C₄×C₆₇	1	16	yes	complemented
353	C₄×D₄	—	4	16	no	—
354	#117	—	4	4	no	—
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	complemented
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	complemented
357	C₂×C₂²⋊C₄	—	4	8	no	—
358	C₂×C₂²⋊C₄	—	4	4	no	—
359	C₂×C₂²⋊C₄	—	4	8	no	—
360	C₂×C₂²⋊C₄	—	4	8	no	—
361	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	16	yes	complemented
362	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
363	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
364	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
365	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
370	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	complemented
377	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	complemented
378	C₂×C₄²	C₂³×C₆₇	1	4	yes	complemented
379	C₂×C₄²	C₂³×C₆₇	1	8	yes	complemented
380	#95	C₂³×C₆₇	1	8	yes	complemented
381	#95	C₂³×C₆₇	1	8	yes	complemented
382	#95	C₂³×C₆₇	1	8	yes	complemented
383	C₂⁴	—	2	8	no	—
384	C₂⁴	—	2	8	no	—
385	C₂⁴	—	2	8	no	—
386	C₂⁴	—	2	8	no	—
387	C₂⁴	—	2	4	no	—
388	C₂⁴	—	2	4	no	—
389	C₂⁴	—	4	32	no	—
390	C₂⁴	—	4	16	no	—
391	C₂⁴	—	4	2	no	—
392	C₂⁴	—	4	4	no	—
393	C₂⁴	—	4	4	no	—
394	C₂⁴	—	4	4	no	—
395	C₂⁴	—	4	4	no	—
396	C₂⁴	—	4	2	no	—
397	C₂⁴	—	4	4	no	—
398	C₂⁴	—	4	4	no	—
399	C₂⁴	—	4	64	no	—
400	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
401	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
402	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
403	C₂⁴	C₂²×C₄×C₆₇	1	8	yes	—
404	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
405	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
406	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
407	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
408	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
409	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
410	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
411	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
412	C₂×D₄	—	2	16	no	—
413	C₂×D₄	—	4	16	no	—
414	C₂×D₄	—	4	2	no	—
415	C₂×D₄	—	4	2	no	—
416	C₂×D₄	—	4	8	no	—
417	C₂×D₄	—	4	2	no	—
418	C₂×D₄	—	4	2	no	—
419	C₂×D₄	—	4	32	no	—
420	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
421	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
422	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
423	C₂×D₄	C₂²×C₄×C₆₇	1	16	yes	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	16	no	—
426	C₂²×C₄	—	4	4	no	—
427	C₂²×C₄	—	4	8	no	—
428	C₂²×C₄	—	4	8	no	—
429	C₂²×C₄	—	4	4	no	—
430	C₂²×C₄	—	4	8	no	—
431	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
432	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
433	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
434	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
435	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
437	C₄⋊C₄	—	4	4	no	—
438	C₄⋊C₄	—	4	8	no	—
439	C₂²⋊C₄	—	2	8	no	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₂²⋊C₄	—	2	16	no	—
442	C₂²⋊C₄	—	4	4	no	—
443	C₂²⋊C₄	—	4	8	no	—
444	C₂²⋊C₄	—	4	4	no	—
445	C₂²⋊C₄	—	4	8	no	—
446	C₂²⋊C₄	—	4	4	no	—
447	C₂²⋊C₄	—	4	4	no	—
448	C₂²⋊C₄	—	4	4	no	—
449	C₂²⋊C₄	—	4	8	no	—
450	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
451	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
452	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
453	C₂²⋊C₄	C₂²×C₄×C₆₇	1	16	yes	—
454	C₄²	—	4	4	no	—
455	C₄²	—	4	8	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	4	no	—
459	C₂³	—	2	16	no	—
460	C₂³	—	2	8	no	—
461	C₂³	—	2	16	no	—
462	C₂³	—	2	16	no	—
463	C₂³	—	2	16	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	4	no	—
466	C₂³	—	2	4	no	—
467	C₂³	—	2	4	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	2	8	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	4	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	8	no	—
475	C₂³	—	4	8	no	—
476	C₂³	—	4	8	no	—
477	C₂³	—	4	4	no	—
478	C₂³	—	4	8	no	—
479	C₂³	—	4	8	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	4	8	no	—
482	C₂³	—	4	8	no	—
483	C₂³	—	4	16	no	—
484	C₂³	—	4	32	no	—
485	C₂³	—	4	32	no	—
486	C₂³	—	4	16	no	—
487	C₂³	—	4	128	no	—
488	C₂³	—	4	16	no	—
489	C₂³	—	8	8	no	—
490	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
491	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
492	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
493	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
494	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
495	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
496	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
497	C₂³	H(5,𝔽₂)×C₆₇	1	8	yes	—
498	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
499	D₄	—	2	32	no	—
500	D₄	—	4	8	no	—
501	D₄	—	4	8	no	—
502	D₄	—	4	16	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	8	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	8	16	no	—
515	C₂×C₄	—	8	8	no	—
516	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
517	C₂×C₄	C₂³×C₄×C₆₇	1	8	yes	—
518	C₂²	—	2	16	no	—
519	C₂²	—	2	8	no	—
520	C₂²	—	2	4	no	—
521	C₂²	—	2	4	no	—
522	C₂²	—	2	4	no	—
523	C₂²	—	2	8	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	4	no	—
532	C₂²	—	4	32	no	—
533	C₂²	—	4	16	no	—
534	C₂²	—	4	16	no	—
535	C₂²	—	4	32	no	—
536	C₂²	—	4	16	no	—
537	C₂²	—	4	4	no	—
538	C₂²	—	4	4	no	—
539	C₂²	—	4	8	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	8	8	no	—
543	C₂²	#518×C₆₇	1	4	yes	—
544	C₂²	#518×C₆₇	1	8	yes	—
545	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
546	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
547	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	4	yes	—
548	C₄	—	2	4	no	—
549	C₄	—	2	8	no	—
550	C₄	—	4	8	no	—
551	C₄	—	4	8	no	—
552	C₂	—	2	4	no	—
553	C₂	—	2	8	no	—
554	C₂	—	4	4	no	—
555	C₂	—	4	2	no	—
556	C₂	—	4	2	no	—
557	C₂	—	4	4	no	—
558	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
559	C₂	C₂×#518×C₆₇	1	2	yes	—
560	C₂	C₆₇×#3532	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#120×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2375	C₂	semidirect
3. C₆₇×#2358	C₂	semidirect
4. C₆₇×#2342	C₂	semidirect
5. D₄×C₂²⋊C₄×C₆₇	C₂	semidirect
6. C₆₇×#2321	C₂	semidirect
7. C₆₇×#2312	C₂	semidirect
8. C₆₇×#2309	C₂	semidirect
9. C₆₇×#2308	C₂	semidirect
10. C₂×#379×C₆₇	C₂	semidirect
14. C₂×#120×C₆₇	C₄	semidirect
18. C₂×C₄×D₄×C₆₇	C₂²	semidirect
19. C₂×#117×C₆₇	C₂²	semidirect
23. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
24. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
25. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
26. #390×C₆₇	C₂²	semidirect
27. #390×C₆₇	C₂²	semidirect
28. #390×C₆₇	C₂²	semidirect
29. #388×C₆₇	C₂²	semidirect
30. #388×C₆₇	C₂²	semidirect
31. #388×C₆₇	C₂²	semidirect
32. #388×C₆₇	C₂²	semidirect
33. #386×C₆₇	C₂²	semidirect
34. #386×C₆₇	C₂²	semidirect
35. #386×C₆₇	C₂²	semidirect
36. #386×C₆₇	C₂²	semidirect
37. #385×C₆₇	C₂²	semidirect
38. #381×C₆₇	C₂²	semidirect
39. #380×C₆₇	C₂²	semidirect
43. #379×C₆₇	C₂²	semidirect
44. #379×C₆₇	C₂²	semidirect
45. #379×C₆₇	C₂²	semidirect
46. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
47. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
50. C₂⁵×C₆₇	C₂×C₄	semidirect
59. C₂²×D₄×C₆₇	C₂×C₄	semidirect
60. C₂²×D₄×C₆₇	C₂×C₄	semidirect
68. C₂³×C₄×C₆₇	C₂³	semidirect
69. #120×C₆₇	C₂×C₄	semidirect
70. #120×C₆₇	C₂×C₄	semidirect
71. #120×C₆₇	C₂×C₄	semidirect
74. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
75. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
80. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
81. C₂×C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
95. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
96. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
97. C₂×C₄²×C₆₇	C₂³	semidirect
98. C₂×C₄²×C₆₇	C₂³	semidirect
99. #95×C₆₇	C₂³	semidirect
100. #95×C₆₇	C₂³	semidirect
101. #95×C₆₇	C₂³	semidirect
271. #34356	C₆₇	direct (N × H)
282. #2375	C₂×C₆₇	semidirect
283. #2358	C₂×C₆₇	semidirect
284. #2342	C₂×C₆₇	semidirect
285. D₄×C₂²⋊C₄	C₂×C₆₇	semidirect
286. #2321	C₂×C₆₇	semidirect
287. #2312	C₂×C₆₇	semidirect
288. #2309	C₂×C₆₇	semidirect
289. #2308	C₂×C₆₇	semidirect
290. C₂×#379	C₂×C₆₇	semidirect
291. C₆₇	#34356	direct (N × H)
295. C₂×#120	C₄×C₆₇	semidirect
299. C₂×C₄×D₄	C₂²×C₆₇	semidirect
300. C₂×#117	C₂²×C₆₇	semidirect
304. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
305. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
306. C₂²×C₂²⋊C₄	C₂²×C₆₇	semidirect
307. #390	C₂²×C₆₇	semidirect
308. #390	C₂²×C₆₇	semidirect
309. #390	C₂²×C₆₇	semidirect
310. #388	C₂²×C₆₇	semidirect
311. #388	C₂²×C₆₇	semidirect
312. #388	C₂²×C₆₇	semidirect
313. #388	C₂²×C₆₇	semidirect
314. #386	C₂²×C₆₇	semidirect
315. #386	C₂²×C₆₇	semidirect
316. #386	C₂²×C₆₇	semidirect
317. #386	C₂²×C₆₇	semidirect
318. #385	C₂²×C₆₇	semidirect
319. #381	C₂²×C₆₇	semidirect
320. #380	C₂²×C₆₇	semidirect
324. #379	C₂²×C₆₇	semidirect
325. #379	C₂²×C₆₇	semidirect
326. #379	C₂²×C₆₇	semidirect
327. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
328. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
331. C₂⁵	C₂×C₄×C₆₇	semidirect
340. C₂²×D₄	C₂×C₄×C₆₇	semidirect
341. C₂²×D₄	C₂×C₄×C₆₇	semidirect
349. C₂³×C₄	C₂³×C₆₇	semidirect
350. #120	C₂×C₄×C₆₇	semidirect
351. #120	C₂×C₄×C₆₇	semidirect
352. #120	C₂×C₄×C₆₇	semidirect
355. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
356. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
361. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
362. C₂×C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
376. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
377. C₂×C₂²⋊C₄	C₂³×C₆₇	semidirect
378. C₂×C₄²	C₂³×C₆₇	semidirect
379. C₂×C₄²	C₂³×C₆₇	semidirect
380. #95	C₂³×C₆₇	semidirect
381. #95	C₂³×C₆₇	semidirect
382. #95	C₂³×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#120×C₆₇	2
C₆₇×#2375	2
C₆₇×#2358	2
C₆₇×#2342	2
D₄×C₂²⋊C₄×C₆₇	2
C₆₇×#2321	2
C₆₇×#2312	2
C₆₇×#2309	2
C₆₇×#2308	2
C₂×#379×C₆₇	2
#34356	67

Subgroups — C67×#34356 #30713607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34356 #30713607