Subgroups — C₂×C₆₇×#2623 #30713830

Order 17152.

3002 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#123×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2623	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#399×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#398×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#386×C₆₇	C₂	1	2	yes	maximal
9	C₂×#385×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#381×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
14	C₂×#123×C₆₇	—	2	8	no	—
15	C₂×#123×C₆₇	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₆₇	—	2	8	no	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
28	#400×C₆₇	C₂²	1	16	yes	complemented
29	#399×C₆₇	C₂²	1	16	yes	complemented
30	#398×C₆₇	C₂²	1	16	yes	complemented
31	#397×C₆₇	C₂²	1	16	yes	complemented
32	#386×C₆₇	C₂²	1	16	yes	—
33	#385×C₆₇	C₂²	1	16	yes	complemented
34	#381×C₆₇	C₂²	1	8	yes	complemented
35	C₂×#95×C₆₇	C₂²	1	2	yes	—
36	C₂×#95×C₆₇	C₂²	1	2	yes	—
37	C₂×#95×C₆₇	C₂²	1	4	yes	—
38	C₂×#95×C₆₇	C₂²	1	4	yes	—
39	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
40	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₆₇	—	2	4	no	—
42	C₂²×D₄×C₆₇	—	2	8	no	—
43	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₆₇	—	2	8	no	—
45	C₂³×C₄×C₆₇	—	2	8	no	—
46	C₂³×C₄×C₆₇	—	4	8	no	—
47	C₂³×C₄×C₆₇	D₄	1	8	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
57	#123×C₆₇	—	2	32	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	32	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
79	#95×C₆₇	C₂³	1	8	yes	—
80	#95×C₆₇	C₂³	1	8	yes	—
81	#95×C₆₇	C₂³	1	16	yes	—
82	#95×C₆₇	C₂³	1	16	yes	—
83	C₂⁴×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	—	2	4	no	—
85	C₂⁴×C₆₇	—	2	2	no	—
86	C₂⁴×C₆₇	—	2	4	no	—
87	C₂⁴×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	—	2	4	no	—
89	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
90	C₂⁴×C₆₇	#41	1	8	yes	—
91	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
92	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
93	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₂×D₄×C₆₇	—	2	8	no	—
95	C₂×D₄×C₆₇	—	2	16	no	—
96	C₂×D₄×C₆₇	—	4	16	no	—
97	C₂²×C₄×C₆₇	—	2	32	no	—
98	C₂²×C₄×C₆₇	—	2	16	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	4	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	16	no	—
104	C₂²×C₄×C₆₇	—	2	8	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	8	no	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	8	no	—
109	C₂²×C₄×C₆₇	—	2	8	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	8	no	—
114	C₂²×C₄×C₆₇	—	4	16	no	—
115	C₂²×C₄×C₆₇	—	4	32	no	—
116	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
117	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
119	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
126	C₄⋊C₄×C₆₇	—	2	16	no	—
127	C₄⋊C₄×C₆₇	—	2	16	no	—
128	C₄⋊C₄×C₆₇	—	2	8	no	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	4	16	no	—
131	C₂²⋊C₄×C₆₇	—	2	8	no	—
132	C₂²⋊C₄×C₆₇	—	2	16	no	—
133	C₂²⋊C₄×C₆₇	—	2	8	no	—
134	C₂²⋊C₄×C₆₇	—	2	8	no	—
135	C₂²⋊C₄×C₆₇	—	4	16	no	—
136	C₂³×C₆₇	—	2	8	no	—
137	C₂³×C₆₇	—	2	8	no	—
138	C₂³×C₆₇	—	2	32	no	—
139	C₂³×C₆₇	—	2	4	no	—
140	C₂³×C₆₇	—	2	4	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	2	4	no	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	2	4	no	—
145	C₂³×C₆₇	—	4	4	no	—
146	C₂³×C₆₇	—	4	4	no	—
147	C₂³×C₆₇	—	4	4	no	—
148	C₂³×C₆₇	—	4	16	no	—
149	C₂³×C₆₇	#123	1	16	yes	—
150	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
151	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
152	C₂³×C₆₇	C₂×#41	1	8	yes	—
153	C₂³×C₆₇	C₂×#41	1	4	yes	—
154	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
155	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
156	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
157	D₄×C₆₇	—	4	16	no	—
158	C₂×C₄×C₆₇	—	2	32	no	—
159	C₂×C₄×C₆₇	—	2	8	no	—
160	C₂×C₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	2	4	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	8	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	8	no	—
170	C₂×C₄×C₆₇	—	2	8	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	8	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	—	4	32	no	—
176	C₂×C₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	4	4	no	—
178	C₂×C₄×C₆₇	—	4	16	no	—
179	C₂×C₄×C₆₇	—	4	4	no	—
180	C₂×C₄×C₆₇	—	4	16	no	—
181	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
182	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
183	C₂²×C₆₇	—	2	32	no	—
184	C₂²×C₆₇	—	2	4	no	—
185	C₂²×C₆₇	—	2	4	no	—
186	C₂²×C₆₇	—	2	16	no	—
187	C₂²×C₆₇	—	4	2	no	—
188	C₂²×C₆₇	—	4	2	no	—
189	C₂²×C₆₇	—	4	2	no	—
190	C₂²×C₆₇	—	4	8	no	—
191	C₂²×C₆₇	—	8	8	no	—
192	C₂²×C₆₇	C₂×#123	1	8	yes	—
193	C₂²×C₆₇	C₂×#123	1	4	yes	—
194	C₂²×C₆₇	#535	1	4	yes	—
195	C₂²×C₆₇	#536	1	4	yes	—
196	C₂²×C₆₇	#539	1	16	yes	—
197	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
198	C₂²×C₆₇	C₂²×#41	1	2	yes	—
199	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
200	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
201	C₄×C₆₇	—	2	4	no	—
202	C₄×C₆₇	—	2	4	no	—
203	C₄×C₆₇	—	4	2	no	—
204	C₄×C₆₇	—	4	2	no	—
205	C₄×C₆₇	—	4	8	no	—
206	C₄×C₆₇	—	4	2	no	—
207	C₄×C₆₇	—	4	8	no	—
208	C₂×#2623	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₆₇	—	2	8	no	—
210	C₂×C₆₇	—	8	2	no	—
211	C₂×C₆₇	#2623	1	8	yes	direct product factor
212	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
213	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
214	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
215	C₂×C₆₇	C₂×#539	1	4	yes	—
216	C₂²×#123	C₂×C₆₇	1	1	yes	characteristic
217	C₂³×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
218	#2623	C₂×C₆₇	1	16	yes	direct product factor
219	C₂×#400	C₂×C₆₇	1	2	yes	complemented
220	C₂×#399	C₂×C₆₇	1	2	yes	complemented
221	C₂×#398	C₂×C₆₇	1	2	yes	complemented
222	C₂×#397	C₂×C₆₇	1	2	yes	complemented
223	C₂×#386	C₂×C₆₇	1	2	yes	—
224	C₂×#385	C₂×C₆₇	1	2	yes	complemented
225	C₂×#381	C₂×C₆₇	1	1	yes	characteristic, complemented
226	C₆₇	C₂×#2623	1	1	yes	characteristic, direct product factor, Sylow
227	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
228	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
229	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
230	C₂×#123	—	2	8	no	—
231	C₂×#123	C₂²×C₆₇	1	8	yes	—
232	C₂²×C₄⋊C₄	—	2	8	no	—
233	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
234	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
235	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
236	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
237	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
238	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
239	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
240	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
241	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
242	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
243	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
244	#400	C₂²×C₆₇	1	16	yes	complemented
245	#399	C₂²×C₆₇	1	16	yes	complemented
246	#398	C₂²×C₆₇	1	16	yes	complemented
247	#397	C₂²×C₆₇	1	16	yes	complemented
248	#386	C₂²×C₆₇	1	16	yes	—
249	#385	C₂²×C₆₇	1	16	yes	complemented
250	#381	C₂²×C₆₇	1	8	yes	complemented
251	C₂×#95	C₂²×C₆₇	1	2	yes	—
252	C₂×#95	C₂²×C₆₇	1	4	yes	—
253	C₂×#95	C₂²×C₆₇	1	4	yes	—
254	C₂×#95	C₂²×C₆₇	1	2	yes	—
255	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
256	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
257	C₂²×D₄	—	2	4	no	—
258	C₂²×D₄	—	2	8	no	—
259	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
260	C₂³×C₄	—	2	8	no	—
261	C₂³×C₄	—	2	8	no	—
262	C₂³×C₄	—	4	8	no	—
263	C₂³×C₄	D₄×C₆₇	1	8	yes	—
264	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
266	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
267	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
268	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
269	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
270	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
271	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
272	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
273	#123	—	2	32	no	—
274	C₂×C₄⋊C₄	—	2	8	no	—
275	C₂×C₄⋊C₄	—	2	8	no	—
276	C₂×C₄⋊C₄	—	2	32	no	—
277	C₂×C₄⋊C₄	—	2	4	no	—
278	C₂×C₄⋊C₄	—	2	4	no	—
279	C₂×C₄⋊C₄	—	4	8	no	—
280	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
281	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
282	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
285	C₂×C₂²⋊C₄	—	2	4	no	—
286	C₂×C₂²⋊C₄	—	2	8	no	—
287	C₂×C₂²⋊C₄	—	2	4	no	—
288	C₂×C₂²⋊C₄	—	2	4	no	—
289	C₂×C₂²⋊C₄	—	4	8	no	—
290	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
293	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
294	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
295	#95	C₂³×C₆₇	1	8	yes	—
296	#95	C₂³×C₆₇	1	8	yes	—
297	#95	C₂³×C₆₇	1	16	yes	—
298	#95	C₂³×C₆₇	1	16	yes	—
299	C₂⁴	—	2	4	no	—
300	C₂⁴	—	2	4	no	—
301	C₂⁴	—	2	2	no	—
302	C₂⁴	—	2	4	no	—
303	C₂⁴	—	2	4	no	—
304	C₂⁴	—	2	4	no	—
305	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
306	C₂⁴	#41×C₆₇	1	8	yes	—
307	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
308	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
309	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
310	C₂×D₄	—	2	8	no	—
311	C₂×D₄	—	2	16	no	—
312	C₂×D₄	—	4	16	no	—
313	C₂²×C₄	—	2	32	no	—
314	C₂²×C₄	—	2	16	no	—
315	C₂²×C₄	—	2	4	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	16	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	8	no	—
323	C₂²×C₄	—	2	4	no	—
324	C₂²×C₄	—	2	8	no	—
325	C₂²×C₄	—	2	8	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	8	no	—
330	C₂²×C₄	—	4	16	no	—
331	C₂²×C₄	—	4	32	no	—
332	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
333	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
335	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
336	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
337	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
339	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
341	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
342	C₄⋊C₄	—	2	16	no	—
343	C₄⋊C₄	—	2	16	no	—
344	C₄⋊C₄	—	2	8	no	—
345	C₄⋊C₄	—	2	8	no	—
346	C₄⋊C₄	—	4	16	no	—
347	C₂²⋊C₄	—	2	8	no	—
348	C₂²⋊C₄	—	2	16	no	—
349	C₂²⋊C₄	—	2	8	no	—
350	C₂²⋊C₄	—	2	8	no	—
351	C₂²⋊C₄	—	4	16	no	—
352	C₂³	—	2	8	no	—
353	C₂³	—	2	8	no	—
354	C₂³	—	2	32	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	2	2	no	—
358	C₂³	—	2	4	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	2	4	no	—
361	C₂³	—	4	4	no	—
362	C₂³	—	4	4	no	—
363	C₂³	—	4	4	no	—
364	C₂³	—	4	16	no	—
365	C₂³	#123×C₆₇	1	16	yes	—
366	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
367	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
368	C₂³	C₂×#41×C₆₇	1	8	yes	—
369	C₂³	C₂×#41×C₆₇	1	4	yes	—
370	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
371	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
372	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
373	D₄	—	4	16	no	—
374	C₂×C₄	—	2	32	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	2	8	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	8	no	—
386	C₂×C₄	—	2	8	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	8	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	4	32	no	—
392	C₂×C₄	—	4	4	no	—
393	C₂×C₄	—	4	16	no	—
394	C₂×C₄	—	4	4	no	—
395	C₂×C₄	—	4	4	no	—
396	C₂×C₄	—	4	16	no	—
397	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
398	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
399	C₂²	—	2	32	no	—
400	C₂²	—	2	4	no	—
401	C₂²	—	2	4	no	—
402	C₂²	—	2	16	no	—
403	C₂²	—	4	2	no	—
404	C₂²	—	4	2	no	—
405	C₂²	—	4	2	no	—
406	C₂²	—	4	8	no	—
407	C₂²	—	8	8	no	—
408	C₂²	C₂×#123×C₆₇	1	8	yes	—
409	C₂²	C₂×#123×C₆₇	1	4	yes	—
410	C₂²	#535×C₆₇	1	4	yes	—
411	C₂²	#536×C₆₇	1	4	yes	—
412	C₂²	#539×C₆₇	1	16	yes	—
413	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
414	C₂²	C₂²×#41×C₆₇	1	2	yes	—
415	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
416	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
417	C₄	—	2	4	no	—
418	C₄	—	2	4	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	8	no	—
421	C₄	—	4	2	no	—
422	C₄	—	4	2	no	—
423	C₄	—	4	8	no	—
424	C₂	—	2	8	no	—
425	C₂	—	8	2	no	—
426	C₂	C₆₇×#2623	1	8	yes	direct product factor
427	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
428	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
429	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
430	C₂	C₂×#539×C₆₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊C₄×C₆₇	C₂	semidirect
3. C₆₇×#2623	C₂	direct (N × H)
4. C₂×#400×C₆₇	C₂	semidirect
5. C₂×#399×C₆₇	C₂	semidirect
6. C₂×#398×C₆₇	C₂	semidirect
7. C₂×#397×C₆₇	C₂	semidirect
9. C₂×#385×C₆₇	C₂	semidirect
10. C₂×#381×C₆₇	C₂	semidirect
22. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
28. #400×C₆₇	C₂²	semidirect
29. #399×C₆₇	C₂²	semidirect
30. #398×C₆₇	C₂²	semidirect
31. #397×C₆₇	C₂²	semidirect
33. #385×C₆₇	C₂²	semidirect
34. #381×C₆₇	C₂²	semidirect
208. C₂×#2623	C₆₇	direct (N × H)
211. C₂×C₆₇	#2623	direct (N × H)
217. C₂³×C₄⋊C₄	C₂×C₆₇	semidirect
218. #2623	C₂×C₆₇	direct (N × H)
219. C₂×#400	C₂×C₆₇	semidirect
220. C₂×#399	C₂×C₆₇	semidirect
221. C₂×#398	C₂×C₆₇	semidirect
222. C₂×#397	C₂×C₆₇	semidirect
224. C₂×#385	C₂×C₆₇	semidirect
225. C₂×#381	C₂×C₆₇	semidirect
226. C₆₇	C₂×#2623	direct (N × H)
238. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
244. #400	C₂²×C₆₇	semidirect
245. #399	C₂²×C₆₇	semidirect
246. #398	C₂²×C₆₇	semidirect
247. #397	C₂²×C₆₇	semidirect
249. #385	C₂²×C₆₇	semidirect
250. #381	C₂²×C₆₇	semidirect
426. C₂	C₆₇×#2623	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#123×C₆₇	2
C₂³×C₄⋊C₄×C₆₇	2
C₆₇×#2623	2
C₂×#400×C₆₇	2
C₂×#399×C₆₇	2
C₂×#398×C₆₇	2
C₂×#397×C₆₇	2
C₂×#386×C₆₇	2
C₂×#385×C₆₇	2
C₂×#381×C₆₇	2
C₂×#2623	67

Subgroups — C2×C67×#2623 #30713830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2623 #30713830