Subgroups — C₂×C₆₇×#2639 #30713814

Order 17152.

3326 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2639	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#394×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#393×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#386×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₂²×#95×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₆₇	—	2	8	no	—
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
17	C₂×#121×C₆₇	—	2	8	no	—
18	C₂×#121×C₆₇	C₂²	1	8	yes	—
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	1	yes	characteristic
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
26	#400×C₆₇	C₂²	1	16	yes	complemented
27	#397×C₆₇	C₂²	1	16	yes	complemented
28	#394×C₆₇	C₂²	1	16	yes	complemented
29	#393×C₆₇	C₂²	1	16	yes	complemented
30	#386×C₆₇	C₂²	1	32	yes	complemented
31	C₂×#95×C₆₇	C₂²	1	2	yes	—
32	C₂×#95×C₆₇	C₂²	1	2	yes	—
33	C₂×#95×C₆₇	C₂²	1	4	yes	—
34	C₂×#95×C₆₇	C₂²	1	4	yes	—
35	C₂×#95×C₆₇	C₂²	1	8	yes	complemented
36	C₂⁵×C₆₇	C₂³	1	2	yes	—
37	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×Q₈×C₆₇	—	2	4	no	—
39	C₂²×Q₈×C₆₇	—	4	4	no	—
40	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
41	C₂²×D₄×C₆₇	—	2	8	no	—
42	C₂²×D₄×C₆₇	—	2	16	no	—
43	C₂²×D₄×C₆₇	—	2	4	no	—
44	C₂²×D₄×C₆₇	—	4	4	no	—
45	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
46	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₆₇	—	2	8	no	—
48	C₂³×C₄×C₆₇	—	4	4	no	—
49	C₂³×C₄×C₆₇	—	4	4	no	—
50	C₂³×C₄×C₆₇	—	4	4	no	—
51	C₂³×C₄×C₆₇	D₄	1	8	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
60	C₂²⋊Q₈×C₆₇	—	2	32	no	—
61	#121×C₆₇	—	2	32	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
70	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
71	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
72	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
73	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
74	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
75	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
76	#95×C₆₇	C₂³	1	8	yes	—
77	#95×C₆₇	C₂³	1	8	yes	—
78	#95×C₆₇	C₂³	1	16	yes	—
79	#95×C₆₇	C₂³	1	16	yes	—
80	C₂⁴×C₆₇	—	2	4	no	—
81	C₂⁴×C₆₇	—	2	2	no	—
82	C₂⁴×C₆₇	—	2	2	no	—
83	C₂⁴×C₆₇	—	2	4	no	—
84	C₂⁴×C₆₇	—	2	8	no	—
85	C₂⁴×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	—	2	8	no	—
87	C₂⁴×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
89	C₂⁴×C₆₇	#41	1	8	yes	—
90	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
91	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
92	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₂×Q₈×C₆₇	—	2	8	no	—
94	C₂×Q₈×C₆₇	—	4	8	no	—
95	C₂×Q₈×C₆₇	—	4	16	no	—
96	C₂×D₄×C₆₇	—	2	16	no	—
97	C₂×D₄×C₆₇	—	2	32	no	—
98	C₂×D₄×C₆₇	—	2	8	no	—
99	C₂×D₄×C₆₇	—	4	8	no	—
100	C₂×D₄×C₆₇	—	4	16	no	—
101	C₂×D₄×C₆₇	—	4	32	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	4	no	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	16	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	8	no	—
113	C₂²×C₄×C₆₇	—	2	8	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	4	16	no	—
116	C₂²×C₄×C₆₇	—	4	8	no	—
117	C₂²×C₄×C₆₇	—	4	8	no	—
118	C₂²×C₄×C₆₇	—	4	8	no	—
119	C₂²×C₄×C₆₇	—	4	32	no	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
123	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
125	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
126	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
129	C₄⋊C₄×C₆₇	—	2	8	no	—
130	C₄⋊C₄×C₆₇	—	2	8	no	—
131	C₄⋊C₄×C₆₇	—	2	16	no	—
132	C₂²⋊C₄×C₆₇	—	2	16	no	—
133	C₂²⋊C₄×C₆₇	—	2	16	no	—
134	C₂²⋊C₄×C₆₇	—	4	16	no	—
135	C₂²⋊C₄×C₆₇	—	4	16	no	—
136	C₂³×C₆₇	—	2	8	no	—
137	C₂³×C₆₇	—	2	8	no	—
138	C₂³×C₆₇	—	2	16	no	—
139	C₂³×C₆₇	—	2	16	no	—
140	C₂³×C₆₇	—	2	4	no	—
141	C₂³×C₆₇	—	2	2	no	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	—	2	4	no	—
144	C₂³×C₆₇	—	2	8	no	—
145	C₂³×C₆₇	—	2	4	no	—
146	C₂³×C₆₇	—	2	8	no	—
147	C₂³×C₆₇	—	2	4	no	—
148	C₂³×C₆₇	—	4	8	no	—
149	C₂³×C₆₇	—	4	8	no	—
150	C₂³×C₆₇	—	4	8	no	—
151	C₂³×C₆₇	—	4	16	no	—
152	C₂³×C₆₇	—	4	16	no	—
153	C₂³×C₆₇	#124	1	16	yes	—
154	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
155	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
156	C₂³×C₆₇	C₂×#41	1	8	yes	—
157	C₂³×C₆₇	C₂×#41	1	4	yes	—
158	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
159	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
160	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
161	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
162	Q₈×C₆₇	—	4	16	no	—
163	D₄×C₆₇	—	4	16	no	—
164	D₄×C₆₇	—	4	32	no	—
165	C₂×C₄×C₆₇	—	2	8	no	—
166	C₂×C₄×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	4	no	—
170	C₂×C₄×C₆₇	—	2	4	no	—
171	C₂×C₄×C₆₇	—	2	4	no	—
172	C₂×C₄×C₆₇	—	2	8	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	2	4	no	—
175	C₂×C₄×C₆₇	—	2	8	no	—
176	C₂×C₄×C₆₇	—	2	8	no	—
177	C₂×C₄×C₆₇	—	2	8	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	4	16	no	—
180	C₂×C₄×C₆₇	—	4	32	no	—
181	C₂×C₄×C₆₇	—	4	8	no	—
182	C₂×C₄×C₆₇	—	4	8	no	—
183	C₂×C₄×C₆₇	—	4	8	no	—
184	C₂×C₄×C₆₇	—	4	16	no	—
185	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
186	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
187	C₂²×C₆₇	—	2	32	no	—
188	C₂²×C₆₇	—	2	4	no	—
189	C₂²×C₆₇	—	2	4	no	—
190	C₂²×C₆₇	—	2	8	no	—
191	C₂²×C₆₇	—	2	8	no	—
192	C₂²×C₆₇	—	4	4	no	—
193	C₂²×C₆₇	—	4	4	no	—
194	C₂²×C₆₇	—	4	4	no	—
195	C₂²×C₆₇	—	4	8	no	—
196	C₂²×C₆₇	—	4	8	no	—
197	C₂²×C₆₇	—	8	16	no	—
198	C₂²×C₆₇	C₂×#124	1	8	yes	—
199	C₂²×C₆₇	C₂×#124	1	4	yes	—
200	C₂²×C₆₇	#534	1	4	yes	—
201	C₂²×C₆₇	#537	1	4	yes	—
202	C₂²×C₆₇	#538	1	8	yes	—
203	C₂²×C₆₇	#539	1	8	yes	—
204	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
205	C₂²×C₆₇	C₂²×#41	1	2	yes	—
206	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
207	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
208	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
209	C₄×C₆₇	—	2	4	no	—
210	C₄×C₆₇	—	2	4	no	—
211	C₄×C₆₇	—	4	4	no	—
212	C₄×C₆₇	—	4	4	no	—
213	C₄×C₆₇	—	4	4	no	—
214	C₄×C₆₇	—	4	8	no	—
215	C₂×#2639	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂×C₆₇	—	2	8	no	—
217	C₂×C₆₇	—	8	4	no	—
218	C₂×C₆₇	#2639	1	8	yes	direct product factor
219	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
220	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
221	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
222	C₂×C₆₇	C₂×#538	1	2	yes	—
223	C₂×C₆₇	C₂×#539	1	2	yes	—
224	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
225	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
226	#2639	C₂×C₆₇	1	16	yes	direct product factor
227	C₂×#400	C₂×C₆₇	1	2	yes	complemented
228	C₂×#397	C₂×C₆₇	1	2	yes	complemented
229	C₂×#394	C₂×C₆₇	1	2	yes	complemented
230	C₂×#393	C₂×C₆₇	1	2	yes	complemented
231	C₂×#386	C₂×C₆₇	1	4	yes	complemented
232	C₂²×#95	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₆₇	C₂×#2639	1	1	yes	characteristic, direct product factor, Sylow
234	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
235	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
236	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
237	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
238	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
239	C₂×C₂²⋊Q₈	—	2	8	no	—
240	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
241	C₂×#121	—	2	8	no	—
242	C₂×#121	C₂²×C₆₇	1	8	yes	—
243	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
244	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
245	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
246	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
247	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
248	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
249	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
250	#400	C₂²×C₆₇	1	16	yes	complemented
251	#397	C₂²×C₆₇	1	16	yes	complemented
252	#394	C₂²×C₆₇	1	16	yes	complemented
253	#393	C₂²×C₆₇	1	16	yes	complemented
254	#386	C₂²×C₆₇	1	32	yes	complemented
255	C₂×#95	C₂²×C₆₇	1	2	yes	—
256	C₂×#95	C₂²×C₆₇	1	4	yes	—
257	C₂×#95	C₂²×C₆₇	1	2	yes	—
258	C₂×#95	C₂²×C₆₇	1	4	yes	—
259	C₂×#95	C₂²×C₆₇	1	8	yes	complemented
260	C₂⁵	C₂³×C₆₇	1	2	yes	—
261	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
262	C₂²×Q₈	—	2	4	no	—
263	C₂²×Q₈	—	4	4	no	—
264	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
265	C₂²×D₄	—	2	8	no	—
266	C₂²×D₄	—	2	16	no	—
267	C₂²×D₄	—	2	4	no	—
268	C₂²×D₄	—	4	4	no	—
269	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
270	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
271	C₂³×C₄	—	2	8	no	—
272	C₂³×C₄	—	4	4	no	—
273	C₂³×C₄	—	4	4	no	—
274	C₂³×C₄	—	4	4	no	—
275	C₂³×C₄	D₄×C₆₇	1	8	yes	—
276	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
277	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
278	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
279	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
280	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
283	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
284	C₂²⋊Q₈	—	2	32	no	—
285	#121	—	2	32	no	—
286	C₂×C₄⋊C₄	—	2	4	no	—
287	C₂×C₄⋊C₄	—	2	4	no	—
288	C₂×C₄⋊C₄	—	2	8	no	—
289	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
291	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
292	C₂×C₂²⋊C₄	—	2	8	no	—
293	C₂×C₂²⋊C₄	—	2	8	no	—
294	C₂×C₂²⋊C₄	—	4	8	no	—
295	C₂×C₂²⋊C₄	—	4	8	no	—
296	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
297	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
298	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
299	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
300	#95	C₂³×C₆₇	1	8	yes	—
301	#95	C₂³×C₆₇	1	8	yes	—
302	#95	C₂³×C₆₇	1	16	yes	—
303	#95	C₂³×C₆₇	1	16	yes	—
304	C₂⁴	—	2	4	no	—
305	C₂⁴	—	2	2	no	—
306	C₂⁴	—	2	2	no	—
307	C₂⁴	—	2	4	no	—
308	C₂⁴	—	2	8	no	—
309	C₂⁴	—	2	4	no	—
310	C₂⁴	—	2	8	no	—
311	C₂⁴	—	2	4	no	—
312	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
313	C₂⁴	#41×C₆₇	1	8	yes	—
314	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
315	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
316	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
317	C₂×Q₈	—	2	8	no	—
318	C₂×Q₈	—	4	8	no	—
319	C₂×Q₈	—	4	16	no	—
320	C₂×D₄	—	2	16	no	—
321	C₂×D₄	—	2	32	no	—
322	C₂×D₄	—	2	8	no	—
323	C₂×D₄	—	4	8	no	—
324	C₂×D₄	—	4	16	no	—
325	C₂×D₄	—	4	32	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	4	no	—
329	C₂²×C₄	—	2	4	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	16	no	—
332	C₂²×C₄	—	2	8	no	—
333	C₂²×C₄	—	2	8	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	8	no	—
336	C₂²×C₄	—	2	8	no	—
337	C₂²×C₄	—	2	8	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	4	16	no	—
340	C₂²×C₄	—	4	8	no	—
341	C₂²×C₄	—	4	8	no	—
342	C₂²×C₄	—	4	8	no	—
343	C₂²×C₄	—	4	32	no	—
344	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
345	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
346	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
347	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
348	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
349	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
350	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
351	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
353	C₄⋊C₄	—	2	8	no	—
354	C₄⋊C₄	—	2	8	no	—
355	C₄⋊C₄	—	2	16	no	—
356	C₂²⋊C₄	—	2	16	no	—
357	C₂²⋊C₄	—	2	16	no	—
358	C₂²⋊C₄	—	4	16	no	—
359	C₂²⋊C₄	—	4	16	no	—
360	C₂³	—	2	8	no	—
361	C₂³	—	2	8	no	—
362	C₂³	—	2	16	no	—
363	C₂³	—	2	16	no	—
364	C₂³	—	2	4	no	—
365	C₂³	—	2	2	no	—
366	C₂³	—	2	2	no	—
367	C₂³	—	2	4	no	—
368	C₂³	—	2	8	no	—
369	C₂³	—	2	4	no	—
370	C₂³	—	2	8	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	4	8	no	—
373	C₂³	—	4	8	no	—
374	C₂³	—	4	8	no	—
375	C₂³	—	4	16	no	—
376	C₂³	—	4	16	no	—
377	C₂³	#124×C₆₇	1	16	yes	—
378	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
379	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
380	C₂³	C₂×#41×C₆₇	1	8	yes	—
381	C₂³	C₂×#41×C₆₇	1	4	yes	—
382	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
383	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
384	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
385	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
386	Q₈	—	4	16	no	—
387	D₄	—	4	16	no	—
388	D₄	—	4	32	no	—
389	C₂×C₄	—	2	8	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	8	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	8	no	—
400	C₂×C₄	—	2	8	no	—
401	C₂×C₄	—	2	8	no	—
402	C₂×C₄	—	2	4	no	—
403	C₂×C₄	—	4	16	no	—
404	C₂×C₄	—	4	32	no	—
405	C₂×C₄	—	4	8	no	—
406	C₂×C₄	—	4	8	no	—
407	C₂×C₄	—	4	8	no	—
408	C₂×C₄	—	4	16	no	—
409	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
410	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
411	C₂²	—	2	32	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	2	8	no	—
415	C₂²	—	2	8	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	4	4	no	—
418	C₂²	—	4	4	no	—
419	C₂²	—	4	8	no	—
420	C₂²	—	4	8	no	—
421	C₂²	—	8	16	no	—
422	C₂²	C₂×#124×C₆₇	1	8	yes	—
423	C₂²	C₂×#124×C₆₇	1	4	yes	—
424	C₂²	#534×C₆₇	1	4	yes	—
425	C₂²	#537×C₆₇	1	4	yes	—
426	C₂²	#538×C₆₇	1	8	yes	—
427	C₂²	#539×C₆₇	1	8	yes	—
428	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
429	C₂²	C₂²×#41×C₆₇	1	2	yes	—
430	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
431	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
432	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
433	C₄	—	2	4	no	—
434	C₄	—	2	4	no	—
435	C₄	—	4	4	no	—
436	C₄	—	4	4	no	—
437	C₄	—	4	4	no	—
438	C₄	—	4	8	no	—
439	C₂	—	2	8	no	—
440	C₂	—	8	4	no	—
441	C₂	C₆₇×#2639	1	8	yes	direct product factor
442	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
443	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
444	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
445	C₂	C₂×#538×C₆₇	1	2	yes	—
446	C₂	C₂×#539×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#121×C₆₇	C₂
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
1. C₂²×C₂²⋊Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2639	C₂	direct (N × H)
4. C₂×#400×C₆₇	C₂	semidirect
5. C₂×#397×C₆₇	C₂	semidirect
6. C₂×#394×C₆₇	C₂	semidirect
7. C₂×#393×C₆₇	C₂	semidirect
8. C₂×#386×C₆₇	C₂	semidirect
9. C₂²×#95×C₆₇	C₂	semidirect
16. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
26. #400×C₆₇	C₂²	semidirect
27. #397×C₆₇	C₂²	semidirect
28. #394×C₆₇	C₂²	semidirect
29. #393×C₆₇	C₂²	semidirect
30. #386×C₆₇	C₂²	semidirect
35. C₂×#95×C₆₇	C₂²	semidirect
215. C₂×#2639	C₆₇	direct (N × H)
218. C₂×C₆₇	#2639	direct (N × H)
224. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
226. #2639	C₂×C₆₇	direct (N × H)
227. C₂×#400	C₂×C₆₇	semidirect
228. C₂×#397	C₂×C₆₇	semidirect
229. C₂×#394	C₂×C₆₇	semidirect
230. C₂×#393	C₂×C₆₇	semidirect
231. C₂×#386	C₂×C₆₇	semidirect
232. C₂²×#95	C₂×C₆₇	semidirect
233. C₆₇	C₂×#2639	direct (N × H)
240. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
250. #400	C₂²×C₆₇	semidirect
251. #397	C₂²×C₆₇	semidirect
252. #394	C₂²×C₆₇	semidirect
253. #393	C₂²×C₆₇	semidirect
254. #386	C₂²×C₆₇	semidirect
259. C₂×#95	C₂²×C₆₇	semidirect
441. C₂	C₆₇×#2639	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×C₂²⋊Q₈×C₆₇	2
C₂²×#121×C₆₇	2
C₆₇×#2639	2
C₂×#400×C₆₇	2
C₂×#397×C₆₇	2
C₂×#394×C₆₇	2
C₂×#393×C₆₇	2
C₂×#386×C₆₇	2
C₂²×#95×C₆₇	2
C₂×#2639	67

Subgroups — C2×C67×#2639 #30713814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2639 #30713814