Subgroups — C₂×C₆₇×#2542 #30714178

Order 17152.

4006 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2542	C₂	1	16	yes	maximal, direct product factor
5	C₂×#399×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#390×C₆₇	C₂	1	2	yes	maximal
7	C₂×#384×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	—	2	8	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
14	C₂×C₂²⋊Q₈×C₆₇	—	2	32	no	—
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	32	yes	complemented
16	C₂×C₄×D₄×C₆₇	—	2	8	no	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	8	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	8	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
24	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
27	#399×C₆₇	C₂²	1	32	yes	complemented
28	#390×C₆₇	C₂²	1	16	yes	—
29	#384×C₆₇	C₂²	1	16	yes	complemented
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
31	C₂×#95×C₆₇	C₂²	1	2	yes	—
32	C₂⁵×C₆₇	C₂³	1	2	yes	—
33	C₂²×Q₈×C₆₇	—	2	16	no	—
34	C₂²×Q₈×C₆₇	—	2	8	no	—
35	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
36	C₂²×D₄×C₆₇	—	2	4	no	—
37	C₂²×D₄×C₆₇	—	2	4	no	—
38	C₂²×D₄×C₆₇	Q₈	1	8	yes	complemented
39	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
40	C₂³×C₄×C₆₇	—	2	8	no	—
41	C₂³×C₄×C₆₇	—	2	16	no	—
42	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	—	2	32	no	—
50	C₂²⋊Q₈×C₆₇	—	2	128	no	—
51	C₄×D₄×C₆₇	—	2	32	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	32	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₆₇	D₄	1	16	yes	complemented
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	32	no	—
64	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	—
66	C₂×C₄²×C₆₇	—	2	8	no	—
67	C₂×C₄²×C₆₇	D₄	1	8	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
70	#95×C₆₇	C₂³	1	8	yes	—
71	C₂⁴×C₆₇	—	2	8	no	—
72	C₂⁴×C₆₇	—	2	4	no	—
73	C₂⁴×C₆₇	—	2	4	no	—
74	C₂⁴×C₆₇	—	2	8	no	—
75	C₂⁴×C₆₇	C₂×Q₈	1	8	yes	—
76	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
77	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
78	C₂×Q₈×C₆₇	—	2	32	no	—
79	C₂×Q₈×C₆₇	—	2	16	no	—
80	C₂×Q₈×C₆₇	—	4	32	no	—
81	C₂×D₄×C₆₇	—	2	32	no	—
82	C₂×D₄×C₆₇	—	2	8	no	—
83	C₂×D₄×C₆₇	—	2	8	no	—
84	C₂×D₄×C₆₇	C₂×Q₈	1	16	yes	complemented
85	C₂²×C₄×C₆₇	—	2	8	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	8	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	2	16	no	—
90	C₂²×C₄×C₆₇	—	2	8	no	—
91	C₂²×C₄×C₆₇	—	2	8	no	—
92	C₂²×C₄×C₆₇	—	2	16	no	—
93	C₂²×C₄×C₆₇	—	2	32	no	—
94	C₂²×C₄×C₆₇	—	4	32	no	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
100	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
101	C₂²×C₄×C₆₇	#41	1	4	yes	—
102	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
103	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
105	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
108	C₄⋊C₄×C₆₇	—	2	8	no	—
109	C₄⋊C₄×C₆₇	—	2	64	no	—
110	C₄⋊C₄×C₆₇	—	2	16	no	—
111	C₄⋊C₄×C₆₇	—	2	16	no	—
112	C₄⋊C₄×C₆₇	C₂×D₄	1	32	yes	complemented
113	C₂²⋊C₄×C₆₇	—	2	16	no	—
114	C₂²⋊C₄×C₆₇	—	2	64	no	—
115	C₄²×C₆₇	—	2	16	no	—
116	C₄²×C₆₇	C₂×D₄	1	16	yes	—
117	C₂³×C₆₇	—	2	16	no	—
118	C₂³×C₆₇	—	2	32	no	—
119	C₂³×C₆₇	—	2	8	no	—
120	C₂³×C₆₇	—	2	4	no	—
121	C₂³×C₆₇	—	2	4	no	—
122	C₂³×C₆₇	—	2	8	no	—
123	C₂³×C₆₇	—	4	8	no	—
124	C₂³×C₆₇	—	4	8	no	—
125	C₂³×C₆₇	—	4	16	no	—
126	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
127	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
128	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
129	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
130	C₂³×C₆₇	C₂×#41	1	2	yes	—
131	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
132	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
133	Q₈×C₆₇	—	4	32	no	—
134	D₄×C₆₇	—	2	32	no	—
135	C₂×C₄×C₆₇	—	2	8	no	—
136	C₂×C₄×C₆₇	—	2	32	no	—
137	C₂×C₄×C₆₇	—	2	8	no	—
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	4	no	—
141	C₂×C₄×C₆₇	—	2	16	no	—
142	C₂×C₄×C₆₇	—	2	8	no	—
143	C₂×C₄×C₆₇	—	2	8	no	—
144	C₂×C₄×C₆₇	—	4	8	no	—
145	C₂×C₄×C₆₇	—	4	16	no	—
146	C₂×C₄×C₆₇	—	4	32	no	—
147	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	complemented
148	C₂×C₄×C₆₇	#127	1	16	yes	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	—
151	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
152	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
153	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
154	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
155	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
156	C₂²×C₆₇	—	2	8	no	—
157	C₂²×C₆₇	—	2	16	no	—
158	C₂²×C₆₇	—	4	4	no	—
159	C₂²×C₆₇	—	4	4	no	—
160	C₂²×C₆₇	—	4	8	no	—
161	C₂²×C₆₇	—	4	32	no	—
162	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
163	C₂²×C₆₇	C₂×#127	1	4	yes	—
164	C₂²×C₆₇	#536	1	4	yes	—
165	C₂²×C₆₇	#547	1	8	yes	—
166	C₂²×C₆₇	D₄×Q₈	1	8	yes	—
167	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
168	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
169	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
170	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
172	C₄×C₆₇	—	2	4	no	—
173	C₄×C₆₇	—	2	16	no	—
174	C₄×C₆₇	—	4	4	no	—
175	C₄×C₆₇	—	4	8	no	—
176	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	complemented
177	C₄×C₆₇	C₂×#127	1	8	yes	—
178	C₂×#2542	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₂×C₆₇	—	4	8	no	—
180	C₂×C₆₇	#2542	1	8	yes	direct product factor
181	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
182	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
183	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
184	C₂×C₆₇	C₂×#547	1	2	yes	—
185	C₂×C₆₇	C₂×D₄×Q₈	1	2	yes	—
186	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
187	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	4	yes	complemented
188	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
189	#2542	C₂×C₆₇	1	16	yes	direct product factor
190	C₂×#399	C₂×C₆₇	1	4	yes	complemented
191	C₂×#390	C₂×C₆₇	1	2	yes	—
192	C₂×#384	C₂×C₆₇	1	2	yes	complemented
193	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
194	C₆₇	C₂×#2542	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
196	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
197	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
198	C₂×C₄⋊Q₈	—	2	8	no	—
199	C₂×C₄⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
200	C₂×C₂²⋊Q₈	—	2	32	no	—
201	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	32	yes	complemented
202	C₂×C₄×D₄	—	2	8	no	—
203	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
204	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
205	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
206	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
207	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
208	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
209	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
210	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
211	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
212	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
213	#399	C₂²×C₆₇	1	32	yes	complemented
214	#390	C₂²×C₆₇	1	16	yes	—
215	#384	C₂²×C₆₇	1	16	yes	complemented
216	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
217	C₂×#95	C₂²×C₆₇	1	2	yes	—
218	C₂⁵	C₂³×C₆₇	1	2	yes	—
219	C₂²×Q₈	—	2	16	no	—
220	C₂²×Q₈	—	2	8	no	—
221	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
222	C₂²×D₄	—	2	4	no	—
223	C₂²×D₄	—	2	4	no	—
224	C₂²×D₄	Q₈×C₆₇	1	8	yes	complemented
225	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
226	C₂³×C₄	—	2	8	no	—
227	C₂³×C₄	—	2	16	no	—
228	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
229	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
230	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
231	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
232	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
233	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
234	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
235	C₄⋊Q₈	—	2	32	no	—
236	C₂²⋊Q₈	—	2	128	no	—
237	C₄×D₄	—	2	32	no	—
238	C₂×C₄⋊C₄	—	2	4	no	—
239	C₂×C₄⋊C₄	—	2	32	no	—
240	C₂×C₄⋊C₄	—	2	8	no	—
241	C₂×C₄⋊C₄	—	2	8	no	—
242	C₂×C₄⋊C₄	D₄×C₆₇	1	16	yes	complemented
243	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
244	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	—
245	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
246	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
247	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
248	C₂×C₂²⋊C₄	—	2	8	no	—
249	C₂×C₂²⋊C₄	—	2	32	no	—
250	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
251	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	—
252	C₂×C₄²	—	2	8	no	—
253	C₂×C₄²	D₄×C₆₇	1	8	yes	—
254	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
255	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
256	#95	C₂³×C₆₇	1	8	yes	—
257	C₂⁴	—	2	8	no	—
258	C₂⁴	—	2	4	no	—
259	C₂⁴	—	2	4	no	—
260	C₂⁴	—	2	8	no	—
261	C₂⁴	C₂×Q₈×C₆₇	1	8	yes	—
262	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
263	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
264	C₂×Q₈	—	2	32	no	—
265	C₂×Q₈	—	2	16	no	—
266	C₂×Q₈	—	4	32	no	—
267	C₂×D₄	—	2	32	no	—
268	C₂×D₄	—	2	8	no	—
269	C₂×D₄	—	2	8	no	—
270	C₂×D₄	C₂×Q₈×C₆₇	1	16	yes	complemented
271	C₂²×C₄	—	2	8	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	8	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	16	no	—
276	C₂²×C₄	—	2	8	no	—
277	C₂²×C₄	—	2	8	no	—
278	C₂²×C₄	—	2	16	no	—
279	C₂²×C₄	—	2	32	no	—
280	C₂²×C₄	—	4	32	no	—
281	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
282	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
283	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
284	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
285	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
286	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
287	C₂²×C₄	#41×C₆₇	1	4	yes	—
288	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
289	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
290	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
292	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
293	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
294	C₄⋊C₄	—	2	8	no	—
295	C₄⋊C₄	—	2	64	no	—
296	C₄⋊C₄	—	2	16	no	—
297	C₄⋊C₄	—	2	16	no	—
298	C₄⋊C₄	C₂×D₄×C₆₇	1	32	yes	complemented
299	C₂²⋊C₄	—	2	16	no	—
300	C₂²⋊C₄	—	2	64	no	—
301	C₄²	—	2	16	no	—
302	C₄²	C₂×D₄×C₆₇	1	16	yes	—
303	C₂³	—	2	16	no	—
304	C₂³	—	2	32	no	—
305	C₂³	—	2	8	no	—
306	C₂³	—	2	4	no	—
307	C₂³	—	2	4	no	—
308	C₂³	—	2	8	no	—
309	C₂³	—	4	8	no	—
310	C₂³	—	4	8	no	—
311	C₂³	—	4	16	no	—
312	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
313	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
314	C₂³	C₂²×Q₈×C₆₇	1	8	yes	—
315	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
316	C₂³	C₂×#41×C₆₇	1	2	yes	—
317	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
318	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
319	Q₈	—	4	32	no	—
320	D₄	—	2	32	no	—
321	C₂×C₄	—	2	8	no	—
322	C₂×C₄	—	2	32	no	—
323	C₂×C₄	—	2	8	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	16	no	—
328	C₂×C₄	—	2	8	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	4	8	no	—
331	C₂×C₄	—	4	16	no	—
332	C₂×C₄	—	4	32	no	—
333	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	complemented
334	C₂×C₄	#127×C₆₇	1	16	yes	—
335	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
336	C₂×C₄	C₂²×D₄×C₆₇	1	16	yes	—
337	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
338	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
339	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
340	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
341	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
342	C₂²	—	2	8	no	—
343	C₂²	—	2	16	no	—
344	C₂²	—	4	32	no	—
345	C₂²	—	4	4	no	—
346	C₂²	—	4	4	no	—
347	C₂²	—	4	8	no	—
348	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
349	C₂²	C₂×#127×C₆₇	1	4	yes	—
350	C₂²	#536×C₆₇	1	4	yes	—
351	C₂²	#547×C₆₇	1	8	yes	—
352	C₂²	D₄×Q₈×C₆₇	1	8	yes	—
353	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
354	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
355	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
356	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
357	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
358	C₄	—	2	4	no	—
359	C₄	—	2	16	no	—
360	C₄	—	4	4	no	—
361	C₄	—	4	8	no	—
362	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	complemented
363	C₄	C₂×#127×C₆₇	1	8	yes	—
364	C₂	—	4	8	no	—
365	C₂	C₆₇×#2542	1	8	yes	direct product factor
366	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
367	C₂	C₂²×#127×C₆₇	1	1	yes	characteristic
368	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
369	C₂	C₂×#547×C₆₇	1	2	yes	—
370	C₂	C₂×D₄×Q₈×C₆₇	1	2	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
1. C₂²×C₄⋊Q₈×C₆₇	C₂	semidirect
2. C₂²×C₂²⋊Q₈×C₆₇	C₂	semidirect
4. C₆₇×#2542	C₂	direct (N × H)
5. C₂×#399×C₆₇	C₂	semidirect
7. C₂×#384×C₆₇	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
13. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
15. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
27. #399×C₆₇	C₂²	semidirect
29. #384×C₆₇	C₂²	semidirect
30. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
38. C₂²×D₄×C₆₇	Q₈	semidirect
56. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
84. C₂×D₄×C₆₇	C₂×Q₈	semidirect
112. C₄⋊C₄×C₆₇	C₂×D₄	semidirect
147. C₂×C₄×C₆₇	C₂²⋊Q₈	semidirect
176. C₄×C₆₇	C₂×C₂²⋊Q₈	semidirect
178. C₂×#2542	C₆₇	direct (N × H)
180. C₂×C₆₇	#2542	direct (N × H)
186. C₂²×C₄⋊Q₈	C₂×C₆₇	semidirect
187. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
189. #2542	C₂×C₆₇	direct (N × H)
190. C₂×#399	C₂×C₆₇	semidirect
192. C₂×#384	C₂×C₆₇	semidirect
193. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
194. C₆₇	C₂×#2542	direct (N × H)
199. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
201. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
213. #399	C₂²×C₆₇	semidirect
215. #384	C₂²×C₆₇	semidirect
216. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
224. C₂²×D₄	Q₈×C₆₇	semidirect
242. C₂×C₄⋊C₄	D₄×C₆₇	semidirect
270. C₂×D₄	C₂×Q₈×C₆₇	semidirect
298. C₄⋊C₄	C₂×D₄×C₆₇	semidirect
333. C₂×C₄	C₂²⋊Q₈×C₆₇	semidirect
362. C₄	C₂×C₂²⋊Q₈×C₆₇	semidirect
365. C₂	C₆₇×#2542	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×C₄⋊Q₈×C₆₇	2
C₂²×C₂²⋊Q₈×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2542	2
C₂×#399×C₆₇	2
C₂×#390×C₆₇	2
C₂×#384×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2542	67

Subgroups — C2×C67×#2542 #30714178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2542 #30714178