Subgroups — C₂×C₆₇×#2502 #30714016

Order 17152.

3094 conjugacy classes of subgroups.

The subgroup lattice has 340 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
2	C₂³×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2502	C₂	1	16	yes	maximal, direct product factor
4	C₂×#398×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#391×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#385×C₆₇	C₂	1	4	yes	maximal
7	C₂×#384×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊Q₈×C₆₇	—	2	8	no	—
13	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₆₇	—	2	8	no	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
24	#398×C₆₇	C₂²	1	32	yes	complemented
25	#391×C₆₇	C₂²	1	16	yes	complemented
26	#385×C₆₇	C₂²	1	32	yes	—
27	#384×C₆₇	C₂²	1	16	yes	complemented
28	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
29	C₂×#95×C₆₇	C₂²	1	2	yes	—
30	C₂×#95×C₆₇	C₂²	1	8	yes	—
31	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×Q₈×C₆₇	—	2	4	no	—
33	C₂²×Q₈×C₆₇	—	2	8	no	—
34	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₆₇	—	2	8	no	—
36	C₂³×C₄×C₆₇	—	2	8	no	—
37	C₂³×C₄×C₆₇	—	4	8	no	—
38	C₂³×C₄×C₆₇	D₄	1	8	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
41	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²⋊Q₈×C₆₇	—	2	32	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	32	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
53	C₂×C₄⋊C₄×C₆₇	—	4	16	no	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	32	yes	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
60	C₂×C₂²⋊C₄×C₆₇	Q₈	1	16	yes	complemented
61	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
63	C₂×C₄²×C₆₇	—	2	8	no	—
64	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
65	#95×C₆₇	C₂³	1	8	yes	—
66	#95×C₆₇	C₂³	1	32	yes	—
67	C₂⁴×C₆₇	—	2	4	no	—
68	C₂⁴×C₆₇	—	2	4	no	—
69	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
70	C₂⁴×C₆₇	C₂×Q₈	1	8	yes	—
71	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
72	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₂×Q₈×C₆₇	—	2	8	no	—
74	C₂×Q₈×C₆₇	—	2	16	no	—
75	C₂×Q₈×C₆₇	—	4	16	no	—
76	C₂²×C₄×C₆₇	—	2	8	no	—
77	C₂²×C₄×C₆₇	—	2	4	no	—
78	C₂²×C₄×C₆₇	—	2	8	no	—
79	C₂²×C₄×C₆₇	—	2	32	no	—
80	C₂²×C₄×C₆₇	—	2	16	no	—
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	—	2	16	no	—
84	C₂²×C₄×C₆₇	—	2	8	no	—
85	C₂²×C₄×C₆₇	—	2	8	no	—
86	C₂²×C₄×C₆₇	—	2	8	no	—
87	C₂²×C₄×C₆₇	—	2	16	no	—
88	C₂²×C₄×C₆₇	—	4	16	no	—
89	C₂²×C₄×C₆₇	—	4	32	no	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₆₇	C₂×Q₈	1	16	yes	—
94	C₂²×C₄×C₆₇	#41	1	8	yes	—
95	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
97	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
98	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
100	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
101	C₄⋊C₄×C₆₇	—	2	16	no	—
102	C₄⋊C₄×C₆₇	—	2	16	no	—
103	C₄⋊C₄×C₆₇	—	2	8	no	—
104	C₄⋊C₄×C₆₇	—	2	16	no	—
105	C₄⋊C₄×C₆₇	—	4	32	no	—
106	C₂²⋊C₄×C₆₇	—	2	16	no	—
107	C₂²⋊C₄×C₆₇	C₂×Q₈	1	32	yes	complemented
108	C₄²×C₆₇	—	2	16	no	—
109	C₂³×C₆₇	—	2	8	no	—
110	C₂³×C₆₇	—	2	8	no	—
111	C₂³×C₆₇	—	2	32	no	—
112	C₂³×C₆₇	—	2	4	no	—
113	C₂³×C₆₇	—	2	4	no	—
114	C₂³×C₆₇	C₄⋊Q₈	1	16	yes	—
115	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
116	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
117	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
118	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
119	C₂³×C₆₇	C₂×#41	1	4	yes	—
120	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
121	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
122	Q₈×C₆₇	—	4	16	no	—
123	C₂×C₄×C₆₇	—	2	8	no	—
124	C₂×C₄×C₆₇	—	2	32	no	—
125	C₂×C₄×C₆₇	—	2	8	no	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	8	no	—
128	C₂×C₄×C₆₇	—	2	32	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	2	4	no	—
131	C₂×C₄×C₆₇	—	2	8	no	—
132	C₂×C₄×C₆₇	—	2	8	no	—
133	C₂×C₄×C₆₇	—	2	8	no	—
134	C₂×C₄×C₆₇	—	2	16	no	—
135	C₂×C₄×C₆₇	—	4	32	no	—
136	C₂×C₄×C₆₇	—	4	8	no	—
137	C₂×C₄×C₆₇	—	4	8	no	—
138	C₂×C₄×C₆₇	—	4	16	no	—
139	C₂×C₄×C₆₇	#123	1	16	yes	—
140	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
141	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
142	C₂×C₄×C₆₇	C₂²×Q₈	1	16	yes	—
143	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
144	C₂²×C₆₇	—	2	4	no	—
145	C₂²×C₆₇	—	2	4	no	—
146	C₂²×C₆₇	—	2	16	no	—
147	C₂²×C₆₇	—	2	32	no	—
148	C₂²×C₆₇	C₂×#123	1	4	yes	—
149	C₂²×C₆₇	C₂×C₄⋊Q₈	1	8	yes	—
150	C₂²×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
151	C₂²×C₆₇	#535	1	4	yes	—
152	C₂²×C₆₇	#554	1	16	yes	—
153	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
154	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
155	C₂²×C₆₇	C₂²×#41	1	2	yes	—
156	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
157	C₄×C₆₇	—	2	4	no	—
158	C₄×C₆₇	—	2	16	no	—
159	C₄×C₆₇	—	4	4	no	—
160	C₄×C₆₇	—	4	4	no	—
161	C₄×C₆₇	—	4	8	no	—
162	C₄×C₆₇	C₂×#123	1	8	yes	—
163	C₂×#2502	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₆₇	—	2	8	no	—
165	C₂×C₆₇	#2502	1	8	yes	direct product factor
166	C₂×C₆₇	C₂²×#123	1	1	yes	characteristic
167	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
168	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
169	C₂×C₆₇	C₂×#554	1	4	yes	—
170	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
171	C₂³×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
172	#2502	C₂×C₆₇	1	16	yes	direct product factor
173	C₂×#398	C₂×C₆₇	1	4	yes	complemented
174	C₂×#391	C₂×C₆₇	1	2	yes	complemented
175	C₂×#385	C₂×C₆₇	1	4	yes	—
176	C₂×#384	C₂×C₆₇	1	2	yes	complemented
177	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
178	C₆₇	C₂×#2502	1	1	yes	characteristic, direct product factor, Sylow
179	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
180	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
181	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
182	C₂×C₂²⋊Q₈	—	2	8	no	—
183	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	—
184	C₂²×C₄⋊C₄	—	2	8	no	—
185	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
186	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
187	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
188	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
189	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
190	C₂²×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
191	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
192	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
193	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
194	#398	C₂²×C₆₇	1	32	yes	complemented
195	#391	C₂²×C₆₇	1	16	yes	complemented
196	#385	C₂²×C₆₇	1	32	yes	—
197	#384	C₂²×C₆₇	1	16	yes	complemented
198	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
199	C₂×#95	C₂²×C₆₇	1	2	yes	—
200	C₂×#95	C₂²×C₆₇	1	8	yes	—
201	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
202	C₂²×Q₈	—	2	4	no	—
203	C₂²×Q₈	—	2	8	no	—
204	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
205	C₂³×C₄	—	2	8	no	—
206	C₂³×C₄	—	2	8	no	—
207	C₂³×C₄	—	4	8	no	—
208	C₂³×C₄	D₄×C₆₇	1	8	yes	—
209	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
210	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
211	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
212	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
213	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
214	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
215	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
216	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
217	C₂²⋊Q₈	—	2	32	no	—
218	C₂×C₄⋊C₄	—	2	8	no	—
219	C₂×C₄⋊C₄	—	2	32	no	—
220	C₂×C₄⋊C₄	—	2	8	no	—
221	C₂×C₄⋊C₄	—	2	4	no	—
222	C₂×C₄⋊C₄	—	2	8	no	—
223	C₂×C₄⋊C₄	—	4	16	no	—
224	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
225	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
226	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
227	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
228	C₂×C₄⋊C₄	C₂³×C₆₇	1	32	yes	—
229	C₂×C₂²⋊C₄	—	2	8	no	—
230	C₂×C₂²⋊C₄	Q₈×C₆₇	1	16	yes	complemented
231	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
232	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
233	C₂×C₄²	—	2	8	no	—
234	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
235	#95	C₂³×C₆₇	1	8	yes	—
236	#95	C₂³×C₆₇	1	32	yes	—
237	C₂⁴	—	2	4	no	—
238	C₂⁴	—	2	4	no	—
239	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
240	C₂⁴	C₂×Q₈×C₆₇	1	8	yes	—
241	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
242	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
243	C₂×Q₈	—	2	8	no	—
244	C₂×Q₈	—	2	16	no	—
245	C₂×Q₈	—	4	16	no	—
246	C₂²×C₄	—	2	8	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	8	no	—
249	C₂²×C₄	—	2	32	no	—
250	C₂²×C₄	—	2	16	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	—	2	16	no	—
254	C₂²×C₄	—	2	8	no	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	8	no	—
257	C₂²×C₄	—	2	16	no	—
258	C₂²×C₄	—	4	16	no	—
259	C₂²×C₄	—	4	32	no	—
260	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
261	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
262	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
263	C₂²×C₄	C₂×Q₈×C₆₇	1	16	yes	—
264	C₂²×C₄	#41×C₆₇	1	8	yes	—
265	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
266	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
267	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
268	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
269	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
270	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
271	C₄⋊C₄	—	2	16	no	—
272	C₄⋊C₄	—	2	16	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	16	no	—
275	C₄⋊C₄	—	4	32	no	—
276	C₂²⋊C₄	—	2	16	no	—
277	C₂²⋊C₄	C₂×Q₈×C₆₇	1	32	yes	complemented
278	C₄²	—	2	16	no	—
279	C₂³	—	2	8	no	—
280	C₂³	—	2	8	no	—
281	C₂³	—	2	32	no	—
282	C₂³	—	2	4	no	—
283	C₂³	—	2	4	no	—
284	C₂³	C₄⋊Q₈×C₆₇	1	16	yes	—
285	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
286	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
287	C₂³	C₂²×Q₈×C₆₇	1	8	yes	—
288	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
289	C₂³	C₂×#41×C₆₇	1	4	yes	—
290	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
291	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
292	Q₈	—	4	16	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	32	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	32	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	2	8	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	—	2	16	no	—
305	C₂×C₄	—	4	32	no	—
306	C₂×C₄	—	4	8	no	—
307	C₂×C₄	—	4	16	no	—
308	C₂×C₄	—	4	8	no	—
309	C₂×C₄	#123×C₆₇	1	16	yes	—
310	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
311	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
312	C₂×C₄	C₂²×Q₈×C₆₇	1	16	yes	—
313	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
314	C₂²	—	2	32	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	16	no	—
318	C₂²	C₂×#123×C₆₇	1	4	yes	—
319	C₂²	C₂×C₄⋊Q₈×C₆₇	1	8	yes	—
320	C₂²	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
321	C₂²	#535×C₆₇	1	4	yes	—
322	C₂²	#554×C₆₇	1	16	yes	—
323	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
324	C₂²	C₂³×Q₈×C₆₇	1	2	yes	—
325	C₂²	C₂²×#41×C₆₇	1	2	yes	—
326	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	16	no	—
329	C₄	—	4	4	no	—
330	C₄	—	4	8	no	—
331	C₄	—	4	4	no	—
332	C₄	C₂×#123×C₆₇	1	8	yes	—
333	C₂	—	2	8	no	—
334	C₂	C₆₇×#2502	1	8	yes	direct product factor
335	C₂	C₂²×#123×C₆₇	1	1	yes	characteristic
336	C₂	C₂²×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
337	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
338	C₂	C₂×#554×C₆₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
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
3. C₆₇×#2502	C₂	direct (N × H)
4. C₂×#398×C₆₇	C₂	semidirect
5. C₂×#391×C₆₇	C₂	semidirect
7. C₂×#384×C₆₇	C₂	semidirect
24. #398×C₆₇	C₂²	semidirect
25. #391×C₆₇	C₂²	semidirect
27. #384×C₆₇	C₂²	semidirect
60. C₂×C₂²⋊C₄×C₆₇	Q₈	semidirect
107. C₂²⋊C₄×C₆₇	C₂×Q₈	semidirect
163. C₂×#2502	C₆₇	direct (N × H)
165. C₂×C₆₇	#2502	direct (N × H)
172. #2502	C₂×C₆₇	direct (N × H)
173. C₂×#398	C₂×C₆₇	semidirect
174. C₂×#391	C₂×C₆₇	semidirect
176. C₂×#384	C₂×C₆₇	semidirect
178. C₆₇	C₂×#2502	direct (N × H)
194. #398	C₂²×C₆₇	semidirect
195. #391	C₂²×C₆₇	semidirect
197. #384	C₂²×C₆₇	semidirect
230. C₂×C₂²⋊C₄	Q₈×C₆₇	semidirect
277. C₂²⋊C₄	C₂×Q₈×C₆₇	semidirect
334. C₂	C₆₇×#2502	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
C₂³×C₄⋊C₄×C₆₇	2
C₆₇×#2502	2
C₂×#398×C₆₇	2
C₂×#391×C₆₇	2
C₂×#385×C₆₇	2
C₂×#384×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#2502	67

Subgroups — C2×C67×#2502 #30714016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2502 #30714016