Subgroups — C₂×C₆₇×#2563 #30714190

Order 17152.

3222 conjugacy classes of subgroups.

The subgroup lattice has 364 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×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2563	C₂	1	16	yes	maximal, direct product factor
4	C₂×#393×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#390×C₆₇	C₂	1	2	yes	maximal
6	C₂×#382×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#380×C₆₇	C₂	1	4	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×C₄×D₄×C₆₇	—	2	8	no	—
15	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
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×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#393×C₆₇	C₂²	1	32	yes	complemented
25	#390×C₆₇	C₂²	1	16	yes	—
26	#382×C₆₇	C₂²	1	16	yes	complemented
27	#380×C₆₇	C₂²	1	32	yes	complemented
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
29	C₂×#95×C₆₇	C₂²	1	2	yes	—
30	C₂×#95×C₆₇	C₂²	1	8	yes	—
31	C₂⁵×C₆₇	C₂³	1	2	yes	—
32	C₂²×Q₈×C₆₇	—	2	8	no	—
33	C₂²×Q₈×C₆₇	—	4	8	no	—
34	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
35	C₂²×D₄×C₆₇	—	2	4	no	—
36	C₂²×D₄×C₆₇	—	2	4	no	—
37	C₂²×D₄×C₆₇	Q₈	1	8	yes	complemented
38	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
39	C₂³×C₄×C₆₇	—	2	8	no	—
40	C₂³×C₄×C₆₇	—	2	16	no	—
41	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₆₇	—	2	32	no	—
49	C₄×D₄×C₆₇	—	2	32	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	4	16	no	—
60	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
61	C₂×C₂²⋊C₄×C₆₇	C₂³	1	32	yes	—
62	C₂×C₄²×C₆₇	—	2	8	no	—
63	C₂×C₄²×C₆₇	D₄	1	8	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
66	#95×C₆₇	C₂³	1	8	yes	—
67	#95×C₆₇	C₂³	1	32	yes	—
68	C₂⁴×C₆₇	—	2	4	no	—
69	C₂⁴×C₆₇	—	2	8	no	—
70	C₂⁴×C₆₇	—	2	4	no	—
71	C₂⁴×C₆₇	—	2	8	no	—
72	C₂⁴×C₆₇	C₂×Q₈	1	8	yes	—
73	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
74	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
75	C₂×Q₈×C₆₇	—	2	16	no	—
76	C₂×Q₈×C₆₇	—	4	16	no	—
77	C₂×Q₈×C₆₇	—	4	32	no	—
78	C₂×D₄×C₆₇	—	2	32	no	—
79	C₂×D₄×C₆₇	—	2	8	no	—
80	C₂×D₄×C₆₇	—	2	8	no	—
81	C₂×D₄×C₆₇	C₂×Q₈	1	16	yes	complemented
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	—	2	4	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	8	no	—
88	C₂²×C₄×C₆₇	—	2	8	no	—
89	C₂²×C₄×C₆₇	—	2	16	no	—
90	C₂²×C₄×C₆₇	—	2	16	no	—
91	C₂²×C₄×C₆₇	—	2	32	no	—
92	C₂²×C₄×C₆₇	—	4	32	no	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
97	C₂²×C₄×C₆₇	#41	1	4	yes	—
98	C₂²×C₄×C₆₇	#41	1	8	yes	—
99	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
100	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
101	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
102	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
105	C₄⋊C₄×C₆₇	—	2	8	no	—
106	C₄⋊C₄×C₆₇	—	2	16	no	—
107	C₄⋊C₄×C₆₇	—	2	16	no	—
108	C₄⋊C₄×C₆₇	—	2	16	no	—
109	C₂²⋊C₄×C₆₇	—	2	16	no	—
110	C₂²⋊C₄×C₆₇	—	4	32	no	—
111	C₄²×C₆₇	—	2	16	no	—
112	C₄²×C₆₇	C₂×D₄	1	16	yes	—
113	C₂³×C₆₇	—	2	16	no	—
114	C₂³×C₆₇	—	2	32	no	—
115	C₂³×C₆₇	—	2	4	no	—
116	C₂³×C₆₇	—	2	8	no	—
117	C₂³×C₆₇	—	2	4	no	—
118	C₂³×C₆₇	—	2	8	no	—
119	C₂³×C₆₇	—	4	8	no	—
120	C₂³×C₆₇	—	4	8	no	—
121	C₂³×C₆₇	—	4	16	no	—
122	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
123	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
124	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
125	C₂³×C₆₇	C₂×#41	1	2	yes	—
126	C₂³×C₆₇	C₂×#41	1	4	yes	—
127	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
128	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
129	Q₈×C₆₇	—	4	32	no	—
130	D₄×C₆₇	—	2	32	no	—
131	C₂×C₄×C₆₇	—	2	8	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	8	no	—
135	C₂×C₄×C₆₇	—	2	8	no	—
136	C₂×C₄×C₆₇	—	2	8	no	—
137	C₂×C₄×C₆₇	—	2	8	no	—
138	C₂×C₄×C₆₇	—	2	8	no	—
139	C₂×C₄×C₆₇	—	2	16	no	—
140	C₂×C₄×C₆₇	—	4	8	no	—
141	C₂×C₄×C₆₇	—	4	16	no	—
142	C₂×C₄×C₆₇	—	4	16	no	—
143	C₂×C₄×C₆₇	—	4	32	no	—
144	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	—
145	C₂×C₄×C₆₇	#124	1	16	yes	—
146	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
147	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
148	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
149	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
150	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
151	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
152	C₂²×C₆₇	—	2	8	no	—
153	C₂²×C₆₇	—	2	16	no	—
154	C₂²×C₆₇	—	4	32	no	—
155	C₂²×C₆₇	—	4	4	no	—
156	C₂²×C₆₇	—	4	4	no	—
157	C₂²×C₆₇	—	4	8	no	—
158	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
159	C₂²×C₆₇	C₂×#124	1	4	yes	—
160	C₂²×C₆₇	#535	1	4	yes	—
161	C₂²×C₆₇	#554	1	8	yes	—
162	C₂²×C₆₇	#555	1	8	yes	—
163	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
164	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
165	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
166	C₂²×C₆₇	C₂²×#41	1	2	yes	—
167	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
168	C₄×C₆₇	—	2	4	no	—
169	C₄×C₆₇	—	4	4	no	—
170	C₄×C₆₇	—	4	8	no	—
171	C₄×C₆₇	—	4	8	no	—
172	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	—
173	C₄×C₆₇	C₂×#124	1	8	yes	—
174	C₂×#2563	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂×C₆₇	—	4	8	no	—
176	C₂×C₆₇	#2563	1	8	yes	direct product factor
177	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
179	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#554	1	2	yes	—
181	C₂×C₆₇	C₂×#555	1	2	yes	—
182	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
183	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
184	#2563	C₂×C₆₇	1	16	yes	direct product factor
185	C₂×#393	C₂×C₆₇	1	4	yes	complemented
186	C₂×#390	C₂×C₆₇	1	2	yes	—
187	C₂×#382	C₂×C₆₇	1	2	yes	complemented
188	C₂×#380	C₂×C₆₇	1	4	yes	complemented
189	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
190	C₆₇	C₂×#2563	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
192	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
193	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
194	C₂×C₄⋊Q₈	—	2	8	no	—
195	C₂×C₄⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
196	C₂×C₄×D₄	—	2	8	no	—
197	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
198	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
199	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
200	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
201	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
202	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
203	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
204	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
205	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
206	#393	C₂²×C₆₇	1	32	yes	complemented
207	#390	C₂²×C₆₇	1	16	yes	—
208	#382	C₂²×C₆₇	1	16	yes	complemented
209	#380	C₂²×C₆₇	1	32	yes	complemented
210	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
211	C₂×#95	C₂²×C₆₇	1	2	yes	—
212	C₂×#95	C₂²×C₆₇	1	8	yes	—
213	C₂⁵	C₂³×C₆₇	1	2	yes	—
214	C₂²×Q₈	—	2	8	no	—
215	C₂²×Q₈	—	4	8	no	—
216	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
217	C₂²×D₄	—	2	4	no	—
218	C₂²×D₄	—	2	4	no	—
219	C₂²×D₄	Q₈×C₆₇	1	8	yes	complemented
220	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
221	C₂³×C₄	—	2	8	no	—
222	C₂³×C₄	—	2	16	no	—
223	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
224	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
225	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
226	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
227	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
228	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
229	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
230	C₄⋊Q₈	—	2	32	no	—
231	C₄×D₄	—	2	32	no	—
232	C₂×C₄⋊C₄	—	2	4	no	—
233	C₂×C₄⋊C₄	—	2	8	no	—
234	C₂×C₄⋊C₄	—	2	8	no	—
235	C₂×C₄⋊C₄	—	2	8	no	—
236	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
237	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
238	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
239	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
240	C₂×C₂²⋊C₄	—	2	8	no	—
241	C₂×C₂²⋊C₄	—	4	16	no	—
242	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
243	C₂×C₂²⋊C₄	C₂³×C₆₇	1	32	yes	—
244	C₂×C₄²	—	2	8	no	—
245	C₂×C₄²	D₄×C₆₇	1	8	yes	—
246	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
247	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
248	#95	C₂³×C₆₇	1	8	yes	—
249	#95	C₂³×C₆₇	1	32	yes	—
250	C₂⁴	—	2	4	no	—
251	C₂⁴	—	2	8	no	—
252	C₂⁴	—	2	4	no	—
253	C₂⁴	—	2	8	no	—
254	C₂⁴	C₂×Q₈×C₆₇	1	8	yes	—
255	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
256	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
257	C₂×Q₈	—	2	16	no	—
258	C₂×Q₈	—	4	16	no	—
259	C₂×Q₈	—	4	32	no	—
260	C₂×D₄	—	2	32	no	—
261	C₂×D₄	—	2	8	no	—
262	C₂×D₄	—	2	8	no	—
263	C₂×D₄	C₂×Q₈×C₆₇	1	16	yes	complemented
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	—	2	8	no	—
267	C₂²×C₄	—	2	8	no	—
268	C₂²×C₄	—	2	8	no	—
269	C₂²×C₄	—	2	8	no	—
270	C₂²×C₄	—	2	8	no	—
271	C₂²×C₄	—	2	16	no	—
272	C₂²×C₄	—	2	16	no	—
273	C₂²×C₄	—	2	32	no	—
274	C₂²×C₄	—	4	32	no	—
275	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
276	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
277	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
278	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
279	C₂²×C₄	#41×C₆₇	1	4	yes	—
280	C₂²×C₄	#41×C₆₇	1	8	yes	—
281	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
282	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
283	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
284	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
285	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
286	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
287	C₄⋊C₄	—	2	8	no	—
288	C₄⋊C₄	—	2	16	no	—
289	C₄⋊C₄	—	2	16	no	—
290	C₄⋊C₄	—	2	16	no	—
291	C₂²⋊C₄	—	2	16	no	—
292	C₂²⋊C₄	—	4	32	no	—
293	C₄²	—	2	16	no	—
294	C₄²	C₂×D₄×C₆₇	1	16	yes	—
295	C₂³	—	2	16	no	—
296	C₂³	—	2	32	no	—
297	C₂³	—	2	4	no	—
298	C₂³	—	2	8	no	—
299	C₂³	—	2	4	no	—
300	C₂³	—	2	8	no	—
301	C₂³	—	4	8	no	—
302	C₂³	—	4	8	no	—
303	C₂³	—	4	16	no	—
304	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
305	C₂³	C₂²×Q₈×C₆₇	1	8	yes	—
306	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
307	C₂³	C₂×#41×C₆₇	1	2	yes	—
308	C₂³	C₂×#41×C₆₇	1	4	yes	—
309	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
310	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
311	Q₈	—	4	32	no	—
312	D₄	—	2	32	no	—
313	C₂×C₄	—	2	8	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	2	8	no	—
318	C₂×C₄	—	2	8	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	8	no	—
321	C₂×C₄	—	2	16	no	—
322	C₂×C₄	—	4	8	no	—
323	C₂×C₄	—	4	16	no	—
324	C₂×C₄	—	4	16	no	—
325	C₂×C₄	—	4	32	no	—
326	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	—
327	C₂×C₄	#124×C₆₇	1	16	yes	—
328	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
329	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
330	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
331	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
332	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
333	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
334	C₂²	—	2	8	no	—
335	C₂²	—	2	16	no	—
336	C₂²	—	4	32	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	8	no	—
340	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
341	C₂²	C₂×#124×C₆₇	1	4	yes	—
342	C₂²	#535×C₆₇	1	4	yes	—
343	C₂²	#554×C₆₇	1	8	yes	—
344	C₂²	#555×C₆₇	1	8	yes	—
345	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
346	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
347	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
348	C₂²	C₂²×#41×C₆₇	1	2	yes	—
349	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
350	C₄	—	2	4	no	—
351	C₄	—	4	4	no	—
352	C₄	—	4	8	no	—
353	C₄	—	4	8	no	—
354	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	—
355	C₄	C₂×#124×C₆₇	1	8	yes	—
356	C₂	—	4	8	no	—
357	C₂	C₆₇×#2563	1	8	yes	direct product factor
358	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
359	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
360	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
361	C₂	C₂×#554×C₆₇	1	2	yes	—
362	C₂	C₂×#555×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
3. C₆₇×#2563	C₂	direct (N × H)
4. C₂×#393×C₆₇	C₂	semidirect
6. C₂×#382×C₆₇	C₂	semidirect
7. C₂×#380×C₆₇	C₂	semidirect
8. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
13. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
24. #393×C₆₇	C₂²	semidirect
26. #382×C₆₇	C₂²	semidirect
27. #380×C₆₇	C₂²	semidirect
28. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
37. C₂²×D₄×C₆₇	Q₈	semidirect
81. C₂×D₄×C₆₇	C₂×Q₈	semidirect
174. C₂×#2563	C₆₇	direct (N × H)
176. C₂×C₆₇	#2563	direct (N × H)
182. C₂²×C₄⋊Q₈	C₂×C₆₇	semidirect
184. #2563	C₂×C₆₇	direct (N × H)
185. C₂×#393	C₂×C₆₇	semidirect
187. C₂×#382	C₂×C₆₇	semidirect
188. C₂×#380	C₂×C₆₇	semidirect
189. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
190. C₆₇	C₂×#2563	direct (N × H)
195. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
206. #393	C₂²×C₆₇	semidirect
208. #382	C₂²×C₆₇	semidirect
209. #380	C₂²×C₆₇	semidirect
210. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
219. C₂²×D₄	Q₈×C₆₇	semidirect
263. C₂×D₄	C₂×Q₈×C₆₇	semidirect
357. C₂	C₆₇×#2563	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×C₄×D₄×C₆₇	2
C₆₇×#2563	2
C₂×#393×C₆₇	2
C₂×#390×C₆₇	2
C₂×#382×C₆₇	2
C₂×#380×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2563	67

Subgroups — C2×C67×#2563 #30714190

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2563 #30714190