Subgroups — C₂×C₆₇×#2581 #30714150

Order 17152.

2614 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
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2581	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	2	yes	maximal
5	C₂×#395×C₆₇	C₂	1	2	yes	maximal
6	C₂×#391×C₆₇	C₂	1	2	yes	maximal
7	C₂×#389×C₆₇	C₂	1	2	yes	maximal
8	C₂×#387×C₆₇	C₂	1	4	yes	maximal
9	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
13	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	—
15	C₂×C₄×Q₈×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	4	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
25	#400×C₆₇	C₂²	1	16	yes	—
26	#395×C₆₇	C₂²	1	16	yes	—
27	#391×C₆₇	C₂²	1	16	yes	—
28	#389×C₆₇	C₂²	1	16	yes	—
29	#387×C₆₇	C₂²	1	32	yes	—
30	#382×C₆₇	C₂²	1	8	yes	—
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	4	yes	—
36	C₂²×Q₈×C₆₇	—	2	4	no	—
37	C₂²×Q₈×C₆₇	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	2	8	no	—
39	C₂²×Q₈×C₆₇	—	2	4	no	—
40	C₂²×Q₈×C₆₇	Q₈	1	8	yes	complemented
41	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
42	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	2	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	2	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₆₇	—	2	32	no	—
52	C₄×Q₈×C₆₇	—	2	32	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
65	C₂×C₄²×C₆₇	—	2	8	no	—
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	8	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
71	#95×C₆₇	C₂³	1	8	yes	—
72	#95×C₆₇	C₂³	1	8	yes	—
73	#95×C₆₇	C₂³	1	16	yes	—
74	#95×C₆₇	C₂³	1	16	yes	—
75	#95×C₆₇	C₂³	1	16	yes	—
76	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×Q₈×C₆₇	—	2	32	no	—
78	C₂×Q₈×C₆₇	—	2	32	no	—
79	C₂×Q₈×C₆₇	—	2	8	no	—
80	C₂×Q₈×C₆₇	—	2	8	no	—
81	C₂×Q₈×C₆₇	—	2	16	no	—
82	C₂×Q₈×C₆₇	—	2	8	no	—
83	C₂×Q₈×C₆₇	C₂×Q₈	1	16	yes	complemented
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	2	8	no	—
93	C₂²×C₄×C₆₇	—	2	8	no	—
94	C₂²×C₄×C₆₇	—	2	8	no	—
95	C₂²×C₄×C₆₇	—	2	8	no	—
96	C₂²×C₄×C₆₇	—	2	8	no	—
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	4	yes	—
100	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
101	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
102	C₂²×C₄×C₆₇	#41	1	8	yes	—
103	C₂²×C₄×C₆₇	#41	1	4	yes	—
104	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
105	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
107	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
112	C₄⋊C₄×C₆₇	—	2	16	no	—
113	C₄⋊C₄×C₆₇	—	2	16	no	—
114	C₄⋊C₄×C₆₇	—	2	8	no	—
115	C₄⋊C₄×C₆₇	—	2	8	no	—
116	C₄⋊C₄×C₆₇	—	2	32	no	—
117	C₄⋊C₄×C₆₇	—	2	16	no	—
118	C₄²×C₆₇	—	2	16	no	—
119	C₄²×C₆₇	—	2	16	no	—
120	C₄²×C₆₇	C₂×D₄	1	16	yes	—
121	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
122	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
123	C₂³×C₆₇	C₂×#41	1	2	yes	—
124	C₂³×C₆₇	C₂×#41	1	4	yes	—
125	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
126	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
127	Q₈×C₆₇	—	2	32	no	—
128	Q₈×C₆₇	—	2	32	no	—
129	C₂×C₄×C₆₇	—	2	16	no	—
130	C₂×C₄×C₆₇	—	2	8	no	—
131	C₂×C₄×C₆₇	—	2	16	no	—
132	C₂×C₄×C₆₇	—	2	8	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	4	no	—
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	2	4	no	—
140	C₂×C₄×C₆₇	—	2	4	no	—
141	C₂×C₄×C₆₇	—	2	8	no	—
142	C₂×C₄×C₆₇	—	2	8	no	—
143	C₂×C₄×C₆₇	—	2	8	no	—
144	C₂×C₄×C₆₇	—	2	8	no	—
145	C₂×C₄×C₆₇	—	2	8	no	—
146	C₂×C₄×C₆₇	—	4	8	no	—
147	C₂×C₄×C₆₇	—	4	8	no	—
148	C₂×C₄×C₆₇	—	4	16	no	—
149	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
151	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
152	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
153	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
154	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
155	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
156	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
157	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
158	C₂²×C₆₇	#525	1	4	yes	—
159	C₂²×C₆₇	#536	1	4	yes	—
160	C₂²×C₆₇	#556	1	8	yes	—
161	C₂²×C₆₇	#557	1	8	yes	—
162	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
163	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
164	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
165	C₂²×C₆₇	C₂²×#41	1	2	yes	—
166	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
167	C₄×C₆₇	—	2	8	no	—
168	C₄×C₆₇	—	2	4	no	—
169	C₄×C₆₇	—	2	8	no	—
170	C₄×C₆₇	—	2	4	no	—
171	C₄×C₆₇	—	4	4	no	—
172	C₄×C₆₇	—	4	4	no	—
173	C₄×C₆₇	—	4	8	no	—
174	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	—
175	C₂×#2581	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	#2581	1	8	yes	direct product factor
177	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
179	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#556	1	2	yes	—
181	C₂×C₆₇	C₂×#557	1	2	yes	—
182	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
183	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
184	#2581	C₂×C₆₇	1	16	yes	direct product factor
185	C₂×#400	C₂×C₆₇	1	2	yes	—
186	C₂×#395	C₂×C₆₇	1	2	yes	—
187	C₂×#391	C₂×C₆₇	1	2	yes	—
188	C₂×#389	C₂×C₆₇	1	2	yes	—
189	C₂×#387	C₂×C₆₇	1	4	yes	—
190	C₂×#382	C₂×C₆₇	1	1	yes	characteristic
191	C₆₇	C₂×#2581	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
193	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×C₄×Q₈	—	2	8	no	—
195	C₂×C₄×Q₈	—	2	8	no	—
196	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	—
197	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	—
198	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
199	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
200	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
201	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
202	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
203	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
204	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
205	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
206	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
207	#400	C₂²×C₆₇	1	16	yes	—
208	#395	C₂²×C₆₇	1	16	yes	—
209	#391	C₂²×C₆₇	1	16	yes	—
210	#389	C₂²×C₆₇	1	16	yes	—
211	#387	C₂²×C₆₇	1	32	yes	—
212	#382	C₂²×C₆₇	1	8	yes	—
213	C₂×#95	C₂²×C₆₇	1	2	yes	—
214	C₂×#95	C₂²×C₆₇	1	4	yes	—
215	C₂×#95	C₂²×C₆₇	1	4	yes	—
216	C₂×#95	C₂²×C₆₇	1	4	yes	—
217	C₂×#95	C₂²×C₆₇	1	2	yes	—
218	C₂²×Q₈	—	2	4	no	—
219	C₂²×Q₈	—	2	4	no	—
220	C₂²×Q₈	—	2	8	no	—
221	C₂²×Q₈	—	2	4	no	—
222	C₂²×Q₈	Q₈×C₆₇	1	8	yes	complemented
223	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
224	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
225	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
226	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
227	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
228	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
229	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
230	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
231	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
232	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
233	C₄×Q₈	—	2	32	no	—
234	C₄×Q₈	—	2	32	no	—
235	C₂×C₄⋊C₄	—	2	8	no	—
236	C₂×C₄⋊C₄	—	2	8	no	—
237	C₂×C₄⋊C₄	—	2	4	no	—
238	C₂×C₄⋊C₄	—	2	4	no	—
239	C₂×C₄⋊C₄	—	2	16	no	—
240	C₂×C₄⋊C₄	—	2	8	no	—
241	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
242	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
243	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
244	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
245	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
246	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
247	C₂×C₄²	—	2	8	no	—
248	C₂×C₄²	—	2	8	no	—
249	C₂×C₄²	D₄×C₆₇	1	8	yes	—
250	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
251	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
252	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
253	#95	C₂³×C₆₇	1	8	yes	—
254	#95	C₂³×C₆₇	1	8	yes	—
255	#95	C₂³×C₆₇	1	16	yes	—
256	#95	C₂³×C₆₇	1	16	yes	—
257	#95	C₂³×C₆₇	1	16	yes	—
258	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
259	C₂×Q₈	—	2	32	no	—
260	C₂×Q₈	—	2	32	no	—
261	C₂×Q₈	—	2	8	no	—
262	C₂×Q₈	—	2	8	no	—
263	C₂×Q₈	—	2	16	no	—
264	C₂×Q₈	—	2	8	no	—
265	C₂×Q₈	C₂×Q₈×C₆₇	1	16	yes	complemented
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	4	no	—
269	C₂²×C₄	—	2	4	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	8	no	—
275	C₂²×C₄	—	2	8	no	—
276	C₂²×C₄	—	2	8	no	—
277	C₂²×C₄	—	2	8	no	—
278	C₂²×C₄	—	2	8	no	—
279	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
280	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
281	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
282	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
283	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
284	C₂²×C₄	#41×C₆₇	1	8	yes	—
285	C₂²×C₄	#41×C₆₇	1	4	yes	—
286	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
287	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
288	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
289	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
290	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
292	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
293	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
294	C₄⋊C₄	—	2	16	no	—
295	C₄⋊C₄	—	2	16	no	—
296	C₄⋊C₄	—	2	8	no	—
297	C₄⋊C₄	—	2	8	no	—
298	C₄⋊C₄	—	2	32	no	—
299	C₄⋊C₄	—	2	16	no	—
300	C₄²	—	2	16	no	—
301	C₄²	—	2	16	no	—
302	C₄²	C₂×D₄×C₆₇	1	16	yes	—
303	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
304	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
305	C₂³	C₂×#41×C₆₇	1	2	yes	—
306	C₂³	C₂×#41×C₆₇	1	4	yes	—
307	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
308	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
309	Q₈	—	2	32	no	—
310	Q₈	—	2	32	no	—
311	C₂×C₄	—	2	16	no	—
312	C₂×C₄	—	2	8	no	—
313	C₂×C₄	—	2	16	no	—
314	C₂×C₄	—	2	8	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	8	no	—
324	C₂×C₄	—	2	8	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	2	8	no	—
327	C₂×C₄	—	2	8	no	—
328	C₂×C₄	—	4	8	no	—
329	C₂×C₄	—	4	8	no	—
330	C₂×C₄	—	4	16	no	—
331	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	—
332	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
333	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
334	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
335	C₂×C₄	C₂²×Q₈×C₆₇	1	8	yes	—
336	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
337	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
338	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
339	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
340	C₂²	#525×C₆₇	1	4	yes	—
341	C₂²	#536×C₆₇	1	4	yes	—
342	C₂²	#556×C₆₇	1	8	yes	—
343	C₂²	#557×C₆₇	1	8	yes	—
344	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
345	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
346	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
347	C₂²	C₂²×#41×C₆₇	1	2	yes	—
348	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
349	C₄	—	2	8	no	—
350	C₄	—	2	4	no	—
351	C₄	—	2	8	no	—
352	C₄	—	2	4	no	—
353	C₄	—	4	4	no	—
354	C₄	—	4	4	no	—
355	C₄	—	4	8	no	—
356	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	—
357	C₂	C₆₇×#2581	1	8	yes	direct product factor
358	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
359	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
360	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
361	C₂	C₂×#556×C₆₇	1	2	yes	—
362	C₂	C₂×#557×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
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×#2581	C₂	direct (N × H)
40. C₂²×Q₈×C₆₇	Q₈	semidirect
83. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
175. C₂×#2581	C₆₇	direct (N × H)
176. C₂×C₆₇	#2581	direct (N × H)
184. #2581	C₂×C₆₇	direct (N × H)
191. C₆₇	C₂×#2581	direct (N × H)
222. C₂²×Q₈	Q₈×C₆₇	semidirect
265. C₂×Q₈	C₂×Q₈×C₆₇	semidirect
357. C₂	C₆₇×#2581	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
C₆₇×#2581	2
C₂×#400×C₆₇	2
C₂×#395×C₆₇	2
C₂×#391×C₆₇	2
C₂×#389×C₆₇	2
C₂×#387×C₆₇	2
C₂×#382×C₆₇	2
C₂×#2581	67

Subgroups — C2×C67×#2581 #30714150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2581 #30714150