Subgroups — C₆₇×#10850 #30690101

Order 17152.

998 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#340×C₆₇	C₂	1	4	yes	maximal
5	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
6	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
7	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
8	C₂²×C₄²×C₆₇	—	2	1	no	—
9	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
11	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
12	C₂×C₄⋊C₈×C₆₇	C₄	1	4	yes	complemented
13	#382×C₆₇	—	2	2	no	—
14	#382×C₆₇	C₄	1	4	yes	complemented
15	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
16	C₄×C₄⋊C₄×C₆₇	C₄	1	4	yes	complemented
17	C₂×#95×C₆₇	—	2	1	no	—
18	C₂×#95×C₆₇	—	2	1	no	—
19	C₂×#95×C₆₇	—	2	1	no	—
20	#340×C₆₇	—	2	8	no	—
21	C₂³×C₄×C₆₇	—	2	2	no	—
22	C₂³×C₄×C₆₇	—	2	2	no	—
23	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₆₇	—	2	4	no	—
27	C₂²×C₈×C₆₇	—	2	2	no	—
28	C₂²×C₈×C₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	—	2	4	no	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
33	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₆₇	Q₈	1	4	yes	—
41	C₂×C₄²×C₆₇	—	2	4	no	—
42	C₂×C₄²×C₆₇	—	4	4	no	—
43	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₆₇	D₄	1	2	yes	—
47	C₂×C₄²×C₆₇	D₄	1	2	yes	—
48	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
49	C₄⋊C₈×C₆₇	—	2	8	no	—
50	#95×C₆₇	—	2	4	no	—
51	#95×C₆₇	—	2	4	no	—
52	#95×C₆₇	—	2	4	no	—
53	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	—	2	4	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	—	2	2	no	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	4	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	—	4	4	no	—
62	C₂²×C₄×C₆₇	—	4	4	no	—
63	C₂²×C₄×C₆₇	—	4	4	no	—
64	C₂²×C₄×C₆₇	—	4	4	no	—
65	C₂²×C₄×C₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₆₇	#41	1	2	yes	—
82	C₂²×C₄×C₆₇	#41	1	2	yes	—
83	C₂×C₈×C₆₇	—	2	4	no	—
84	C₂×C₈×C₆₇	—	2	8	no	—
85	C₂×C₈×C₆₇	—	4	4	no	—
86	C₂×C₈×C₆₇	—	4	2	no	—
87	C₂×C₈×C₆₇	—	4	2	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	2	8	no	—
92	C₄⋊C₄×C₆₇	—	4	2	no	—
93	C₄⋊C₄×C₆₇	—	4	2	no	—
94	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
95	C₄²×C₆₇	—	4	8	no	—
96	C₄²×C₆₇	C₄²	1	4	yes	—
97	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
98	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
99	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
100	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₆₇	#121	1	2	yes	—
104	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
105	C₂³×C₆₇	#135	1	4	yes	—
106	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
107	C₂³×C₆₇	#137	1	2	yes	—
108	C₂×C₄×C₆₇	—	2	4	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	4	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	2	8	no	—
116	C₂×C₄×C₆₇	—	4	4	no	—
117	C₂×C₄×C₆₇	—	4	4	no	—
118	C₂×C₄×C₆₇	—	4	8	no	—
119	C₂×C₄×C₆₇	—	4	8	no	—
120	C₂×C₄×C₆₇	—	4	4	no	—
121	C₂×C₄×C₆₇	—	4	4	no	—
122	C₂×C₄×C₆₇	—	4	4	no	—
123	C₂×C₄×C₆₇	—	8	4	no	—
124	C₂×C₄×C₆₇	—	8	4	no	—
125	C₂×C₄×C₆₇	#95	1	4	yes	—
126	C₂×C₄×C₆₇	#95	1	4	yes	—
127	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#117	1	2	yes	—
133	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₆₇	#121	1	2	yes	—
138	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
140	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
141	C₂×C₄×C₆₇	#125	1	2	yes	—
142	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
144	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
145	C₈×C₆₇	—	4	4	no	—
146	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
147	C₂²×C₆₇	#380	1	2	yes	—
148	C₂²×C₆₇	#389	1	2	yes	—
149	C₂²×C₆₇	#390	1	1	yes	characteristic
150	C₂²×C₆₇	#391	1	1	yes	characteristic
151	C₂²×C₆₇	#416	1	2	yes	—
152	C₂²×C₆₇	#416	1	2	yes	—
153	C₂²×C₆₇	#417	1	2	yes	—
154	C₂²×C₆₇	#418	1	1	yes	characteristic
155	C₂²×C₆₇	#419	1	1	yes	characteristic
156	C₂²×C₆₇	#427	1	2	yes	—
157	C₂²×C₆₇	#428	1	2	yes	—
158	C₂²×C₆₇	#463	1	4	yes	—
159	C₂²×C₆₇	#464	1	4	yes	—
160	C₂²×C₆₇	#478	1	4	yes	—
161	C₂²×C₆₇	#479	1	4	yes	—
162	C₄×C₆₇	—	2	4	no	—
163	C₄×C₆₇	—	2	4	no	—
164	C₄×C₆₇	—	4	8	no	—
165	C₄×C₆₇	—	8	4	no	—
166	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
167	C₄×C₆₇	#382	1	1	yes	characteristic
168	C₄×C₆₇	#384	1	1	yes	characteristic
169	C₄×C₆₇	#389	1	2	yes	—
170	C₄×C₆₇	#390	1	1	yes	characteristic
171	C₄×C₆₇	#391	1	1	yes	characteristic
172	#10850	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
174	C₂×C₆₇	#1735	1	1	yes	characteristic
175	C₂×C₆₇	#1736	1	1	yes	characteristic
176	C₂×C₆₇	#1804	1	2	yes	—
177	C₂×C₆₇	#1805	1	2	yes	—
178	C₂×C₆₇	#1949	1	4	yes	—
179	C₂×C₆₇	#1983	1	2	yes	—
180	C₂×C₆₇	#1989	1	2	yes	—
181	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
182	C₂×#382	C₂×C₆₇	1	1	yes	characteristic
183	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
184	C₂×#340	C₂×C₆₇	1	4	yes	—
185	C₆₇	#10850	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
187	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
188	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
189	C₂²×C₄²	—	2	1	no	—
190	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	2	2	no	—
192	C₂×C₄⋊C₈	—	2	2	no	—
193	C₂×C₄⋊C₈	C₄×C₆₇	1	4	yes	complemented
194	#382	—	2	2	no	—
195	#382	C₄×C₆₇	1	4	yes	complemented
196	C₄×C₄⋊C₄	—	2	2	no	—
197	C₄×C₄⋊C₄	C₄×C₆₇	1	4	yes	complemented
198	C₂×#95	—	2	1	no	—
199	C₂×#95	—	2	1	no	—
200	C₂×#95	—	2	1	no	—
201	#340	—	2	8	no	—
202	C₂³×C₄	—	2	2	no	—
203	C₂³×C₄	—	2	2	no	—
204	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
205	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
206	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
207	C₂²×C₈	—	2	4	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	2	4	no	—
211	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
219	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
220	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
221	C₂×C₄⋊C₄	Q₈×C₆₇	1	4	yes	—
222	C₂×C₄²	—	2	4	no	—
223	C₂×C₄²	—	4	4	no	—
224	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
225	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
226	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
227	C₂×C₄²	D₄×C₆₇	1	2	yes	—
228	C₂×C₄²	D₄×C₆₇	1	2	yes	—
229	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
230	C₄⋊C₈	—	2	8	no	—
231	#95	—	2	4	no	—
232	#95	—	2	4	no	—
233	#95	—	2	4	no	—
234	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
249	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
250	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
251	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
252	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
253	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
254	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
256	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
260	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
261	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
262	C₂²×C₄	#41×C₆₇	1	2	yes	—
263	C₂²×C₄	#41×C₆₇	1	2	yes	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	2	8	no	—
266	C₂×C₈	—	4	4	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
276	C₄²	—	4	8	no	—
277	C₄²	C₄²×C₆₇	1	4	yes	—
278	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
279	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
280	C₄²	C₄⋊C₄×C₆₇	1	4	yes	—
281	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
282	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
283	C₂³	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
284	C₂³	#121×C₆₇	1	2	yes	—
285	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
286	C₂³	#135×C₆₇	1	4	yes	—
287	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
288	C₂³	#137×C₆₇	1	2	yes	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	8	4	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	#95×C₆₇	1	4	yes	—
307	C₂×C₄	#95×C₆₇	1	4	yes	—
308	C₂×C₄	C₂×C₄²×C₆₇	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
310	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
313	C₂×C₄	#117×C₆₇	1	2	yes	—
314	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
315	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
316	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
317	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
318	C₂×C₄	#121×C₆₇	1	2	yes	—
319	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
320	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
321	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
322	C₂×C₄	#125×C₆₇	1	2	yes	—
323	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
324	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
325	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
326	C₈	—	4	4	no	—
327	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
328	C₂²	#380×C₆₇	1	2	yes	—
329	C₂²	#389×C₆₇	1	2	yes	—
330	C₂²	#390×C₆₇	1	1	yes	characteristic
331	C₂²	#391×C₆₇	1	1	yes	characteristic
332	C₂²	#416×C₆₇	1	2	yes	—
333	C₂²	#416×C₆₇	1	2	yes	—
334	C₂²	#417×C₆₇	1	2	yes	—
335	C₂²	#418×C₆₇	1	1	yes	characteristic
336	C₂²	#419×C₆₇	1	1	yes	characteristic
337	C₂²	#427×C₆₇	1	2	yes	—
338	C₂²	#428×C₆₇	1	2	yes	—
339	C₂²	#463×C₆₇	1	4	yes	—
340	C₂²	#464×C₆₇	1	4	yes	—
341	C₂²	#478×C₆₇	1	4	yes	—
342	C₂²	#479×C₆₇	1	4	yes	—
343	C₄	—	2	4	no	—
344	C₄	—	2	4	no	—
345	C₄	—	4	8	no	—
346	C₄	—	8	4	no	—
347	C₄	C₄×C₄⋊C₄×C₆₇	1	2	yes	—
348	C₄	#382×C₆₇	1	1	yes	characteristic
349	C₄	#384×C₆₇	1	1	yes	characteristic
350	C₄	#389×C₆₇	1	2	yes	—
351	C₄	#390×C₆₇	1	1	yes	characteristic
352	C₄	#391×C₆₇	1	1	yes	characteristic
353	C₂	C₆₇×C₄⋊#95	1	1	yes	characteristic
354	C₂	C₆₇×#1735	1	1	yes	characteristic
355	C₂	C₆₇×#1736	1	1	yes	characteristic
356	C₂	C₆₇×#1804	1	2	yes	—
357	C₂	C₆₇×#1805	1	2	yes	—
358	C₂	C₆₇×#1949	1	4	yes	—
359	C₂	C₆₇×#1983	1	2	yes	—
360	C₂	C₆₇×#1989	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
12. C₂×C₄⋊C₈×C₆₇	C₄	semidirect
14. #382×C₆₇	C₄	semidirect
16. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
172. #10850	C₆₇	direct (N × H)
185. C₆₇	#10850	direct (N × H)
193. C₂×C₄⋊C₈	C₄×C₆₇	semidirect
195. #382	C₄×C₆₇	semidirect
197. C₄×C₄⋊C₄	C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊C₈×C₆₇	2
C₂×#382×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#340×C₆₇	2
#10850	67

Subgroups — C67×#10850 #30690101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10850 #30690101