Subgroups — C₆₇×#20435 #30699686

Order 17152.

1166 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#424×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2296	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1771	C₂	1	4	yes	maximal, complemented
7	C₈×C₄⋊C₄×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#424×C₆₇	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
19	#405×C₆₇	—	2	4	no	—
20	#405×C₆₇	C₄	1	8	yes	complemented
21	C₂×#97×C₆₇	C₂²	1	2	yes	—
22	C₂×#97×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	—	2	4	no	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	#389×C₆₇	C₄	1	4	yes	—
29	#385×C₆₇	C₄	1	4	yes	—
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
31	#336×C₆₇	C₂²	1	4	yes	—
32	#336×C₆₇	C₂²	1	4	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₆₇	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	—	2	2	no	—
38	C₂×M₁₆×C₆₇	—	2	2	no	—
39	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
40	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₆₇	D₄	1	4	yes	—
46	C₂²×C₈×C₆₇	Q₈	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	#117×C₆₇	—	2	2	no	—
53	#117×C₆₇	—	2	2	no	—
54	#117×C₆₇	C₂×C₄	1	4	yes	—
55	#117×C₆₇	C₂×C₄	1	4	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
67	C₄⋊C₈×C₆₇	C₂×C₄	1	16	yes	complemented
68	#97×C₆₇	—	2	4	no	—
69	#97×C₆₇	C₂×C₄	1	8	yes	complemented
70	C₄×C₈×C₆₇	—	2	4	no	—
71	C₄×C₈×C₆₇	—	2	4	no	—
72	C₄×C₈×C₆₇	—	2	4	no	—
73	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	complemented
74	#95×C₆₇	C₂×C₄	1	8	yes	—
75	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	8	no	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₆₇	#41	1	2	yes	—
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
89	M₁₆×C₆₇	—	2	8	no	—
90	M₁₆×C₆₇	—	2	8	no	—
91	C₂×C₈×C₆₇	—	2	8	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	C₄⋊C₄	1	16	yes	complemented
99	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	2	16	no	—
110	C₄⋊C₄×C₆₇	C₄²	1	8	yes	—
111	C₂²⋊C₄×C₆₇	—	2	4	no	—
112	C₂²⋊C₄×C₆₇	C₄²	1	8	yes	—
113	C₄²×C₆₇	—	2	4	no	—
114	C₄²×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	—	2	2	no	—
116	C₄²×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
119	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
120	C₂³×C₆₇	—	2	4	no	—
121	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
122	C₂³×C₆₇	C₄×D₄	1	2	yes	—
123	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
124	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₆₇	—	2	16	no	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	8	no	—
128	C₂×C₄×C₆₇	—	2	4	no	—
129	C₂×C₄×C₆₇	—	2	4	no	—
130	C₂×C₄×C₆₇	—	2	8	no	—
131	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
132	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
134	C₂×C₄×C₆₇	#117	1	4	yes	—
135	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
136	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
138	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	#131	1	8	yes	—
140	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₈×C₆₇	—	2	4	no	—
147	C₈×C₆₇	—	2	4	no	—
148	C₈×C₆₇	—	2	4	no	—
149	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
150	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
151	C₂²×C₆₇	—	2	4	no	—
152	C₂²×C₆₇	—	2	4	no	—
153	C₂²×C₆₇	—	2	4	no	—
154	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
155	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
156	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₆₇	C₂×#131	1	4	yes	—
159	C₄×C₆₇	—	2	8	no	—
160	C₄×C₆₇	—	2	8	no	—
161	C₄×C₆₇	—	4	4	no	—
162	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
163	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
165	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
167	#20435	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₆₇	—	2	4	no	—
169	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₆₇	C₄×#131	1	2	yes	—
171	C₂×C₆₇	#2929	1	2	yes	—
172	C₂×C₆₇	#2960	1	2	yes	—
173	C₂×#424	C₂×C₆₇	1	1	yes	characteristic
174	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
175	C₂×#405	C₂×C₆₇	1	1	yes	characteristic
176	C₂²×C₄×C₈	C₂×C₆₇	1	1	yes	characteristic
177	#2296	C₂×C₆₇	1	1	yes	characteristic
178	#1771	C₂×C₆₇	1	4	yes	complemented
179	C₈×C₄⋊C₄	C₂×C₆₇	1	4	yes	complemented
180	#1729	C₂×C₆₇	1	1	yes	characteristic
181	#1728	C₂×C₆₇	1	1	yes	characteristic
182	C₆₇	#20435	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
184	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
185	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
186	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
187	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
188	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
189	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
190	#424	C₄×C₆₇	1	8	yes	complemented
191	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
192	#405	—	2	4	no	—
193	#405	C₄×C₆₇	1	8	yes	complemented
194	C₂×#97	C₂²×C₆₇	1	2	yes	—
195	C₂×#97	C₂²×C₆₇	1	2	yes	—
196	C₂×C₄×C₈	—	2	4	no	—
197	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
198	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
199	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
200	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
201	#389	C₄×C₆₇	1	4	yes	—
202	#385	C₄×C₆₇	1	4	yes	—
203	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
204	#336	C₂²×C₆₇	1	4	yes	—
205	#336	C₂²×C₆₇	1	4	yes	—
206	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
207	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂×M₁₆	—	2	2	no	—
212	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
213	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
217	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
218	C₂²×C₈	D₄×C₆₇	1	4	yes	—
219	C₂²×C₈	Q₈×C₆₇	1	4	yes	—
220	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
221	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
222	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
223	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
225	#117	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	C₂×C₄×C₆₇	1	4	yes	—
228	#117	C₂×C₄×C₆₇	1	4	yes	—
229	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	—
231	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
232	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
233	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
234	C₂×C₄²	—	2	2	no	—
235	C₂×C₄²	—	2	2	no	—
236	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
237	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
238	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
239	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
240	C₄⋊C₈	C₂×C₄×C₆₇	1	16	yes	complemented
241	#97	—	2	4	no	—
242	#97	C₂×C₄×C₆₇	1	8	yes	complemented
243	C₄×C₈	—	2	4	no	—
244	C₄×C₈	—	2	4	no	—
245	C₄×C₈	—	2	4	no	—
246	C₄×C₈	C₂×C₄×C₆₇	1	8	yes	complemented
247	#95	C₂×C₄×C₆₇	1	8	yes	—
248	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	8	no	—
251	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
252	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
253	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
254	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
255	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
256	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
257	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
258	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
259	C₂²×C₄	#41×C₆₇	1	2	yes	—
260	C₂²×C₄	#41×C₆₇	1	2	yes	—
261	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
262	M₁₆	—	2	8	no	—
263	M₁₆	—	2	8	no	—
264	C₂×C₈	—	2	8	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₄⋊C₄×C₆₇	1	16	yes	complemented
272	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
273	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
274	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
275	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
277	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
278	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
279	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
280	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	16	no	—
283	C₄⋊C₄	C₄²×C₆₇	1	8	yes	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₂²⋊C₄	C₄²×C₆₇	1	8	yes	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
292	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
293	C₂³	—	2	4	no	—
294	C₂³	C₂×C₄²×C₆₇	1	2	yes	—
295	C₂³	C₄×D₄×C₆₇	1	2	yes	—
296	C₂³	C₄×Q₈×C₆₇	1	2	yes	—
297	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
298	C₂×C₄	—	2	16	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	8	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	8	no	—
304	C₂×C₄	C₂×C₄²×C₆₇	1	8	yes	—
305	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
306	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
307	C₂×C₄	#117×C₆₇	1	4	yes	—
308	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
309	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
310	C₂×C₄	C₄×Q₈×C₆₇	1	4	yes	—
311	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
312	C₂×C₄	#131×C₆₇	1	8	yes	—
313	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
316	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
317	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
318	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	2	4	no	—
321	C₈	—	2	4	no	—
322	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
323	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	4	no	—
327	C₂²	C₄×C₄⋊C₄×C₆₇	1	4	yes	—
328	C₂²	C₂²×C₄²×C₆₇	1	1	yes	characteristic
329	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
330	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
331	C₂²	C₂×#131×C₆₇	1	4	yes	—
332	C₄	—	2	8	no	—
333	C₄	—	2	8	no	—
334	C₄	—	4	4	no	—
335	C₄	C₄×C₄⋊C₄×C₆₇	1	4	yes	—
336	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
337	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
338	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
339	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
340	C₂	—	2	4	no	—
341	C₂	C₂×C₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
342	C₂	C₄×#131×C₆₇	1	2	yes	—
343	C₂	C₆₇×#2929	1	2	yes	—
344	C₂	C₆₇×#2960	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₆₇	C₂³
commutator	C₂²	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₆₇×#1771	C₂	semidirect
7. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
17. #424×C₆₇	C₄	semidirect
20. #405×C₆₇	C₄	semidirect
67. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
69. #97×C₆₇	C₂×C₄	semidirect
73. C₄×C₈×C₆₇	C₂×C₄	semidirect
98. C₂×C₈×C₆₇	C₄⋊C₄	semidirect
167. #20435	C₆₇	direct (N × H)
178. #1771	C₂×C₆₇	semidirect
179. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
182. C₆₇	#20435	direct (N × H)
190. #424	C₄×C₆₇	semidirect
193. #405	C₄×C₆₇	semidirect
240. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
242. #97	C₂×C₄×C₆₇	semidirect
246. C₄×C₈	C₂×C₄×C₆₇	semidirect
271. C₂×C₈	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
C₂×#424×C₆₇	2
C₂×#405×C₆₇	2
C₂²×C₄×C₈×C₆₇	2
C₆₇×#2296	2
C₆₇×#1771	2
C₈×C₄⋊C₄×C₆₇	2
C₆₇×#1729	2
C₆₇×#1728	2
#20435	67

Subgroups — C67×#20435 #30699686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20435 #30699686