Subgroups — C₆₇×#22735 #30701986

Order 17152.

740 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1940	C₂	1	2	yes	maximal
5	C₆₇×#1939	C₂	1	2	yes	maximal
6	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1832	C₂	1	2	yes	maximal
9	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1768	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#446×C₆₇	—	2	4	no	—
14	C₈×Q₈×C₆₇	—	2	4	no	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×#97×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
27	#391×C₆₇	C₄	1	1	yes	characteristic
28	#391×C₆₇	C₄	1	1	yes	characteristic
29	#389×C₆₇	C₄	1	1	yes	characteristic
30	#389×C₆₇	C₄	1	1	yes	characteristic
31	#387×C₆₇	C₄	1	2	yes	—
32	#387×C₆₇	C₄	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	#376×C₆₇	C₂²	1	2	yes	—
36	C₄³×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	1	yes	characteristic
38	#336×C₆₇	C₂²	1	1	yes	characteristic
39	#336×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	#336×C₆₇	C₂²	1	2	yes	—
42	#336×C₆₇	C₂²	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₆₇	D₄	1	4	yes	—
51	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
67	C₄⋊C₈×C₆₇	—	2	2	no	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	4	no	—
70	C₄⋊C₈×C₆₇	—	2	4	no	—
71	C₄⋊C₈×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₈×C₆₇	—	2	2	no	—
74	C₄⋊C₈×C₆₇	—	2	4	no	—
75	#97×C₆₇	—	2	4	no	—
76	C₄×C₈×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	—	2	4	no	—
78	C₄×C₈×C₆₇	—	2	2	no	—
79	#95×C₆₇	C₂×C₄	1	2	yes	—
80	#95×C₆₇	C₂×C₄	1	2	yes	—
81	#95×C₆₇	C₂×C₄	1	2	yes	—
82	#95×C₆₇	C₂×C₄	1	2	yes	—
83	#95×C₆₇	C₂×C₄	1	2	yes	—
84	#95×C₆₇	C₂×C₄	1	2	yes	—
85	#95×C₆₇	C₂×C₄	1	2	yes	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	2	no	—
88	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₆₇	—	2	1	no	—
101	C₂×C₈×C₆₇	—	2	1	no	—
102	C₂×C₈×C₆₇	—	2	1	no	—
103	C₂×C₈×C₆₇	—	2	1	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	2	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	2	2	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	#41	1	2	yes	—
117	C₂×C₈×C₆₇	#41	1	2	yes	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	4	2	no	—
122	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
123	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	—	2	1	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	—	2	2	no	—
131	C₄²×C₆₇	—	2	1	no	—
132	C₄²×C₆₇	—	2	1	no	—
133	C₄²×C₆₇	—	4	1	no	—
134	C₄²×C₆₇	C₂×D₄	1	2	yes	—
135	C₄²×C₆₇	C₂×D₄	1	4	yes	—
136	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
137	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
138	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
139	Q₈×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	1	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	2	1	no	—
156	C₂×C₄×C₆₇	—	2	1	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	1	no	—
161	C₂×C₄×C₆₇	—	2	1	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
165	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
166	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄×C₆₇	#131	1	2	yes	—
168	C₂×C₄×C₆₇	#131	1	2	yes	—
169	C₂×C₄×C₆₇	#131	1	2	yes	—
170	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
172	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
176	C₈×C₆₇	—	2	2	no	—
177	C₈×C₆₇	—	2	2	no	—
178	C₈×C₆₇	—	4	2	no	—
179	C₈×C₆₇	—	4	1	no	—
180	C₈×C₆₇	—	4	1	no	—
181	C₈×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
184	C₂²×C₆₇	#519	1	1	yes	characteristic
185	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
186	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
187	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
188	C₂²×C₆₇	#568	1	1	yes	characteristic
189	C₄×C₆₇	—	2	4	no	—
190	C₄×C₆₇	—	2	2	no	—
191	C₄×C₆₇	—	2	4	no	—
192	C₄×C₆₇	—	2	2	no	—
193	C₄×C₆₇	—	4	2	no	—
194	C₄×C₆₇	—	4	1	no	—
195	C₄×C₆₇	—	4	2	no	—
196	C₄×C₆₇	—	4	1	no	—
197	C₄×C₆₇	—	4	1	no	—
198	C₄×C₆₇	#424	1	4	yes	—
199	C₄×C₆₇	#548	1	1	yes	characteristic
200	C₄×C₆₇	#548	1	1	yes	characteristic
201	C₄×C₆₇	#557	1	1	yes	characteristic
202	C₄×C₆₇	#557	1	1	yes	characteristic
203	#22735	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂×C₆₇	#2351	1	1	yes	characteristic
205	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
206	C₂×C₆₇	#2906	1	1	yes	characteristic
207	C₂×C₆₇	#2966	1	1	yes	characteristic
208	C₂×C₆₇	#2969	1	1	yes	characteristic
209	C₂×C₆₇	#2985	1	1	yes	characteristic
210	C₂×C₆₇	#2986	1	1	yes	characteristic
211	C₂×#446	C₂×C₆₇	1	1	yes	characteristic
212	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic
213	#2295	C₂×C₆₇	1	1	yes	characteristic
214	#1940	C₂×C₆₇	1	2	yes	—
215	#1939	C₂×C₆₇	1	2	yes	—
216	#1939	C₂×C₆₇	1	1	yes	characteristic
217	#1939	C₂×C₆₇	1	1	yes	characteristic
218	#1832	C₂×C₆₇	1	2	yes	—
219	#1831	C₂×C₆₇	1	1	yes	characteristic
220	#1768	C₂×C₆₇	1	2	yes	—
221	C₄×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic
222	C₆₇	#22735	1	1	yes	characteristic, direct product factor, Sylow
223	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
224	#446	—	2	4	no	—
225	C₈×Q₈	—	2	4	no	—
226	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
229	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
230	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
232	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
233	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#97	C₂²×C₆₇	1	2	yes	—
235	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
237	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
238	#391	C₄×C₆₇	1	1	yes	characteristic
239	#391	C₄×C₆₇	1	1	yes	characteristic
240	#389	C₄×C₆₇	1	1	yes	characteristic
241	#389	C₄×C₆₇	1	1	yes	characteristic
242	#387	C₄×C₆₇	1	2	yes	—
243	#387	C₄×C₆₇	1	2	yes	—
244	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
245	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
246	#376	C₂²×C₆₇	1	2	yes	—
247	C₄³	C₂²×C₆₇	1	1	yes	characteristic
248	#336	C₂²×C₆₇	1	2	yes	—
249	#336	C₂²×C₆₇	1	2	yes	—
250	#336	C₂²×C₆₇	1	1	yes	characteristic
251	#336	C₂²×C₆₇	1	2	yes	—
252	#336	C₂²×C₆₇	1	2	yes	—
253	#336	C₂²×C₆₇	1	1	yes	characteristic
254	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
255	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
256	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
261	C₄×Q₈	D₄×C₆₇	1	4	yes	—
262	C₄×Q₈	Q₈×C₆₇	1	4	yes	—
263	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
264	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
268	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
269	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
270	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
271	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
272	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
276	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
278	C₄⋊C₈	—	2	2	no	—
279	C₄⋊C₈	—	2	4	no	—
280	C₄⋊C₈	—	2	4	no	—
281	C₄⋊C₈	—	2	2	no	—
282	C₄⋊C₈	—	2	2	no	—
283	C₄⋊C₈	—	2	4	no	—
284	C₄⋊C₈	—	2	2	no	—
285	C₄⋊C₈	—	2	4	no	—
286	#97	—	2	4	no	—
287	C₄×C₈	—	2	2	no	—
288	C₄×C₈	—	2	4	no	—
289	C₄×C₈	—	2	2	no	—
290	#95	C₂×C₄×C₆₇	1	2	yes	—
291	#95	C₂×C₄×C₆₇	1	2	yes	—
292	#95	C₂×C₄×C₆₇	1	2	yes	—
293	#95	C₂×C₄×C₆₇	1	2	yes	—
294	#95	C₂×C₄×C₆₇	1	2	yes	—
295	#95	C₂×C₄×C₆₇	1	2	yes	—
296	#95	C₂×C₄×C₆₇	1	2	yes	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	C₄⋊C₄×C₆₇	1	4	yes	—
300	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
302	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	#41×C₆₇	1	2	yes	—
328	C₂×C₈	#41×C₆₇	1	2	yes	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
334	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	4	1	no	—
345	C₄²	C₂×D₄×C₆₇	1	2	yes	—
346	C₄²	C₂×D₄×C₆₇	1	4	yes	—
347	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
348	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
349	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
350	Q₈	—	2	8	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
375	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
376	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
377	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
378	C₂×C₄	#131×C₆₇	1	2	yes	—
379	C₂×C₄	#131×C₆₇	1	2	yes	—
380	C₂×C₄	#131×C₆₇	1	2	yes	—
381	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
382	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
385	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
386	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
387	C₈	—	2	2	no	—
388	C₈	—	2	2	no	—
389	C₈	—	4	1	no	—
390	C₈	—	4	2	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	2	no	—
393	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
394	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
395	C₂²	#519×C₆₇	1	1	yes	characteristic
396	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
397	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
398	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
399	C₂²	#568×C₆₇	1	1	yes	characteristic
400	C₄	—	2	4	no	—
401	C₄	—	2	2	no	—
402	C₄	—	2	4	no	—
403	C₄	—	2	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	#424×C₆₇	1	4	yes	—
410	C₄	#548×C₆₇	1	1	yes	characteristic
411	C₄	#548×C₆₇	1	1	yes	characteristic
412	C₄	#557×C₆₇	1	1	yes	characteristic
413	C₄	#557×C₆₇	1	1	yes	characteristic
414	C₂	C₆₇×#2351	1	1	yes	characteristic
415	C₂	C₂×#424×C₆₇	1	1	yes	characteristic
416	C₂	C₆₇×#2906	1	1	yes	characteristic
417	C₂	C₆₇×#2966	1	1	yes	characteristic
418	C₂	C₆₇×#2969	1	1	yes	characteristic
419	C₂	C₆₇×#2985	1	1	yes	characteristic
420	C₂	C₆₇×#2986	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
203. #22735	C₆₇	direct (N × H)
222. C₆₇	#22735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#446×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₆₇×#2295	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1832	2
C₆₇×#1831	2
C₆₇×#1768	2
C₄×C₄⋊C₈×C₆₇	2
#22735	67

Subgroups — C67×#22735 #30701986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22735 #30701986