Subgroups — C₆₇×#21848 #30701099

Order 17152.

1170 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1984	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1954	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1869	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1798	C₂	1	4	yes	maximal, complemented
8	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
9	#529×C₆₇	—	2	4	no	—
10	#529×C₆₇	—	2	4	no	—
11	C₂×#126×C₆₇	C₄	1	2	yes	—
12	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
14	C₂×#123×C₆₇	C₄	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#431×C₆₇	—	2	4	no	—
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×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	8	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	8	yes	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	—	2	4	no	—
39	C₂²×C₈×C₆₇	—	4	4	no	—
40	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
42	#126×C₆₇	—	2	8	no	—
43	#125×C₆₇	—	2	4	no	—
44	#124×C₆₇	—	2	4	no	—
45	#123×C₆₇	—	2	8	no	—
46	C₂²⋊Q₈×C₆₇	—	2	4	no	—
47	#121×C₆₇	—	2	4	no	—
48	C₄×Q₈×C₆₇	—	2	2	no	—
49	C₄×Q₈×C₆₇	—	2	2	no	—
50	C₄×D₄×C₆₇	—	2	4	no	—
51	C₄×D₄×C₆₇	—	2	4	no	—
52	C₄×D₄×C₆₇	—	2	2	no	—
53	C₄×D₄×C₆₇	—	2	2	no	—
54	#117×C₆₇	—	2	4	no	—
55	#117×C₆₇	—	2	4	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	D₄	1	4	yes	—
70	C₂×C₄²×C₆₇	D₄	1	4	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₆₇	—	2	4	no	—
75	C₄⋊C₈×C₆₇	—	2	4	no	—
76	C₄⋊C₈×C₆₇	—	2	4	no	—
77	C₂²⋊C₈×C₆₇	—	2	4	no	—
78	C₂²⋊C₈×C₆₇	—	4	8	no	—
79	C₄×C₈×C₆₇	—	2	4	no	—
80	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	4	no	—
84	C₂×D₄×C₆₇	—	2	4	no	—
85	C₂×D₄×C₆₇	—	2	8	no	—
86	C₂×D₄×C₆₇	—	2	2	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	4	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	2	2	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
97	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₆₇	—	2	4	no	—
110	C₂×C₈×C₆₇	—	2	4	no	—
111	C₂×C₈×C₆₇	—	2	8	no	—
112	C₂×C₈×C₆₇	—	2	4	no	—
113	C₂×C₈×C₆₇	—	2	4	no	—
114	C₂×C₈×C₆₇	—	4	8	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₂²⋊C₄×C₆₇	—	2	4	no	—
122	C₂²⋊C₄×C₆₇	—	2	4	no	—
123	C₂²⋊C₄×C₆₇	—	2	4	no	—
124	C₂²⋊C₄×C₆₇	—	2	4	no	—
125	C₂²⋊C₄×C₆₇	—	2	4	no	—
126	C₄²×C₆₇	—	2	2	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
129	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
130	C₄²×C₆₇	C₂×D₄	1	2	yes	—
131	C₄²×C₆₇	C₂×D₄	1	2	yes	—
132	C₄²×C₆₇	C₂×D₄	1	2	yes	—
133	C₄²×C₆₇	C₂×D₄	1	2	yes	—
134	C₂³×C₆₇	—	2	2	no	—
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	—	2	4	no	—
138	C₂³×C₆₇	—	2	4	no	—
139	C₂³×C₆₇	—	2	4	no	—
140	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
141	C₂³×C₆₇	#131	1	4	yes	—
142	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	Q₈×C₆₇	—	4	4	no	—
144	D₄×C₆₇	—	4	8	no	—
145	D₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	8	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	8	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	2	no	—
158	C₂×C₄×C₆₇	—	2	4	no	—
159	C₂×C₄×C₆₇	—	2	4	no	—
160	C₂×C₄×C₆₇	—	2	4	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	4	no	—
164	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
167	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂×C₄×C₆₇	#131	1	4	yes	—
170	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
173	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
174	C₈×C₆₇	—	4	4	no	—
175	C₈×C₆₇	—	4	4	no	—
176	C₂²×C₆₇	—	2	2	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	8	no	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	—	4	2	no	—
181	C₂²×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	—	4	8	no	—
183	C₂²×C₆₇	#408	1	4	yes	—
184	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₆₇	#520	1	2	yes	—
186	C₂²×C₆₇	C₂×#131	1	2	yes	—
187	C₂²×C₆₇	C₂×#131	1	2	yes	—
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	2	2	no	—
190	C₄×C₆₇	—	2	2	no	—
191	C₄×C₆₇	—	2	2	no	—
192	C₄×C₆₇	—	4	2	no	—
193	C₄×C₆₇	—	4	2	no	—
194	C₄×C₆₇	—	4	2	no	—
195	C₄×C₆₇	#408	1	4	yes	—
196	C₄×C₆₇	#535	1	1	yes	characteristic
197	C₄×C₆₇	#536	1	1	yes	characteristic
198	C₄×C₆₇	#537	1	2	yes	—
199	#21848	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₆₇	—	2	4	no	—
201	C₂×C₆₇	—	8	2	no	—
202	C₂×C₆₇	#2311	1	1	yes	characteristic
203	C₂×C₆₇	C₂×#408	1	1	yes	characteristic
204	C₂×C₆₇	C₂×#408	1	1	yes	characteristic
205	C₂×C₆₇	#2994	1	4	yes	—
206	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
207	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
208	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#1984	C₂×C₆₇	1	2	yes	complemented
210	#1954	C₂×C₆₇	1	2	yes	complemented
211	#1869	C₂×C₆₇	1	4	yes	complemented
212	#1798	C₂×C₆₇	1	4	yes	complemented
213	C₆₇	#21848	1	1	yes	characteristic, direct product factor, Sylow
214	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
215	#529	—	2	4	no	—
216	#529	—	2	4	no	—
217	C₂×#126	C₄×C₆₇	1	2	yes	—
218	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
219	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
220	C₂×#123	C₄×C₆₇	1	2	yes	—
221	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
222	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
223	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
225	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×#117	C₂²×C₆₇	1	2	yes	—
227	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
228	#431	—	2	4	no	—
229	C₂×C₄⋊C₈	—	2	4	no	—
230	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
231	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
232	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
233	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
234	C₂×C₂²⋊C₈	C₂²×C₆₇	1	8	yes	—
235	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
236	#336	C₂²×C₆₇	1	8	yes	—
237	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
238	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
239	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
240	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
241	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
242	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
243	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
244	C₂²×C₈	—	2	4	no	—
245	C₂²×C₈	—	4	4	no	—
246	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
247	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
248	#126	—	2	8	no	—
249	#125	—	2	4	no	—
250	#124	—	2	4	no	—
251	#123	—	2	8	no	—
252	C₂²⋊Q₈	—	2	4	no	—
253	#121	—	2	4	no	—
254	C₄×Q₈	—	2	2	no	—
255	C₄×Q₈	—	2	2	no	—
256	C₄×D₄	—	2	4	no	—
257	C₄×D₄	—	2	4	no	—
258	C₄×D₄	—	2	2	no	—
259	C₄×D₄	—	2	2	no	—
260	#117	—	2	4	no	—
261	#117	—	2	4	no	—
262	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	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	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
267	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
268	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
270	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
271	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
272	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
274	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
275	C₂×C₄²	D₄×C₆₇	1	4	yes	—
276	C₂×C₄²	D₄×C₆₇	1	4	yes	—
277	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
278	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
279	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
280	C₄⋊C₈	—	2	4	no	—
281	C₄⋊C₈	—	2	4	no	—
282	C₄⋊C₈	—	2	4	no	—
283	C₂²⋊C₈	—	2	4	no	—
284	C₂²⋊C₈	—	4	8	no	—
285	C₄×C₈	—	2	4	no	—
286	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
287	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	4	no	—
290	C₂×D₄	—	2	4	no	—
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	4	no	—
302	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
303	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
304	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
312	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
314	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
315	C₂×C₈	—	2	4	no	—
316	C₂×C₈	—	2	4	no	—
317	C₂×C₈	—	2	4	no	—
318	C₂×C₈	—	2	8	no	—
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	4	8	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	4	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	4	no	—
327	C₂²⋊C₄	—	2	4	no	—
328	C₂²⋊C₄	—	2	4	no	—
329	C₂²⋊C₄	—	2	4	no	—
330	C₂²⋊C₄	—	2	4	no	—
331	C₂²⋊C₄	—	2	4	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
335	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
336	C₄²	C₂×D₄×C₆₇	1	2	yes	—
337	C₄²	C₂×D₄×C₆₇	1	2	yes	—
338	C₄²	C₂×D₄×C₆₇	1	2	yes	—
339	C₄²	C₂×D₄×C₆₇	1	2	yes	—
340	C₂³	—	2	2	no	—
341	C₂³	—	2	2	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	4	no	—
344	C₂³	—	2	4	no	—
345	C₂³	—	2	4	no	—
346	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
347	C₂³	#131×C₆₇	1	4	yes	—
348	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
349	Q₈	—	4	4	no	—
350	D₄	—	4	8	no	—
351	D₄	—	4	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	8	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	4	no	—
360	C₂×C₄	—	2	8	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
371	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
372	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
373	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
374	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
375	C₂×C₄	#131×C₆₇	1	4	yes	—
376	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
378	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
379	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
380	C₈	—	4	4	no	—
381	C₈	—	4	4	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	8	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	4	8	no	—
389	C₂²	#408×C₆₇	1	4	yes	—
390	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
391	C₂²	#520×C₆₇	1	2	yes	—
392	C₂²	C₂×#131×C₆₇	1	2	yes	—
393	C₂²	C₂×#131×C₆₇	1	2	yes	—
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	2	2	no	—
397	C₄	—	2	2	no	—
398	C₄	—	4	2	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	2	no	—
401	C₄	#408×C₆₇	1	4	yes	—
402	C₄	#535×C₆₇	1	1	yes	characteristic
403	C₄	#536×C₆₇	1	1	yes	characteristic
404	C₄	#537×C₆₇	1	2	yes	—
405	C₂	—	2	4	no	—
406	C₂	—	8	2	no	—
407	C₂	C₆₇×#2311	1	1	yes	characteristic
408	C₂	C₂×#408×C₆₇	1	1	yes	characteristic
409	C₂	C₂×#408×C₆₇	1	1	yes	characteristic
410	C₂	C₆₇×#2994	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#121×C₆₇	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
2. C₂×#431×C₆₇	C₂	semidirect
3. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
4. C₆₇×#1984	C₂	semidirect
5. C₆₇×#1954	C₂	semidirect
6. C₆₇×#1869	C₂	semidirect
7. C₆₇×#1798	C₂	semidirect
199. #21848	C₆₇	direct (N × H)
207. C₂×#431	C₂×C₆₇	semidirect
208. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
209. #1984	C₂×C₆₇	semidirect
210. #1954	C₂×C₆₇	semidirect
211. #1869	C₂×C₆₇	semidirect
212. #1798	C₂×C₆₇	semidirect
213. C₆₇	#21848	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×#529×C₆₇	2
C₂×#431×C₆₇	2
C₂²×C₄⋊C₈×C₆₇	2
C₆₇×#1984	2
C₆₇×#1954	2
C₆₇×#1869	2
C₆₇×#1798	2
#21848	67

Subgroups — C67×#21848 #30701099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21848 #30701099