Subgroups — C₂×C₆₇×#1535 #30696909

Order 17152.

986 conjugacy classes of subgroups.

The subgroup lattice has 464 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×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1535	C₂	1	8	yes	maximal, direct product factor
5	C₂×#330×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#329×C₆₇	C₂	1	2	yes	maximal, complemented
7	#529×C₆₇	—	2	2	no	—
8	#529×C₆₇	—	2	2	no	—
9	#529×C₆₇	C₄	1	4	yes	—
10	#529×C₆₇	C₂²	1	4	yes	—
11	C₂×#126×C₆₇	—	2	1	no	—
12	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#123×C₆₇	—	2	1	no	—
15	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
20	C₂×#117×C₆₇	—	2	1	no	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#432×C₆₇	—	2	2	no	—
23	#432×C₆₇	—	2	2	no	—
24	#432×C₆₇	C₂²	1	4	yes	complemented
25	#432×C₆₇	C₂²	1	4	yes	complemented
26	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
27	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
28	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
29	C₂×C₂²⋊C₈×C₆₇	—	2	1	no	—
30	C₂×#97×C₆₇	C₂²	1	2	yes	—
31	C₂×#97×C₆₇	C₂²	1	2	yes	—
32	#330×C₆₇	C₂²	1	8	yes	complemented
33	#329×C₆₇	C₂²	1	8	yes	complemented
34	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	—	2	1	no	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	#126×C₆₇	—	2	4	no	—
44	#126×C₆₇	—	2	4	no	—
45	#125×C₆₇	—	2	1	no	—
46	#125×C₆₇	—	2	1	no	—
47	#125×C₆₇	C₂×C₄	1	2	yes	—
48	#125×C₆₇	C₂³	1	2	yes	—
49	#124×C₆₇	—	2	1	no	—
50	#124×C₆₇	—	2	1	no	—
51	#124×C₆₇	C₂×C₄	1	2	yes	—
52	#124×C₆₇	C₂³	1	2	yes	—
53	#123×C₆₇	—	2	4	no	—
54	#123×C₆₇	—	2	4	no	—
55	C₂²⋊Q₈×C₆₇	—	2	2	no	—
56	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
57	#121×C₆₇	—	2	2	no	—
58	#121×C₆₇	C₂×C₄	1	4	yes	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
61	C₄×D₄×C₆₇	—	2	4	no	—
62	C₄×D₄×C₆₇	—	2	4	no	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
65	#117×C₆₇	—	2	4	no	—
66	#117×C₆₇	—	2	4	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	—	2	1	no	—
79	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
81	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	D₄	1	2	yes	—
84	C₂×C₄²×C₆₇	D₄	1	2	yes	—
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
87	C₄⋊C₈×C₆₇	—	2	4	no	—
88	C₄⋊C₈×C₆₇	—	2	4	no	—
89	C₂²⋊C₈×C₆₇	—	2	4	no	—
90	C₂²⋊C₈×C₆₇	—	2	4	no	—
91	#97×C₆₇	—	2	2	no	—
92	#97×C₆₇	—	2	2	no	—
93	#97×C₆₇	C₂³	1	4	yes	—
94	#97×C₆₇	C₂³	1	4	yes	—
95	C₂⁴×C₆₇	—	2	1	no	—
96	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×Q₈×C₆₇	—	2	1	no	—
98	C₂×Q₈×C₆₇	—	4	2	no	—
99	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂×D₄×C₆₇	—	2	4	no	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂×D₄×C₆₇	—	4	4	no	—
104	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	C₂²×C₄×C₆₇	—	4	2	no	—
111	C₂²×C₄×C₆₇	—	4	2	no	—
112	C₂²×C₄×C₆₇	—	4	4	no	—
113	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	4	no	—
127	C₂×C₈×C₆₇	—	4	4	no	—
128	C₂×C₈×C₆₇	—	4	4	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	2	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂²⋊C₄×C₆₇	—	2	4	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	2	2	no	—
142	C₂²⋊C₄×C₆₇	—	2	2	no	—
143	C₂²⋊C₄×C₆₇	—	2	4	no	—
144	C₂²⋊C₄×C₆₇	—	2	2	no	—
145	C₂²⋊C₄×C₆₇	—	2	2	no	—
146	C₄²×C₆₇	—	2	4	no	—
147	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
148	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
149	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
150	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
151	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄²×C₆₇	C₂×D₄	1	2	yes	—
153	C₄²×C₆₇	C₂×D₄	1	2	yes	—
154	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
157	C₂³×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	—	2	2	no	—
160	C₂³×C₆₇	—	2	2	no	—
161	C₂³×C₆₇	—	4	2	no	—
162	C₂³×C₆₇	—	4	2	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	Q₈×C₆₇	—	4	2	no	—
167	Q₈×C₆₇	—	4	2	no	—
168	D₄×C₆₇	—	4	2	no	—
169	D₄×C₆₇	—	4	2	no	—
170	D₄×C₆₇	—	4	8	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	2	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	8	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
193	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₈×C₆₇	—	4	4	no	—
200	C₈×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	2	2	no	—
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	2	no	—
204	C₂²×C₆₇	—	4	4	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	4	2	no	—
207	C₂²×C₆₇	—	4	2	no	—
208	C₂²×C₆₇	—	8	4	no	—
209	C₂²×C₆₇	#413	1	2	yes	—
210	C₂²×C₆₇	#421	1	2	yes	—
211	C₂²×C₆₇	#421	1	2	yes	—
212	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
213	C₄×C₆₇	—	2	2	no	—
214	C₄×C₆₇	—	2	1	no	—
215	C₄×C₆₇	—	2	1	no	—
216	C₄×C₆₇	—	2	1	no	—
217	C₄×C₆₇	—	2	1	no	—
218	C₄×C₆₇	—	2	2	no	—
219	C₄×C₆₇	—	2	1	no	—
220	C₄×C₆₇	—	2	1	no	—
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	4	2	no	—
223	C₄×C₆₇	—	4	2	no	—
224	C₄×C₆₇	—	4	2	no	—
225	C₂×#1535	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂×C₆₇	—	4	2	no	—
227	C₂×C₆₇	—	8	2	no	—
228	C₂×C₆₇	#1535	1	4	yes	direct product factor
229	C₂×C₆₇	C₂×#413	1	1	yes	characteristic
230	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
231	C₂×C₆₇	C₂×#421	1	1	yes	characteristic
232	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
233	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
234	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#1535	C₂×C₆₇	1	8	yes	direct product factor
236	C₂×#330	C₂×C₆₇	1	2	yes	complemented
237	C₂×#329	C₂×C₆₇	1	2	yes	complemented
238	C₆₇	C₂×#1535	1	1	yes	characteristic, direct product factor, Sylow
239	#529	—	2	2	no	—
240	#529	—	2	2	no	—
241	#529	C₄×C₆₇	1	4	yes	—
242	#529	C₂²×C₆₇	1	4	yes	—
243	C₂×#126	—	2	1	no	—
244	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#123	—	2	1	no	—
247	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
248	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
249	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
250	C₂×C₄×D₄	—	2	1	no	—
251	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
252	C₂×#117	—	2	1	no	—
253	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
254	#432	—	2	2	no	—
255	#432	—	2	2	no	—
256	#432	C₂²×C₆₇	1	4	yes	complemented
257	#432	C₂²×C₆₇	1	4	yes	complemented
258	C₂×C₄⋊C₈	—	2	1	no	—
259	C₂×C₄⋊C₈	—	2	1	no	—
260	C₂×C₂²⋊C₈	—	2	1	no	—
261	C₂×C₂²⋊C₈	—	2	1	no	—
262	C₂×#97	C₂²×C₆₇	1	2	yes	—
263	C₂×#97	C₂²×C₆₇	1	2	yes	—
264	#330	C₂²×C₆₇	1	8	yes	complemented
265	#329	C₂²×C₆₇	1	8	yes	complemented
266	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
267	C₂²×D₄	—	2	1	no	—
268	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
269	C₂³×C₄	—	2	1	no	—
270	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
271	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
272	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
273	C₂²×C₈	—	2	2	no	—
274	C₂²×C₈	—	2	2	no	—
275	#126	—	2	4	no	—
276	#126	—	2	4	no	—
277	#125	—	2	1	no	—
278	#125	—	2	1	no	—
279	#125	C₂×C₄×C₆₇	1	2	yes	—
280	#125	C₂³×C₆₇	1	2	yes	—
281	#124	—	2	1	no	—
282	#124	—	2	1	no	—
283	#124	C₂×C₄×C₆₇	1	2	yes	—
284	#124	C₂³×C₆₇	1	2	yes	—
285	#123	—	2	4	no	—
286	#123	—	2	4	no	—
287	C₂²⋊Q₈	—	2	2	no	—
288	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
289	#121	—	2	2	no	—
290	#121	C₂×C₄×C₆₇	1	4	yes	—
291	C₄×Q₈	—	2	2	no	—
292	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
293	C₄×D₄	—	2	4	no	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	—	2	2	no	—
296	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
297	#117	—	2	4	no	—
298	#117	—	2	4	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
305	C₂×C₂²⋊C₄	—	2	1	no	—
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	2	1	no	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
313	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
315	C₂×C₄²	D₄×C₆₇	1	2	yes	—
316	C₂×C₄²	D₄×C₆₇	1	2	yes	—
317	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
319	C₄⋊C₈	—	2	4	no	—
320	C₄⋊C₈	—	2	4	no	—
321	C₂²⋊C₈	—	2	4	no	—
322	C₂²⋊C₈	—	2	4	no	—
323	#97	—	2	2	no	—
324	#97	—	2	2	no	—
325	#97	C₂³×C₆₇	1	4	yes	—
326	#97	C₂³×C₆₇	1	4	yes	—
327	C₂⁴	—	2	1	no	—
328	C₂⁴	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	—	4	2	no	—
335	C₂×D₄	—	4	4	no	—
336	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	4	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
346	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
347	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
348	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
349	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
350	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
351	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
357	C₂×C₈	—	2	4	no	—
358	C₂×C₈	—	2	4	no	—
359	C₂×C₈	—	4	4	no	—
360	C₂×C₈	—	4	4	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
370	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	2	2	no	—
374	C₂²⋊C₄	—	2	2	no	—
375	C₂²⋊C₄	—	2	4	no	—
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₄²	—	2	4	no	—
379	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
380	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
381	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
382	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
383	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
384	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
385	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
386	C₄²	C₂×D₄×C₆₇	1	2	yes	—
387	C₄²	C₂×D₄×C₆₇	1	2	yes	—
388	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
389	C₂³	—	2	2	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
397	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
398	Q₈	—	4	2	no	—
399	Q₈	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	8	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	8	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
423	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
424	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
425	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
426	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
427	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
428	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
430	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
431	C₈	—	4	4	no	—
432	C₈	—	4	4	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	4	4	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	8	4	no	—
441	C₂²	#413×C₆₇	1	2	yes	—
442	C₂²	#421×C₆₇	1	2	yes	—
443	C₂²	#421×C₆₇	1	2	yes	—
444	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
445	C₄	—	2	1	no	—
446	C₄	—	2	2	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	2	2	no	—
451	C₄	—	2	1	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₂	—	4	2	no	—
458	C₂	—	8	2	no	—
459	C₂	C₆₇×#1535	1	4	yes	direct product factor
460	C₂	C₂×#413×C₆₇	1	1	yes	characteristic
461	C₂	C₂×#421×C₆₇	1	1	yes	characteristic
462	C₂	C₂×#421×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#432×C₆₇	C₂	semidirect
3. C₂×#432×C₆₇	C₂	semidirect
4. C₆₇×#1535	C₂	direct (N × H)
5. C₂×#330×C₆₇	C₂	semidirect
6. C₂×#329×C₆₇	C₂	semidirect
24. #432×C₆₇	C₂²	semidirect
25. #432×C₆₇	C₂²	semidirect
32. #330×C₆₇	C₂²	semidirect
33. #329×C₆₇	C₂²	semidirect
225. C₂×#1535	C₆₇	direct (N × H)
228. C₂×C₆₇	#1535	direct (N × H)
233. C₂×#432	C₂×C₆₇	semidirect
234. C₂×#432	C₂×C₆₇	semidirect
235. #1535	C₂×C₆₇	direct (N × H)
236. C₂×#330	C₂×C₆₇	semidirect
237. C₂×#329	C₂×C₆₇	semidirect
238. C₆₇	C₂×#1535	direct (N × H)
256. #432	C₂²×C₆₇	semidirect
257. #432	C₂²×C₆₇	semidirect
264. #330	C₂²×C₆₇	semidirect
265. #329	C₂²×C₆₇	semidirect
459. C₂	C₆₇×#1535	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×#432×C₆₇	2
C₆₇×#1535	2
C₂×#330×C₆₇	2
C₂×#329×C₆₇	2
C₂×#1535	67

Subgroups — C2×C67×#1535 #30696909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1535 #30696909