Subgroups — C₂×C₆₇×#1500 #30696780

Order 17152.

1182 conjugacy classes of subgroups.

The subgroup lattice has 492 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	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1500	C₂	1	8	yes	maximal, direct product factor
5	C₂×#328×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#327×C₆₇	C₂	1	2	yes	maximal
7	#529×C₆₇	—	2	2	no	—
8	#529×C₆₇	—	2	2	no	—
9	#529×C₆₇	C₄	1	4	yes	complemented
10	#529×C₆₇	C₂²	1	4	yes	—
11	C₂×#126×C₆₇	—	2	1	no	—
12	C₂×#125×C₆₇	C₄	1	1	yes	characteristic, complemented
13	C₂×#124×C₆₇	C₄	1	1	yes	characteristic, complemented
14	C₂×#123×C₆₇	—	2	1	no	—
15	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
18	C₂×C₄×D₄×C₆₇	—	2	1	no	—
19	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic, complemented
20	C₂×#117×C₆₇	—	2	1	no	—
21	C₂²×C₄²×C₆₇	—	2	1	no	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#431×C₆₇	—	2	2	no	—
24	#431×C₆₇	C₂²	1	4	yes	complemented
25	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	—	2	1	no	—
28	#376×C₆₇	—	2	2	no	—
29	#376×C₆₇	C₂²	1	4	yes	complemented
30	C₂×#95×C₆₇	—	2	1	no	—
31	C₂×#95×C₆₇	C₂²	1	2	yes	—
32	#328×C₆₇	C₂²	1	8	yes	complemented
33	#327×C₆₇	C₂²	1	8	yes	—
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	2	no	—
38	C₂³×C₄×C₆₇	—	2	1	no	—
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	—	2	2	no	—
43	#126×C₆₇	—	2	4	no	—
44	#126×C₆₇	—	2	4	no	—
45	#125×C₆₇	—	2	2	no	—
46	#125×C₆₇	C₂×C₄	1	4	yes	complemented
47	#124×C₆₇	—	2	2	no	—
48	#124×C₆₇	C₂×C₄	1	4	yes	complemented
49	#123×C₆₇	—	2	4	no	—
50	#123×C₆₇	—	2	4	no	—
51	C₂²⋊Q₈×C₆₇	—	2	1	no	—
52	C₂²⋊Q₈×C₆₇	—	2	1	no	—
53	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
54	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
55	#121×C₆₇	—	2	1	no	—
56	#121×C₆₇	—	2	1	no	—
57	#121×C₆₇	C₂×C₄	1	2	yes	—
58	#121×C₆₇	C₂³	1	2	yes	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
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	complemented
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	8	no	—
79	C₂×C₄²×C₆₇	—	2	1	no	—
80	C₂×C₄²×C₆₇	—	2	4	no	—
81	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
83	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	D₄	1	2	yes	—
85	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	D₄	1	2	yes	—
87	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
89	C₄⋊C₈×C₆₇	—	2	4	no	—
90	C₂²⋊C₈×C₆₇	—	2	2	no	—
91	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
92	C₄×C₈×C₆₇	—	2	4	no	—
93	#95×C₆₇	—	2	4	no	—
94	#95×C₆₇	—	2	2	no	—
95	#95×C₆₇	C₂³	1	4	yes	—
96	C₂⁴×C₆₇	—	2	1	no	—
97	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂×Q₈×C₆₇	—	2	1	no	—
99	C₂×Q₈×C₆₇	—	4	2	no	—
100	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	—	4	2	no	—
105	C₂×D₄×C₆₇	—	4	4	no	—
106	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₆₇	—	2	4	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	8	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	1	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	4	2	no	—
116	C₂²×C₄×C₆₇	—	4	4	no	—
117	C₂²×C₄×C₆₇	—	4	4	no	—
118	C₂²×C₄×C₆₇	—	4	4	no	—
119	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
124	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
133	C₂×C₈×C₆₇	—	2	4	no	—
134	C₂×C₈×C₆₇	—	4	4	no	—
135	C₄⋊C₄×C₆₇	—	2	2	no	—
136	C₄⋊C₄×C₆₇	—	2	4	no	—
137	C₄⋊C₄×C₆₇	—	2	4	no	—
138	C₄⋊C₄×C₆₇	—	2	4	no	—
139	C₄⋊C₄×C₆₇	—	2	1	no	—
140	C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
143	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂²⋊C₄×C₆₇	—	2	2	no	—
145	C₂²⋊C₄×C₆₇	—	2	2	no	—
146	C₂²⋊C₄×C₆₇	—	2	2	no	—
147	C₂²⋊C₄×C₆₇	—	2	4	no	—
148	C₂²⋊C₄×C₆₇	—	2	4	no	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₂²⋊C₄×C₆₇	—	2	2	no	—
151	C₄²×C₆₇	—	2	16	no	—
152	C₄²×C₆₇	—	2	4	no	—
153	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
154	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
155	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
156	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
157	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
158	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
159	C₄²×C₆₇	C₂×D₄	1	2	yes	—
160	C₄²×C₆₇	C₂×D₄	1	2	yes	—
161	C₂³×C₆₇	—	2	2	no	—
162	C₂³×C₆₇	—	2	2	no	—
163	C₂³×C₆₇	—	2	2	no	—
164	C₂³×C₆₇	—	4	2	no	—
165	C₂³×C₆₇	—	4	2	no	—
166	C₂³×C₆₇	—	4	2	no	—
167	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂³×C₆₇	#135	1	4	yes	—
170	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
171	Q₈×C₆₇	—	4	2	no	—
172	Q₈×C₆₇	—	4	2	no	—
173	D₄×C₆₇	—	4	2	no	—
174	D₄×C₆₇	—	4	2	no	—
175	D₄×C₆₇	—	4	4	no	—
176	D₄×C₆₇	—	4	4	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	4	no	—
186	C₂×C₄×C₆₇	—	2	8	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	4	1	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	1	no	—
193	C₂×C₄×C₆₇	—	4	1	no	—
194	C₂×C₄×C₆₇	—	4	4	no	—
195	C₂×C₄×C₆₇	—	4	16	no	—
196	C₂×C₄×C₆₇	—	4	4	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	8	no	—
201	C₂×C₄×C₆₇	#104	1	4	yes	—
202	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
205	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
206	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
207	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₈×C₆₇	—	4	4	no	—
211	C₂²×C₆₇	—	2	1	no	—
212	C₂²×C₆₇	—	2	1	no	—
213	C₂²×C₆₇	—	2	1	no	—
214	C₂²×C₆₇	—	2	1	no	—
215	C₂²×C₆₇	—	2	8	no	—
216	C₂²×C₆₇	—	4	2	no	—
217	C₂²×C₆₇	—	4	2	no	—
218	C₂²×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	8	4	no	—
220	C₂²×C₆₇	#410	1	2	yes	—
221	C₂²×C₆₇	#416	1	2	yes	—
222	C₂²×C₆₇	#416	1	2	yes	—
223	C₂²×C₆₇	C₂×#104	1	2	yes	—
224	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₆₇	C₂×#135	1	2	yes	—
226	C₄×C₆₇	—	2	1	no	—
227	C₄×C₆₇	—	2	1	no	—
228	C₄×C₆₇	—	2	1	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	2	no	—
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	4	8	no	—
234	C₄×C₆₇	—	4	2	no	—
235	C₄×C₆₇	—	4	2	no	—
236	C₄×C₆₇	—	4	2	no	—
237	C₄×C₆₇	C₂×#104	1	4	yes	—
238	C₂×#1500	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂×C₆₇	—	2	2	no	—
240	C₂×C₆₇	—	2	2	no	—
241	C₂×C₆₇	—	8	2	no	—
242	C₂×C₆₇	#1500	1	4	yes	direct product factor
243	C₂×C₆₇	C₂×#410	1	1	yes	characteristic
244	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
245	C₂×C₆₇	C₂²×#104	1	1	yes	characteristic
246	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
247	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
248	C₂×#376	C₂×C₆₇	1	1	yes	characteristic, complemented
249	#1500	C₂×C₆₇	1	8	yes	direct product factor
250	C₂×#328	C₂×C₆₇	1	2	yes	complemented
251	C₂×#327	C₂×C₆₇	1	2	yes	—
252	C₆₇	C₂×#1500	1	1	yes	characteristic, direct product factor, Sylow
253	#529	—	2	2	no	—
254	#529	—	2	2	no	—
255	#529	C₄×C₆₇	1	4	yes	complemented
256	#529	C₂²×C₆₇	1	4	yes	—
257	C₂×#126	—	2	1	no	—
258	C₂×#125	C₄×C₆₇	1	1	yes	characteristic, complemented
259	C₂×#124	C₄×C₆₇	1	1	yes	characteristic, complemented
260	C₂×#123	—	2	1	no	—
261	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
262	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
263	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic, complemented
264	C₂×C₄×D₄	—	2	1	no	—
265	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic, complemented
266	C₂×#117	—	2	1	no	—
267	C₂²×C₄²	—	2	1	no	—
268	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
269	#431	—	2	2	no	—
270	#431	C₂²×C₆₇	1	4	yes	complemented
271	C₂×C₄⋊C₈	—	2	1	no	—
272	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
273	C₂×C₄×C₈	—	2	1	no	—
274	#376	—	2	2	no	—
275	#376	C₂²×C₆₇	1	4	yes	complemented
276	C₂×#95	—	2	1	no	—
277	C₂×#95	C₂²×C₆₇	1	2	yes	—
278	#328	C₂²×C₆₇	1	8	yes	complemented
279	#327	C₂²×C₆₇	1	8	yes	—
280	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
281	C₂²×D₄	—	2	1	no	—
282	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
283	C₂³×C₄	—	2	2	no	—
284	C₂³×C₄	—	2	1	no	—
285	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
286	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂²×C₈	—	2	2	no	—
289	#126	—	2	4	no	—
290	#126	—	2	4	no	—
291	#125	—	2	2	no	—
292	#125	C₂×C₄×C₆₇	1	4	yes	complemented
293	#124	—	2	2	no	—
294	#124	C₂×C₄×C₆₇	1	4	yes	complemented
295	#123	—	2	4	no	—
296	#123	—	2	4	no	—
297	C₂²⋊Q₈	—	2	1	no	—
298	C₂²⋊Q₈	—	2	1	no	—
299	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
300	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
301	#121	—	2	1	no	—
302	#121	—	2	1	no	—
303	#121	C₂×C₄×C₆₇	1	2	yes	—
304	#121	C₂³×C₆₇	1	2	yes	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	—	2	4	no	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
311	#117	—	2	4	no	—
312	#117	—	2	4	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
318	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
319	C₂×C₂²⋊C₄	—	2	1	no	—
320	C₂×C₂²⋊C₄	—	2	1	no	—
321	C₂×C₂²⋊C₄	—	2	1	no	—
322	C₂×C₂²⋊C₄	—	2	1	no	—
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₄²	—	2	8	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	4	no	—
327	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
328	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
330	C₂×C₄²	D₄×C₆₇	1	2	yes	—
331	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
333	C₂×C₄²	D₄×C₆₇	1	2	yes	—
334	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
335	C₄⋊C₈	—	2	4	no	—
336	C₂²⋊C₈	—	2	2	no	—
337	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
338	C₄×C₈	—	2	4	no	—
339	#95	—	2	4	no	—
340	#95	—	2	2	no	—
341	#95	C₂³×C₆₇	1	4	yes	—
342	C₂⁴	—	2	1	no	—
343	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
344	C₂×Q₈	—	2	1	no	—
345	C₂×Q₈	—	4	2	no	—
346	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	4	2	no	—
351	C₂×D₄	—	4	4	no	—
352	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	8	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	4	no	—
360	C₂²×C₄	—	2	4	no	—
361	C₂²×C₄	—	4	2	no	—
362	C₂²×C₄	—	4	4	no	—
363	C₂²×C₄	—	4	4	no	—
364	C₂²×C₄	—	4	4	no	—
365	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
368	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
371	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
372	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
374	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
375	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
378	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
379	C₂×C₈	—	2	4	no	—
380	C₂×C₈	—	4	4	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
389	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
390	C₂²⋊C₄	—	2	2	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	2	2	no	—
393	C₂²⋊C₄	—	2	4	no	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	2	4	no	—
396	C₂²⋊C₄	—	2	2	no	—
397	C₄²	—	2	16	no	—
398	C₄²	—	2	4	no	—
399	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
400	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
401	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
402	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
403	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
404	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
405	C₄²	C₂×D₄×C₆₇	1	2	yes	—
406	C₄²	C₂×D₄×C₆₇	1	2	yes	—
407	C₂³	—	2	2	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	2	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
414	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
415	C₂³	#135×C₆₇	1	4	yes	—
416	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
417	Q₈	—	4	2	no	—
418	Q₈	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	2	no	—
421	D₄	—	4	4	no	—
422	D₄	—	4	4	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	16	no	—
441	C₂×C₄	—	4	4	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	8	no	—
447	C₂×C₄	#104×C₆₇	1	4	yes	—
448	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
450	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
451	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
452	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
453	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
454	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
455	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
456	C₈	—	4	4	no	—
457	C₂²	—	2	8	no	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	1	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	4	2	no	—
465	C₂²	—	8	4	no	—
466	C₂²	#410×C₆₇	1	2	yes	—
467	C₂²	#416×C₆₇	1	2	yes	—
468	C₂²	#416×C₆₇	1	2	yes	—
469	C₂²	C₂×#104×C₆₇	1	2	yes	—
470	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
471	C₂²	C₂×#135×C₆₇	1	2	yes	—
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	2	1	no	—
475	C₄	—	2	1	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	2	no	—
479	C₄	—	4	8	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	4	2	no	—
483	C₄	C₂×#104×C₆₇	1	4	yes	—
484	C₂	—	2	2	no	—
485	C₂	—	2	2	no	—
486	C₂	—	8	2	no	—
487	C₂	C₆₇×#1500	1	4	yes	direct product factor
488	C₂	C₂×#410×C₆₇	1	1	yes	characteristic
489	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
490	C₂	C₂²×#104×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₄²×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. C₂×#376×C₆₇	C₂	semidirect
4. C₆₇×#1500	C₂	direct (N × H)
5. C₂×#328×C₆₇	C₂	semidirect
9. #529×C₆₇	C₄	semidirect
12. C₂×#125×C₆₇	C₄	semidirect
13. C₂×#124×C₆₇	C₄	semidirect
17. C₂×C₄×Q₈×C₆₇	C₄	semidirect
19. C₂×C₄×D₄×C₆₇	C₄	semidirect
24. #431×C₆₇	C₂²	semidirect
29. #376×C₆₇	C₂²	semidirect
32. #328×C₆₇	C₂²	semidirect
46. #125×C₆₇	C₂×C₄	semidirect
48. #124×C₆₇	C₂×C₄	semidirect
60. C₄×Q₈×C₆₇	C₂×C₄	semidirect
64. C₄×D₄×C₆₇	C₂×C₄	semidirect
238. C₂×#1500	C₆₇	direct (N × H)
242. C₂×C₆₇	#1500	direct (N × H)
247. C₂×#431	C₂×C₆₇	semidirect
248. C₂×#376	C₂×C₆₇	semidirect
249. #1500	C₂×C₆₇	direct (N × H)
250. C₂×#328	C₂×C₆₇	semidirect
252. C₆₇	C₂×#1500	direct (N × H)
255. #529	C₄×C₆₇	semidirect
258. C₂×#125	C₄×C₆₇	semidirect
259. C₂×#124	C₄×C₆₇	semidirect
263. C₂×C₄×Q₈	C₄×C₆₇	semidirect
265. C₂×C₄×D₄	C₄×C₆₇	semidirect
270. #431	C₂²×C₆₇	semidirect
275. #376	C₂²×C₆₇	semidirect
278. #328	C₂²×C₆₇	semidirect
292. #125	C₂×C₄×C₆₇	semidirect
294. #124	C₂×C₄×C₆₇	semidirect
306. C₄×Q₈	C₂×C₄×C₆₇	semidirect
310. C₄×D₄	C₂×C₄×C₆₇	semidirect
487. C₂	C₆₇×#1500	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
C₂×#376×C₆₇	2
C₆₇×#1500	2
C₂×#328×C₆₇	2
C₂×#327×C₆₇	2
C₂×#1500	67

Subgroups — C2×C67×#1500 #30696780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1500 #30696780