Subgroups — C₇×C₈×#120 #13198654

Order 1792.

1786 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×C₈×D₄	C₂	1	3	yes	maximal, complemented
3	C₄×C₇×#120	C₂	1	1	yes	maximal, characteristic
4	C₇×#1869	C₂	1	3	yes	maximal, complemented
5	C₇×#1850	C₂	1	3	yes	maximal, complemented
6	C₇×#1779	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×C₈×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
8	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂³×C₇×C₈	—	2	12	no	—
10	C₂³×C₇×C₈	C₂²	1	3	yes	—
11	C₂³×C₇×C₈	C₂²	1	3	yes	—
12	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₇×#120	C₄	1	1	yes	characteristic
14	C₂×C₄×C₇×D₄	C₂²	1	3	yes	—
15	C₇×C₈×D₄	—	2	12	no	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	complemented
17	C₂×C₇×C₂²⋊C₈	C₂²	1	3	yes	—
18	C₂×C₇×C₂²⋊C₈	C₂²	1	3	yes	—
19	C₂×C₇×C₂²⋊C₈	C₂²	1	3	yes	complemented
20	C₂×C₄×C₇×C₈	C₂²	1	3	yes	complemented
21	C₇×#386	C₄	1	3	yes	—
22	C₇×#385	C₄	1	3	yes	—
23	C₇×#379	C₄	1	1	yes	characteristic
24	C₄×C₇×C₂²⋊C₄	C₂²	1	3	yes	—
25	C₇×#336	C₂²	1	6	yes	complemented
26	C₈×#120	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂⁵×C₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×D₄	C₂×C₄	1	3	yes	—
29	C₂³×C₄×C₇	—	2	6	no	—
30	C₂³×C₄×C₇	—	2	6	no	—
31	C₂³×C₄×C₇	C₂×C₄	1	3	yes	—
32	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₇	C₂³	1	3	yes	—
34	C₂²×C₇×C₈	—	2	32	no	—
35	C₂²×C₇×C₈	—	2	6	no	—
36	C₂²×C₇×C₈	—	2	12	no	—
37	C₂²×C₇×C₈	—	2	12	no	—
38	C₂²×C₇×C₈	—	2	6	no	—
39	C₂²×C₇×C₈	D₄	1	12	yes	complemented
40	C₂²×C₇×C₈	C₂³	1	3	yes	—
41	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇×C₈	C₂³	1	3	yes	—
43	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
44	C₇×#120	C₈	1	8	yes	direct product factor
45	C₄×C₇×D₄	—	2	6	no	—
46	C₄×C₇×D₄	—	2	6	no	—
47	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
48	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	3	yes	—
49	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	3	yes	—
50	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	3	yes	—
51	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	3	yes	—
52	C₂×C₄²×C₇	C₂³	1	3	yes	—
53	C₇×C₄⋊C₈	—	2	6	no	—
54	C₇×C₂²⋊C₈	—	2	6	no	—
55	C₇×C₂²⋊C₈	—	2	6	no	—
56	C₇×C₂²⋊C₈	—	2	6	no	—
57	C₄×C₇×C₈	—	2	6	no	—
58	C₇×#95	C₂×C₄	1	6	yes	—
59	C₂⁴×C₈	C₂×C₇	1	1	yes	characteristic, complemented
60	C₂×C₈×D₄	C₂×C₇	1	3	yes	complemented
61	C₄×#120	C₂×C₇	1	1	yes	characteristic
62	#1869	C₂×C₇	1	3	yes	complemented
63	#1850	C₂×C₇	1	3	yes	complemented
64	#1779	C₂×C₇	1	1	yes	characteristic, complemented
65	C₈×C₂²⋊C₄	C₂×C₇	1	3	yes	complemented
66	C₂⁴×C₇	—	2	6	no	—
67	C₂⁴×C₇	—	2	6	no	—
68	C₂⁴×C₇	C₂×C₈	1	4	yes	complemented
69	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
70	C₂⁴×C₇	C₂²×C₄	1	3	yes	—
71	C₂×C₇×D₄	—	2	6	no	—
72	C₂×C₇×D₄	—	2	6	no	—
73	C₂×C₇×D₄	C₂×C₈	1	12	yes	complemented
74	C₂²×C₄×C₇	—	2	8	no	—
75	C₂²×C₄×C₇	—	2	24	no	—
76	C₂²×C₄×C₇	—	2	6	no	—
77	C₂²×C₄×C₇	—	2	12	no	—
78	C₂²×C₄×C₇	—	2	6	no	—
79	C₂²×C₄×C₇	—	2	6	no	—
80	C₂²×C₄×C₇	—	2	3	no	—
81	C₂²×C₄×C₇	—	2	3	no	—
82	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
83	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
85	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
86	C₂²×C₄×C₇	C₂×D₄	1	6	yes	—
87	C₂²×C₄×C₇	#41	1	6	yes	—
88	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₇×C₈	—	2	24	no	—
90	C₂×C₇×C₈	—	2	6	no	—
91	C₂×C₇×C₈	—	2	24	no	—
92	C₂×C₇×C₈	—	2	3	no	—
93	C₂×C₇×C₈	—	2	3	no	—
94	C₂×C₇×C₈	—	2	6	no	—
95	C₂×C₇×C₈	—	4	4	no	—
96	C₂×C₇×C₈	C₂×D₄	1	6	yes	—
97	C₂×C₇×C₈	C₂×D₄	1	6	yes	—
98	C₇×C₄⋊C₄	—	2	6	no	—
99	C₇×C₂²⋊C₄	—	2	6	no	—
100	C₇×C₂²⋊C₄	—	2	6	no	—
101	C₇×C₂²⋊C₄	—	2	6	no	—
102	C₇×C₂²⋊C₄	C₂×C₈	1	12	yes	complemented
103	C₄²×C₇	—	2	3	no	—
104	C₄²×C₇	—	2	3	no	—
105	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
106	C₂³×C₈	—	2	12	no	—
107	C₂³×C₈	C₂²×C₇	1	3	yes	—
108	C₂³×C₈	C₂²×C₇	1	3	yes	—
109	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
110	C₂×#120	C₄×C₇	1	1	yes	characteristic
111	C₂×C₄×D₄	C₂²×C₇	1	3	yes	—
112	C₈×D₄	—	2	12	no	—
113	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	complemented
114	C₂×C₂²⋊C₈	C₂²×C₇	1	3	yes	—
115	C₂×C₂²⋊C₈	C₂²×C₇	1	3	yes	—
116	C₂×C₂²⋊C₈	C₂²×C₇	1	3	yes	complemented
117	C₂×C₄×C₈	C₂²×C₇	1	3	yes	complemented
118	#386	C₄×C₇	1	3	yes	—
119	#385	C₄×C₇	1	3	yes	—
120	#379	C₄×C₇	1	1	yes	characteristic
121	C₄×C₂²⋊C₄	C₂²×C₇	1	3	yes	—
122	#336	C₂²×C₇	1	6	yes	complemented
123	C₂³×C₇	—	2	8	no	—
124	C₂³×C₇	—	2	24	no	—
125	C₂³×C₇	—	2	6	no	—
126	C₂³×C₇	—	2	6	no	—
127	C₂³×C₇	—	2	24	no	—
128	C₂³×C₇	—	2	3	no	—
129	C₂³×C₇	—	2	3	no	—
130	C₂³×C₇	C₄×D₄	1	6	yes	—
131	C₂³×C₇	C₂²×C₈	1	6	yes	—
132	C₂³×C₇	C₂²×C₈	1	2	yes	complemented
133	C₂³×C₇	#131	1	6	yes	—
134	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
135	C₇×D₄	—	2	24	no	—
136	C₂×C₄×C₇	—	2	24	no	—
137	C₂×C₄×C₇	—	2	3	no	—
138	C₂×C₄×C₇	—	2	6	no	—
139	C₂×C₄×C₇	—	2	3	no	—
140	C₂×C₄×C₇	—	2	6	no	—
141	C₂×C₄×C₇	—	2	6	no	—
142	C₂×C₄×C₇	—	2	12	no	—
143	C₂×C₄×C₇	—	2	3	no	—
144	C₂×C₄×C₇	—	2	3	no	—
145	C₂×C₄×C₇	—	2	6	no	—
146	C₂×C₄×C₇	—	4	1	no	—
147	C₂×C₄×C₇	—	4	3	no	—
148	C₂×C₄×C₇	C₄×D₄	1	6	yes	—
149	C₂×C₄×C₇	C₂²×C₈	1	6	yes	complemented
150	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
151	C₂×C₄×C₇	C₂×#41	1	3	yes	—
152	C₇×C₈	—	2	6	no	—
153	C₇×C₈	—	4	1	no	—
154	C₇×C₈	—	4	3	no	—
155	C₇×C₈	#120	1	4	yes	direct product factor
156	C₂⁵	C₂×C₄×C₇	1	1	yes	characteristic
157	C₂²×D₄	C₂×C₄×C₇	1	3	yes	—
158	C₂³×C₄	—	2	6	no	—
159	C₂³×C₄	—	2	6	no	—
160	C₂³×C₄	C₂×C₄×C₇	1	3	yes	—
161	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
162	C₂³×C₄	C₂³×C₇	1	3	yes	—
163	C₂²×C₈	—	2	32	no	—
164	C₂²×C₈	—	2	6	no	—
165	C₂²×C₈	—	2	12	no	—
166	C₂²×C₈	—	2	12	no	—
167	C₂²×C₈	—	2	6	no	—
168	C₂²×C₈	C₇×D₄	1	12	yes	complemented
169	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₇	1	3	yes	—
171	C₂²×C₈	C₂³×C₇	1	3	yes	—
172	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
173	#120	C₇×C₈	1	8	yes	direct product factor
174	C₄×D₄	—	2	6	no	—
175	C₄×D₄	—	2	6	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	3	yes	—
178	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	3	yes	—
179	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	3	yes	—
180	C₂×C₂²⋊C₄	C₂×C₄×C₇	1	3	yes	—
181	C₂×C₄²	C₂³×C₇	1	3	yes	—
182	C₄⋊C₈	—	2	6	no	—
183	C₂²⋊C₈	—	2	6	no	—
184	C₂²⋊C₈	—	2	6	no	—
185	C₂²⋊C₈	—	2	6	no	—
186	C₄×C₈	—	2	6	no	—
187	#95	C₂×C₄×C₇	1	6	yes	—
188	C₂²×C₇	—	2	32	no	—
189	C₂²×C₇	—	2	12	no	—
190	C₂²×C₇	—	2	6	no	—
191	C₂²×C₇	—	2	12	no	—
192	C₂²×C₇	—	2	6	no	—
193	C₂²×C₇	—	2	6	no	—
194	C₂²×C₇	—	4	3	no	—
195	C₂²×C₇	—	4	1	no	—
196	C₂²×C₇	C₈×D₄	1	12	yes	complemented
197	C₂²×C₇	C₂×C₄×D₄	1	3	yes	—
198	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
199	C₂²×C₇	C₂×#131	1	3	yes	—
200	C₄×C₇	—	2	6	no	—
201	C₄×C₇	—	2	6	no	—
202	C₄×C₇	—	4	3	no	—
203	C₄×C₇	—	4	1	no	—
204	C₄×C₇	C₂×#120	1	1	yes	characteristic
205	C₄×C₇	#525	1	3	yes	—
206	C₂⁴	—	2	6	no	—
207	C₂⁴	—	2	6	no	—
208	C₂⁴	C₂×C₇×C₈	1	4	yes	complemented
209	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
210	C₂⁴	C₂²×C₄×C₇	1	3	yes	—
211	C₂×D₄	—	2	6	no	—
212	C₂×D₄	—	2	6	no	—
213	C₂×D₄	C₂×C₇×C₈	1	12	yes	complemented
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	24	no	—
216	C₂²×C₄	—	2	6	no	—
217	C₂²×C₄	—	2	12	no	—
218	C₂²×C₄	—	2	6	no	—
219	C₂²×C₄	—	2	6	no	—
220	C₂²×C₄	—	2	3	no	—
221	C₂²×C₄	—	2	3	no	—
222	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
223	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
225	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
226	C₂²×C₄	C₂×C₇×D₄	1	6	yes	—
227	C₂²×C₄	C₇×#41	1	6	yes	—
228	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
229	C₂×C₈	—	2	24	no	—
230	C₂×C₈	—	2	6	no	—
231	C₂×C₈	—	2	24	no	—
232	C₂×C₈	—	2	3	no	—
233	C₂×C₈	—	2	3	no	—
234	C₂×C₈	—	2	6	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	C₂×C₇×D₄	1	6	yes	—
237	C₂×C₈	C₂×C₇×D₄	1	6	yes	—
238	C₄⋊C₄	—	2	6	no	—
239	C₂²⋊C₄	—	2	6	no	—
240	C₂²⋊C₄	—	2	6	no	—
241	C₂²⋊C₄	—	2	6	no	—
242	C₂²⋊C₄	C₂×C₇×C₈	1	12	yes	complemented
243	C₄²	—	2	3	no	—
244	C₄²	—	2	3	no	—
245	C₂×C₇	—	2	12	no	—
246	C₂×C₇	—	4	2	no	—
247	C₂×C₇	C₄×#120	1	1	yes	characteristic
248	C₂×C₇	C₂×C₈×D₄	1	3	yes	—
249	C₂×C₇	#2929	1	3	yes	—
250	C₂³	—	2	8	no	—
251	C₂³	—	2	24	no	—
252	C₂³	—	2	6	no	—
253	C₂³	—	2	6	no	—
254	C₂³	—	2	24	no	—
255	C₂³	—	2	3	no	—
256	C₂³	—	2	3	no	—
257	C₂³	C₄×C₇×D₄	1	6	yes	—
258	C₂³	C₂²×C₇×C₈	1	6	yes	—
259	C₂³	C₂²×C₇×C₈	1	2	yes	complemented
260	C₂³	C₇×#131	1	6	yes	—
261	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
262	D₄	—	2	24	no	—
263	C₂×C₄	—	2	24	no	—
264	C₂×C₄	—	2	3	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	2	3	no	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	6	no	—
269	C₂×C₄	—	2	12	no	—
270	C₂×C₄	—	2	3	no	—
271	C₂×C₄	—	2	3	no	—
272	C₂×C₄	—	2	6	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	3	no	—
275	C₂×C₄	C₄×C₇×D₄	1	6	yes	—
276	C₂×C₄	C₂²×C₇×C₈	1	6	yes	complemented
277	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
278	C₂×C₄	C₂×C₇×#41	1	3	yes	—
279	C₈	—	2	6	no	—
280	C₈	—	4	1	no	—
281	C₈	—	4	3	no	—
282	C₈	C₇×#120	1	4	yes	direct product factor
283	C₇	C₈×#120	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	32	no	—
285	C₂²	—	2	12	no	—
286	C₂²	—	2	6	no	—
287	C₂²	—	2	12	no	—
288	C₂²	—	2	6	no	—
289	C₂²	—	2	6	no	—
290	C₂²	—	4	3	no	—
291	C₂²	—	4	1	no	—
292	C₂²	C₇×C₈×D₄	1	12	yes	complemented
293	C₂²	C₂×C₄×C₇×D₄	1	3	yes	—
294	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
295	C₂²	C₂×C₇×#131	1	3	yes	—
296	C₄	—	2	6	no	—
297	C₄	—	2	6	no	—
298	C₄	—	4	3	no	—
299	C₄	—	4	1	no	—
300	C₄	C₂×C₇×#120	1	1	yes	characteristic
301	C₄	C₇×#525	1	3	yes	—
302	C₂	—	2	12	no	—
303	C₂	—	4	2	no	—
304	C₂	C₄×C₇×#120	1	1	yes	characteristic
305	C₂	C₂×C₇×C₈×D₄	1	3	yes	—
306	C₂	C₇×#2929	1	3	yes	—

	subgroup	G/H
center	C₂²×C₇×C₈	C₂³
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂³×C₄
prime radical	C₂×C₇×#120	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₇×C₈	C₂	semidirect
2. C₂×C₇×C₈×D₄	C₂	semidirect
4. C₇×#1869	C₂	semidirect
5. C₇×#1850	C₂	semidirect
6. C₇×#1779	C₂	semidirect
7. C₇×C₈×C₂²⋊C₄	C₂	semidirect
12. C₂³×C₇×C₈	C₂²	semidirect
16. C₂×C₇×C₄⋊C₈	C₂²	semidirect
19. C₂×C₇×C₂²⋊C₈	C₂²	semidirect
20. C₂×C₄×C₇×C₈	C₂²	semidirect
25. C₇×#336	C₂²	semidirect
26. C₈×#120	C₇	direct (N × H)
39. C₂²×C₇×C₈	D₄	semidirect
44. C₇×#120	C₈	direct (N × H)
59. C₂⁴×C₈	C₂×C₇	semidirect
60. C₂×C₈×D₄	C₂×C₇	semidirect
62. #1869	C₂×C₇	semidirect
63. #1850	C₂×C₇	semidirect
64. #1779	C₂×C₇	semidirect
65. C₈×C₂²⋊C₄	C₂×C₇	semidirect
68. C₂⁴×C₇	C₂×C₈	semidirect
73. C₂×C₇×D₄	C₂×C₈	semidirect
102. C₇×C₂²⋊C₄	C₂×C₈	semidirect
109. C₂³×C₈	C₂²×C₇	semidirect
113. C₂×C₄⋊C₈	C₂²×C₇	semidirect
116. C₂×C₂²⋊C₈	C₂²×C₇	semidirect
117. C₂×C₄×C₈	C₂²×C₇	semidirect
122. #336	C₂²×C₇	semidirect
132. C₂³×C₇	C₂²×C₈	semidirect
149. C₂×C₄×C₇	C₂²×C₈	semidirect
155. C₇×C₈	#120	direct (N × H)
168. C₂²×C₈	C₇×D₄	semidirect
173. #120	C₇×C₈	direct (N × H)
196. C₂²×C₇	C₈×D₄	semidirect
208. C₂⁴	C₂×C₇×C₈	semidirect
213. C₂×D₄	C₂×C₇×C₈	semidirect
242. C₂²⋊C₄	C₂×C₇×C₈	semidirect
259. C₂³	C₂²×C₇×C₈	semidirect
276. C₂×C₄	C₂²×C₇×C₈	semidirect
282. C₈	C₇×#120	direct (N × H)
283. C₇	C₈×#120	direct (N × H)
292. C₂²	C₇×C₈×D₄	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₇×C₈	2
C₂×C₇×C₈×D₄	2
C₄×C₇×#120	2
C₇×#1869	2
C₇×#1850	2
C₇×#1779	2
C₇×C₈×C₂²⋊C₄	2
C₈×#120	7

Subgroups — C7×C8×#120 #13198654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₈×#120 #13198654