Subgroups — C₇×#3240 #11742549

Order 896.

328 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#529	C₂	1	1	yes	maximal, characteristic
3	C₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#501	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#489	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#485	C₂	1	2	yes	maximal, complemented
8	C₇×#482	C₂	1	2	yes	maximal
9	C₇×#479	C₂	1	2	yes	maximal, complemented
10	C₇×#478	C₂	1	2	yes	maximal, complemented
11	C₇×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₇×#126	—	2	1	no	—
15	C₇×#125	C₂²	1	2	yes	—
16	C₇×#125	C₂²	1	1	yes	characteristic
17	C₇×#124	C₂²	1	1	yes	characteristic
18	C₇×#123	—	2	1	no	—
19	C₇×C₂²⋊Q₈	—	2	1	no	—
20	C₇×C₂²⋊Q₈	—	2	1	no	—
21	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₇×#121	C₂²	1	1	yes	characteristic
23	C₄×C₇×Q₈	—	2	1	no	—
24	C₄×C₇×Q₈	—	2	1	no	—
25	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
26	C₄×C₇×D₄	—	2	1	no	—
27	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
28	C₇×#117	—	2	1	no	—
29	C₇×#117	C₂²	1	2	yes	—
30	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
31	C₇×#107	C₂²	1	2	yes	—
32	C₇×#107	C₂²	1	2	yes	—
33	C₇×#106	C₂²	1	2	yes	—
34	C₇×#106	C₂²	1	2	yes	—
35	C₇×C₄⋊C₈	C₂²	1	2	yes	—
36	C₇×#103	C₂²	1	2	yes	—
37	C₇×#103	C₂²	1	2	yes	—
38	C₇×#102	C₂²	1	2	yes	—
39	C₇×#102	C₂²	1	2	yes	—
40	C₇×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₇×#97	C₂²	1	1	yes	characteristic
42	C₇×#97	C₂²	1	1	yes	characteristic
43	#3240	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×Q₈	—	2	1	no	—
45	C₂×C₇×Q₈	—	2	1	no	—
46	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
47	C₂×C₇×D₄	—	2	1	no	—
48	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	2	1	no	—
50	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
53	C₂×C₇×C₈	C₂³	1	2	yes	—
54	C₂×C₇×C₈	C₂³	1	2	yes	—
55	C₇×C₄⋊C₄	—	2	1	no	—
56	C₇×C₄⋊C₄	—	2	1	no	—
57	C₇×C₄⋊C₄	—	2	1	no	—
58	C₇×C₄⋊C₄	—	2	2	no	—
59	C₇×C₄⋊C₄	—	2	2	no	—
60	C₇×C₄⋊C₄	—	2	1	no	—
61	C₇×C₄⋊C₄	—	2	1	no	—
62	C₇×C₄⋊C₄	—	2	1	no	—
63	C₇×C₄⋊C₄	C₂³	1	2	yes	—
64	C₇×C₄⋊C₄	C₂³	1	2	yes	—
65	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₇×C₂²⋊C₄	—	2	2	no	—
68	C₇×C₂²⋊C₄	—	2	1	no	—
69	C₇×C₂²⋊C₄	—	2	1	no	—
70	C₇×C₂²⋊C₄	—	2	1	no	—
71	C₇×C₂²⋊C₄	—	2	1	no	—
72	C₇×C₂²⋊C₄	—	2	1	no	—
73	C₄²×C₇	—	2	1	no	—
74	C₄²×C₇	—	2	2	no	—
75	C₄²×C₇	D₄	1	1	yes	characteristic
76	C₄²×C₇	D₄	1	1	yes	characteristic
77	C₄²×C₇	C₂³	1	1	yes	characteristic
78	C₄²×C₇	C₂³	1	1	yes	characteristic
79	#533	C₂×C₇	1	1	yes	characteristic, complemented
80	#529	C₂×C₇	1	1	yes	characteristic
81	#501	C₂×C₇	1	1	yes	characteristic, complemented
82	#501	C₂×C₇	1	1	yes	characteristic, complemented
83	#489	C₂×C₇	1	1	yes	characteristic, complemented
84	#489	C₂×C₇	1	1	yes	characteristic, complemented
85	#485	C₂×C₇	1	2	yes	complemented
86	#482	C₂×C₇	1	2	yes	—
87	#479	C₂×C₇	1	2	yes	complemented
88	#478	C₂×C₇	1	2	yes	complemented
89	#432	C₂×C₇	1	1	yes	characteristic, complemented
90	C₂³×C₇	—	2	1	no	—
91	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
92	C₇×Q₈	—	2	2	no	—
93	C₇×Q₈	—	4	2	no	—
94	C₇×Q₈	—	4	2	no	—
95	C₇×D₄	—	2	2	no	—
96	C₇×D₄	—	4	2	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	1	no	—
99	C₂×C₄×C₇	—	2	1	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	2	no	—
102	C₂×C₄×C₇	—	2	1	no	—
103	C₂×C₄×C₇	—	4	1	no	—
104	C₂×C₄×C₇	—	4	1	no	—
105	C₂×C₄×C₇	—	4	2	no	—
106	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
112	C₇×C₈	—	2	2	no	—
113	C₇×C₈	—	2	2	no	—
114	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
115	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
116	#126	—	2	1	no	—
117	#125	C₂²×C₇	1	1	yes	characteristic
118	#125	C₂²×C₇	1	2	yes	—
119	#124	C₂²×C₇	1	1	yes	characteristic
120	#123	—	2	1	no	—
121	C₂²⋊Q₈	—	2	1	no	—
122	C₂²⋊Q₈	—	2	1	no	—
123	C₂²⋊Q₈	C₂²×C₇	1	1	yes	characteristic
124	#121	C₂²×C₇	1	1	yes	characteristic
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	—	2	1	no	—
127	C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
128	C₄×D₄	—	2	1	no	—
129	C₄×D₄	C₂²×C₇	1	1	yes	characteristic
130	#117	—	2	1	no	—
131	#117	C₂²×C₇	1	2	yes	—
132	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
133	#107	C₂²×C₇	1	2	yes	—
134	#107	C₂²×C₇	1	2	yes	—
135	#106	C₂²×C₇	1	2	yes	—
136	#106	C₂²×C₇	1	2	yes	—
137	C₄⋊C₈	C₂²×C₇	1	2	yes	—
138	#103	C₂²×C₇	1	2	yes	—
139	#103	C₂²×C₇	1	2	yes	—
140	#102	C₂²×C₇	1	2	yes	—
141	#102	C₂²×C₇	1	2	yes	—
142	C₂²⋊C₈	C₂²×C₇	1	2	yes	—
143	#97	C₂²×C₇	1	1	yes	characteristic
144	#97	C₂²×C₇	1	1	yes	characteristic
145	C₂²×C₇	—	2	1	no	—
146	C₂²×C₇	—	2	1	no	—
147	C₂²×C₇	—	2	1	no	—
148	C₂²×C₇	—	4	1	no	—
149	C₂²×C₇	—	4	1	no	—
150	C₂²×C₇	—	4	1	no	—
151	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₇	—	2	1	no	—
153	C₄×C₇	—	2	1	no	—
154	C₄×C₇	—	2	1	no	—
155	C₄×C₇	—	2	1	no	—
156	C₄×C₇	—	2	1	no	—
157	C₄×C₇	—	4	1	no	—
158	C₄×C₇	—	4	1	no	—
159	C₄×C₇	—	4	1	no	—
160	C₄×C₇	—	4	2	no	—
161	C₄×C₇	—	4	2	no	—
162	C₄×C₇	2₋¹⁺⁴	1	2	yes	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	C₂³×C₇	1	1	yes	characteristic
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	C₂³×C₇	1	1	yes	characteristic
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
170	C₂²×C₄	C₇×D₄	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₇	1	2	yes	—
173	C₂×C₈	C₂³×C₇	1	2	yes	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	C₂³×C₇	1	2	yes	—
183	C₄⋊C₄	C₂³×C₇	1	2	yes	—
184	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₇	1	1	yes	characteristic
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₇×D₄	1	1	yes	characteristic
195	C₄²	C₇×D₄	1	1	yes	characteristic
196	C₄²	C₂³×C₇	1	1	yes	characteristic
197	C₄²	C₂³×C₇	1	1	yes	characteristic
198	C₂×C₇	—	4	1	no	—
199	C₂×C₇	—	8	1	no	—
200	C₂×C₇	#536	1	1	yes	characteristic
201	C₂×C₇	#575	1	1	yes	characteristic
202	C₂×C₇	#575	1	1	yes	characteristic
203	C₂³	—	2	1	no	—
204	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
205	Q₈	—	2	2	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
220	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
221	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
223	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
224	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₇	#3240	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	1	no	—
233	C₂²	—	4	1	no	—
234	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	C₇×2₋¹⁺⁴	1	2	yes	—
246	C₂	—	4	1	no	—
247	C₂	—	8	1	no	—
248	C₂	C₇×#536	1	1	yes	characteristic
249	C₂	C₇×#575	1	1	yes	characteristic
250	C₂	C₇×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₇×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#533	C₂	semidirect
3. C₇×#501	C₂	semidirect
4. C₇×#501	C₂	semidirect
5. C₇×#489	C₂	semidirect
6. C₇×#489	C₂	semidirect
7. C₇×#485	C₂	semidirect
9. C₇×#479	C₂	semidirect
10. C₇×#478	C₂	semidirect
11. C₇×#432	C₂	semidirect
43. #3240	C₇	direct (N × H)
79. #533	C₂×C₇	semidirect
81. #501	C₂×C₇	semidirect
82. #501	C₂×C₇	semidirect
83. #489	C₂×C₇	semidirect
84. #489	C₂×C₇	semidirect
85. #485	C₂×C₇	semidirect
87. #479	C₂×C₇	semidirect
88. #478	C₂×C₇	semidirect
89. #432	C₂×C₇	semidirect
227. C₇	#3240	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#533	2
C₇×#529	2
C₇×#501	2
C₇×#489	2
C₇×#485	2
C₇×#482	2
C₇×#479	2
C₇×#478	2
C₇×#432	2
#3240	7

Subgroups — C7×#3240 #11742549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3240 #11742549