Subgroups — C₇×#3260 #11742569

Order 896.

346 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#540	C₂	1	1	yes	maximal, characteristic
3	C₇×#479	C₂	1	2	yes	maximal, complemented
4	C₇×#478	C₂	1	2	yes	maximal, complemented
5	C₇×#468	C₂	1	2	yes	maximal, complemented
6	C₇×#465	C₂	1	2	yes	maximal, complemented
7	C₇×#461	C₂	1	2	yes	maximal, complemented
8	C₇×#460	C₂	1	2	yes	maximal, complemented
9	C₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
11	C₂×C₇×SD₁₆	C₂²	1	2	yes	—
12	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₇×#127	C₂²	1	1	yes	characteristic
16	C₇×#126	—	2	2	no	—
17	C₇×#125	—	2	2	no	—
18	C₇×#125	C₂²	1	1	yes	characteristic
19	C₇×#125	C₂²	1	1	yes	characteristic
20	C₇×#123	—	2	2	no	—
21	C₇×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₇×#121	—	2	2	no	—
23	C₇×#121	C₂²	1	2	yes	—
24	C₄×C₇×Q₈	C₂²	1	2	yes	—
25	C₄×C₇×D₄	C₂²	1	2	yes	—
26	C₇×#117	C₂²	1	1	yes	characteristic
27	C₇×#107	C₂²	1	2	yes	—
28	C₇×#106	C₂²	1	2	yes	—
29	C₇×C₄⋊C₈	C₂²	1	2	yes	—
30	C₇×C₄⋊C₈	C₂²	1	2	yes	—
31	C₇×#103	C₂²	1	2	yes	—
32	C₇×#103	C₂²	1	2	yes	—
33	C₇×#102	C₂²	1	2	yes	—
34	C₇×#102	C₂²	1	2	yes	—
35	#3260	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₇×Q₈	C₂³	1	2	yes	—
37	C₂×C₇×D₄	—	2	2	no	—
38	C₂×C₇×D₄	—	2	2	no	—
39	C₂×C₇×D₄	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	—	2	2	no	—
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₇×SD₁₆	—	4	2	no	—
43	C₇×SD₁₆	—	4	2	no	—
44	C₇×M₁₆	—	4	1	no	—
45	C₂×C₇×C₈	—	4	1	no	—
46	C₂×C₇×C₈	C₂³	1	2	yes	—
47	C₂×C₇×C₈	C₂³	1	2	yes	—
48	C₇×C₄⋊C₄	—	2	2	no	—
49	C₇×C₄⋊C₄	—	2	2	no	—
50	C₇×C₄⋊C₄	—	2	2	no	—
51	C₇×C₄⋊C₄	—	2	2	no	—
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	D₄	1	2	yes	—
54	C₇×C₄⋊C₄	C₂³	1	2	yes	—
55	C₇×C₄⋊C₄	C₂³	1	2	yes	—
56	C₇×C₂²⋊C₄	—	2	2	no	—
57	C₇×C₂²⋊C₄	—	2	2	no	—
58	C₇×C₂²⋊C₄	—	2	2	no	—
59	C₇×C₂²⋊C₄	D₄	1	2	yes	—
60	C₄²×C₇	—	2	2	no	—
61	C₄²×C₇	C₂³	1	1	yes	characteristic
62	C₄²×C₇	C₂³	1	1	yes	characteristic
63	#541	C₂×C₇	1	1	yes	characteristic, complemented
64	#540	C₂×C₇	1	1	yes	characteristic
65	#479	C₂×C₇	1	2	yes	complemented
66	#478	C₂×C₇	1	2	yes	complemented
67	#468	C₂×C₇	1	2	yes	complemented
68	#465	C₂×C₇	1	2	yes	complemented
69	#461	C₂×C₇	1	2	yes	complemented
70	#460	C₂×C₇	1	2	yes	complemented
71	#424	C₂×C₇	1	1	yes	characteristic, complemented
72	C₂³×C₇	—	2	2	no	—
73	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
74	C₇×Q₈	—	2	2	no	—
75	C₇×Q₈	—	2	2	no	—
76	C₇×D₄	—	2	2	no	—
77	C₇×D₄	—	2	2	no	—
78	C₇×D₄	—	8	2	no	—
79	C₇×D₄	—	8	2	no	—
80	C₂×C₄×C₇	—	2	1	no	—
81	C₂×C₄×C₇	—	2	1	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	8	2	no	—
86	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₇×C₈	—	2	2	no	—
91	C₇×C₈	—	2	2	no	—
92	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
93	C₂×SD₁₆	C₂²×C₇	1	2	yes	—
94	C₂×M₁₆	C₂²×C₇	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
96	C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
97	#127	C₂²×C₇	1	1	yes	characteristic
98	#126	—	2	2	no	—
99	#125	—	2	2	no	—
100	#125	C₂²×C₇	1	1	yes	characteristic
101	#125	C₂²×C₇	1	1	yes	characteristic
102	#123	—	2	2	no	—
103	C₂²⋊Q₈	C₂²×C₇	1	2	yes	—
104	#121	—	2	2	no	—
105	#121	C₂²×C₇	1	2	yes	—
106	C₄×Q₈	C₂²×C₇	1	2	yes	—
107	C₄×D₄	C₂²×C₇	1	2	yes	—
108	#117	C₂²×C₇	1	1	yes	characteristic
109	#107	C₂²×C₇	1	2	yes	—
110	#106	C₂²×C₇	1	2	yes	—
111	C₄⋊C₈	C₂²×C₇	1	2	yes	—
112	C₄⋊C₈	C₂²×C₇	1	2	yes	—
113	#103	C₂²×C₇	1	2	yes	—
114	#103	C₂²×C₇	1	2	yes	—
115	#102	C₂²×C₇	1	2	yes	—
116	#102	C₂²×C₇	1	2	yes	—
117	C₂²×C₇	—	2	1	no	—
118	C₂²×C₇	—	2	2	no	—
119	C₂²×C₇	—	4	2	no	—
120	C₂²×C₇	—	4	2	no	—
121	C₂²×C₇	—	4	2	no	—
122	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₇	—	2	1	no	—
124	C₄×C₇	—	2	2	no	—
125	C₄×C₇	—	2	2	no	—
126	C₄×C₇	—	4	2	no	—
127	C₄×C₇	—	4	2	no	—
128	C₄×C₇	—	4	2	no	—
129	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
130	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
131	C₂×Q₈	C₂³×C₇	1	2	yes	—
132	C₂×D₄	—	2	2	no	—
133	C₂×D₄	—	2	2	no	—
134	C₂×D₄	C₂³×C₇	1	2	yes	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
137	SD₁₆	—	4	2	no	—
138	SD₁₆	—	4	2	no	—
139	M₁₆	—	4	1	no	—
140	C₂×C₈	—	4	1	no	—
141	C₂×C₈	C₂³×C₇	1	2	yes	—
142	C₂×C₈	C₂³×C₇	1	2	yes	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	C₇×D₄	1	2	yes	—
149	C₄⋊C₄	C₂³×C₇	1	2	yes	—
150	C₄⋊C₄	C₂³×C₇	1	2	yes	—
151	C₂²⋊C₄	—	2	2	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	C₇×D₄	1	2	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	C₂³×C₇	1	1	yes	characteristic
157	C₄²	C₂³×C₇	1	1	yes	characteristic
158	C₂×C₇	—	4	1	no	—
159	C₂×C₇	—	8	2	no	—
160	C₂×C₇	#537	1	1	yes	characteristic
161	C₂×C₇	#577	1	2	yes	—
162	C₂³	—	2	2	no	—
163	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
164	Q₈	—	2	2	no	—
165	Q₈	—	2	2	no	—
166	D₄	—	2	2	no	—
167	D₄	—	2	2	no	—
168	D₄	—	8	2	no	—
169	D₄	—	8	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	8	2	no	—
176	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
177	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
178	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
179	C₂×C₄	C₂⁴×C₇	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₇	#3260	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
196	C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	8	2	no	—
199	C₂	C₇×#537	1	1	yes	characteristic
200	C₂	C₇×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇
prime radical	C₇×#540	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#541	C₂	semidirect
3. C₇×#479	C₂	semidirect
4. C₇×#478	C₂	semidirect
5. C₇×#468	C₂	semidirect
6. C₇×#465	C₂	semidirect
7. C₇×#461	C₂	semidirect
8. C₇×#460	C₂	semidirect
9. C₇×#424	C₂	semidirect
35. #3260	C₇	direct (N × H)
63. #541	C₂×C₇	semidirect
65. #479	C₂×C₇	semidirect
66. #478	C₂×C₇	semidirect
67. #468	C₂×C₇	semidirect
68. #465	C₂×C₇	semidirect
69. #461	C₂×C₇	semidirect
70. #460	C₂×C₇	semidirect
71. #424	C₂×C₇	semidirect
182. C₇	#3260	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×#541	2
C₇×#540	2
C₇×#479	2
C₇×#478	2
C₇×#468	2
C₇×#465	2
C₇×#461	2
C₇×#460	2
C₇×#424	2
#3260	7

Subgroups — C7×#3260 #11742569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#3260 #11742569