Group #11730516 — subgroups

Subgroups — C₇×#2047 #11730516

Order 896.

256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#424	C₂	1	1	yes	maximal, characteristic
3	C₇×#419	C₂	1	2	yes	maximal
4	C₂×C₇×#101	C₂	1	2	yes	maximal
5	C₇×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×C₇×Q₁₆	—	2	4	no	—
8	C₂×C₇×Q₁₆	—	2	2	no	—
9	C₂×C₇×M₁₆	C₂²	1	2	yes	—
10	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×#117	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊C₈	—	2	2	no	—
14	C₇×#103	—	2	4	no	—
15	C₇×#101	—	2	4	no	—
16	#2047	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₇×Q₈	—	2	2	no	—
18	C₂×C₇×Q₈	—	2	2	no	—
19	C₂×C₇×Q₈	—	2	2	no	—
20	C₂×C₇×Q₈	—	2	2	no	—
21	C₂×C₇×Q₈	—	2	2	no	—
22	C₂×C₇×Q₈	D₄	1	4	yes	—
23	C₂²×C₄×C₇	—	2	2	no	—
24	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×Q₁₆	—	2	8	no	—
26	C₇×M₁₆	—	2	2	no	—
27	C₇×M₁₆	—	2	2	no	—
28	C₇×M₁₆	—	2	2	no	—
29	C₂×C₇×C₈	—	2	2	no	—
30	C₂×C₇×C₈	—	2	2	no	—
31	C₂×C₇×C₈	—	2	1	no	—
32	C₂×C₇×C₈	D₄	1	2	yes	—
33	C₇×C₄⋊C₄	D₄	1	2	yes	—
34	C₇×C₂²⋊C₄	—	2	1	no	—
35	C₄²×C₇	—	2	1	no	—
36	C₂²×Q₁₆	C₂×C₇	1	1	yes	characteristic
37	#424	C₂×C₇	1	1	yes	characteristic
38	#419	C₂×C₇	1	2	yes	—
39	C₂×#101	C₂×C₇	1	2	yes	—
40	#341	C₂×C₇	1	1	yes	characteristic
41	C₂³×C₇	#41	1	1	yes	characteristic
42	C₇×Q₈	—	2	4	no	—
43	C₇×Q₈	—	2	4	no	—
44	C₇×Q₈	—	4	2	no	—
45	C₇×Q₈	—	4	2	no	—
46	C₂×C₄×C₇	—	2	2	no	—
47	C₂×C₄×C₇	—	2	4	no	—
48	C₂×C₄×C₇	—	4	2	no	—
49	C₂×C₄×C₇	—	4	2	no	—
50	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇	#41	1	2	yes	—
54	C₇×C₈	—	2	2	no	—
55	C₇×C₈	—	4	1	no	—
56	C₇×C₈	—	4	2	no	—
57	C₂²×Q₈	C₂²×C₇	1	2	yes	—
58	C₂×Q₁₆	—	2	4	no	—
59	C₂×Q₁₆	—	2	2	no	—
60	C₂×M₁₆	C₂²×C₇	1	2	yes	—
61	C₂×M₁₆	C₂²×C₇	1	1	yes	characteristic
62	C₂²×C₈	C₂²×C₇	1	1	yes	characteristic
63	#117	C₂²×C₇	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	#103	—	2	4	no	—
66	#101	—	2	4	no	—
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	#121	1	1	yes	characteristic
69	C₂²×C₇	#123	1	2	yes	—
70	C₄×C₇	—	4	4	no	—
71	C₄×C₇	—	4	2	no	—
72	C₄×C₇	#120	1	1	yes	characteristic
73	C₄×C₇	#121	1	2	yes	—
74	C₄×C₇	#124	1	1	yes	characteristic
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	C₇×D₄	1	4	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
83	Q₁₆	—	2	8	no	—
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	C₇×D₄	1	2	yes	—
91	C₄⋊C₄	C₇×D₄	1	2	yes	—
92	C₂²⋊C₄	—	2	1	no	—
93	C₄²	—	2	1	no	—
94	C₂×C₇	—	2	2	no	—
95	C₂×C₇	#394	1	1	yes	characteristic
96	C₂×C₇	#473	1	2	yes	—
97	C₂³	C₇×#41	1	1	yes	characteristic
98	Q₈	—	2	4	no	—
99	Q₈	—	2	4	no	—
100	Q₈	—	4	2	no	—
101	Q₈	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
107	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
108	C₂×C₄	C₂×C₇×D₄	1	1	yes	characteristic
109	C₂×C₄	C₇×#41	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	4	1	no	—
112	C₈	—	4	2	no	—
113	C₇	#2047	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²	—	2	2	no	—
115	C₂²	C₇×#121	1	1	yes	characteristic
116	C₂²	C₇×#123	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	4	4	no	—
119	C₄	C₇×#120	1	1	yes	characteristic
120	C₄	C₇×#121	1	2	yes	—
121	C₄	C₇×#124	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	C₇×#394	1	1	yes	characteristic
124	C₂	C₇×#473	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇×Q₁₆

C₂

yes

maximal, characteristic

C₇×#424

C₂

yes

maximal, characteristic

C₇×#419

C₂

yes

maximal

C₂×C₇×#101

C₂

yes

maximal

C₇×#341

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₂×C₇×M₁₆

C₂²

yes

—

C₂×C₇×M₁₆

C₂²

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

characteristic

C₇×#117

C₂²

yes

characteristic

C₇×C₄⋊C₈

—

C₇×#103

—

C₇×#101

—

#2047

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

D₄

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₇×Q₁₆

—

C₇×M₁₆

—

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₇×C₄⋊C₄

D₄

yes

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₂²×Q₁₆

C₂×C₇

yes

characteristic

#424

C₂×C₇

yes

characteristic

#419

C₂×C₇

yes

—

C₂×#101

C₂×C₇

yes

—

#341

C₂×C₇

yes

characteristic

C₂³×C₇

#41

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₂²×Q₈

C₂²×C₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂²×C₇

yes

—

C₂×M₁₆

C₂²×C₇

yes

characteristic

C₂²×C₈

C₂²×C₇

yes

characteristic

#117

C₂²×C₇

yes

characteristic

C₄⋊C₈

—

#103

—

#101

—

C₂²×C₇

—

C₂²×C₇

#121

yes

characteristic

C₂²×C₇

#123

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#120

yes

characteristic

C₄×C₇

#121

yes

—

C₄×C₇

#124

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₇×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₇

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₇×D₄

yes

—

C₄⋊C₄

C₇×D₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₂×C₇

—

C₂×C₇

#394

yes

characteristic

C₂×C₇

#473

yes

—

C₂³

C₇×#41

yes

characteristic

Q₈

—

Q₈

—

100

Q₈

—

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₇×D₄

yes

—

107

C₂×C₄

C₂×C₇×D₄

yes

characteristic

108

C₂×C₄

C₂×C₇×D₄

yes

characteristic

109

C₂×C₄

C₇×#41

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₇

#2047

yes

characteristic, direct product factor, Sylow

114

C₂²

—

115

C₂²

C₇×#121

yes

characteristic

116

C₂²

C₇×#123

yes

—

117

C₄

—

118

C₄

—

119

C₄

C₇×#120

yes

characteristic

120

C₄

C₇×#121

yes

—

121

C₄

C₇×#124

yes

characteristic

122

C₂

—

123

C₂

C₇×#394

yes

characteristic

124

C₂

C₇×#473

yes

—

	subgroup	G/H
center, socle	C₂²×C₇	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
prime radical	C₂³×C₇	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

#121

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₇

C₂³×C₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ G

N	H ≅ G/N	type
16. #2047	C₇	direct (N × H)
113. C₇	#2047	direct (N × H)

H ≅ G/N

type

16. #2047

C₇

direct (N × H)

113. C₇

#2047

direct (N × H)

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

subgroup	index
C₂²×C₇×Q₁₆	2
C₇×#424	2
C₇×#419	2
C₂×C₇×#101	2
C₇×#341	2
#2047	7

subgroup

index

C₂²×C₇×Q₁₆

C₂×C₇×#101

Subgroups — C₇×#2047 #11730516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#2047 #11730516

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2047 #11730516