Group #11741891 — subgroups

Subgroups — C₇×#2582 #11741891

Order 896.

484 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 248 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	C₂×C₄×C₇×Q₈	C₂	1	2	yes	maximal
2	C₇×#398	C₂	1	4	yes	maximal
3	C₇×#391	C₂	1	4	yes	maximal
4	C₇×#389	C₂	1	4	yes	maximal
5	C₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	2	yes	—
7	C₄×C₇×Q₈	—	2	8	no	—
8	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄²×C₇	C₂²	1	4	yes	—
14	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
15	C₇×#95	C₂²	1	8	yes	—
16	C₇×#95	C₂²	1	4	yes	—
17	#2582	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂×C₇×Q₈	—	2	4	no	—
19	C₂×C₇×Q₈	—	2	4	no	—
20	C₂×C₇×Q₈	Q₈	1	8	yes	complemented
21	C₂²×C₄×C₇	C₂³	1	4	yes	—
22	C₂²×C₄×C₇	C₂³	1	4	yes	—
23	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₇	C₂³	1	2	yes	—
26	C₇×C₄⋊C₄	—	2	8	no	—
27	C₇×C₄⋊C₄	—	2	4	no	—
28	C₇×C₄⋊C₄	—	2	8	no	—
29	C₇×C₄⋊C₄	—	4	2	no	—
30	C₇×C₄⋊C₄	—	4	4	no	—
31	C₄²×C₇	—	2	8	no	—
32	C₄²×C₇	D₄	1	4	yes	—
33	C₂×C₄×Q₈	C₂×C₇	1	2	yes	—
34	#398	C₂×C₇	1	4	yes	—
35	#391	C₂×C₇	1	4	yes	—
36	#389	C₂×C₇	1	4	yes	—
37	#384	C₂×C₇	1	1	yes	characteristic
38	C₂³×C₇	C₂⁴	1	1	yes	characteristic
39	C₇×Q₈	—	2	16	no	—
40	C₂×C₄×C₇	—	2	4	no	—
41	C₂×C₄×C₇	—	2	4	no	—
42	C₂×C₄×C₇	—	2	1	no	—
43	C₂×C₄×C₇	—	2	1	no	—
44	C₂×C₄×C₇	—	2	4	no	—
45	C₂×C₄×C₇	—	2	8	no	—
46	C₂×C₄×C₇	—	2	4	no	—
47	C₂×C₄×C₇	—	2	4	no	—
48	C₂×C₄×C₇	—	2	4	no	—
49	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
51	C₂×C₄×C₇	C₂×Q₈	1	8	yes	—
52	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₇	#41	1	4	yes	—
54	C₂²×Q₈	C₂²×C₇	1	2	yes	—
55	C₄×Q₈	—	2	8	no	—
56	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
57	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
58	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
59	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
60	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
61	C₂×C₄²	C₂²×C₇	1	4	yes	—
62	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
63	#95	C₂²×C₇	1	8	yes	—
64	#95	C₂²×C₇	1	4	yes	—
65	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₇	C₂²×Q₈	1	2	yes	—
67	C₂²×C₇	C₂×#41	1	2	yes	—
68	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
69	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
70	C₄×C₇	—	2	8	no	—
71	C₄×C₇	—	2	2	no	—
72	C₄×C₇	—	4	4	no	—
73	C₄×C₇	—	4	4	no	—
74	C₄×C₇	C₂²⋊Q₈	1	8	yes	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₇×Q₈	1	8	yes	complemented
78	C₂²×C₄	C₂³×C₇	1	4	yes	—
79	C₂²×C₄	C₂³×C₇	1	4	yes	—
80	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
81	C₂²×C₄	C₂³×C₇	1	4	yes	—
82	C₂²×C₄	C₂³×C₇	1	2	yes	—
83	C₄⋊C₄	—	2	8	no	—
84	C₄⋊C₄	—	2	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	4	4	no	—
88	C₄²	—	2	8	no	—
89	C₄²	C₇×D₄	1	4	yes	—
90	C₂×C₇	C₂×C₂²⋊Q₈	1	2	yes	—
91	C₂×C₇	#537	1	1	yes	characteristic
92	C₂×C₇	#557	1	2	yes	—
93	C₂×C₇	Q₈²	1	1	yes	characteristic
94	C₂×C₇	#559	1	1	yes	characteristic
95	C₂³	C₂⁴×C₇	1	1	yes	characteristic
96	Q₈	—	2	16	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
107	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
108	C₂×C₄	C₂×C₇×Q₈	1	8	yes	—
109	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
110	C₂×C₄	C₇×#41	1	4	yes	—
111	C₇	#2582	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
113	C₂²	C₂²×C₇×Q₈	1	2	yes	—
114	C₂²	C₂×C₇×#41	1	2	yes	—
115	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
117	C₄	—	2	8	no	—
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	4	4	no	—
121	C₄	C₇×C₂²⋊Q₈	1	8	yes	—
122	C₂	C₂×C₇×C₂²⋊Q₈	1	2	yes	—
123	C₂	C₇×#537	1	1	yes	characteristic
124	C₂	C₇×#557	1	2	yes	—
125	C₂	C₇×Q₈²	1	1	yes	characteristic
126	C₂	C₇×#559	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₇×Q₈

C₂

yes

maximal

C₇×#398

C₂

yes

maximal

C₇×#391

C₂

yes

maximal

C₇×#389

C₂

yes

maximal

C₇×#384

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

—

C₄×C₇×Q₈

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₇×#95

C₂²

yes

—

C₇×#95

C₂²

yes

—

#2582

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

Q₈

yes

complemented

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₄²×C₇

D₄

yes

—

C₂×C₄×Q₈

C₂×C₇

yes

—

#398

C₂×C₇

yes

—

#391

C₂×C₇

yes

—

#389

C₂×C₇

yes

—

#384

C₂×C₇

yes

characteristic

C₂³×C₇

C₂⁴

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

#41

yes

—

C₂²×Q₈

C₂²×C₇

yes

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₄²

C₂²×C₇

yes

—

C₂×C₄²

C₂²×C₇

yes

characteristic

#95

C₂²×C₇

yes

—

#95

C₂²×C₇

yes

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

C₂²×Q₈

yes

—

C₂²×C₇

C₂×#41

yes

—

C₂²×C₇

H(5,𝔽₂)

yes

characteristic

C₂²×C₇

2₋¹⁺⁴

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²⋊Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₇×Q₈

yes

complemented

C₂²×C₄

C₂³×C₇

yes

—

C₂²×C₄

C₂³×C₇

yes

—

C₂²×C₄

C₂³×C₇

yes

characteristic

C₂²×C₄

C₂³×C₇

yes

—

C₂²×C₄

C₂³×C₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₇×D₄

yes

—

C₂×C₇

C₂×C₂²⋊Q₈

yes

—

C₂×C₇

#537

yes

characteristic

C₂×C₇

#557

yes

—

C₂×C₇

Q₈²

yes

characteristic

C₂×C₇

#559

yes

characteristic

C₂³

C₂⁴×C₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

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

—

108

C₂×C₄

C₂×C₇×Q₈

yes

—

109

C₂×C₄

C₂×C₇×Q₈

yes

—

110

C₂×C₄

C₇×#41

yes

—

111

C₇

#2582

yes

characteristic, direct product factor, Sylow

112

C₂²

C₂²×C₇×D₄

yes

characteristic

113

C₂²

C₂²×C₇×Q₈

yes

—

114

C₂²

C₂×C₇×#41

yes

—

115

C₂²

C₇×H(5,𝔽₂)

yes

characteristic

116

C₂²

C₇×2₋¹⁺⁴

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₇×C₂²⋊Q₈

yes

—

122

C₂

C₂×C₇×C₂²⋊Q₈

yes

—

123

C₂

C₇×#537

yes

characteristic

124

C₂

C₇×#557

yes

—

125

C₂

C₇×Q₈²

yes

characteristic

126

C₂

C₇×#559

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₇

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₇ ◁ G

N	H ≅ G/N	type
17. #2582	C₇	direct (N × H)
20. C₂×C₇×Q₈	Q₈	semidirect
77. C₂×Q₈	C₇×Q₈	semidirect
111. C₇	#2582	direct (N × H)

H ≅ G/N

type

17. #2582

C₇

direct (N × H)

20. C₂×C₇×Q₈

Q₈

semidirect

77. C₂×Q₈

C₇×Q₈

semidirect

111. C₇

#2582

direct (N × H)

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

subgroup	index
C₂×C₄×C₇×Q₈	2
C₇×#398	2
C₇×#391	2
C₇×#389	2
C₇×#384	2
#2582	7

subgroup

index

C₂×C₄×C₇×Q₈

Subgroups — C₇×#2582 #11741891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#2582 #11741891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2582 #11741891