Group #11731788 — subgroups

Subgroups — #11731788

Order 896.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×Dic₁₄	C₂	1	2	yes	maximal
2	C₇×#384	C₂	1	1	yes	maximal, characteristic
3	#89757	C₂	1	4	yes	maximal
4	#89755	C₂	1	4	yes	maximal
5	#89746	C₂	1	4	yes	maximal
6	C₂²×Dic₁₄	C₂²	1	2	yes	—
7	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇	C₂²	1	4	yes	—
12	C₂×Dic₇⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄×Dic₇	C₂²	1	4	yes	—
14	C₄×Dic₁₄	—	2	8	no	—
15	C₇⋊#95	C₂²	1	4	yes	—
16	C₇⋊#95	C₂²	1	8	yes	—
17	#2582	—	7	1	no	maximal, Sylow
18	C₂²×C₄×C₇	C₂³	1	4	yes	—
19	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₇	C₂³	1	2	yes	—
21	C₂²×Dic₇	C₂³	1	4	yes	—
22	C₂²×Dic₇	C₂³	1	4	yes	—
23	C₂×Dic₁₄	—	2	4	no	—
24	C₂×Dic₁₄	—	2	4	no	—
25	C₂×Dic₁₄	Q₈	1	8	yes	complemented
26	C₇×C₄⋊C₄	—	2	8	no	—
27	C₇×C₄⋊C₄	—	4	2	no	—
28	C₄²×C₇	D₄	1	4	yes	—
29	C₄⋊Dic₇	—	2	4	no	—
30	C₄⋊Dic₇	—	4	4	no	—
31	Dic₇⋊C₄	—	2	8	no	—
32	C₄×Dic₇	—	2	8	no	—
33	C₂×C₄×Q₈	—	7	2	no	—
34	#398	—	7	4	no	—
35	#391	—	7	4	no	—
36	#389	—	7	4	no	—
37	#384	D₇	1	1	yes	characteristic
38	C₂³×C₇	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₇	—	2	1	no	—
40	C₂×C₄×C₇	—	2	1	no	—
41	C₂×C₄×C₇	—	2	8	no	—
42	C₂×C₄×C₇	—	2	4	no	—
43	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₇	#41	1	4	yes	—
47	C₂×Dic₇	—	2	4	no	—
48	C₂×Dic₇	—	2	4	no	—
49	C₂×Dic₇	—	2	4	no	—
50	C₂×Dic₇	—	2	4	no	—
51	C₂×Dic₇	—	2	4	no	—
52	C₂×Dic₇	C₂×Q₈	1	8	yes	—
53	Dic₁₄	—	2	16	no	—
54	C₂²×Q₈	—	7	2	no	—
55	C₄×Q₈	—	14	8	no	—
56	C₂×C₄⋊C₄	—	7	4	no	—
57	C₂×C₄⋊C₄	—	7	2	no	—
58	C₂×C₄⋊C₄	—	7	4	no	—
59	C₂×C₄⋊C₄	C₂×D₇	1	4	yes	—
60	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
61	C₂×C₄²	—	7	4	no	—
62	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
63	#95	—	7	8	no	—
64	#95	—	7	4	no	—
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	2	no	—
71	C₄×C₇	—	4	4	no	—
72	C₄×C₇	C₂²⋊Q₈	1	8	yes	—
73	Dic₇	—	2	8	no	—
74	Dic₇	—	4	4	no	—
75	C₂×Q₈	—	7	8	no	—
76	C₂×Q₈	—	14	4	no	—
77	C₂×Q₈	—	14	4	no	—
78	C₂²×C₄	—	7	4	no	—
79	C₂²×C₄	—	7	4	no	—
80	C₂²×C₄	C₂²×D₇	1	4	yes	—
81	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₇	1	2	yes	—
83	C₄⋊C₄	—	2	8	no	—
84	C₄⋊C₄	—	4	2	no	—
85	C₄⋊C₄	—	14	8	no	—
86	C₄⋊C₄	—	14	4	no	—
87	C₄⋊C₄	—	28	4	no	—
88	C₄²	—	14	8	no	—
89	C₄²	D₂₈	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	C₂³×D₇	1	1	yes	characteristic
96	Q₈	—	14	16	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	7	8	no	—
102	C₂×C₄	—	14	4	no	—
103	C₂×C₄	—	14	4	no	—
104	C₂×C₄	—	14	4	no	—
105	C₂×C₄	—	14	4	no	—
106	C₂×C₄	—	14	4	no	—
107	C₂×C₄	C₂×D₂₈	1	2	yes	—
108	C₂×C₄	C₂×D₂₈	1	4	yes	—
109	C₂×C₄	#1159	1	4	yes	—
110	C₂×C₄	Q₈×D₇	1	4	yes	—
111	C₇	#2582	1	1	yes	characteristic, complemented, Sylow
112	C₂²	C₂²×D₂₈	1	1	yes	characteristic
113	C₂²	C₂×#1159	1	2	yes	—
114	C₂²	#6548	1	1	yes	characteristic
115	C₂²	C₂×Q₈×D₇	1	2	yes	—
116	C₂²	#6551	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	4	4	no	—
119	C₄	—	14	8	no	—
120	C₄	—	28	4	no	—
121	C₄	#6460	1	8	yes	—
122	C₂	C₂×#6460	1	2	yes	—
123	C₂	#90201	1	1	yes	characteristic
124	C₂	#90404	1	1	yes	characteristic
125	C₂	#90412	1	2	yes	—
126	C₂	#90413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₄

C₂

yes

maximal

C₇×#384

C₂

yes

maximal, characteristic

#89757

C₂

yes

maximal

#89755

C₂

yes

maximal

#89746

C₂

yes

maximal

C₂²×Dic₁₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₇

C₂²

yes

—

C₄×Dic₁₄

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2582

—

maximal, Sylow

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×Dic₁₄

Q₈

yes

complemented

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

C₄×Dic₇

—

C₂×C₄×Q₈

—

—

—

—

D₇

yes

characteristic

C₂³×C₇

C₂⁴

yes

characteristic

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

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

C₂×Q₈

yes

—

Dic₁₄

—

C₂²×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇

yes

characteristic

#95

—

#95

—

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

C₂²⋊Q₈

yes

—

Dic₇

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₂₈

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

C₂³×D₇

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

—

107

C₂×C₄

C₂×D₂₈

yes

—

108

C₂×C₄

C₂×D₂₈

yes

—

109

C₂×C₄

#1159

yes

—

110

C₂×C₄

Q₈×D₇

yes

—

111

C₇

#2582

yes

characteristic, complemented, Sylow

112

C₂²

C₂²×D₂₈

yes

characteristic

113

C₂²

C₂×#1159

yes

—

114

C₂²

#6548

yes

characteristic

115

C₂²

C₂×Q₈×D₇

yes

—

116

C₂²

#6551

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#6460

yes

—

122

C₂

C₂×#6460

yes

—

123

C₂

#90201

yes

characteristic

124

C₂

#90404

yes

characteristic

125

C₂

#90412

yes

—

126

C₂

#90413

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#384	C₂
hypercenter	#384	D₇
hypocenter	C₇	#2582
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, prime radical, cosocle

C₂³×C₇

C₂⁴

C₂

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
25. C₂×Dic₁₄	Q₈	semidirect
111. C₇	#2582	semidirect

H ≅ G/N

type

25. C₂×Dic₁₄

Q₈

semidirect

111. C₇

#2582

semidirect

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

subgroup	index
C₂×C₄×Dic₁₄	2
C₇×#384	2
#89757	2
#89755	2
#89746	2
#2582	7

subgroup

index

C₂×C₄×Dic₁₄

Subgroups — #11731788

Conjugacy classes of subgroups

Complements

Maximal subgroups