Group #11731819 — subgroups

Subgroups — #11731819

Order 896.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6439	C₂	1	2	yes	maximal, complemented
2	C₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
3	#89766	C₂	1	6	yes	maximal, complemented
4	#89763	C₂	1	3	yes	maximal
5	#89755	C₂	1	3	yes	maximal, complemented
6	C₂²×D₂₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
8	C₂×C₇×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₇)⋊C₄	C₂²	1	6	yes	—
11	C₂×(C₂×D₇)⋊C₄	C₂²	1	6	yes	—
12	C₂×C₄⋊Dic₇	C₂²	1	3	yes	—
13	C₂×Dic₇⋊C₄	C₂²	1	6	yes	—
14	#6439	—	2	8	no	—
15	C₇⋊#95	C₂²	1	3	yes	—
16	C₇⋊#95	C₂²	1	3	yes	—
17	#2823	—	7	1	no	maximal, Sylow
18	C₂³×D₇	C₂³	1	2	yes	—
19	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₇	C₂³	1	3	yes	—
21	C₂²×C₄×C₇	C₂³	1	3	yes	—
22	C₂²×Dic₇	C₂³	1	6	yes	—
23	C₂×D₂₈	—	4	3	no	—
24	C₇×C₄⋊C₄	—	4	3	no	—
25	C₇×C₄⋊C₄	—	4	3	no	—
26	C₄²×C₇	D₄	1	4	yes	complemented
27	(C₂×D₇)⋊C₄	—	2	12	no	—
28	(C₂×D₇)⋊C₄	—	4	6	no	—
29	C₄⋊Dic₇	—	4	3	no	—
30	Dic₇⋊C₄	—	2	12	no	—
31	C₂×#126	—	7	2	no	—
32	#398	—	7	3	no	—
33	#396	—	7	6	no	—
34	#394	—	7	3	no	—
35	#384	D₇	1	1	yes	characteristic, complemented
36	C₂³×C₇	C₂⁴	1	1	yes	characteristic
37	C₂²×D₇	—	2	2	no	—
38	C₂²×D₇	—	2	6	no	—
39	C₂²×D₇	—	2	6	no	—
40	C₂×C₄×C₇	—	2	3	no	—
41	C₂×C₄×C₇	—	2	3	no	—
42	C₂×C₄×C₇	—	2	3	no	—
43	C₂×C₄×C₇	—	2	6	no	—
44	C₂×C₄×C₇	—	2	3	no	—
45	C₂×C₄×C₇	C₂×D₄	1	6	yes	—
46	C₂×Dic₇	—	2	6	no	—
47	C₂×Dic₇	—	2	6	no	—
48	C₂×Dic₇	—	2	6	no	—
49	D₂₈	—	4	4	no	—
50	C₂²×D₄	—	7	1	no	—
51	#126	—	14	8	no	—
52	C₂×C₄⋊C₄	—	7	6	no	—
53	C₂×C₄⋊C₄	—	7	3	no	—
54	C₂×C₄⋊C₄	C₂×D₇	1	3	yes	—
55	C₂×C₄⋊C₄	C₂×D₇	1	3	yes	—
56	C₂×C₂²⋊C₄	—	7	6	no	—
57	C₂×C₂²⋊C₄	—	7	6	no	—
58	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
59	#95	—	7	3	no	—
60	#95	—	7	3	no	—
61	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
62	C₂²×C₇	H(5,𝔽₂)	1	3	yes	—
63	C₂²×C₇	2₋¹⁺⁴	1	3	yes	—
64	C₂×D₇	—	4	6	no	—
65	C₂×D₇	—	4	2	no	—
66	C₂×D₇	—	4	6	no	—
67	C₄×C₇	—	2	6	no	—
68	C₄×C₇	—	4	1	no	—
69	C₄×C₇	—	4	3	no	—
70	Dic₇	—	4	6	no	—
71	C₂⁴	—	7	2	no	—
72	C₂×D₄	—	28	3	no	—
73	C₂²×C₄	—	7	6	no	—
74	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
75	C₂²×C₄	C₂²×D₇	1	3	yes	—
76	C₂²×C₄	C₂²×D₇	1	3	yes	—
77	C₄⋊C₄	—	4	3	no	—
78	C₄⋊C₄	—	4	3	no	—
79	C₄⋊C₄	—	14	12	no	—
80	C₄⋊C₄	—	28	3	no	—
81	C₂²⋊C₄	—	14	12	no	—
82	C₂²⋊C₄	—	28	6	no	—
83	C₄²	D₂₈	1	4	yes	complemented
84	C₂×C₇	#537	1	3	yes	—
85	C₂×C₇	#560	1	1	yes	characteristic
86	C₂×C₇	#563	1	3	yes	—
87	D₇	—	8	2	no	—
88	C₂³	—	14	2	no	—
89	C₂³	—	14	6	no	—
90	C₂³	—	14	6	no	—
91	C₂³	C₂³×D₇	1	1	yes	characteristic
92	D₄	—	28	4	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	6	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	14	6	no	—
99	C₂×C₄	—	14	6	no	—
100	C₂×C₄	—	14	6	no	—
101	C₂×C₄	C₂×D₂₈	1	6	yes	—
102	C₇	#2823	1	1	yes	characteristic, complemented, Sylow
103	C₂²	—	28	2	no	—
104	C₂²	—	28	6	no	—
105	C₂²	—	28	6	no	—
106	C₂²	C₂²×D₂₈	1	1	yes	characteristic
107	C₂²	#6548	1	3	yes	—
108	C₂²	#6551	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	3	no	—
112	C₄	—	28	6	no	—
113	C₂	—	56	2	no	—
114	C₂	#90201	1	3	yes	—
115	C₂	#90411	1	1	yes	characteristic
116	C₂	#90426	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6439

C₂

yes

maximal, complemented

C₇×#384

C₂

yes

maximal, characteristic, complemented

#89766

C₂

yes

maximal, complemented

#89763

C₂

yes

maximal

#89755

C₂

yes

maximal, complemented

C₂²×D₂₈

C₂²

yes

characteristic

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₇)⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

C₂²

yes

—

#6439

—

C₇⋊#95

C₂²

yes

—

C₇⋊#95

C₂²

yes

—

#2823

—

maximal, Sylow

C₂³×D₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×C₄×C₇

C₂³

yes

—

C₂²×C₄×C₇

C₂³

yes

—

C₂²×Dic₇

C₂³

yes

—

C₂×D₂₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

complemented

(C₂×D₇)⋊C₄

—

(C₂×D₇)⋊C₄

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

—

—

—

—

D₇

yes

characteristic, complemented

C₂³×C₇

C₂⁴

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

D₂₈

—

C₂²×D₄

—

#126

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇

yes

characteristic

#95

—

#95

—

C₂²×C₇

C₂²×D₄

yes

characteristic

C₂²×C₇

H(5,𝔽₂)

yes

—

C₂²×C₇

2₋¹⁺⁴

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

C₂⁴

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂²×C₄

C₂²×D₇

yes

—

C₂²×C₄

C₂²×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

D₂₈

yes

complemented

C₂×C₇

#537

yes

—

C₂×C₇

#560

yes

characteristic

C₂×C₇

#563

yes

—

D₇

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₇

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×D₂₈

yes

—

102

C₇

#2823

yes

characteristic, complemented, Sylow

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²×D₂₈

yes

characteristic

107

C₂²

#6548

yes

—

108

C₂²

#6551

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₂

—

114

C₂

#90201

yes

—

115

C₂

#90411

yes

characteristic

116

C₂

#90426

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁴
Fitting	C₇×#384	C₂
prime radical	C₂²×D₂₈	C₂²
hypercenter	#384	D₇
hypocenter	C₇	#2823
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁴

Fitting

C₇×#384

C₂

prime radical

C₂²×D₂₈

C₂²

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#6439	C₂	semidirect
2. C₇×#384	C₂	semidirect
3. #89766	C₂	semidirect
5. #89755	C₂	semidirect
26. C₄²×C₇	D₄	semidirect
35. #384	D₇	semidirect
83. C₄²	D₂₈	semidirect
102. C₇	#2823	semidirect

H ≅ G/N

type

1. C₂×#6439

C₂

semidirect

2. C₇×#384

C₂

semidirect

3. #89766

C₂

semidirect

5. #89755

C₂

semidirect

26. C₄²×C₇

D₄

semidirect

35. #384

D₇

semidirect

83. C₄²

D₂₈

semidirect

102. C₇

#2823

semidirect

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

subgroup	index
C₂×#6439	2
C₇×#384	2
#89766	2
#89763	2
#89755	2
#2823	7

subgroup

index

Subgroups — #11731819

Conjugacy classes of subgroups

Complements

Maximal subgroups