Group #13256873 — subgroups

Subgroups — #13256873

Order 1792.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₇×#2845	C₂	1	1	yes	maximal, characteristic, complemented
2	#11727307	C₂	1	3	yes	maximal
3	#11725436	C₂	1	3	yes	maximal, complemented
4	C₂×#6437	C₂²	1	1	yes	characteristic
5	C₇×#398	—	2	6	no	—
6	C₇×#382	C₂²	1	3	yes	—
7	C₂×#6383	—	2	6	no	—
8	C₂×#6377	C₂²	1	3	yes	—
9	#10360	—	7	1	no	maximal, Sylow
10	C₂²×D₂₈	—	2	3	no	—
11	C₂×C₇×C₄⋊C₄	—	2	6	no	—
12	C₂×C₇×C₄⋊C₄	D₄	1	6	yes	—
13	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
14	C₂²×C₇⋊C₈	—	2	3	no	—
15	#6437	—	2	3	no	—
16	#6437	Q₈	1	2	yes	—
17	C₇×#95	—	2	2	no	—
18	C₇×#95	—	2	6	no	—
19	#6383	—	4	12	no	—
20	#6377	—	2	6	no	—
21	#2845	D₇	1	1	yes	characteristic, complemented
22	#1961	—	7	3	no	—
23	#1295	—	7	3	no	—
24	C₂³×D₇	—	4	1	no	—
25	C₂²×C₄×C₇	—	2	6	no	—
26	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
27	C₂×D₂₈	—	2	6	no	—
28	C₂×D₂₈	—	4	6	no	—
29	C₂×D₂₈	—	4	6	no	—
30	C₂×D₂₈	—	4	3	no	—
31	C₇×C₄⋊C₄	—	4	6	no	—
32	C₇×C₄⋊C₄	—	8	6	no	—
33	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₇	#41	1	3	yes	—
35	C₂×C₇⋊C₈	—	2	6	no	—
36	C₂×C₇⋊C₈	—	4	3	no	—
37	C₂×C₇⋊C₈	—	4	3	no	—
38	C₂×#127	—	7	1	no	—
39	C₂×#102	—	14	6	no	—
40	C₂×#97	—	7	3	no	—
41	#398	—	2	6	no	—
42	#382	C₂×D₇	1	3	yes	—
43	C₂³×C₇	#120	1	1	yes	characteristic
44	C₂²×D₇	—	4	2	no	—
45	C₂²×D₇	—	8	3	no	—
46	C₂²×D₇	—	8	3	no	—
47	C₂×C₄×C₇	—	2	3	no	—
48	C₂×C₄×C₇	—	2	3	no	—
49	C₂×C₄×C₇	—	4	6	no	—
50	C₂×C₄×C₇	—	4	6	no	—
51	C₂×C₄×C₇	—	4	6	no	—
52	C₂×C₄×C₇	C₂²⋊Q₈	1	3	yes	—
53	C₂×C₄×C₇	#124	1	3	yes	—
54	D₂₈	—	4	12	no	—
55	D₂₈	—	8	6	no	—
56	C₇⋊C₈	—	4	6	no	—
57	C₂²×D₄	—	14	3	no	—
58	C₂²×C₈	—	14	3	no	—
59	#127	—	7	2	no	—
60	#127	—	14	3	no	—
61	C₂×C₄⋊C₄	—	2	6	no	—
62	C₂×C₄⋊C₄	C₂²⋊D₇	1	6	yes	—
63	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
64	#102	—	28	12	no	—
65	#97	—	14	6	no	—
66	#95	—	2	2	no	—
67	#95	—	2	6	no	—
68	C₂²×C₇	#393	1	1	yes	characteristic
69	C₂²×C₇	#453	1	3	yes	—
70	C₂²×C₇	#454	1	3	yes	—
71	C₂×D₇	—	8	6	no	—
72	C₂×D₇	—	16	3	no	—
73	C₂×D₇	—	16	1	no	—
74	C₄×C₇	—	2	3	no	—
75	C₄×C₇	—	2	3	no	—
76	C₄×C₇	—	8	6	no	—
77	C₂⁴	—	28	1	no	—
78	C₂×D₄	—	14	6	no	—
79	C₂×D₄	—	28	6	no	—
80	C₂×D₄	—	28	6	no	—
81	C₂×D₄	—	28	3	no	—
82	C₂²×C₄	—	2	6	no	—
83	C₂²×C₄	C₂×C₂²⋊D₇	1	3	yes	—
84	C₂×C₈	—	14	6	no	—
85	C₂×C₈	—	28	3	no	—
86	C₂×C₈	—	28	3	no	—
87	C₄⋊C₄	—	4	6	no	—
88	C₄⋊C₄	—	8	6	no	—
89	C₄²	Q₈×D₇	1	1	yes	characteristic
90	C₄²	#1161	1	3	yes	—
91	C₂×C₇	#1656	1	3	yes	—
92	C₂×C₇	#1658	1	1	yes	characteristic
93	C₂×C₇	#2027	1	3	yes	—
94	D₇	—	16	2	no	—
95	C₂³	—	28	2	no	—
96	C₂³	—	56	3	no	—
97	C₂³	—	56	3	no	—
98	C₂³	#6516	1	1	yes	characteristic
99	D₄	—	28	12	no	—
100	D₄	—	56	6	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	4	6	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	#6509	1	3	yes	—
107	C₂×C₄	#6510	1	3	yes	—
108	C₈	—	28	6	no	—
109	C₇	#10360	1	1	yes	characteristic, complemented, Sylow
110	C₂²	—	56	6	no	—
111	C₂²	—	112	3	no	—
112	C₂²	—	112	1	no	—
113	C₂²	#90003	1	1	yes	characteristic
114	C₂²	#90013	1	3	yes	—
115	C₂²	#90017	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	112	2	no	—
120	C₂	#11729657	1	3	yes	—
121	C₂	#11729706	1	3	yes	—
122	C₂	#11729714	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2845

C₂

yes

maximal, characteristic, complemented

#11727307

C₂

yes

maximal

#11725436

C₂

yes

maximal, complemented

C₂×#6437

C₂²

yes

characteristic

C₇×#398

—

C₇×#382

C₂²

yes

—

C₂×#6383

—

C₂×#6377

C₂²

yes

—

#10360

—

maximal, Sylow

C₂²×D₂₈

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂²×C₇⋊C₈

—

#6437

—

#6437

Q₈

yes

—

C₇×#95

—

C₇×#95

—

#6383

—

#6377

—

#2845

D₇

yes

characteristic, complemented

#1961

—

#1295

—

C₂³×D₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#127

—

C₂×#102

—

C₂×#97

—

#398

—

#382

C₂×D₇

yes

—

C₂³×C₇

#120

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

C₂²⋊Q₈

yes

—

C₂×C₄×C₇

#124

yes

—

D₂₈

—

D₂₈

—

C₇⋊C₈

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₇

yes

—

C₂×C₄²

C₂²×D₇

yes

—

—

—

—

C₂²×C₇

#393

yes

characteristic

C₂²×C₇

#453

yes

—

C₂²×C₇

#454

yes

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₇

yes

characteristic

C₄²

#1161

yes

—

C₂×C₇

#1656

yes

—

C₂×C₇

#1658

yes

characteristic

C₂×C₇

#2027

yes

—

D₇

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#6516

yes

characteristic

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#6509

yes

—

107

C₂×C₄

#6510

yes

—

108

C₈

—

109

C₇

#10360

yes

characteristic, complemented, Sylow

110

C₂²

—

111

C₂²

—

112

—

112

C₂²

—

112

—

113

C₂²

#90003

yes

characteristic

114

C₂²

#90013

yes

—

115

C₂²

#90017

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

112

—

120

C₂

#11729657

yes

—

121

C₂

#11729706

yes

—

122

C₂

#11729714

yes

characteristic

	subgroup	G/H
center	C₂³	#6516
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2845	C₂
socle	C₂³×C₇	#120
prime radical	C₂×#6437	C₂²
hypercenter	#2845	D₇
hypocenter	C₇	#10360
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#6516

commutator, cosocle

C₂×C₄²×C₇

C₂³

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂³×C₇

C₂²

D₇

C₇

G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
1. C₇×#2845	C₂	semidirect
3. #11725436	C₂	semidirect
21. #2845	D₇	semidirect
109. C₇	#10360	semidirect

H ≅ G/N

type

1. C₇×#2845

C₂

semidirect

3. #11725436

C₂

semidirect

21. #2845

D₇

semidirect

109. C₇

#10360

semidirect

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

subgroup	index
C₇×#2845	2
#11727307	2
#11725436	2
#10360	7

subgroup

index

Subgroups — #13256873

Conjugacy classes of subgroups

Complements

Maximal subgroups