Group #13247481 — subgroups

Subgroups — #13247481

Order 1792.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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×#1450	C₂	1	1	yes	maximal, characteristic
2	#11727545	C₂	1	1	yes	maximal, characteristic
3	#11727517	C₂	1	1	yes	maximal, characteristic
4	#11725544	C₂	1	4	yes	maximal
5	C₇×#431	C₂²	1	2	yes	—
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	#89708	C₂²	1	2	yes	—
8	#89696	C₂²	1	2	yes	—
9	C₇×#322	C₄	1	2	yes	complemented
10	C₇×C₈²	C₄	1	2	yes	complemented
11	#89347	—	2	2	no	—
12	#89346	—	2	2	no	—
13	#8791	—	7	1	no	maximal, Sylow
14	C₂²×C₇×C₈	D₄	1	2	yes	—
15	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇	—	2	2	no	—
17	C₂×#1136	—	2	2	no	—
18	C₇×C₄⋊C₈	—	2	2	no	—
19	C₇×C₂²⋊C₈	—	2	2	no	—
20	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
21	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₇×C₈	Q₈	1	2	yes	—
24	C₇⋊#95	—	2	4	no	—
25	C₄⋊(C₇⋊C₈)	—	2	4	no	—
26	#1837	—	7	1	no	—
27	#1835	—	7	1	no	—
28	#1450	D₇	1	1	yes	characteristic
29	#1286	—	7	4	no	—
30	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₇	#41	1	2	yes	—
32	C₂²×Dic₇	—	4	2	no	—
33	C₂×C₇×C₈	—	2	4	no	—
34	C₂×C₇×C₈	—	2	4	no	—
35	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
37	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₇	#41	1	2	yes	—
40	C₄⋊Dic₇	—	4	4	no	—
41	#1136	—	4	4	no	—
42	C₂×C₇⋊C₈	—	4	2	no	—
43	#431	C₂×D₇	1	2	yes	—
44	#423	—	7	2	no	—
45	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
46	#382	—	7	2	no	—
47	#341	—	14	2	no	—
48	#340	—	14	2	no	—
49	#322	Dic₇	1	2	yes	complemented
50	C₈²	Dic₇	1	2	yes	complemented
51	C₂³×C₇	#124	1	1	yes	characteristic
52	C₂×C₄×C₇	—	4	2	no	—
53	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇	#117	1	2	yes	—
55	C₂×C₄×C₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₇	#125	1	2	yes	—
57	C₂×C₄×C₇	#135	1	4	yes	—
58	C₂×Dic₇	—	8	4	no	—
59	C₂×Dic₇	—	8	2	no	—
60	C₇×C₈	—	2	2	no	—
61	C₇×C₈	—	2	2	no	—
62	C₇×C₈	—	4	4	no	—
63	C₇⋊C₈	—	8	2	no	—
64	C₂×M₁₆	—	14	2	no	—
65	C₂²×C₈	D₂₈	1	2	yes	—
66	C₂×C₄⋊C₄	—	14	2	no	—
67	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	14	4	no	—
70	C₂²⋊C₈	—	2	2	no	—
71	C₄×C₈	Dic₁₄	1	2	yes	—
72	C₄×C₈	C₂×Dic₇	1	4	yes	—
73	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
74	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
75	#95	—	14	4	no	—
76	C₂²×C₇	—	2	2	no	—
77	C₂²×C₇	#382	1	1	yes	characteristic
78	C₂²×C₇	#488	1	2	yes	—
79	C₄×C₇	—	2	2	no	—
80	C₄×C₇	—	2	2	no	—
81	C₄×C₇	#427	1	2	yes	—
82	C₄×C₇	#488	1	2	yes	—
83	Dic₇	—	16	2	no	—
84	C₂²×C₄	—	28	2	no	—
85	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
86	C₂²×C₄	#1157	1	2	yes	—
87	M₁₆	—	28	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	28	2	no	—
91	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
92	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
93	C₄⋊C₄	—	28	4	no	—
94	C₄²	C₂×Dic₁₄	1	1	yes	characteristic
95	C₄²	#1157	1	2	yes	—
96	C₄²	C₂²×Dic₇	1	1	yes	characteristic
97	C₂×C₇	—	2	2	no	—
98	C₂×C₇	#1833	1	2	yes	—
99	C₂×C₇	#1835	1	1	yes	characteristic
100	C₂³	#6438	1	1	yes	characteristic
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	56	4	no	—
103	C₂×C₄	—	56	2	no	—
104	C₂×C₄	#6433	1	2	yes	—
105	C₂×C₄	#6438	1	1	yes	characteristic
106	C₂×C₄	#6467	1	4	yes	—
107	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
108	C₂×C₄	#6489	1	2	yes	—
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	4	4	no	—
112	C₈	—	56	2	no	—
113	C₇	#8791	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	2	2	no	—
115	C₂²	#89469	1	2	yes	—
116	C₂²	#89708	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	112	2	no	—
120	C₄	#89469	1	2	yes	—
121	C₄	#89878	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#11727517	1	1	yes	characteristic
124	C₂	#11727650	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1450

C₂

yes

maximal, characteristic

#11727545

C₂

yes

maximal, characteristic

#11727517

C₂

yes

maximal, characteristic

#11725544

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

#89708

C₂²

yes

—

#89696

C₂²

yes

—

C₇×#322

C₄

yes

complemented

C₇×C₈²

C₄

yes

—

—

—

C₂²×C₇×C₈

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×#1136

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

Q₈

yes

—

C₇⋊#95

—

C₄⋊(C₇⋊C₈)

—

#1837

—

#1835

—

#1450

D₇

yes

characteristic

#1286

—

C₂²×C₄×C₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄²×C₇

#41

yes

—

C₄⋊Dic₇

—

#1136

—

C₂×C₇⋊C₈

—

#431

C₂×D₇

yes

—

#423

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

—

Dic₇

yes

complemented

C₈²

Dic₇

yes

complemented

C₂³×C₇

#124

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#124

yes

characteristic

C₂×C₄×C₇

#125

yes

—

C₂×C₄×C₇

#135

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

D₂₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₄

yes

—

C₄×C₈

C₂×Dic₇

yes

—

C₄×C₈

C₂×Dic₇

yes

characteristic

C₄×C₈

C₂×Dic₇

yes

characteristic

#95

—

C₂²×C₇

—

C₂²×C₇

#382

yes

characteristic

C₂²×C₇

#488

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#427

yes

—

C₄×C₇

#488

yes

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₈

yes

characteristic

C₂²×C₄

#1157

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₇

yes

—

C₂×C₈

C₄⋊Dic₇

yes

—

C₄⋊C₄

—

C₄²

C₂×Dic₁₄

yes

characteristic

C₄²

#1157

yes

—

C₄²

C₂²×Dic₇

yes

characteristic

C₂×C₇

—

C₂×C₇

#1833

yes

—

C₂×C₇

#1835

yes

characteristic

100

C₂³

#6438

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#6433

yes

—

105

C₂×C₄

#6438

yes

characteristic

106

C₂×C₄

#6467

yes

—

107

C₂×C₄

C₂×C₄⋊Dic₇

yes

characteristic

108

C₂×C₄

#6489

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₇

#8791

yes

characteristic, complemented, Sylow

114

C₂²

—

115

C₂²

#89469

yes

—

116

C₂²

#89708

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

112

—

120

C₄

#89469

yes

—

121

C₄

#89878

yes

—

122

C₂

—

123

C₂

#11727517

yes

characteristic

124

C₂

#11727650

yes

—

	subgroup	G/H
center	C₂²	#89708
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1450	C₂
socle	C₂²×C₇	#382
prime radical	C₂³×C₇	#124
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1450	D₇
hypocenter	C₇	#8791
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#89708

commutator

C₄²×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
9. C₇×#322	C₄	semidirect
10. C₇×C₈²	C₄	semidirect
49. #322	Dic₇	semidirect
50. C₈²	Dic₇	semidirect
113. C₇	#8791	semidirect

H ≅ G/N

type

9. C₇×#322

C₄

semidirect

10. C₇×C₈²

C₄

semidirect

49. #322

Dic₇

semidirect

50. C₈²

Dic₇

semidirect

113. C₇

#8791

semidirect

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

subgroup	index
C₇×#1450	2
#11727545	2
#11727517	2
#11725544	2
#8791	7

subgroup

index

Subgroups — #13247481

Conjugacy classes of subgroups

Complements

Maximal subgroups