Group #13506362 — subgroups

Subgroups — #13506362

Order 1792.

1006 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 459 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×#89452	C₂	1	4	yes	maximal
2	#11730756	C₂	1	2	yes	maximal
3	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	#11727518	C₂	1	8	yes	maximal
5	C₂×#6432	—	2	4	no	—
6	C₂×#6432	C₂²	1	4	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
9	C₄³×C₇	C₂²	1	1	yes	characteristic
10	C₂×#6391	C₂²	1	16	yes	—
11	#89707	—	2	4	no	—
12	#89707	C₂²	1	8	yes	—
13	#89452	—	2	16	no	—
14	#33310	—	7	1	no	maximal, Sylow
15	C₂²×Dic₁₄	—	2	4	no	—
16	C₂²×C₇×C₈	C₂³	1	4	yes	—
17	C₂×C₄²×C₇	D₄	1	4	yes	—
18	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂³	1	2	yes	—
20	C₂×C₄⋊Dic₇	—	2	16	no	—
21	C₂×C₄⋊Dic₇	—	2	4	no	—
22	C₂×C₄⋊Dic₇	C₂³	1	8	yes	—
23	#6432	—	2	16	no	—
24	#6432	—	4	16	no	—
25	C₄×C₇×C₈	D₄	1	8	yes	—
26	C₄×C₇×C₈	Q₈	1	16	yes	complemented
27	#6391	—	2	32	no	—
28	C₂×#498	—	7	4	no	—
29	#2868	—	7	2	no	—
30	#1844	—	7	8	no	—
31	C₄²×C₈	D₇	1	1	yes	characteristic
32	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
33	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
34	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
35	C₂²×Dic₇	—	2	8	no	—
36	C₂×Dic₁₄	—	4	8	no	—
37	C₂×Dic₁₄	—	4	16	no	—
38	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
39	C₂×C₇×C₈	C₂×Q₈	1	16	yes	—
40	C₄²×C₇	SD₁₆	1	16	yes	—
41	C₄²×C₇	C₂×D₄	1	4	yes	—
42	C₄²×C₇	C₂×Q₈	1	8	yes	—
43	C₄⋊Dic₇	—	2	16	no	—
44	C₄⋊Dic₇	—	4	8	no	—
45	C₄⋊Dic₇	—	4	32	no	—
46	C₂×C₄⋊Q₈	—	7	4	no	—
47	C₂×C₄⋊Q₈	—	14	4	no	—
48	#498	—	14	16	no	—
49	C₂×#106	—	7	16	no	—
50	C₂×C₄×C₈	C₂×D₇	1	4	yes	—
51	C₂×C₄×C₈	C₂×D₇	1	2	yes	—
52	#384	—	7	8	no	—
53	#384	—	14	4	no	—
54	C₄³	C₂×D₇	1	1	yes	characteristic
55	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇	#127	1	4	yes	—
57	C₂×C₄×C₇	C₄⋊Q₈	1	8	yes	—
58	C₂×C₄×C₇	C₂×SD₁₆	1	8	yes	—
59	C₂×C₄×C₇	C₂×SD₁₆	1	16	yes	—
60	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
61	C₂×C₄×C₇	C₂²×Q₈	1	4	yes	—
62	C₂×Dic₇	—	4	8	no	—
63	C₂×Dic₇	—	4	16	no	—
64	Dic₁₄	—	8	16	no	—
65	C₇×C₈	C₄⋊Q₈	1	16	yes	complemented
66	C₂²×Q₈	—	14	4	no	—
67	C₂²×C₈	C₂²×D₇	1	4	yes	—
68	C₄⋊Q₈	—	14	16	no	—
69	C₄⋊Q₈	—	28	16	no	—
70	C₂×C₄⋊C₄	—	7	8	no	—
71	C₂×C₄⋊C₄	—	14	4	no	—
72	C₂×C₄⋊C₄	—	14	16	no	—
73	C₂×C₄²	D₂₈	1	4	yes	—
74	C₂×C₄²	C₂²×D₇	1	2	yes	—
75	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
76	#106	—	14	32	no	—
77	C₄×C₈	Dic₁₄	1	16	yes	complemented
78	C₄×C₈	D₂₈	1	8	yes	—
79	C₂²×C₇	C₂×#127	1	1	yes	characteristic
80	C₂²×C₇	C₂×C₄⋊Q₈	1	2	yes	—
81	C₂²×C₇	C₂²×SD₁₆	1	4	yes	—
82	C₄×C₇	#492	1	8	yes	—
83	C₄×C₇	#498	1	16	yes	complemented
84	C₄×C₇	C₂×C₄⋊Q₈	1	4	yes	—
85	Dic₇	—	8	8	no	—
86	C₂×Q₈	—	28	8	no	—
87	C₂×Q₈	—	28	16	no	—
88	C₂²×C₄	—	14	8	no	—
89	C₂²×C₄	C₂×D₂₈	1	4	yes	—
90	C₂²×C₄	C₂×D₂₈	1	2	yes	—
91	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
92	C₂×C₈	C₂×Dic₁₄	1	16	yes	—
93	C₂×C₈	C₂×D₂₈	1	8	yes	—
94	C₄⋊C₄	—	14	16	no	—
95	C₄⋊C₄	—	28	8	no	—
96	C₄⋊C₄	—	28	32	no	—
97	C₄²	#1132	1	16	yes	—
98	C₄²	C₂×Dic₁₄	1	8	yes	—
99	C₄²	C₂×D₂₈	1	4	yes	—
100	C₂×C₇	#2868	1	1	yes	characteristic
101	C₂×C₇	C₂×#492	1	2	yes	—
102	C₂×C₇	C₂×#498	1	4	yes	—
103	C₂³	C₂²×D₂₈	1	1	yes	characteristic
104	Q₈	—	56	16	no	—
105	C₂×C₄	—	28	8	no	—
106	C₂×C₄	—	28	16	no	—
107	C₂×C₄	#6432	1	8	yes	—
108	C₂×C₄	#6437	1	4	yes	—
109	C₂×C₄	C₂×#1132	1	16	yes	—
110	C₂×C₄	C₂×#1132	1	8	yes	—
111	C₂×C₄	C₂²×Dic₁₄	1	4	yes	—
112	C₂×C₄	C₂²×D₂₈	1	2	yes	—
113	C₈	#6432	1	16	yes	complemented
114	C₇	#33310	1	1	yes	characteristic, complemented, Sylow
115	C₂²	C₂×#6432	1	2	yes	—
116	C₂²	C₂×#6437	1	1	yes	characteristic
117	C₂²	C₂²×#1132	1	4	yes	—
118	C₄	—	56	8	no	—
119	C₄	#89452	1	16	yes	complemented
120	C₄	#89465	1	8	yes	—
121	C₄	C₂×#6432	1	4	yes	—
122	C₂	#11730756	1	1	yes	characteristic
123	C₂	C₂×#89452	1	4	yes	—
124	C₂	C₂×#89465	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#89452

C₂

yes

maximal

#11730756

C₂

yes

maximal

C₄²×C₇×C₈

C₂

yes

maximal, characteristic

#11727518

C₂

yes

maximal

C₂×#6432

—

C₂×#6432

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

—

C₄³×C₇

C₂²

yes

characteristic

C₂×#6391

C₂²

yes

—

#89707

—

#89707

C₂²

yes

—

#89452

—

#33310

—

maximal, Sylow

C₂²×Dic₁₄

—

C₂²×C₇×C₈

C₂³

yes

—

C₂×C₄²×C₇

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄²×C₇

C₂³

yes

—

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

C₂³

yes

—

#6432

—

#6432

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

Q₈

yes

—

—

—

—

C₄²×C₈

D₇

yes

characteristic

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×C₄×C₇

C₂⁴

yes

characteristic

C₂²×Dic₇

—

C₂×Dic₁₄

—

C₂×Dic₁₄

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₂×C₇×C₈

C₂×Q₈

yes

—

C₄²×C₇

SD₁₆

yes

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

C₂×Q₈

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

#498

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₇

yes

—

C₂×C₄×C₈

C₂×D₇

yes

—

#384

—

#384

—

C₄³

C₂×D₇

yes

characteristic

C₂³×C₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₇

#127

yes

—

C₂×C₄×C₇

C₄⋊Q₈

yes

—

C₂×C₄×C₇

C₂×SD₁₆

yes

—

C₂×C₄×C₇

C₂×SD₁₆

yes

—

C₂×C₄×C₇

C₂²×D₄

yes

—

C₂×C₄×C₇

C₂²×Q₈

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

C₇×C₈

C₄⋊Q₈

yes

complemented

C₂²×Q₈

—

C₂²×C₈

C₂²×D₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₂₈

yes

—

C₂×C₄²

C₂²×D₇

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

#106

—

C₄×C₈

Dic₁₄

yes

complemented

C₄×C₈

D₂₈

yes

—

C₂²×C₇

C₂×#127

yes

characteristic

C₂²×C₇

C₂×C₄⋊Q₈

yes

—

C₂²×C₇

C₂²×SD₁₆

yes

—

C₄×C₇

#492

yes

—

C₄×C₇

#498

yes

complemented

C₄×C₇

C₂×C₄⋊Q₈

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₂₈

yes

—

C₂²×C₄

C₂×D₂₈

yes

—

C₂²×C₄

C₂³×D₇

yes

characteristic

C₂×C₈

C₂×Dic₁₄

yes

—

C₂×C₈

C₂×D₂₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#1132

yes

—

C₄²

C₂×Dic₁₄

yes

—

C₄²

C₂×D₂₈

yes

—

100

C₂×C₇

#2868

yes

characteristic

101

C₂×C₇

C₂×#492

yes

—

102

C₂×C₇

C₂×#498

yes

—

103

C₂³

C₂²×D₂₈

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#6432

yes

—

108

C₂×C₄

#6437

yes

—

109

C₂×C₄

C₂×#1132

yes

—

110

C₂×C₄

C₂×#1132

yes

—

111

C₂×C₄

C₂²×Dic₁₄

yes

—

112

C₂×C₄

C₂²×D₂₈

yes

—

113

C₈

#6432

yes

complemented

114

C₇

#33310

yes

characteristic, complemented, Sylow

115

C₂²

C₂×#6432

yes

—

116

C₂²

C₂×#6437

yes

characteristic

117

C₂²

C₂²×#1132

yes

—

118

C₄

—

119

C₄

#89452

yes

complemented

120

C₄

#89465

yes

—

121

C₄

C₂×#6432

yes

—

122

C₂

#11730756

yes

characteristic

123

C₂

C₂×#89452

yes

—

124

C₂

C₂×#89465

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₂₈
commutator, cosocle	C₂²×C₄×C₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₄²×C₇×C₈	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	C₄²×C₈	D₇
hypocenter	C₇	#33310
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

subgroup

G/H

center

C₂³

C₂²×D₂₈

commutator, cosocle

C₂²×C₄×C₇

C₂⁴

Frattini

C₂²×C₄

C₂³×D₇

Fitting

C₄²×C₇×C₈

C₂

socle, prime radical

C₂³×C₇

C₂²×D₄

hypercenter

C₄²×C₈

D₇

hypocenter

C₇

#33310

LCS

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

N	H ≅ G/N	type
26. C₄×C₇×C₈	Q₈	semidirect
65. C₇×C₈	C₄⋊Q₈	semidirect
77. C₄×C₈	Dic₁₄	semidirect
83. C₄×C₇	#498	semidirect
113. C₈	#6432	semidirect
114. C₇	#33310	semidirect
119. C₄	#89452	semidirect

H ≅ G/N

type

26. C₄×C₇×C₈

Q₈

semidirect

65. C₇×C₈

C₄⋊Q₈

semidirect

77. C₄×C₈

Dic₁₄

semidirect

83. C₄×C₇

#498

semidirect

113. C₈

#6432

semidirect

114. C₇

#33310

semidirect

119. C₄

#89452

semidirect

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

subgroup	index
C₂×#89452	2
#11730756	2
C₄²×C₇×C₈	2
#11727518	2
#33310	7

subgroup

index

C₄²×C₇×C₈

Subgroups — #13506362

Conjugacy classes of subgroups

Complements

Maximal subgroups