Group #13268509 — subgroups

Subgroups — #13268509

Order 1792.

310 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2247	C₂	1	1	yes	maximal, characteristic
2	#11729066	C₂	1	2	yes	maximal
3	#11729052	C₂	1	2	yes	maximal
4	#11726820	C₂	1	1	yes	maximal, characteristic
5	C₁₆×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₃₂	—	2	2	no	—
7	C₇×#494	C₂²	1	2	yes	—
8	C₂×#6403	—	2	4	no	—
9	#89850	—	2	4	no	—
10	C₂×Dic₅₆	—	2	2	no	—
11	#89471	C₂²	1	2	yes	—
12	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	#12516	—	7	1	no	maximal, Sylow
16	C₂×C₇×Q₁₆	—	2	2	no	—
17	C₂×C₇×Q₁₆	—	2	2	no	—
18	C₇×C₄⋊Q₈	—	2	2	no	—
19	C₂×Dic₂₈	—	2	2	no	—
20	C₂×Dic₂₈	—	2	2	no	—
21	#6432	—	2	2	no	—
22	C₇×Q₃₂	—	4	4	no	—
23	C₂×C₇×C₁₆	D₄	1	2	yes	—
24	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
25	#6403	—	4	8	no	—
26	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
27	#6381	—	4	8	no	—
28	C₄×C₇⋊C₈	D₄	1	2	yes	—
29	Dic₅₆	—	4	4	no	—
30	#2247	—	7	2	no	—
31	#2247	—	7	1	no	—
32	#2247	D₇	1	1	yes	characteristic
33	#1713	—	7	2	no	—
34	C₈×C₁₆	—	7	1	no	—
35	C₂×C₇×Q₈	—	4	2	no	—
36	C₂×Dic₁₄	—	4	2	no	—
37	C₇×Q₁₆	—	4	4	no	—
38	C₇×Q₁₆	—	4	4	no	—
39	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
41	C₇×C₄⋊C₄	—	4	4	no	—
42	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₇	—	4	4	no	—
44	C₂×C₇⋊C₈	D₈	1	4	yes	—
45	Dic₂₈	—	4	4	no	—
46	Dic₂₈	—	4	4	no	—
47	C₇×C₁₆	D₈	1	4	yes	—
48	C₇⋊C₁₆	D₈	1	4	yes	—
49	C₂×Q₃₂	—	2	2	no	—
50	C₂×Q₃₂	—	14	4	no	—
51	C₂×Q₃₂	—	14	2	no	—
52	#500	—	14	4	no	—
53	#494	—	7	2	no	—
54	#494	C₂×D₇	1	2	yes	—
55	C₄×C₁₆	—	7	1	no	—
56	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
57	C₈²	—	7	1	no	—
58	C₇×Q₈	—	8	2	no	—
59	C₇×Q₈	—	8	2	no	—
60	C₂×C₄×C₇	—	8	2	no	—
61	C₂×C₄×C₇	#127	1	1	yes	characteristic
62	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
63	C₂×Dic₇	—	8	2	no	—
64	Dic₁₄	—	8	2	no	—
65	Dic₁₄	—	8	2	no	—
66	C₇×C₈	C₂×D₈	1	2	yes	—
67	C₇×C₈	C₂×D₈	1	2	yes	—
68	C₇⋊C₈	Q₃₂	1	8	yes	complemented
69	C₂×Q₁₆	—	2	2	no	—
70	C₂×Q₁₆	—	2	2	no	—
71	C₂×Q₁₆	—	14	2	no	—
72	C₂×Q₁₆	—	14	2	no	—
73	C₄⋊Q₈	—	2	2	no	—
74	C₄⋊Q₈	—	14	2	no	—
75	Q₃₂	—	4	4	no	—
76	Q₃₂	—	28	8	no	—
77	Q₃₂	—	28	4	no	—
78	C₂×C₁₆	—	7	2	no	—
79	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
80	#107	—	28	8	no	—
81	C₄×C₈	—	7	2	no	—
82	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
83	C₂²×C₇	#493	1	1	yes	characteristic
84	C₄×C₇	—	16	2	no	—
85	C₄×C₇	#493	1	1	yes	characteristic
86	C₄×C₇	#493	1	1	yes	characteristic
87	C₄×C₇	C₂×Q₃₂	1	4	yes	—
88	Dic₇	—	16	2	no	—
89	C₂×Q₈	—	4	2	no	—
90	C₂×Q₈	—	28	2	no	—
91	Q₁₆	—	4	4	no	—
92	Q₁₆	—	4	4	no	—
93	Q₁₆	—	28	4	no	—
94	Q₁₆	—	28	4	no	—
95	C₂×C₈	—	7	4	no	—
96	C₂×C₈	D₄×D₇	1	1	yes	characteristic
97	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
98	C₄⋊C₄	—	4	4	no	—
99	C₄⋊C₄	—	28	4	no	—
100	C₄²	D₄×D₇	1	1	yes	characteristic
101	C₁₆	—	7	4	no	—
102	C₁₆	#1141	1	4	yes	—
103	C₂×C₇	#1712	1	1	yes	characteristic
104	C₂×C₇	#2247	1	2	yes	—
105	Q₈	—	8	2	no	—
106	Q₈	—	8	2	no	—
107	Q₈	—	56	2	no	—
108	Q₈	—	56	2	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	—	56	2	no	—
111	C₂×C₄	D₇×D₈	1	2	yes	—
112	C₂×C₄	#6503	1	1	yes	characteristic
113	C₈	—	7	8	no	—
114	C₈	D₇×D₈	1	2	yes	—
115	C₈	C₂×#1141	1	2	yes	—
116	C₇	#12516	1	1	yes	characteristic, complemented, Sylow
117	C₂²	#89923	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	112	2	no	—
120	C₄	D₇×Q₃₂	1	4	yes	—
121	C₄	#89845	1	1	yes	characteristic
122	C₄	#89923	1	1	yes	characteristic
123	C₂	#11729030	1	1	yes	characteristic
124	C₂	#11729253	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2247

C₂

yes

maximal, characteristic

#11729066

C₂

yes

maximal

#11729052

C₂

yes

maximal

#11726820

C₂

yes

maximal, characteristic

C₁₆×C₇⋊C₈

C₂

yes

maximal, characteristic

C₂×C₇×Q₃₂

—

C₇×#494

C₂²

yes

—

C₂×#6403

—

#89850

—

C₂×Dic₅₆

—

#89471

C₂²

yes

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₄×C₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₇⋊C₈

C₂²

yes

characteristic

#12516

—

maximal, Sylow

C₂×C₇×Q₁₆

—

C₂×C₇×Q₁₆

—

C₇×C₄⋊Q₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

#6432

—

C₇×Q₃₂

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6403

—

C₂×C₇⋊C₁₆

D₄

yes

—

#6381

—

C₄×C₇⋊C₈

D₄

yes

—

Dic₅₆

—

#2247

—

#2247

—

#2247

D₇

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂×C₇×Q₈

—

C₂×Dic₁₄

—

C₇×Q₁₆

—

C₇×Q₁₆

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₂×C₇⋊C₈

D₈

yes

—

Dic₂₈

—

Dic₂₈

—

C₇×C₁₆

D₈

yes

—

C₇⋊C₁₆

D₈

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

#500

—

#494

—

#494

C₂×D₇

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇

yes

characteristic

C₈²

—

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

C₂×D₈

yes

—

C₇×C₈

C₂×D₈

yes

—

C₇⋊C₈

Q₃₂

yes

complemented

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇

yes

—

#107

—

C₄×C₈

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

C₂×Q₃₂

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₇

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

D₄×D₇

yes

characteristic

101

C₁₆

—

102

C₁₆

#1141

yes

—

103

C₂×C₇

#1712

yes

characteristic

104

C₂×C₇

#2247

yes

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

D₇×D₈

yes

—

112

C₂×C₄

#6503

yes

characteristic

113

C₈

—

114

C₈

D₇×D₈

yes

—

115

C₈

C₂×#1141

yes

—

116

C₇

#12516

yes

characteristic, complemented, Sylow

117

C₂²

#89923

yes

characteristic

118

C₄

—

119

C₄

—

112

—

120

C₄

D₇×Q₃₂

yes

—

121

C₄

#89845

yes

characteristic

122

C₄

#89923

yes

characteristic

123

C₂

#11729030

yes

characteristic

124

C₂

#11729253

yes

—

	subgroup	G/H
center	C₂²	#89923
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#2247	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#2247	D₇
hypocenter	C₇	#12516
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#89923

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
68. C₇⋊C₈	Q₃₂	semidirect
116. C₇	#12516	semidirect

H ≅ G/N

type

68. C₇⋊C₈

Q₃₂

semidirect

116. C₇

#12516

semidirect

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

subgroup	index
C₇×#2247	2
#11729066	2
#11729052	2
#11726820	2
C₁₆×C₇⋊C₈	2
#12516	7

subgroup

index

C₁₆×C₇⋊C₈

Subgroups — #13268509

Conjugacy classes of subgroups

Complements

Maximal subgroups