Group #13268538 — subgroups

Subgroups — #13268538

Order 1792.

262 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 103 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×#1713	C₂	1	1	yes	maximal, characteristic
2	#11729139	C₂	1	4	yes	maximal
3	#11725819	C₂	1	1	yes	maximal, characteristic
4	C₈×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₇×#500	—	2	1	no	—
6	C₇×#500	C₂²	1	2	yes	—
7	C₇×#494	—	2	1	no	—
8	#89471	—	2	1	no	—
9	#89454	—	2	1	no	—
10	#89454	C₂²	1	2	yes	—
11	C₇×C₈²	C₂²	1	1	yes	characteristic
12	#89285	—	2	8	no	—
13	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
14	#12523	—	7	1	no	maximal, Sylow
15	C₂×C₇×Q₁₆	—	4	2	no	—
16	C₇×C₄⋊Q₈	—	2	2	no	—
17	C₂×Dic₂₈	—	4	2	no	—
18	#6432	—	2	2	no	—
19	C₇×#107	—	2	4	no	—
20	C₇×#107	—	4	2	no	—
21	C₄×C₇×C₈	D₄	1	1	yes	characteristic
22	C₄×C₇×C₈	D₄	1	1	yes	characteristic
23	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
24	#6392	—	2	4	no	—
25	#6392	—	4	2	no	—
26	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
27	#2254	—	7	4	no	—
28	#1713	—	7	1	no	—
29	#1713	D₇	1	1	yes	characteristic
30	C₈×C₁₆	—	7	1	no	—
31	C₂×C₇×Q₈	—	4	2	no	—
32	C₂×Dic₁₄	—	4	2	no	—
33	C₇×Q₁₆	—	8	4	no	—
34	C₂×C₇×C₈	D₈	1	2	yes	—
35	C₂×C₇×C₈	D₈	1	2	yes	—
36	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
37	C₇×C₄⋊C₄	—	4	2	no	—
38	C₇×C₄⋊C₄	—	4	2	no	—
39	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₇	—	4	2	no	—
41	C₄⋊Dic₇	—	4	2	no	—
42	Dic₂₈	—	8	4	no	—
43	C₇⋊C₁₆	Q₁₆	1	8	yes	complemented
44	#500	—	2	1	no	—
45	#500	—	7	2	no	—
46	#500	—	14	1	no	—
47	#500	C₂×D₇	1	2	yes	—
48	#494	—	2	1	no	—
49	#494	—	14	1	no	—
50	#367	—	14	8	no	—
51	C₄×C₁₆	—	7	2	no	—
52	C₈²	C₂×D₇	1	1	yes	characteristic
53	C₇×Q₈	—	8	4	no	—
54	C₂×C₄×C₇	—	8	2	no	—
55	C₂×C₄×C₇	#127	1	1	yes	characteristic
56	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₇	—	8	2	no	—
59	Dic₁₄	—	8	4	no	—
60	C₇×C₈	SD₃₂	1	4	yes	—
61	C₇×C₈	SD₃₂	1	4	yes	—
62	C₇×C₈	C₂×Q₁₆	1	4	yes	—
63	C₂×Q₁₆	—	4	2	no	—
64	C₂×Q₁₆	—	28	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	14	2	no	—
67	C₂×C₁₆	—	7	4	no	—
68	#107	—	2	4	no	—
69	#107	—	4	2	no	—
70	#107	—	14	4	no	—
71	#107	—	28	2	no	—
72	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
73	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
74	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
75	C₂²×C₇	#493	1	1	yes	characteristic
76	C₄×C₇	—	16	2	no	—
77	C₄×C₇	#494	1	2	yes	—
78	C₄×C₇	C₂×SD₃₂	1	2	yes	—
79	C₄×C₇	C₂×SD₃₂	1	2	yes	—
80	Dic₇	—	16	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	28	2	no	—
83	Q₁₆	—	8	4	no	—
84	Q₁₆	—	56	4	no	—
85	C₂×C₈	#1141	1	2	yes	—
86	C₂×C₈	#1141	1	2	yes	—
87	C₂×C₈	D₄×D₇	1	2	yes	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	28	2	no	—
91	C₄⋊C₄	—	28	2	no	—
92	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
93	C₁₆	—	7	8	no	—
94	C₂×C₇	#1713	1	1	yes	characteristic
95	C₂×C₇	#2248	1	1	yes	characteristic
96	C₂×C₇	#2248	1	1	yes	characteristic
97	Q₈	—	8	4	no	—
98	Q₈	—	56	4	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	—	56	2	no	—
101	C₂×C₄	C₂×#1141	1	1	yes	characteristic
102	C₂×C₄	C₂×#1141	1	1	yes	characteristic
103	C₂×C₄	#6503	1	1	yes	characteristic
104	C₈	#6401	1	4	yes	—
105	C₈	#6402	1	4	yes	—
106	C₈	D₇×Q₁₆	1	4	yes	—
107	C₇	#12523	1	1	yes	characteristic, complemented, Sylow
108	C₂²	#89845	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	112	2	no	—
111	C₄	C₂×#6401	1	2	yes	—
112	C₄	C₂×#6402	1	2	yes	—
113	C₄	#89953	1	2	yes	—
114	C₂	#11729045	1	1	yes	characteristic
115	C₂	#11729052	1	1	yes	characteristic
116	C₂	#11729067	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1713

C₂

yes

maximal, characteristic

#11729139

C₂

yes

maximal

#11725819

C₂

yes

maximal, characteristic

C₈×C₇⋊C₁₆

C₂

yes

maximal, characteristic

C₇×#500

—

C₇×#500

C₂²

yes

—

—

—

—

C₂²

yes

—

C₇×C₈²

C₂²

yes

characteristic

#89285

—

C₄×C₇⋊C₁₆

C₂²

yes

—

#12523

—

maximal, Sylow

C₂×C₇×Q₁₆

—

C₇×C₄⋊Q₈

—

C₂×Dic₂₈

—

#6432

—

C₇×#107

—

C₇×#107

—

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

characteristic

#6392

—

#6392

—

C₂×C₇⋊C₁₆

D₄

yes

—

#2254

—

#1713

—

#1713

D₇

yes

characteristic

C₈×C₁₆

—

C₂×C₇×Q₈

—

C₂×Dic₁₄

—

C₇×Q₁₆

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₂₈

—

C₇⋊C₁₆

Q₁₆

yes

—

—

—

C₂×D₇

yes

—

#494

—

#494

—

#367

—

C₄×C₁₆

—

C₈²

C₂×D₇

yes

characteristic

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

characteristic

C₂×Dic₇

—

Dic₁₄

—

C₇×C₈

SD₃₂

yes

—

C₇×C₈

SD₃₂

yes

—

C₇×C₈

C₂×Q₁₆

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₇

yes

characteristic

C₄×C₈

C₂²⋊D₇

yes

characteristic

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₄×C₇

—

C₄×C₇

#494

yes

—

C₄×C₇

C₂×SD₃₂

yes

—

C₄×C₇

C₂×SD₃₂

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#1141

yes

—

C₂×C₈

#1141

yes

—

C₂×C₈

D₄×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₇

yes

characteristic

C₁₆

—

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×#1141

yes

characteristic

102

C₂×C₄

C₂×#1141

yes

characteristic

103

C₂×C₄

#6503

yes

characteristic

104

C₈

#6401

yes

—

105

C₈

#6402

yes

—

106

C₈

D₇×Q₁₆

yes

—

107

C₇

#12523

yes

characteristic, complemented, Sylow

108

C₂²

#89845

yes

characteristic

109

C₄

—

110

C₄

—

112

—

111

C₄

C₂×#6401

yes

—

112

C₄

C₂×#6402

yes

—

113

C₄

#89953

yes

—

114

C₂

#11729045

yes

characteristic

115

C₂

#11729052

yes

characteristic

116

C₂

#11729067

yes

characteristic

	subgroup	G/H
center	C₂²	#89845
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1713	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#1713	D₇
hypocenter	C₇	#12523
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#89845

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 ◁ C₈² ◁ #1713

N	H ≅ G/N	type
43. C₇⋊C₁₆	Q₁₆	semidirect
107. C₇	#12523	semidirect

H ≅ G/N

type

43. C₇⋊C₁₆

Q₁₆

semidirect

107. C₇

#12523

semidirect

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

subgroup	index
C₇×#1713	2
#11729139	2
#11725819	2
C₈×C₇⋊C₁₆	2
#12523	7

subgroup

index

C₈×C₇⋊C₁₆

Subgroups — #13268538

Conjugacy classes of subgroups

Complements

Maximal subgroups