Group #13268472 — subgroups

Subgroups — #13268472

Order 1792.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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×#2254	C₂	1	1	yes	maximal, characteristic
2	#11729136	C₂	1	2	yes	maximal
3	#11729114	C₂	1	2	yes	maximal
4	#11726820	C₂	1	1	yes	maximal, characteristic
5	#11725304	C₂	1	1	yes	maximal, characteristic
6	C₇×#500	C₂²	1	2	yes	—
7	#89853	—	2	2	no	—
8	#89851	—	2	2	no	—
9	C₂×Dic₅₆	—	2	2	no	—
10	#89471	C₂²	1	2	yes	—
11	C₇×#367	—	2	2	no	—
12	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	#89287	—	2	4	no	—
14	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	#12507	—	7	1	no	maximal, Sylow
17	C₇×C₄⋊Q₈	—	2	2	no	—
18	C₂×Dic₂₈	—	2	2	no	—
19	C₂×Dic₂₈	—	2	2	no	—
20	#6432	—	2	2	no	—
21	C₂×C₇×C₁₆	D₄	1	2	yes	—
22	C₇×#107	—	2	2	no	—
23	C₇×#107	—	2	2	no	—
24	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
25	#6409	—	4	4	no	—
26	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
27	#6382	—	4	4	no	—
28	C₄×C₇⋊C₈	D₄	1	2	yes	—
29	Dic₅₆	—	4	4	no	—
30	#2254	D₇	1	1	yes	characteristic
31	#2247	—	7	1	no	—
32	#2241	—	7	2	no	—
33	#1707	—	7	2	no	—
34	#1311	—	7	1	no	—
35	C₂×C₇×Q₈	—	4	2	no	—
36	C₂×Dic₁₄	—	4	2	no	—
37	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
39	C₇×C₄⋊C₄	—	4	2	no	—
40	C₇×C₄⋊C₄	—	4	2	no	—
41	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₇	—	4	2	no	—
43	C₄⋊Dic₇	—	4	2	no	—
44	C₂×C₇⋊C₈	D₈	1	4	yes	—
45	Dic₂₈	—	4	4	no	—
46	Dic₂₈	—	4	4	no	—
47	C₇×C₁₆	SD₁₆	1	4	yes	—
48	C₇⋊C₁₆	—	2	2	no	—
49	C₂×Q₃₂	—	14	2	no	—
50	#500	C₂×D₇	1	2	yes	—
51	#498	—	14	2	no	—
52	#494	—	7	2	no	—
53	#487	—	14	2	no	—
54	#367	—	2	2	no	—
55	#358	—	14	4	no	—
56	C₄×C₁₆	—	7	1	no	—
57	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
58	C₈²	—	7	1	no	—
59	C₇×Q₈	—	8	4	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×SD₁₆	1	2	yes	—
67	C₇×C₈	#135	1	2	yes	—
68	C₇⋊C₈	—	2	4	no	—
69	C₂×Q₁₆	—	14	2	no	—
70	C₂×Q₁₆	—	14	2	no	—
71	C₄⋊Q₈	—	2	2	no	—
72	C₄⋊Q₈	—	14	2	no	—
73	Q₃₂	—	28	4	no	—
74	C₂×C₁₆	—	7	2	no	—
75	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#106	—	28	4	no	—
79	#103	—	28	4	no	—
80	C₄×C₈	—	7	2	no	—
81	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
82	C₂²×C₇	#493	1	1	yes	characteristic
83	C₄×C₇	—	16	2	no	—
84	C₄×C₇	#492	1	1	yes	characteristic
85	C₄×C₇	#495	1	1	yes	characteristic
86	C₄×C₇	#510	1	4	yes	—
87	Dic₇	—	16	2	no	—
88	C₂×Q₈	—	4	2	no	—
89	C₂×Q₈	—	28	2	no	—
90	Q₁₆	—	28	4	no	—
91	Q₁₆	—	28	4	no	—
92	C₂×C₈	—	7	4	no	—
93	C₂×C₈	D₄×D₇	1	1	yes	characteristic
94	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
95	C₄⋊C₄	—	4	2	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	28	2	no	—
98	C₄⋊C₄	—	28	2	no	—
99	C₄²	D₄×D₇	1	1	yes	characteristic
100	C₁₆	—	14	2	no	—
101	C₁₆	#1143	1	4	yes	—
102	C₂×C₇	#1706	1	1	yes	characteristic
103	C₂×C₇	#2251	1	2	yes	—
104	Q₈	—	8	4	no	—
105	Q₈	—	56	2	no	—
106	Q₈	—	56	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	56	2	no	—
109	C₂×C₄	D₇×D₈	1	2	yes	—
110	C₂×C₄	#6503	1	1	yes	characteristic
111	C₈	—	14	4	no	—
112	C₈	#6482	1	2	yes	—
113	C₈	C₂×#1143	1	2	yes	—
114	C₇	#12507	1	1	yes	characteristic, complemented, Sylow
115	C₂²	#89923	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	112	2	no	—
118	C₄	#89687	1	4	yes	—
119	C₄	#89857	1	1	yes	characteristic
120	C₄	#89963	1	1	yes	characteristic
121	C₂	#11729113	1	1	yes	characteristic
122	C₂	#11729236	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2254

C₂

yes

maximal, characteristic

#11729136

C₂

yes

maximal

#11729114

C₂

yes

maximal

#11726820

C₂

yes

maximal, characteristic

#11725304

C₂

yes

maximal, characteristic

C₇×#500

C₂²

yes

—

#89853

—

#89851

—

C₂×Dic₅₆

—

#89471

C₂²

yes

—

C₇×#367

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

#89287

—

C₄×C₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₇⋊C₈

C₂²

yes

characteristic

#12507

—

maximal, Sylow

C₇×C₄⋊Q₈

—

C₂×Dic₂₈

—

C₂×Dic₂₈

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₇×#107

—

C₇×#107

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6409

—

C₂×C₇⋊C₁₆

D₄

yes

—

#6382

—

C₄×C₇⋊C₈

D₄

yes

—

Dic₅₆

—

#2254

D₇

yes

—

—

—

—

C₂×C₇×Q₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₂×C₇⋊C₈

D₈

yes

—

Dic₂₈

—

Dic₂₈

—

C₇×C₁₆

SD₁₆

yes

—

C₇⋊C₁₆

—

C₂×Q₃₂

—

#500

C₂×D₇

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇

yes

characteristic

C₈²

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

Dic₁₄

—

Dic₁₄

—

C₇×C₈

C₂×SD₁₆

yes

—

C₇×C₈

#135

yes

—

C₇⋊C₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

Dic₇

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

D₄×D₇

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₄×D₇

yes

characteristic

100

C₁₆

—

101

C₁₆

#1143

yes

—

102

C₂×C₇

#1706

yes

characteristic

103

C₂×C₇

#2251

yes

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

D₇×D₈

yes

—

110

C₂×C₄

#6503

yes

characteristic

111

C₈

—

112

C₈

#6482

yes

—

113

C₈

C₂×#1143

yes

—

114

C₇

#12507

yes

characteristic, complemented, Sylow

115

C₂²

#89923

yes

characteristic

116

C₄

—

117

C₄

—

112

—

118

C₄

#89687

yes

—

119

C₄

#89857

yes

characteristic

120

C₄

#89963

yes

characteristic

121

C₂

#11729113

yes

characteristic

122

C₂

#11729236

yes

—

	subgroup	G/H
center	C₂²	#89923
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#2254	C₂
socle, prime radical	C₂²×C₇	#493
hypercenter	#2254	D₇
hypocenter	C₇	#12507
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

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 ◁ #2254

N	H ≅ G/N	type
114. C₇	#12507	semidirect

H ≅ G/N

type

114. C₇

#12507

semidirect

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

subgroup	index
C₇×#2254	2
#11729136	2
#11729114	2
#11726820	2
#11725304	2
#12507	7

subgroup

index

Subgroups — #13268472

Conjugacy classes of subgroups

Complements

Maximal subgroups