Group #13268536 — subgroups

Subgroups — #13268536

Order 1792.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#11729138	C₂	1	2	yes	maximal
3	#11729115	C₂	1	2	yes	maximal
4	#11726800	C₂	1	1	yes	maximal, characteristic
5	C₁₆×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₇×#500	C₂²	1	2	yes	—
7	#89865	—	2	2	no	—
8	#89853	—	2	2	no	—
9	#89454	C₂²	1	2	yes	—
10	C₇×#367	—	2	2	no	—
11	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	#89300	—	2	2	no	—
13	#89284	—	2	4	no	—
14	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
16	#12522	—	7	1	no	maximal, Sylow
17	C₇×C₄⋊Q₈	—	2	2	no	—
18	#6432	—	2	2	no	—
19	C₂×C₇×C₁₆	D₄	1	2	yes	—
20	C₇×#107	—	2	2	no	—
21	C₇×#107	—	2	2	no	—
22	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
23	#6409	—	4	4	no	—
24	#6392	—	2	2	no	—
25	#6392	—	2	2	no	—
26	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
27	#6384	—	4	4	no	—
28	C₄×C₇⋊C₈	D₄	1	2	yes	—
29	#2254	—	7	1	no	—
30	#2254	D₇	1	1	yes	characteristic
31	#2252	—	7	2	no	—
32	#1710	—	7	2	no	—
33	C₈×C₁₆	—	7	1	no	—
34	C₂×C₇×Q₈	—	4	2	no	—
35	C₂×Dic₁₄	—	4	2	no	—
36	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
38	C₇×C₄⋊C₄	—	4	2	no	—
39	C₇×C₄⋊C₄	—	4	2	no	—
40	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₇	—	4	2	no	—
42	C₄⋊Dic₇	—	4	2	no	—
43	C₂×C₇⋊C₈	D₈	1	4	yes	—
44	C₇×C₁₆	Q₁₆	1	4	yes	—
45	C₇⋊C₁₆	Q₁₆	1	4	yes	—
46	#500	—	7	2	no	—
47	#500	C₂×D₇	1	2	yes	—
48	#487	—	14	2	no	—
49	#487	—	14	2	no	—
50	#367	—	2	2	no	—
51	#367	—	14	2	no	—
52	#366	—	14	4	no	—
53	C₄×C₁₆	—	7	1	no	—
54	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
55	C₈²	—	7	1	no	—
56	C₇×Q₈	—	8	4	no	—
57	C₂×C₄×C₇	—	8	2	no	—
58	C₂×C₄×C₇	#127	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
60	C₂×Dic₇	—	8	2	no	—
61	Dic₁₄	—	8	4	no	—
62	C₇×C₈	C₂×Q₁₆	1	2	yes	—
63	C₇×C₈	C₂×Q₁₆	1	2	yes	—
64	C₇⋊C₈	—	2	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	14	2	no	—
67	C₂×C₁₆	—	7	2	no	—
68	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
69	#107	—	2	2	no	—
70	#107	—	2	2	no	—
71	#107	—	14	2	no	—
72	#107	—	14	2	no	—
73	#103	—	28	4	no	—
74	#103	—	28	4	no	—
75	C₄×C₈	—	7	2	no	—
76	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
77	C₂²×C₇	#493	1	1	yes	characteristic
78	C₄×C₇	—	16	2	no	—
79	C₄×C₇	#494	1	1	yes	characteristic
80	C₄×C₇	#494	1	1	yes	characteristic
81	C₄×C₇	#508	1	4	yes	—
82	Dic₇	—	16	2	no	—
83	C₂×Q₈	—	4	2	no	—
84	C₂×Q₈	—	28	2	no	—
85	C₂×C₈	—	7	4	no	—
86	C₂×C₈	D₄×D₇	1	1	yes	characteristic
87	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
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	D₄×D₇	1	1	yes	characteristic
93	C₁₆	—	7	4	no	—
94	C₁₆	#1144	1	4	yes	—
95	C₂×C₇	#1713	1	1	yes	characteristic
96	C₂×C₇	#2249	1	2	yes	—
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	D₇×D₈	1	2	yes	—
102	C₂×C₄	#6503	1	1	yes	characteristic
103	C₈	—	14	4	no	—
104	C₈	D₇×Q₁₆	1	2	yes	—
105	C₈	C₂×#1144	1	2	yes	—
106	C₇	#12522	1	1	yes	characteristic, complemented, Sylow
107	C₂²	#89923	1	1	yes	characteristic
108	C₄	—	16	2	no	—
109	C₄	—	112	2	no	—
110	C₄	#89688	1	4	yes	—
111	C₄	#89864	1	1	yes	characteristic
112	C₄	#89953	1	1	yes	characteristic
113	C₂	#11729116	1	1	yes	characteristic
114	C₂	#11729234	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2254

C₂

yes

maximal, characteristic

#11729138

C₂

yes

maximal

#11729115

C₂

yes

maximal

#11726800

C₂

yes

maximal, characteristic

C₁₆×C₇⋊C₈

C₂

yes

maximal, characteristic

C₇×#500

C₂²

yes

—

#89865

—

#89853

—

#89454

C₂²

yes

—

C₇×#367

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

#89300

—

#89284

—

C₄×C₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₇⋊C₈

C₂²

yes

characteristic

#12522

—

maximal, Sylow

C₇×C₄⋊Q₈

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₇×#107

—

C₇×#107

—

C₄×C₇×C₈

C₂³

yes

—

—

—

C₂×C₇⋊C₁₆

D₄

yes

—

#6384

—

C₄×C₇⋊C₈

D₄

yes

—

#2254

—

#2254

D₇

yes

characteristic

#2252

—

#1710

—

C₈×C₁₆

—

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

—

C₇×C₁₆

Q₁₆

yes

—

C₇⋊C₁₆

Q₁₆

yes

—

#500

—

#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

—

C₇×C₈

C₂×Q₁₆

yes

—

C₇×C₈

C₂×Q₁₆

yes

—

C₇⋊C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

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

—

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

C₁₆

—

C₁₆

#1144

yes

—

C₂×C₇

#1713

yes

characteristic

C₂×C₇

#2249

yes

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

D₇×D₈

yes

—

102

C₂×C₄

#6503

yes

characteristic

103

C₈

—

104

C₈

D₇×Q₁₆

yes

—

105

C₈

C₂×#1144

yes

—

106

C₇

#12522

yes

characteristic, complemented, Sylow

107

C₂²

#89923

yes

characteristic

108

C₄

—

109

C₄

—

112

—

110

C₄

#89688

yes

—

111

C₄

#89864

yes

characteristic

112

C₄

#89953

yes

characteristic

113

C₂

#11729116

yes

characteristic

114

C₂

#11729234

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	#12522
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
106. C₇	#12522	semidirect

H ≅ G/N

type

106. C₇

#12522

semidirect

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

subgroup	index
C₇×#2254	2
#11729138	2
#11729115	2
#11726800	2
C₁₆×C₇⋊C₈	2
#12522	7

subgroup

index

C₁₆×C₇⋊C₈

Subgroups — #13268536

Conjugacy classes of subgroups

Complements

Maximal subgroups