Group #11729496 — subgroups

Subgroups — #11729496

Order 896.

228 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₇×#125	C₂	1	1	yes	maximal, characteristic
2	#89696	C₂	1	1	yes	maximal, characteristic
3	C₄×#1136	C₂	1	1	yes	maximal, characteristic
4	#89325	C₂	1	4	yes	maximal
5	C₇×#125	—	2	1	no	—
6	C₇×#125	—	2	1	no	—
7	C₇×#125	—	2	2	no	—
8	C₂×C₇×C₄⋊C₄	C₄	1	4	yes	—
9	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×#1136	C₂²	1	2	yes	—
12	C₂×#1136	C₂²	1	2	yes	—
13	C₄⋊(C₇⋊C₈)	—	2	2	no	—
14	#6377	—	2	1	no	—
15	C₄×C₇⋊C₈	—	2	1	no	—
16	#1998	—	7	1	no	maximal, Sylow
17	C₂²×C₄×C₇	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₇×C₄⋊C₄	—	2	4	no	—
21	C₇×C₄⋊C₄	—	2	4	no	—
22	C₇×C₄⋊C₄	—	2	2	no	—
23	C₇×C₄⋊C₄	—	2	2	no	—
24	C₄²×C₇	D₄	1	1	yes	characteristic
25	C₄²×C₇	D₄	1	1	yes	characteristic
26	C₄²×C₇	D₄	1	2	yes	—
27	#1136	—	2	2	no	—
28	#1136	—	2	4	no	—
29	#1136	Q₈	1	4	yes	—
30	C₂×C₇⋊C₈	—	2	2	no	—
31	C₂×C₇⋊C₈	—	2	2	no	—
32	C₂×#125	D₇	1	1	yes	characteristic
33	#423	—	7	1	no	—
34	C₄×M₁₆	—	7	1	no	—
35	#343	—	7	4	no	—
36	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇	—	2	4	no	—
38	C₂×C₄×C₇	—	4	2	no	—
39	C₂×C₄×C₇	—	4	4	no	—
40	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₇	#41	1	2	yes	—
46	C₇⋊C₈	—	2	4	no	—
47	C₇⋊C₈	—	4	2	no	—
48	C₂×M₁₆	—	7	2	no	—
49	C₂×M₁₆	—	7	2	no	—
50	#125	—	2	1	no	—
51	#125	—	2	1	no	—
52	#125	—	2	2	no	—
53	C₂×C₄⋊C₄	Dic₇	1	4	yes	—
54	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
55	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
56	C₄⋊C₈	—	14	2	no	—
57	#97	—	14	1	no	—
58	C₄×C₈	—	14	1	no	—
59	C₂²×C₇	—	2	2	no	—
60	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₇	C₄×Q₈	1	2	yes	—
62	C₄×C₇	—	2	2	no	—
63	C₄×C₇	—	2	2	no	—
64	C₄×C₇	—	4	4	no	—
65	C₄×C₇	C₂²⋊Q₈	1	2	yes	—
66	C₄×C₇	#124	1	1	yes	characteristic
67	C₄×C₇	C₄⋊Q₈	1	1	yes	characteristic
68	C₂²×C₄	C₂×Dic₇	1	4	yes	—
69	C₂²×C₄	C₂×Dic₇	1	2	yes	—
70	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
71	M₁₆	—	7	4	no	—
72	M₁₆	—	14	2	no	—
73	M₁₆	—	14	4	no	—
74	C₂×C₈	—	14	2	no	—
75	C₂×C₈	—	14	2	no	—
76	C₄⋊C₄	—	2	4	no	—
77	C₄⋊C₄	—	2	4	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	2	2	no	—
80	C₄²	C₂²⋊D₇	1	1	yes	characteristic
81	C₄²	C₂²⋊D₇	1	1	yes	characteristic
82	C₄²	C₂²⋊D₇	1	2	yes	—
83	C₂×C₇	—	2	2	no	—
84	C₂×C₇	#391	1	1	yes	characteristic
85	C₂×C₇	#413	1	2	yes	—
86	C₂³	C₂²×Dic₇	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	4	no	—
90	C₂×C₄	C₂²⋊Dic₇	1	2	yes	—
91	C₂×C₄	C₂²⋊Dic₇	1	2	yes	—
92	C₂×C₄	Q₈×D₇	1	2	yes	—
93	C₂×C₄	#1161	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
96	C₈	—	14	4	no	—
97	C₈	—	28	2	no	—
98	C₇	#1998	1	1	yes	characteristic, complemented, Sylow
99	C₂²	—	2	2	no	—
100	C₂²	Q₈×Dic₇	1	2	yes	—
101	C₂²	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#6507	1	1	yes	characteristic
106	C₄	#6509	1	2	yes	—
107	C₄	#6510	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#90002	1	1	yes	characteristic
110	C₂	#90009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×#125

C₂

yes

maximal, characteristic

#89696

C₂

yes

maximal, characteristic

C₄×#1136

C₂

yes

maximal, characteristic

#89325

C₂

yes

—

—

—

C₂×C₇×C₄⋊C₄

C₄

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×#1136

C₂²

yes

—

C₂×#1136

C₂²

yes

—

C₄⋊(C₇⋊C₈)

—

#6377

—

C₄×C₇⋊C₈

—

#1998

—

maximal, Sylow

C₂²×C₄×C₇

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂×C₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

D₄

yes

characteristic

C₄²×C₇

D₄

yes

characteristic

C₄²×C₇

D₄

yes

—

#1136

—

#1136

—

#1136

Q₈

yes

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#125

D₇

yes

characteristic

#423

—

C₄×M₁₆

—

#343

—

C₂³×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

#41

yes

—

C₇⋊C₈

—

C₇⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₇

yes

—

C₂×C₄⋊C₄

C₂×D₇

yes

—

C₂×C₄²

C₂×D₇

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂²×C₇

—

C₂²×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇

C₄×Q₈

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²⋊Q₈

yes

—

C₄×C₇

#124

yes

characteristic

C₄×C₇

C₄⋊Q₈

yes

characteristic

C₂²×C₄

C₂×Dic₇

yes

—

C₂²×C₄

C₂×Dic₇

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₇

yes

characteristic

C₄²

C₂²⋊D₇

yes

characteristic

C₄²

C₂²⋊D₇

yes

—

C₂×C₇

—

C₂×C₇

#391

yes

characteristic

C₂×C₇

#413

yes

—

C₂³

C₂²×Dic₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₇

yes

—

C₂×C₄

C₂²⋊Dic₇

yes

—

C₂×C₄

Q₈×D₇

yes

—

C₂×C₄

#1161

yes

—

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

C₈

—

C₈

—

C₇

#1998

yes

characteristic, complemented, Sylow

C₂²

—

100

C₂²

Q₈×Dic₇

yes

—

101

C₂²

C₂×C₂²⋊Dic₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#6507

yes

characteristic

106

C₄

#6509

yes

—

107

C₄

#6510

yes

characteristic

108

C₂

—

109

C₂

#90002

yes

characteristic

110

C₂

#90009

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇
commutator, prime radical	C₂³×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×#125	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×#125	D₇
hypocenter	C₇	#1998
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇

commutator, prime radical

C₂³×C₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₂×C₇×#125

C₂

socle

C₂²×C₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₇

C₂³

D₇

C₇

G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
98. C₇	#1998	semidirect

H ≅ G/N

type

98. C₇

#1998

semidirect

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

subgroup	index
C₂×C₇×#125	2
#89696	2
C₄×#1136	2
#89325	2
#1998	7

subgroup

index

C₂×C₇×#125

Subgroups — #11729496

Conjugacy classes of subgroups

Complements

Maximal subgroups