Group #11727581 — subgroups

Subgroups — #11727581

Order 896.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#6433	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#97	C₂	1	1	yes	maximal, characteristic
3	#89707	C₂	1	1	yes	maximal, characteristic
4	#89346	C₂	1	4	yes	maximal
5	C₂²×C₇×C₈	C₂²	1	2	yes	—
6	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₇	—	2	2	no	—
8	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
10	C₂×Dic₇⋊C₄	—	2	2	no	—
11	#6433	—	2	2	no	—
12	#6433	C₄	1	4	yes	complemented
13	C₇×#97	—	2	2	no	—
14	#1992	—	7	1	no	maximal, Sylow
15	C₂²×C₄×C₇	D₄	1	2	yes	—
16	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₇	—	2	2	no	—
18	C₂²×Dic₇	—	2	2	no	—
19	C₂×C₇×C₈	—	2	2	no	—
20	C₂×C₇×C₈	—	2	2	no	—
21	C₂×C₇×C₈	Q₈	1	4	yes	—
22	C₄²×C₇	C₂×C₄	1	2	yes	—
23	C₄²×C₇	D₄	1	2	yes	—
24	C₄⋊Dic₇	—	2	2	no	—
25	C₄⋊Dic₇	—	2	4	no	—
26	C₄⋊Dic₇	—	4	4	no	—
27	C₄⋊Dic₇	C₂×C₄	1	4	yes	—
28	Dic₇⋊C₄	—	2	8	no	—
29	C₂×#125	—	7	1	no	—
30	C₂×#97	D₇	1	1	yes	characteristic
31	#384	—	7	1	no	—
32	#340	—	7	4	no	—
33	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇	—	2	2	no	—
35	C₂×C₄×C₇	—	2	2	no	—
36	C₂×C₄×C₇	C₂²⋊C₄	1	4	yes	—
37	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₇	#41	1	2	yes	—
41	C₂×Dic₇	—	2	4	no	—
42	C₂×Dic₇	—	4	2	no	—
43	C₂×Dic₇	—	4	2	no	—
44	C₂×Dic₇	—	4	2	no	—
45	C₂×Dic₇	—	4	4	no	—
46	C₇×C₈	—	2	4	no	—
47	C₂²×C₈	C₂×D₇	1	2	yes	—
48	#125	—	7	4	no	—
49	#125	—	14	2	no	—
50	C₂×C₄⋊C₄	—	7	2	no	—
51	C₂×C₄⋊C₄	—	7	2	no	—
52	C₂×C₄⋊C₄	—	14	2	no	—
53	C₂×C₄⋊C₄	—	14	2	no	—
54	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
55	#97	—	2	2	no	—
56	C₂²×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₇	#124	1	1	yes	characteristic
58	C₂²×C₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂²×C₇	AGL(1,ℤ/8)	1	2	yes	—
60	C₂²×C₇	#137	1	2	yes	—
61	C₄×C₇	—	2	4	no	—
62	C₄×C₇	C₄×Q₈	1	2	yes	—
63	C₄×C₇	C₂²⋊Q₈	1	2	yes	—
64	Dic₇	—	4	4	no	—
65	Dic₇	—	8	2	no	—
66	C₂²×C₄	—	14	2	no	—
67	C₂²×C₄	—	14	2	no	—
68	C₂²×C₄	D₂₈	1	2	yes	—
69	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	Dic₁₄	1	4	yes	—
73	C₄⋊C₄	—	7	4	no	—
74	C₄⋊C₄	—	14	8	no	—
75	C₄⋊C₄	—	14	2	no	—
76	C₄⋊C₄	—	14	4	no	—
77	C₄⋊C₄	—	28	4	no	—
78	C₄²	C₄×D₇	1	2	yes	—
79	C₄²	C₂²⋊D₇	1	2	yes	—
80	C₂×C₇	#391	1	1	yes	characteristic
81	C₂×C₇	#417	1	2	yes	—
82	C₂×C₇	#490	1	1	yes	characteristic
83	C₂×C₇	#491	1	1	yes	characteristic
84	C₂×C₇	#501	1	2	yes	—
85	C₂³	C₂×D₂₈	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	14	4	no	—
89	C₂×C₄	—	28	2	no	—
90	C₂×C₄	—	28	2	no	—
91	C₂×C₄	—	28	2	no	—
92	C₂×C₄	—	28	4	no	—
93	C₂×C₄	(C₂×D₇)⋊C₄	1	4	yes	—
94	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
95	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
96	C₂×C₄	#1157	1	2	yes	—
97	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
98	C₈	—	2	4	no	—
99	C₇	#1992	1	1	yes	characteristic, complemented, Sylow
100	C₂²	#6432	1	1	yes	characteristic
101	C₂²	#6438	1	1	yes	characteristic
102	C₂²	#6471	1	2	yes	—
103	C₂²	#6472	1	2	yes	—
104	C₂²	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
105	C₄	—	2	4	no	—
106	C₄	—	28	4	no	—
107	C₄	—	56	2	no	—
108	C₄	C₄×Dic₁₄	1	2	yes	—
109	C₄	#6487	1	2	yes	—
110	C₂	#89474	1	2	yes	—
111	C₂	#89475	1	1	yes	characteristic
112	C₂	#89485	1	1	yes	characteristic
113	C₂	#89701	1	1	yes	characteristic
114	C₂	#89905	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#6433

C₂

yes

maximal, characteristic

C₂×C₇×#97

C₂

yes

maximal, characteristic

#89707

C₂

yes

maximal, characteristic

#89346

C₂

yes

maximal

C₂²×C₇×C₈

C₂²

yes

—

C₂×C₄²×C₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×C₄⋊Dic₇

C₂²

yes

—

C₂×Dic₇⋊C₄

—

#6433

—

#6433

C₄

yes

complemented

C₇×#97

—

#1992

—

maximal, Sylow

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂²×Dic₇

—

C₂²×Dic₇

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

Q₈

yes

—

C₄²×C₇

C₂×C₄

yes

—

C₄²×C₇

D₄

yes

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

—

C₄⋊Dic₇

C₂×C₄

yes

—

Dic₇⋊C₄

—

C₂×#125

—

C₂×#97

D₇

yes

characteristic

#384

—

#340

—

C₂³×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊C₄

yes

—

C₂×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₇

C₂×D₄

yes

characteristic

C₂×C₄×C₇

C₂×Q₈

yes

—

C₂×C₄×C₇

#41

yes

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₂×Dic₇

—

C₇×C₈

—

C₂²×C₈

C₂×D₇

yes

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇

yes

characteristic

#97

—

C₂²×C₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇

#124

yes

characteristic

C₂²×C₇

C₄⋊Q₈

yes

characteristic

C₂²×C₇

AGL(1,ℤ/8)

yes

—

C₂²×C₇

#137

yes

—

C₄×C₇

—

C₄×C₇

C₄×Q₈

yes

—

C₄×C₇

C₂²⋊Q₈

yes

—

Dic₇

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₂₈

yes

—

C₂²×C₄

C₂²×D₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₇

yes

—

C₄²

C₂²⋊D₇

yes

—

C₂×C₇

#391

yes

characteristic

C₂×C₇

#417

yes

—

C₂×C₇

yes

C₂×C₇

yes

C₂×C₇

yes

—

C₂³

C₂×D₂₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₇)⋊C₄

yes

—

C₂×C₄

C₂×Dic₁₄

yes

—

C₂×C₄

C₂×C₄×D₇

yes

characteristic

C₂×C₄

#1157

yes

—

C₂×C₄

C₂×C₂²⋊D₇

yes

characteristic

C₈

—

C₇

#1992

yes

characteristic, complemented, Sylow

100

C₂²

#6432

yes

characteristic

101

C₂²

#6438

yes

characteristic

102

C₂²

#6471

yes

—

103

C₂²

#6472

yes

—

104

C₂²

C₂×(C₂×D₇)⋊C₄

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

C₄×Dic₁₄

yes

—

109

C₄

#6487

yes

—

110

C₂

#89474

yes

—

111

C₂

#89475

yes

characteristic

112

C₂

#89485

yes

characteristic

113

C₂

#89701

yes

characteristic

114

C₂

#89905

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂×C₇×#97	C₂
socle, prime radical	C₂³×C₇	C₂×D₄
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂×#97	D₇
hypocenter	C₇	#1992
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₈

commutator

C₂×C₄×C₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇

Fitting

C₂×C₇×#97

C₂

socle, prime radical

C₂³×C₇

C₂×D₄

cosocle

C₂²×C₄×C₇

C₂³

hypercenter

C₂×#97

D₇

hypocenter

C₇

#1992

LCS

G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. #6433	C₄	semidirect
99. C₇	#1992	semidirect

H ≅ G/N

type

12. #6433

C₄

semidirect

99. C₇

#1992

semidirect

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

subgroup	index
C₂×#6433	2
C₂×C₇×#97	2
#89707	2
#89346	2
#1992	7

subgroup

index

C₂×C₇×#97

Subgroups — #11727581

Conjugacy classes of subgroups

Complements

Maximal subgroups