Group #22586285 — subgroups

Subgroups — #22586285

Order 9088.

134 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 37 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	#17985868	C₂	1	1	yes	maximal, characteristic
2	#17985725	C₂	1	1	yes	maximal, characteristic
3	#324×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₇₁	C₄	1	1	yes	characteristic
5	C₂×Dic₇₁⋊C₄	—	2	1	no	—
6	C₂×Dic₇₁⋊C₄	—	2	1	no	—
7	C₂×C₄×Dic₇₁	C₄	1	1	yes	characteristic
8	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²⋊C₈×C₇₁	—	2	1	no	—
11	C₂²×Dic₇₁	—	2	1	no	—
12	C₂²×Dic₇₁	—	2	1	no	—
13	C₂×C₇₁⋊C₈	—	4	1	no	—
14	C₄⋊Dic₇₁	—	4	1	no	—
15	Dic₇₁⋊C₄	—	4	2	no	—
16	Dic₇₁⋊C₄	—	4	2	no	—
17	C₄×Dic₇₁	—	2	1	no	—
18	C₄×Dic₇₁	C₈	1	2	yes	complemented
19	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
22	C₂×C₈×C₇₁	—	4	1	no	—
23	C₄⋊C₄×C₇₁	—	4	1	no	—
24	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₇₁	—	2	1	no	—
26	C₂×C₄×C₇₁	—	2	1	no	—
27	C₂×C₄×C₇₁	—	2	1	no	—
28	C₂×C₄×C₇₁	—	2	1	no	—
29	C₂×C₄×C₇₁	—	4	1	no	—
30	C₂×C₄×C₇₁	—	4	1	no	—
31	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
33	C₂×Dic₇₁	—	2	2	no	—
34	C₂×Dic₇₁	—	2	2	no	—
35	C₂×Dic₇₁	—	2	2	no	—
36	C₂×Dic₇₁	—	4	1	no	—
37	C₂×Dic₇₁	—	4	1	no	—
38	C₂×Dic₇₁	—	4	1	no	—
39	C₈×C₇₁	—	8	1	no	—
40	C₇₁⋊C₈	—	8	1	no	—
41	C₂²×C₇₁	—	2	1	no	—
42	C₂²×C₇₁	—	2	1	no	—
43	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
44	C₂²×C₇₁	#99	1	1	yes	characteristic
45	C₂²×C₇₁	#101	1	1	yes	characteristic
46	C₄×C₇₁	—	2	1	no	—
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	—	4	1	no	—
49	C₄×C₇₁	—	4	1	no	—
50	Dic₇₁	—	2	4	no	—
51	Dic₇₁	—	8	1	no	—
52	C₂×C₇₁	—	2	1	no	—
53	C₂×C₇₁	—	2	1	no	—
54	C₂×C₇₁	#324	1	1	yes	characteristic
55	C₂×C₇₁	#353	1	1	yes	characteristic
56	C₂×C₇₁	#356	1	1	yes	characteristic
57	#1325	—	71	1	no	maximal, Sylow
58	C₇₁	#1325	1	1	yes	characteristic, complemented, Sylow
59	#384	—	71	1	no	—
60	#324	—	71	1	no	—
61	#324	D₇₁	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	71	1	no	—
63	C₂×C₄⋊C₄	—	142	1	no	—
64	C₂×C₄⋊C₄	—	142	1	no	—
65	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
66	C₂×C₄²	—	71	1	no	—
67	C₂²⋊C₈	—	2	1	no	—
68	C₂²⋊C₈	—	142	1	no	—
69	C₂²×C₄	—	142	1	no	—
70	C₂²×C₄	—	142	1	no	—
71	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
72	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
74	C₂×C₈	—	4	1	no	—
75	C₂×C₈	—	284	1	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	284	1	no	—
78	C₄⋊C₄	—	284	2	no	—
79	C₄⋊C₄	—	284	2	no	—
80	C₄²	—	71	2	no	—
81	C₄²	—	142	1	no	—
82	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	1	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	—	284	1	no	—
90	C₂×C₄	—	284	1	no	—
91	C₂×C₄	—	284	1	no	—
92	C₂×C₄	—	142	2	no	—
93	C₂×C₄	—	142	2	no	—
94	C₂×C₄	—	142	2	no	—
95	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
96	C₂×C₄	#11765848	1	1	yes	characteristic
97	C₈	—	8	1	no	—
98	C₈	—	568	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
102	C₂²	#14556042	1	1	yes	characteristic
103	C₂²	#14556046	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	4	1	no	—
108	C₄	—	568	1	no	—
109	C₄	—	142	4	no	—
110	C₂	—	2	1	no	—
111	C₂	—	2	1	no	—
112	C₂	#17985724	1	1	yes	characteristic
113	C₂	#17985791	1	1	yes	characteristic
114	C₂	#17985797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985868

C₂

yes

maximal, characteristic

#17985725

C₂

yes

maximal, characteristic

#324×C₇₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₁

C₄

yes

characteristic

C₂×Dic₇₁⋊C₄

—

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₄

yes

characteristic

C₂²⋊(C₇₁⋊C₈)

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂²⋊C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₈

yes

complemented

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊C₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#384

—

#324

—

#324

D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

284

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄²

—

C₄²

—

142

—

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

C₈×D₇₁

yes

characteristic

C₂×C₄

#11765848

yes

characteristic

C₈

—

C₈

—

568

—

C₂²

—

100

C₂²

—

101

C₂²

(C₂×D₇₁)⋊C₈

yes

characteristic

102

C₂²

#14556042

yes

characteristic

103

C₂²

#14556046

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

568

—

109

C₄

—

142

—

110

C₂

—

111

C₂

—

112

C₂

#17985724

yes

characteristic

113

C₂

#17985791

yes

characteristic

114

C₂

#17985797

yes

characteristic

	subgroup	G/H
center	C₂²	#14556042
commutator	C₂×C₄×C₇₁	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₇₁
Fitting	#324×C₇₁	C₂
socle	C₂²×C₇₁	#99
hypercenter	#324	D₇₁
hypocenter	C₇₁	#1325
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #324

subgroup

G/H

center

C₂²

#14556042

commutator

C₂×C₄×C₇₁

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₇₁

Fitting

#324×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #324

N	H ≅ G/N	type
18. C₄×Dic₇₁	C₈	semidirect
58. C₇₁	#1325	semidirect

H ≅ G/N

type

18. C₄×Dic₇₁

C₈

semidirect

58. C₇₁

#1325

semidirect

Subgroups — #22586285

Conjugacy classes of subgroups

Complements