Group #22586369 — subgroups

Subgroups — #22586369

Order 9088.

140 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 35 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	#17985919	C₂	1	1	yes	maximal, characteristic
2	#17985725	C₂	1	1	yes	maximal, characteristic
3	#343×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₄₂	C₄	1	1	yes	characteristic
5	C₂×Dic₇₁⋊C₄	—	2	1	no	—
6	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₇₁⋊#95	—	2	1	no	—
8	C₇₁⋊#95	C₄	1	1	yes	characteristic
9	C₂×M₁₆×C₇₁	—	2	1	no	—
10	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₄₂	—	4	1	no	—
12	C₂×Dic₁₄₂	—	4	2	no	—
13	C₂²×Dic₇₁	—	2	1	no	—
14	C₂²×Dic₇₁	—	2	1	no	—
15	C₂×C₇₁⋊C₈	—	4	1	no	—
16	Dic₇₁⋊C₄	—	4	2	no	—
17	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
20	M₁₆×C₇₁	—	2	2	no	—
21	M₁₆×C₇₁	—	4	1	no	—
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	SD₁₆	1	1	yes	characteristic
32	C₂×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
33	C₂×Dic₇₁	—	2	2	no	—
34	C₂×Dic₇₁	—	2	2	no	—
35	C₂×Dic₇₁	—	4	1	no	—
36	C₂×Dic₇₁	—	4	1	no	—
37	C₂×Dic₇₁	—	4	1	no	—
38	C₂×Dic₇₁	—	4	1	no	—
39	Dic₁₄₂	—	4	2	no	—
40	Dic₁₄₂	—	4	2	no	—
41	C₈×C₇₁	—	4	2	no	—
42	C₇₁⋊C₈	—	8	1	no	—
43	C₂²×C₇₁	—	2	1	no	—
44	C₂²×C₇₁	—	2	1	no	—
45	C₂²×C₇₁	#99	1	1	yes	characteristic
46	C₂²×C₇₁	#103	1	1	yes	characteristic
47	C₂²×C₇₁	#104	1	1	yes	characteristic
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	—	4	1	no	—
51	C₄×C₇₁	—	4	1	no	—
52	Dic₇₁	—	4	2	no	—
53	Dic₇₁	—	4	2	no	—
54	C₂×C₇₁	—	2	1	no	—
55	C₂×C₇₁	—	2	1	no	—
56	C₂×C₇₁	#328	1	1	yes	characteristic
57	C₂×C₇₁	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
58	C₂×C₇₁	#356	1	1	yes	characteristic
59	#1354	—	71	1	no	maximal, Sylow
60	C₇₁	#1354	1	1	yes	characteristic, complemented, Sylow
61	#395	—	71	1	no	—
62	#343	D₇₁	1	1	yes	characteristic
63	#324	—	71	1	no	—
64	C₂²×Q₈	—	71	1	no	—
65	C₂×M₁₆	—	2	1	no	—
66	C₂×C₄⋊C₄	—	142	1	no	—
67	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
68	C₂²⋊C₈	—	142	1	no	—
69	#95	—	71	1	no	—
70	#95	—	142	1	no	—
71	C₂×Q₈	—	284	1	no	—
72	C₂×Q₈	—	284	2	no	—
73	C₂²×C₄	—	142	1	no	—
74	C₂²×C₄	—	142	1	no	—
75	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
76	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
78	M₁₆	—	2	2	no	—
79	M₁₆	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	284	1	no	—
82	C₄⋊C₄	—	4	1	no	—
83	C₄⋊C₄	—	284	2	no	—
84	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
85	Q₈	—	284	2	no	—
86	Q₈	—	284	2	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	284	1	no	—
94	C₂×C₄	—	284	1	no	—
95	C₂×C₄	—	284	1	no	—
96	C₂×C₄	—	284	1	no	—
97	C₂×C₄	—	142	2	no	—
98	C₂×C₄	—	142	2	no	—
99	C₂×C₄	#11765853	1	1	yes	characteristic
100	C₂×C₄	#11765855	1	1	yes	characteristic
101	C₈	—	4	2	no	—
102	C₈	—	568	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#14556042	1	1	yes	characteristic
106	C₂²	#14556051	1	1	yes	characteristic
107	C₂²	#14556056	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	1	no	—
112	C₄	—	284	2	no	—
113	C₄	—	284	2	no	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#17985735	1	1	yes	characteristic
117	C₂	#17985787	1	1	yes	characteristic
118	C₂	#17985797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985919

C₂

yes

maximal, characteristic

#17985725

C₂

yes

maximal, characteristic

#343×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₄

yes

characteristic

C₂×Dic₇₁⋊C₄

—

C₂²⋊(C₇₁⋊C₈)

—

C₇₁⋊#95

—

C₇₁⋊#95

C₄

yes

characteristic

C₂×M₁₆×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

Dic₇₁⋊C₄

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

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

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#328

yes

characteristic

C₂×C₇₁

AΣL(1,𝔽₁₆)

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#395

—

#343

D₇₁

yes

characteristic

#324

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

characteristic

C₂²⋊C₈

—

142

—

#95

—

#95

—

142

—

C₂×Q₈

—

284

—

C₂×Q₈

—

284

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

284

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

Q₈

—

284

—

Q₈

—

284

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

#11765853

yes

characteristic

100

C₂×C₄

#11765855

yes

characteristic

101

C₈

—

102

C₈

—

568

—

103

C₂²

—

104

C₂²

—

105

C₂²

#14556042

yes

characteristic

106

C₂²

#14556051

yes

characteristic

107

C₂²

#14556056

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

284

—

113

C₄

—

284

—

114

C₂

—

115

C₂

—

116

C₂

#17985735

yes

characteristic

117

C₂

#17985787

yes

characteristic

118

C₂

#17985797

yes

characteristic

	subgroup	G/H
center	C₂²	#14556042
commutator	C₂²×C₄×C₇₁	C₂×C₄
Frattini	C₂×C₄⋊C₄	C₂×D₇₁
Fitting	#343×C₇₁	C₂
socle	C₂²×C₇₁	#99
hypercenter	#343	D₇₁
hypocenter	C₇₁	#1354
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #343

subgroup

G/H

center

C₂²

#14556042

commutator

C₂²×C₄×C₇₁

C₂×C₄

Frattini

C₂×C₄⋊C₄

C₂×D₇₁

Fitting

#343×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #343

N	H ≅ G/N	type
60. C₇₁	#1354	semidirect

H ≅ G/N

type

60. C₇₁

#1354

semidirect

Subgroups — #22586369

Conjugacy classes of subgroups

Complements