Group #22589312 — subgroups

Subgroups — #22589312

Order 9088.

276 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 105 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×#14556128	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556039	C₂	1	1	yes	maximal, characteristic
3	#17985761	C₂	1	4	yes	maximal
4	#382×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₂	—	2	1	no	—
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	#14556128	—	2	2	no	—
8	#14556128	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₇₁	—	2	1	no	—
10	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
11	#14556039	—	2	2	no	—
12	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
14	#95×C₇₁	—	2	1	no	—
15	#95×C₇₁	—	2	1	no	—
16	C₂×Dic₁₄₂	—	2	4	no	—
17	C₂×Dic₁₄₂	—	4	4	no	—
18	C₂²×Dic₇₁	—	2	2	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₂×C₇₁⋊C₈	Q₈	1	4	yes	complemented
22	C₄⋊Dic₇₁	—	2	4	no	—
23	C₄⋊Dic₇₁	—	2	2	no	—
24	C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₇₁	—	2	2	no	—
26	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	—	2	4	no	—
30	C₄²×C₇₁	C₂×C₄	1	2	yes	—
31	C₄²×C₇₁	D₄	1	2	yes	—
32	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	—	2	2	no	—
34	C₂×C₄×C₇₁	—	2	2	no	—
35	C₂×C₄×C₇₁	—	4	2	no	—
36	C₂×C₄×C₇₁	—	4	2	no	—
37	C₂×C₄×C₇₁	—	4	2	no	—
38	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₇₁	#41	1	2	yes	—
44	C₂×Dic₇₁	—	2	4	no	—
45	C₂×Dic₇₁	—	4	4	no	—
46	Dic₁₄₂	—	4	8	no	—
47	C₇₁⋊C₈	—	2	4	no	—
48	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₇₁	#124	1	1	yes	characteristic
50	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₇₁	#137	1	2	yes	—
53	C₄×C₇₁	—	2	2	no	—
54	C₄×C₇₁	—	2	2	no	—
55	C₄×C₇₁	—	8	2	no	—
56	C₄×C₇₁	C₄×Q₈	1	2	yes	—
57	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
58	Dic₇₁	—	4	4	no	—
59	C₂×C₇₁	#391	1	1	yes	characteristic
60	C₂×C₇₁	#418	1	1	yes	characteristic
61	C₂×C₇₁	#419	1	1	yes	characteristic
62	C₂×C₇₁	#489	1	2	yes	—
63	C₂×C₇₁	#501	1	2	yes	—
64	#1993	—	71	1	no	maximal, Sylow
65	C₇₁	#1993	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₄⋊Q₈	—	71	1	no	—
67	C₂×#97	—	71	1	no	—
68	#382	D₇₁	1	1	yes	characteristic
69	#340	—	71	4	no	—
70	C₂²×Q₈	—	142	1	no	—
71	C₂²×C₈	—	71	2	no	—
72	C₄⋊Q₈	—	71	4	no	—
73	C₄⋊Q₈	—	142	2	no	—
74	C₂×C₄⋊C₄	—	71	2	no	—
75	C₂×C₄⋊C₄	—	142	1	no	—
76	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
77	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
78	#97	—	142	2	no	—
79	#95	—	2	1	no	—
80	#95	—	2	1	no	—
81	C₂×Q₈	—	284	4	no	—
82	C₂×Q₈	—	142	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	142	2	no	—
85	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
88	C₂×C₈	—	71	4	no	—
89	C₂×C₈	—	142	2	no	—
90	C₂×C₈	—	142	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	71	4	no	—
93	C₄⋊C₄	—	142	4	no	—
94	C₄⋊C₄	—	142	2	no	—
95	C₄²	C₄×D₇₁	1	2	yes	—
96	C₄²	D₂₈₄	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
98	Q₈	—	284	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	284	4	no	—
105	C₂×C₄	—	142	4	no	—
106	C₂×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
107	C₂×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₇₁	1	2	yes	—
111	C₂×C₄	#11765864	1	2	yes	—
112	C₈	—	142	4	no	—
113	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
114	C₂²	#14556115	1	1	yes	characteristic
115	C₂²	#14556126	1	1	yes	characteristic
116	C₂²	#14556153	1	2	yes	—
117	C₂²	#14556158	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	2	no	—
121	C₄	—	284	4	no	—
122	C₄	#14556093	1	2	yes	—
123	C₄	#14556108	1	2	yes	—
124	C₂	#17985904	1	1	yes	characteristic
125	C₂	#17986003	1	1	yes	characteristic
126	C₂	#17986007	1	1	yes	characteristic
127	C₂	#17986358	1	2	yes	—
128	C₂	#17986409	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556128

C₂

yes

maximal, characteristic

C₂×#14556039

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal

#382×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

#14556128

—

#14556128

C₄

yes

complemented

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

#14556039

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#95×C₇₁

—

#95×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₈

yes

complemented

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

#124

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

AGL(1,ℤ/8)

yes

—

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

C₂²⋊Q₈

yes

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#501

yes

—

#1993

—

maximal, Sylow

C₇₁

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

D₇₁

yes

characteristic

#340

—

C₂²×Q₈

—

142

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#97

—

142

—

#95

—

#95

—

C₂×Q₈

—

284

—

C₂×Q₈

—

142

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

C₄×D₇₁

yes

—

C₄²

D₂₈₄

yes

—

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

Q₈

—

284

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

284

—

105

C₂×C₄

—

142

—

106

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

—

107

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

—

108

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

109

C₂×C₄

C₂×D₂₈₄

yes

characteristic

110

C₂×C₄

Q₈×D₇₁

yes

—

111

C₂×C₄

#11765864

yes

—

112

C₈

—

142

—

113

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

114

C₂²

#14556115

yes

characteristic

115

C₂²

#14556126

yes

characteristic

116

C₂²

#14556153

yes

—

117

C₂²

#14556158

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

284

—

122

C₄

#14556093

yes

—

123

C₄

#14556108

yes

—

124

C₂

#17985904

yes

characteristic

125

C₂

#17986003

yes

characteristic

126

C₂

#17986007

yes

characteristic

127

C₂

#17986358

yes

—

128

C₂

#17986409

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#382×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	#382	D₇₁
hypocenter	C₇₁	#1993
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#382×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #14556128	C₄	semidirect
21. C₂×C₇₁⋊C₈	Q₈	semidirect
65. C₇₁	#1993	semidirect

H ≅ G/N

type

8. #14556128

C₄

semidirect

21. C₂×C₇₁⋊C₈

Q₈

semidirect

65. C₇₁

#1993

semidirect

Subgroups — #22589312

Conjugacy classes of subgroups

Complements