Group #22590201 — subgroups

Subgroups — #22590201

Order 9088.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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×#14556072	C₂	1	1	yes	maximal, characteristic
2	#17986093	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985824	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985804	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985774	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
9	#14556093	—	2	1	no	—
10	D₄×Dic₇₁	—	2	1	no	—
11	#14556072	C₄	1	4	yes	complemented
12	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#14556062	C₂²	1	1	yes	characteristic
14	#14556051	—	2	1	no	—
15	#14556049	—	2	1	no	—
16	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	#107×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₄₂	—	2	1	no	—
21	C₂²×Dic₇₁	—	4	1	no	—
22	Dic₂₈₄	—	2	1	no	—
23	Dic₂₈₄	C₂×C₄	1	2	yes	complemented
24	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Dic₇₁	—	2	1	no	—
26	Dic₇₁⋊C₄	—	4	1	no	—
27	C₂²⋊Dic₇₁	—	4	1	no	—
28	C₄×Dic₇₁	—	4	1	no	—
29	C₄×Dic₇₁	D₄	1	1	yes	characteristic
30	C₇₁⋊C₁₆	C₂×C₄	1	2	yes	complemented
31	C₂×D₄×C₇₁	—	2	1	no	—
32	D₈×C₇₁	—	2	1	no	—
33	D₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₁₆×C₇₁	—	2	1	no	—
37	C₂³×C₇₁	—	4	1	no	—
38	D₄×C₇₁	—	2	2	no	—
39	D₄×C₇₁	—	4	1	no	—
40	C₂×C₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₇₁	—	4	1	no	—
43	C₂×Dic₇₁	—	4	1	no	—
44	C₂×Dic₇₁	—	8	2	no	—
45	C₂×Dic₇₁	D₈	1	1	yes	characteristic
46	C₂×Dic₇₁	D₈	1	1	yes	characteristic
47	Dic₁₄₂	—	2	2	no	—
48	Dic₁₄₂	—	4	1	no	—
49	C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₇₁	#41	1	1	yes	characteristic
51	C₇₁⋊C₈	—	2	1	no	—
52	C₂²×C₇₁	—	4	2	no	—
53	C₂²×C₇₁	—	8	1	no	—
54	C₂²×C₇₁	—	8	1	no	—
55	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
56	C₄×C₇₁	—	8	1	no	—
57	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
58	C₄×C₇₁	#135	1	1	yes	characteristic
59	Dic₇₁	—	2	1	no	—
60	Dic₇₁	—	2	1	no	—
61	Dic₇₁	—	4	2	no	—
62	Dic₇₁	—	8	1	no	—
63	C₂×C₇₁	—	8	2	no	—
64	C₂×C₇₁	C₄×D₈	1	1	yes	characteristic
65	C₂×C₇₁	#509	1	1	yes	characteristic
66	C₂×C₇₁	#510	1	1	yes	characteristic
67	#2175	—	71	1	no	maximal, Sylow
68	C₇₁	#2175	1	1	yes	characteristic, complemented, Sylow
69	C₂×SD₃₂	—	71	1	no	—
70	C₄×Q₁₆	—	71	1	no	—
71	C₄×D₈	—	71	1	no	—
72	#366	—	71	1	no	—
73	#358	—	71	1	no	—
74	#357	D₇₁	1	1	yes	characteristic
75	#346	—	71	1	no	—
76	C₂×Q₁₆	—	71	1	no	—
77	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
78	C₄×Q₈	—	142	1	no	—
79	C₄×D₄	—	142	1	no	—
80	SD₃₂	—	71	4	no	—
81	C₂×C₁₆	—	71	1	no	—
82	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
83	#107	—	71	1	no	—
84	#107	C₂×D₇₁	1	1	yes	characteristic
85	#103	—	142	1	no	—
86	#102	—	142	1	no	—
87	C₄×C₈	—	71	1	no	—
88	C₂×Q₈	—	142	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂²×C₄	—	284	1	no	—
91	Q₁₆	—	71	2	no	—
92	Q₁₆	—	142	1	no	—
93	D₈	—	2	1	no	—
94	D₈	C₄×D₇₁	1	2	yes	—
95	C₂×C₈	—	71	1	no	—
96	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	284	1	no	—
99	C₄⋊C₄	—	142	1	no	—
100	C₂²⋊C₄	—	284	1	no	—
101	C₄²	—	284	1	no	—
102	C₄²	—	71	1	no	—
103	C₁₆	—	2	1	no	—
104	C₁₆	—	71	2	no	—
105	C₂³	—	4	1	no	—
106	Q₈	—	284	1	no	—
107	Q₈	—	142	2	no	—
108	D₄	—	2	2	no	—
109	D₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	284	1	no	—
112	C₂×C₄	—	284	1	no	—
113	C₂×C₄	—	568	2	no	—
114	C₂×C₄	—	71	1	no	—
115	C₂×C₄	—	71	1	no	—
116	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
117	C₈	—	142	1	no	—
118	C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
119	C₈	#11765864	1	1	yes	characteristic
120	C₂²	—	4	2	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	D₈×D₇₁	1	1	yes	characteristic
124	C₄	—	8	1	no	—
125	C₄	—	284	2	no	—
126	C₄	—	568	1	no	—
127	C₄	—	142	1	no	—
128	C₄	—	142	1	no	—
129	C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
130	C₄	#14556149	1	1	yes	characteristic
131	C₂	—	8	2	no	—
132	C₂	#17986079	1	1	yes	characteristic
133	C₂	#17986433	1	1	yes	characteristic
134	C₂	#17986438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556072

C₂

yes

maximal, characteristic

#17986093

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₁

C₂

yes

maximal, characteristic

#17985824

C₂

yes

maximal, characteristic, complemented

#17985804

C₂

yes

maximal, characteristic, complemented

#17985774

C₂

yes

maximal, characteristic, complemented

#357×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

characteristic

#14556093

—

D₄×Dic₇₁

—

#14556072

C₄

yes

complemented

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556051

—

#14556049

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×D₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

Dic₂₈₄

—

Dic₂₈₄

C₂×C₄

yes

complemented

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

C₂×C₄

yes

complemented

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂²×C₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

#135

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₄×D₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₄×Q₁₆

—

C₄×D₈

—

#366

—

#358

—

#357

D₇₁

yes

characteristic

#346

—

C₂×Q₁₆

—

C₂×D₈

C₂×D₇₁

yes

characteristic

C₄×Q₈

—

142

—

C₄×D₄

—

142

—

SD₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#107

C₂×D₇₁

yes

characteristic

#103

—

142

—

#102

—

142

—

C₄×C₈

—

C₂×Q₈

—

142

—

C₂×D₄

—

C₂²×C₄

—

284

—

Q₁₆

—

Q₁₆

—

142

—

D₈

—

D₈

C₄×D₇₁

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

100

C₂²⋊C₄

—

284

—

101

C₄²

—

284

—

102

C₄²

—

103

C₁₆

—

104

C₁₆

—

105

C₂³

—

106

Q₈

—

284

—

107

Q₈

—

142

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

284

—

113

C₂×C₄

—

568

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

D₄×D₇₁

yes

characteristic

117

C₈

—

142

—

118

C₈

C₂×C₄×D₇₁

yes

characteristic

119

C₈

#11765864

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

D₈×D₇₁

yes

characteristic

124

C₄

—

125

C₄

—

284

—

126

C₄

—

568

—

127

C₄

—

142

—

128

C₄

—

142

—

129

C₄

(C₂²⋊D₇₁)⋊C₄

yes

characteristic

130

C₄

#14556149

yes

characteristic

131

C₂

—

132

C₂

#17986079

yes

characteristic

133

C₂

#17986433

yes

characteristic

134

C₂

#17986438

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₈×C₇₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#357×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#357	D₇₁
hypocenter	C₇₁	#2175
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₈×C₇₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#357×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
2. #17986093	C₂	semidirect
4. #17985824	C₂	semidirect
5. #17985804	C₂	semidirect
6. #17985774	C₂	semidirect
11. #14556072	C₄	semidirect
23. Dic₂₈₄	C₂×C₄	semidirect
30. C₇₁⋊C₁₆	C₂×C₄	semidirect
68. C₇₁	#2175	semidirect

H ≅ G/N

type

2. #17986093

C₂

semidirect

4. #17985824

C₂

semidirect

5. #17985804

C₂

semidirect

6. #17985774

C₂

semidirect

11. #14556072

C₄

semidirect

23. Dic₂₈₄

C₂×C₄

semidirect

30. C₇₁⋊C₁₆

C₂×C₄

semidirect

68. C₇₁

#2175

semidirect

Subgroups — #22590201

Conjugacy classes of subgroups

Complements