Group #22588113 — subgroups

Subgroups — C₂²⋊C₄×C₇₁⋊C₈ #22588113

Order 9088.

384 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂³×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₇₁⋊C₈	C₂	1	2	yes	maximal, complemented
4	#17985754	C₂	1	2	yes	maximal, complemented
5	C₄×C₂²⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	—	2	4	no	—
7	C₂²×C₇₁⋊C₈	C₄	1	4	yes	complemented
8	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
9	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²⋊(C₇₁⋊C₈)	C₄	1	4	yes	complemented
12	C₄×C₇₁⋊C₈	—	2	4	no	—
13	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₇₁	C₄	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂²	1	2	yes	—
16	#95×C₇₁	C₄	1	2	yes	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	2	8	no	—
21	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
22	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₇₁⋊C₈	D₄	1	4	yes	—
25	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	—	2	2	no	—
27	C₂²×C₄×C₇₁	—	2	2	no	—
28	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
33	C₂²⋊C₄×C₇₁	C₈	1	8	yes	complemented
34	C₄²×C₇₁	—	2	2	no	—
35	C₄²×C₇₁	—	2	2	no	—
36	C₂³×C₇₁	—	2	2	no	—
37	C₂³×C₇₁	—	2	2	no	—
38	C₂³×C₇₁	C₄²	1	2	yes	—
39	C₂³×C₇₁	C₂×C₈	1	4	yes	—
40	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	—	2	4	no	—
42	C₂×C₄×C₇₁	—	2	2	no	—
43	C₂×C₄×C₇₁	—	2	4	no	—
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	—	2	4	no	—
47	C₂×C₄×C₇₁	C₄²	1	2	yes	—
48	C₂×C₄×C₇₁	C₂×C₈	1	8	yes	complemented
49	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇₁	#41	1	2	yes	—
53	C₇₁⋊C₈	—	2	2	no	—
54	C₇₁⋊C₈	—	2	4	no	—
55	C₇₁⋊C₈	C₂²⋊C₄	1	4	yes	direct product factor
56	C₂²×C₇₁	—	2	2	no	—
57	C₂²×C₇₁	—	2	2	no	—
58	C₂²×C₇₁	—	2	8	no	—
59	C₂²×C₇₁	C₄×C₈	1	4	yes	complemented
60	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
61	C₂²×C₇₁	C₄×D₄	1	2	yes	—
62	C₂²×C₇₁	C₂²×C₈	1	2	yes	—
63	C₂²×C₇₁	#131	1	2	yes	—
64	C₄×C₇₁	—	2	2	no	—
65	C₄×C₇₁	—	2	8	no	—
66	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₇₁	#117	1	1	yes	characteristic
68	C₄×C₇₁	C₄×D₄	1	2	yes	—
69	C₂×C₇₁	—	2	4	no	—
70	C₂×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂×C₇₁	#405	1	1	yes	characteristic
73	C₂×C₇₁	C₈×D₄	1	4	yes	—
74	C₈×C₂²⋊C₄	—	71	1	no	maximal, Sylow
75	C₇₁	C₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₈	—	71	1	no	—
77	C₂×C₂²⋊C₈	—	71	1	no	—
78	C₂×C₄×C₈	—	71	2	no	—
79	C₄×C₂²⋊C₄	D₇₁	1	1	yes	characteristic
80	#336	—	71	2	no	—
81	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
82	C₂²×C₈	—	71	1	no	—
83	C₂²×C₈	—	71	4	no	—
84	C₂²×C₈	—	71	1	no	—
85	C₂²×C₈	—	71	2	no	—
86	C₂²×C₈	—	142	4	no	—
87	C₂×C₂²⋊C₄	Dic₇₁	1	2	yes	—
88	C₂×C₄²	C₂×D₇₁	1	2	yes	—
89	C₂²⋊C₈	—	71	4	no	—
90	C₄×C₈	—	142	4	no	—
91	#95	Dic₇₁	1	2	yes	—
92	C₂⁴	C₂×Dic₇₁	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	C₄×D₇₁	1	2	yes	—
96	C₂²×C₄	C₄×D₇₁	1	2	yes	—
97	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
98	C₂²×C₄	C₂×Dic₇₁	1	4	yes	—
99	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
100	C₂×C₈	—	71	2	no	—
101	C₂×C₈	—	71	4	no	—
102	C₂×C₈	—	71	4	no	—
103	C₂×C₈	—	71	2	no	—
104	C₂×C₈	—	142	8	no	—
105	C₂×C₈	—	142	2	no	—
106	C₂×C₈	—	142	2	no	—
107	C₂×C₈	—	142	2	no	—
108	C₂²⋊C₄	C₇₁⋊C₈	1	8	yes	direct product factor
109	C₄²	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	C₄×Dic₇₁	1	2	yes	—
114	C₂³	C₂×C₇₁⋊C₈	1	4	yes	—
115	C₂³	C₂²×Dic₇₁	1	1	yes	characteristic
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
123	C₂×C₄	C₂×C₇₁⋊C₈	1	8	yes	complemented
124	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
126	C₂×C₄	D₄×D₇₁	1	2	yes	—
127	C₂×C₄	#11765864	1	2	yes	—
128	C₈	—	71	4	no	—
129	C₈	—	142	2	no	—
130	C₈	—	142	4	no	—
131	C₂²	—	2	8	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	C₄×C₇₁⋊C₈	1	4	yes	complemented
135	C₂²	C₂×C₄×Dic₇₁	1	1	yes	characteristic
136	C₂²	D₄×Dic₇₁	1	2	yes	—
137	C₂²	C₂²×C₇₁⋊C₈	1	2	yes	—
138	C₂²	#14556135	1	2	yes	—
139	C₄	—	2	8	no	—
140	C₄	—	2	2	no	—
141	C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
142	C₄	#14556084	1	1	yes	characteristic
143	C₄	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
144	C₂	—	2	4	no	—
145	C₂	C₂²⋊C₄×Dic₇₁	1	1	yes	characteristic
146	C₂	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
147	C₂	#17985955	1	1	yes	characteristic
148	C₂	D₄×C₇₁⋊C₈	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, complemented

#17985754

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

C₄

yes

complemented

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²⋊(C₇₁⋊C₈)

C₄

yes

complemented

C₄×C₇₁⋊C₈

—

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

—

#95×C₇₁

C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂⁴×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂²⋊C₄×C₇₁

C₈

yes

complemented

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₄²

yes

—

C₂³×C₇₁

C₂×C₈

yes

—

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

C₄²

yes

—

C₂×C₄×C₇₁

C₂×C₈

yes

complemented

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₂²⋊C₄

yes

direct product factor

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×C₈

yes

complemented

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

—

C₂²×C₇₁

C₂²×C₈

yes

—

C₂²×C₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

—

C₂×C₇₁

—

C₂×C₇₁

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₇₁

C₂×C₄×C₈

yes

characteristic

C₂×C₇₁

#405

yes

characteristic

C₂×C₇₁

C₈×D₄

yes

—

C₈×C₂²⋊C₄

—

maximal, Sylow

C₇₁

C₈×C₂²⋊C₄

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₄×C₂²⋊C₄

D₇₁

yes

characteristic

#336

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂×C₂²⋊C₄

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

—

C₂²⋊C₈

—

C₄×C₈

—

142

—

#95

Dic₇₁

yes

—

C₂⁴

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₁

yes

—

C₂²×C₄

C₄×D₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

142

—

105

C₂×C₈

—

142

—

106

C₂×C₈

—

142

—

107

C₂×C₈

—

142

—

108

C₂²⋊C₄

C₇₁⋊C₈

yes

direct product factor

109

C₄²

—

110

C₄²

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₄×Dic₇₁

yes

—

114

C₂³

C₂×C₇₁⋊C₈

yes

—

115

C₂³

C₂²×Dic₇₁

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₄×Dic₇₁

yes

—

123

C₂×C₄

C₂×C₇₁⋊C₈

yes

complemented

124

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

125

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

126

C₂×C₄

D₄×D₇₁

yes

—

127

C₂×C₄

#11765864

yes

—

128

C₈

—

129

C₈

—

142

—

130

C₈

—

142

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₄×C₇₁⋊C₈

yes

complemented

135

C₂²

C₂×C₄×Dic₇₁

yes

characteristic

136

C₂²

D₄×Dic₇₁

yes

—

137

C₂²

C₂²×C₇₁⋊C₈

yes

—

138

C₂²

#14556135

yes

—

139

C₄

—

140

C₄

—

141

C₄

C₂²⋊C₄×D₇₁

yes

characteristic

142

C₄

#14556084

yes

characteristic

143

C₄

(C₂²⋊D₇₁)⋊C₄

yes

—

144

C₂

—

145

C₂

C₂²⋊C₄×Dic₇₁

yes

characteristic

146

C₂

C₂×C₄×C₇₁⋊C₈

yes

characteristic

147

C₂

#17985955

yes

characteristic

148

C₂

D₄×C₇₁⋊C₈

yes

—

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

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂×C₇₁

C₂×C₄×C₈

Fitting

C₄×C₂²⋊C₄×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

hypercenter

C₄×C₂²⋊C₄

D₇₁

hypocenter

C₇₁

C₈×C₂²⋊C₄

LCS

G ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
3. C₂×C₄×C₇₁⋊C₈	C₂	semidirect
4. #17985754	C₂	semidirect
7. C₂²×C₇₁⋊C₈	C₄	semidirect
11. C₂²⋊(C₇₁⋊C₈)	C₄	semidirect
33. C₂²⋊C₄×C₇₁	C₈	semidirect
48. C₂×C₄×C₇₁	C₂×C₈	semidirect
55. C₇₁⋊C₈	C₂²⋊C₄	direct (N × H)
59. C₂²×C₇₁	C₄×C₈	semidirect
75. C₇₁	C₈×C₂²⋊C₄	semidirect
108. C₂²⋊C₄	C₇₁⋊C₈	direct (N × H)
123. C₂×C₄	C₂×C₇₁⋊C₈	semidirect
134. C₂²	C₄×C₇₁⋊C₈	semidirect

H ≅ G/N

type

3. C₂×C₄×C₇₁⋊C₈

C₂

semidirect

4. #17985754

C₂

semidirect

7. C₂²×C₇₁⋊C₈

C₄

semidirect

11. C₂²⋊(C₇₁⋊C₈)

C₄

semidirect

33. C₂²⋊C₄×C₇₁

C₈

semidirect

48. C₂×C₄×C₇₁

C₂×C₈

semidirect

55. C₇₁⋊C₈

C₂²⋊C₄

direct (N × H)

59. C₂²×C₇₁

C₄×C₈

semidirect

75. C₇₁

C₈×C₂²⋊C₄

semidirect

108. C₂²⋊C₄

C₇₁⋊C₈

direct (N × H)

123. C₂×C₄

C₂×C₇₁⋊C₈

semidirect

134. C₂²

C₄×C₇₁⋊C₈

semidirect

Subgroups — C₂²⋊C₄×C₇₁⋊C₈ #22588113

Conjugacy classes of subgroups

Complements

Subgroups — C22⋊C4×C71⋊C8 #22588113

Conjugacy classes of subgroups

Complements

Subgroups — C₂²⋊C₄×C₇₁⋊C₈ #22588113