Group #22596908 — subgroups

Subgroups — #22596908

Order 9088.

484 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 297 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×#14556135	C₂	1	2	yes	maximal
2	#17986026	C₂	1	6	yes	maximal, complemented
3	#17986023	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#14556135	—	2	8	no	—
7	C₂×#11765849	C₂²	1	6	yes	—
8	C₂²×C₇₁⋊C₈	C₂²	1	6	yes	—
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	3	yes	—
11	C₄⋊(C₇₁⋊C₈)	C₂²	1	6	yes	—
12	C₄⋊(C₇₁⋊C₈)	C₂²	1	6	yes	—
13	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
15	C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
16	C₄×D₄×C₇₁	C₄	1	3	yes	—
17	C₄×D₄×C₇₁	C₄	1	3	yes	—
18	#117×C₇₁	C₂²	1	3	yes	—
19	C₂×C₄²×C₇₁	C₂²	1	3	yes	—
20	#11765849	—	2	12	no	—
21	C₂×C₇₁⋊C₈	—	2	12	no	—
22	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
23	C₂×C₇₁⋊C₈	C₂³	1	6	yes	—
24	#41×C₇₁	D₄	1	4	yes	—
25	#41×C₇₁	Q₈	1	4	yes	—
26	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₇₁	C₂×C₄	1	3	yes	—
28	C₂²×C₄×C₇₁	C₂×C₄	1	3	yes	—
29	C₂²×C₄×C₇₁	C₂×C₄	1	3	yes	—
30	C₂²×C₄×C₇₁	C₂³	1	3	yes	—
31	C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
32	C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
33	C₂²⋊C₄×C₇₁	C₂×C₄	1	6	yes	—
34	C₄²×C₇₁	C₂×C₄	1	3	yes	—
35	C₄²×C₇₁	C₂×C₄	1	3	yes	—
36	C₄²×C₇₁	C₂³	1	1	yes	characteristic
37	C₄²×C₇₁	C₂³	1	3	yes	—
38	C₂³×C₇₁	C₂²×C₄	1	3	yes	—
39	Q₈×C₇₁	C₄⋊C₄	1	4	yes	—
40	D₄×C₇₁	C₄⋊C₄	1	12	yes	—
41	C₂×C₄×C₇₁	—	2	6	no	—
42	C₂×C₄×C₇₁	—	4	3	no	—
43	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₄	1	6	yes	—
49	C₂×C₄×C₇₁	C₂×Q₈	1	6	yes	—
50	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
51	C₇₁⋊C₈	—	2	2	no	—
52	C₇₁⋊C₈	—	2	6	no	—
53	C₂²×C₇₁	—	2	3	no	—
54	C₂²×C₇₁	—	2	3	no	—
55	C₂²×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
56	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
57	C₄×C₇₁	—	2	3	no	—
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	—	2	3	no	—
60	C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
61	C₄×C₇₁	#131	1	2	yes	complemented
62	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₇₁	—	2	6	no	—
65	C₂×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₇₁	C₂×#131	1	1	yes	characteristic
67	C₂×C₇₁	#568	1	1	yes	characteristic
68	#2906	—	71	1	no	maximal, Sylow
69	C₇₁	#2906	1	1	yes	characteristic, complemented, Sylow
70	C₂×#131	—	71	2	no	—
71	C₄×#41	D₇₁	1	1	yes	characteristic
72	#424	—	71	6	no	—
73	#423	—	71	3	no	—
74	C₂×C₄⋊C₈	—	71	3	no	—
75	C₂×#41	C₂×D₇₁	1	1	yes	characteristic
76	#131	—	142	8	no	—
77	C₂×M₁₆	—	71	6	no	—
78	C₂²×C₈	—	71	6	no	—
79	C₄×Q₈	Dic₇₁	1	1	yes	characteristic
80	C₄×Q₈	Dic₇₁	1	1	yes	characteristic
81	C₄×D₄	Dic₇₁	1	3	yes	—
82	C₄×D₄	Dic₇₁	1	3	yes	—
83	#117	C₂×D₇₁	1	3	yes	—
84	C₂×C₄²	C₂×D₇₁	1	3	yes	—
85	C₄⋊C₈	—	71	1	no	—
86	C₄⋊C₈	—	71	3	no	—
87	C₄⋊C₈	—	71	6	no	—
88	C₄⋊C₈	—	71	6	no	—
89	#41	Dic₁₄₂	1	4	yes	—
90	#41	D₂₈₄	1	4	yes	—
91	C₂×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
92	C₂×D₄	C₂×Dic₇₁	1	3	yes	—
93	C₂²×C₄	C₂×Dic₇₁	1	3	yes	—
94	C₂²×C₄	C₂×Dic₇₁	1	3	yes	—
95	C₂²×C₄	C₂²×D₇₁	1	3	yes	—
96	M₁₆	—	142	12	no	—
97	C₂×C₈	—	71	2	no	—
98	C₂×C₈	—	71	6	no	—
99	C₂×C₈	—	142	12	no	—
100	C₄⋊C₄	C₂×Dic₇₁	1	3	yes	—
101	C₄⋊C₄	C₂×Dic₇₁	1	3	yes	—
102	C₂²⋊C₄	C₂×Dic₇₁	1	6	yes	—
103	C₄²	C₂×Dic₇₁	1	3	yes	—
104	C₄²	C₂×Dic₇₁	1	3	yes	—
105	C₄²	C₂²×D₇₁	1	1	yes	characteristic
106	C₄²	C₂²×D₇₁	1	3	yes	—
107	C₂³	C₂²×Dic₇₁	1	3	yes	—
108	Q₈	C₄⋊Dic₇₁	1	4	yes	—
109	D₄	C₄⋊Dic₇₁	1	12	yes	—
110	C₂×C₄	—	2	6	no	—
111	C₂×C₄	—	4	3	no	—
112	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
114	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
115	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
116	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
117	C₂×C₄	C₂×D₂₈₄	1	6	yes	—
118	C₂×C₄	C₂×Dic₁₄₂	1	6	yes	—
119	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
120	C₈	—	142	2	no	—
121	C₈	—	142	6	no	—
122	C₂²	—	2	3	no	—
123	C₂²	—	2	3	no	—
124	C₂²	C₂×C₄⋊Dic₇₁	1	6	yes	—
125	C₂²	C₂³×Dic₇₁	1	1	yes	characteristic
126	C₄	—	2	3	no	—
127	C₄	—	2	1	no	—
128	C₄	—	2	3	no	—
129	C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
130	C₄	#14556135	1	2	yes	complemented
131	C₄	C₂²×D₂₈₄	1	1	yes	characteristic
132	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
133	C₂	—	2	6	no	—
134	C₂	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
135	C₂	C₂×#14556135	1	1	yes	characteristic
136	C₂	#17986766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556135

C₂

yes

maximal

#17986026

C₂

yes

maximal, complemented

#17986023

C₂

yes

maximal, complemented

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, complemented

C₄×#41×C₇₁

C₂

yes

maximal, characteristic

#14556135

—

C₂×#11765849

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂×#41×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

C₄

yes

characteristic

C₄×Q₈×C₇₁

C₄

yes

characteristic

C₄×D₄×C₇₁

C₄

yes

—

C₄×D₄×C₇₁

C₄

yes

—

#117×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

—

#41×C₇₁

D₄

yes

—

#41×C₇₁

Q₈

yes

—

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂×D₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂²⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂²×C₄

yes

—

Q₈×C₇₁

C₄⋊C₄

yes

—

D₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

—

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

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

—

C₂²×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₄×C₇₁

#131

yes

complemented

C₄×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

C₂²×C₄⋊C₄

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#131

—

C₄×#41

D₇₁

yes

characteristic

#424

—

#423

—

C₂×C₄⋊C₈

—

C₂×#41

C₂×D₇₁

yes

characteristic

#131

—

142

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

Dic₇₁

yes

characteristic

C₄×Q₈

Dic₇₁

yes

characteristic

C₄×D₄

Dic₇₁

yes

—

C₄×D₄

Dic₇₁

yes

—

#117

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#41

Dic₁₄₂

yes

—

#41

D₂₈₄

yes

—

C₂×Q₈

C₂×Dic₇₁

yes

characteristic

C₂×D₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

100

C₄⋊C₄

C₂×Dic₇₁

yes

—

101

C₄⋊C₄

C₂×Dic₇₁

yes

—

102

C₂²⋊C₄

C₂×Dic₇₁

yes

—

103

C₄²

C₂×Dic₇₁

yes

—

104

C₄²

C₂×Dic₇₁

yes

—

105

C₄²

C₂²×D₇₁

yes

characteristic

106

C₄²

C₂²×D₇₁

yes

—

107

C₂³

C₂²×Dic₇₁

yes

—

108

Q₈

C₄⋊Dic₇₁

yes

—

109

D₄

C₄⋊Dic₇₁

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

113

C₂×C₄

C₂²×Dic₇₁

yes

—

114

C₂×C₄

C₂²×Dic₇₁

yes

—

115

C₂×C₄

C₂²×Dic₇₁

yes

—

116

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

117

C₂×C₄

C₂×D₂₈₄

yes

—

118

C₂×C₄

C₂×Dic₁₄₂

yes

—

119

C₂×C₄

C₂³×D₇₁

yes

characteristic

120

C₈

—

142

—

121

C₈

—

142

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₄⋊Dic₇₁

yes

—

125

C₂²

C₂³×Dic₇₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₂×C₄⋊Dic₇₁

yes

—

130

C₄

#14556135

yes

complemented

131

C₄

C₂²×D₂₈₄

yes

characteristic

132

C₄

C₂²×Dic₁₄₂

yes

characteristic

133

C₂

—

134

C₂

C₂²×C₄⋊Dic₇₁

yes

characteristic

135

C₂

C₂×#14556135

yes

characteristic

136

C₂

#17986766

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	C₄×#41×C₇₁	C₂
hypercenter	C₄×#41	D₇₁
hypocenter	C₇₁	#2906
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₁

commutator, socle

C₂²×C₇₁

C₂³×C₄

Fitting

C₄×#41×C₇₁

C₂

hypercenter

C₄×#41

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
2. #17986026	C₂	semidirect
3. #17986023	C₂	semidirect
4. C₂×C₄⋊(C₇₁⋊C₈)	C₂	semidirect
61. C₄×C₇₁	#131	semidirect
69. C₇₁	#2906	semidirect
130. C₄	#14556135	semidirect

H ≅ G/N

type

2. #17986026

C₂

semidirect

3. #17986023

C₂

semidirect

4. C₂×C₄⋊(C₇₁⋊C₈)

C₂

semidirect

61. C₄×C₇₁

#131

semidirect

69. C₇₁

#2906

semidirect

130. C₄

#14556135

semidirect

Subgroups — #22596908

Conjugacy classes of subgroups

Complements