Group #22601438 — subgroups

Subgroups — C₂×#17986415 #22601438

Order 9088.

368 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 213 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×#14556134	C₂	1	1	yes	maximal, characteristic
2	#17986415	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556067	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×D₇₁	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11765865	C₄	1	1	yes	characteristic
8	#14556134	C₄	1	4	yes	—
9	#14556134	C₂²	1	4	yes	—
10	C₂×#11765849	C₄	1	1	yes	characteristic
11	C₂×#11765848	C₄	1	2	yes	—
12	C₂×C₈×D₇₁	C₂²	1	2	yes	—
13	#14556068	C₂²	1	4	yes	complemented
14	#14556067	C₂²	1	8	yes	complemented
15	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	—
16	C₁₆×D₇₁	C₂²	1	8	yes	complemented
17	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₇₁	C₂²	1	4	yes	complemented
21	#11765865	C₈	1	4	yes	—
22	#11765865	C₂×C₄	1	4	yes	—
23	C₂×Dic₁₄₂	C₈	1	1	yes	characteristic
24	C₂×D₂₈₄	C₈	1	1	yes	characteristic
25	C₂×C₂²⋊D₇₁	C₈	1	2	yes	—
26	C₂×C₄×D₇₁	C₂×C₄	1	2	yes	—
27	#11765849	C₂×C₄	1	4	yes	—
28	#11765848	C₂×C₄	1	8	yes	—
29	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
30	C₈×D₇₁	C₂×C₄	1	4	yes	—
31	C₈×D₇₁	C₂³	1	4	yes	—
32	C₇₁⋊C₁₆	C₂³	1	4	yes	—
33	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
36	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₂³	1	2	yes	—
38	C₁₆×C₇₁	C₂³	1	2	yes	—
39	C₁₆×C₇₁	C₂³	1	2	yes	—
40	C₂³×C₇₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₇₁	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂²×C₄	1	2	yes	—
46	C₂²⋊D₇₁	C₂×C₈	1	8	yes	—
47	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
48	D₂₈₄	C₂×C₈	1	4	yes	—
49	C₄×D₇₁	C₂×C₈	1	4	yes	—
50	C₄×D₇₁	C₂²×C₄	1	4	yes	—
51	Dic₁₄₂	C₂×C₈	1	4	yes	—
52	C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₈×C₇₁	C₂²×C₄	1	2	yes	—
54	C₈×C₇₁	C₂⁴	1	1	yes	characteristic
55	C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
56	C₂²×C₇₁	—	2	2	no	—
57	C₂²×C₇₁	C₂²×C₈	1	2	yes	—
58	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
59	C₂×D₇₁	—	2	4	no	—
60	C₂×D₇₁	C₂²×C₈	1	4	yes	—
61	C₄×C₇₁	C₂²×C₈	1	2	yes	—
62	C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
64	Dic₇₁	C₂²×C₈	1	4	yes	—
65	C₂×C₇₁	—	2	2	no	—
66	C₂×C₇₁	#504	1	2	yes	complemented
67	C₂×C₇₁	C₂³×C₈	1	1	yes	characteristic
68	D₇₁	—	2	4	no	—
69	C₂×#504	—	71	1	no	maximal, Sylow
70	C₇₁	C₂×#504	1	1	yes	characteristic, complemented, Sylow
71	C₂×#131	—	71	1	no	—
72	#504	—	71	8	no	—
73	C₂×M₃₂	—	71	1	no	—
74	C₂×M₃₂	—	71	2	no	—
75	C₂²×C₁₆	—	71	2	no	—
76	C₂²×C₁₆	D₇₁	1	1	yes	characteristic, complemented
77	C₂×#41	—	71	1	no	—
78	#131	—	71	4	no	—
79	#131	—	71	4	no	—
80	C₂×M₁₆	—	71	1	no	—
81	C₂×M₁₆	—	71	2	no	—
82	C₂²×C₈	—	71	2	no	—
83	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
84	M₃₂	—	71	4	no	—
85	M₃₂	—	71	8	no	—
86	C₂×C₁₆	—	71	2	no	—
87	C₂×C₁₆	—	71	8	no	—
88	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
89	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
90	C₂×C₁₆	C₂×D₇₁	1	4	yes	complemented
91	#41	—	71	4	no	—
92	#41	—	71	4	no	—
93	C₂×Q₈	—	71	1	no	—
94	C₂×D₄	—	71	2	no	—
95	C₂×D₄	—	71	1	no	—
96	C₂²×C₄	—	71	2	no	—
97	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
98	M₁₆	—	71	4	no	—
99	M₁₆	—	71	8	no	—
100	C₂×C₈	—	71	2	no	—
101	C₂×C₈	—	71	4	no	—
102	C₂×C₈	—	71	4	no	—
103	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
104	C₂×C₈	C₄×D₇₁	1	2	yes	—
105	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
106	C₂×C₈	C₂²×D₇₁	1	2	yes	—
107	C₁₆	—	71	4	no	—
108	C₁₆	C₂²×D₇₁	1	2	yes	—
109	C₁₆	C₂²×D₇₁	1	2	yes	—
110	C₂³	—	71	2	no	—
111	C₂³	C₈×D₇₁	1	1	yes	characteristic
112	Q₈	—	71	4	no	—
113	D₄	—	71	8	no	—
114	D₄	—	71	4	no	—
115	C₂×C₄	—	71	2	no	—
116	C₂×C₄	—	71	4	no	—
117	C₂×C₄	—	71	4	no	—
118	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
119	C₂×C₄	C₈×D₇₁	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄×D₇₁	1	2	yes	—
122	C₈	—	71	4	no	—
123	C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
124	C₈	C₂×C₄×D₇₁	1	2	yes	—
125	C₈	C₂³×D₇₁	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	—	71	4	no	—
128	C₂²	—	142	4	no	—
129	C₂²	C₂×C₈×D₇₁	1	2	yes	—
130	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
131	C₄	—	71	4	no	—
132	C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
133	C₄	C₂×C₈×D₇₁	1	2	yes	—
134	C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	142	4	no	—
137	C₂	#17986415	1	2	yes	direct product factor
138	C₂	C₂²×C₈×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556134

C₂

yes

maximal, characteristic

#17986415

C₂

yes

maximal, direct product factor

C₂×#14556068

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

C₂

yes

maximal, complemented

C₂×C₁₆×D₇₁

C₂

yes

maximal, complemented

C₂²×C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765865

C₄

yes

characteristic

#14556134

C₄

yes

—

#14556134

C₂²

yes

—

C₂×#11765849

C₄

yes

characteristic

C₂×#11765848

C₄

yes

—

C₂×C₈×D₇₁

C₂²

yes

—

#14556068

C₂²

yes

complemented

#14556067

C₂²

yes

complemented

C₂×C₇₁⋊C₁₆

C₂²

yes

—

C₁₆×D₇₁

C₂²

yes

complemented

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

complemented

#11765865

C₈

yes

—

#11765865

C₂×C₄

yes

—

C₂×Dic₁₄₂

C₈

yes

characteristic

C₂×D₂₈₄

C₈

yes

characteristic

C₂×C₂²⋊D₇₁

C₈

yes

—

C₂×C₄×D₇₁

C₂×C₄

yes

—

#11765849

C₂×C₄

yes

—

#11765848

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₈×D₇₁

C₂×C₄

yes

—

C₈×D₇₁

C₂³

yes

—

C₇₁⋊C₁₆

C₂³

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

—

C₁₆×C₇₁

C₂³

yes

—

C₁₆×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂×C₈

yes

characteristic

C₂²×D₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂²⋊D₇₁

C₂×C₈

yes

—

C₂×Dic₇₁

C₂×C₈

yes

—

D₂₈₄

C₂×C₈

yes

—

C₄×D₇₁

C₂×C₈

yes

—

C₄×D₇₁

C₂²×C₄

yes

—

Dic₁₄₂

C₂×C₈

yes

—

C₈×C₇₁

C₂²×C₄

yes

characteristic

C₈×C₇₁

C₂²×C₄

yes

—

C₈×C₇₁

C₂⁴

yes

characteristic

C₇₁⋊C₈

C₂²×C₄

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×C₈

yes

—

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

C₂²×C₈

yes

—

C₄×C₇₁

C₂²×C₈

yes

—

C₄×C₇₁

C₂²×C₈

yes

characteristic

C₄×C₇₁

C₂³×C₄

yes

characteristic

Dic₇₁

C₂²×C₈

yes

—

C₂×C₇₁

—

C₂×C₇₁

#504

yes

complemented

C₂×C₇₁

C₂³×C₈

yes

characteristic

D₇₁

—

C₂×#504

—

maximal, Sylow

C₇₁

C₂×#504

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

D₇₁

yes

characteristic, complemented

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

C₂×C₁₆

C₂×D₇₁

yes

characteristic

C₂×C₁₆

C₂×D₇₁

yes

complemented

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₄×D₇₁

yes

characteristic

104

C₂×C₈

C₄×D₇₁

yes

—

105

C₂×C₈

C₂²×D₇₁

yes

characteristic

106

C₂×C₈

C₂²×D₇₁

yes

—

107

C₁₆

—

108

C₁₆

C₂²×D₇₁

yes

—

109

C₁₆

C₂²×D₇₁

yes

—

110

C₂³

—

111

C₂³

C₈×D₇₁

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₈×D₇₁

yes

characteristic

119

C₂×C₄

C₈×D₇₁

yes

—

120

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

121

C₂×C₄

C₂×C₄×D₇₁

yes

—

122

C₈

—

123

C₈

C₂×C₄×D₇₁

yes

characteristic

124

C₈

C₂×C₄×D₇₁

yes

—

125

C₈

C₂³×D₇₁

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

142

—

129

C₂²

C₂×C₈×D₇₁

yes

—

130

C₂²

C₂×C₈×D₇₁

yes

characteristic

131

C₄

—

132

C₄

C₂×C₈×D₇₁

yes

characteristic

133

C₄

C₂×C₈×D₇₁

yes

—

134

C₄

C₂²×C₄×D₇₁

yes

characteristic

135

C₂

—

136

C₂

—

142

—

137

C₂

#17986415

yes

direct product factor

138

C₂

C₂²×C₈×D₇₁

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₇₁
commutator	C₂×C₇₁	C₂³×C₈
Frattini	C₈	C₂³×D₇₁
Fitting	C₂²×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₈
hypercenter	C₂²×C₁₆	D₇₁
hypocenter	C₇₁	C₂×#504
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₁₆

C₂×D₇₁

commutator

C₂×C₇₁

C₂³×C₈

Frattini

C₈

C₂³×D₇₁

Fitting

C₂²×C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×C₈

hypercenter

C₂²×C₁₆

D₇₁

C₇₁

G ▷ C₂×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #17986415	C₂	direct (N × H)
3. C₂×#14556068	C₂	semidirect
4. C₂×#14556067	C₂	semidirect
5. C₂×C₁₆×D₇₁	C₂	semidirect
6. C₂²×C₁₆×C₇₁	C₂	semidirect
13. #14556068	C₂²	semidirect
14. #14556067	C₂²	semidirect
16. C₁₆×D₇₁	C₂²	semidirect
20. C₂×C₁₆×C₇₁	C₂²	semidirect
66. C₂×C₇₁	#504	semidirect
70. C₇₁	C₂×#504	semidirect
76. C₂²×C₁₆	D₇₁	semidirect
90. C₂×C₁₆	C₂×D₇₁	semidirect
137. C₂	#17986415	direct (N × H)

H ≅ G/N

type

2. #17986415

C₂

direct (N × H)

3. C₂×#14556068

C₂

semidirect

4. C₂×#14556067

C₂

semidirect

5. C₂×C₁₆×D₇₁

C₂

semidirect

6. C₂²×C₁₆×C₇₁

C₂

semidirect

13. #14556068

C₂²

semidirect

14. #14556067

C₂²

semidirect

16. C₁₆×D₇₁

C₂²

semidirect

20. C₂×C₁₆×C₇₁

C₂²

semidirect

66. C₂×C₇₁

#504

semidirect

70. C₇₁

C₂×#504

semidirect

76. C₂²×C₁₆

D₇₁

semidirect

90. C₂×C₁₆

C₂×D₇₁

semidirect

137. C₂

#17986415

direct (N × H)

Subgroups — C₂×#17986415 #22601438

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17986415 #22601438

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17986415 #22601438