Group #22601735 — subgroups

Subgroups — C₂×#17986625 #22601735

Order 9088.

1296 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1296 conjugacy classes of subgroups — too many to draw. Only its 565 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	#17986625	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14556107	C₂	1	4	yes	maximal, complemented
3	C₂×#14556105	C₂	1	8	yes	maximal, complemented
4	C₂²×C₂²⋊Dic₇₁	C₂	1	2	yes	maximal
5	C₂²×C₂²⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₂	C₂²	1	4	yes	complemented
7	C₂³×Dic₇₁	C₂²	1	4	yes	—
8	#14556107	C₂²	1	32	yes	complemented
9	#14556105	C₂²	1	64	yes	complemented
10	C₂×C₄⋊Dic₇₁	C₂²	1	4	yes	complemented
11	C₂×Dic₇₁⋊C₄	C₂²	1	8	yes	complemented
12	C₂×C₂²⋊Dic₇₁	C₂²	1	16	yes	—
13	C₂×C₂²⋊Dic₇₁	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₇₁	C₂²	1	4	yes	—
15	C₂⁵×C₇₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₇₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₇₁	C₂²	1	8	yes	—
18	C₂×C₂²⋊C₄×C₇₁	C₂²	1	4	yes	—
19	C₂×Dic₁₄₂	—	4	8	no	—
20	C₂×Dic₁₄₂	C₂³	1	16	yes	complemented
21	C₂²×Dic₇₁	—	4	8	no	—
22	C₂²×Dic₇₁	C₂³	1	8	yes	—
23	C₂²×Dic₇₁	C₂³	1	16	yes	—
24	C₄⋊Dic₇₁	C₂³	1	16	yes	complemented
25	Dic₇₁⋊C₄	C₂³	1	32	yes	complemented
26	C₂²⋊Dic₇₁	C₂³	1	16	yes	—
27	C₂²⋊Dic₇₁	C₂³	1	16	yes	—
28	C₂⁴×C₇₁	—	4	4	no	—
29	C₂⁴×C₇₁	Q₈	1	8	yes	complemented
30	C₂⁴×C₇₁	C₂³	1	2	yes	—
31	C₂⁴×C₇₁	C₂³	1	1	yes	characteristic
32	C₂⁴×C₇₁	C₂³	1	4	yes	—
33	C₂²×C₄×C₇₁	—	4	4	no	—
34	C₂²×C₄×C₇₁	C₂³	1	4	yes	—
35	C₂²×C₄×C₇₁	C₂³	1	8	yes	—
36	C₂²⋊C₄×C₇₁	C₂³	1	16	yes	—
37	C₂³×C₇₁	—	2	4	no	—
38	C₂³×C₇₁	—	2	8	no	—
39	C₂³×C₇₁	—	4	16	no	—
40	C₂³×C₇₁	—	4	8	no	—
41	C₂³×C₇₁	C₂×Q₈	1	8	yes	—
42	C₂³×C₇₁	C₂×Q₈	1	4	yes	—
43	C₂³×C₇₁	C₂×Q₈	1	16	yes	complemented
44	C₂³×C₇₁	C₂⁴	1	4	yes	—
45	C₂³×C₇₁	C₂⁴	1	2	yes	—
46	C₂³×C₇₁	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₇₁	—	2	8	no	—
48	C₂×C₄×C₇₁	—	4	8	no	—
49	C₂×C₄×C₇₁	C₂⁴	1	8	yes	—
50	C₂×Dic₇₁	—	2	16	no	—
51	C₂×Dic₇₁	—	4	16	no	—
52	C₂×Dic₇₁	C₂⁴	1	16	yes	—
53	Dic₁₄₂	—	4	16	no	—
54	C₂²×C₇₁	—	2	8	no	—
55	C₂²×C₇₁	—	2	16	no	—
56	C₂²×C₇₁	—	2	8	no	—
57	C₂²×C₇₁	—	2	4	no	—
58	C₂²×C₇₁	—	4	16	no	—
59	C₂²×C₇₁	C₂²×Q₈	1	8	yes	—
60	C₂²×C₇₁	C₂²×Q₈	1	4	yes	—
61	C₂²×C₇₁	C₂²×Q₈	1	2	yes	—
62	C₂²×C₇₁	H(5,𝔽₂)	1	4	yes	—
63	C₂²×C₇₁	C₂⁵	1	1	yes	characteristic
64	C₄×C₇₁	—	2	8	no	—
65	Dic₇₁	—	2	16	no	—
66	C₂×C₇₁	—	2	8	no	—
67	C₂×C₇₁	—	2	4	no	—
68	C₂×C₇₁	#543	1	4	yes	complemented
69	C₂×C₇₁	C₂³×Q₈	1	1	yes	characteristic
70	C₂×C₇₁	C₂×H(5,𝔽₂)	1	2	yes	—
71	C₂×#543	—	71	1	no	maximal, Sylow
72	C₇₁	C₂×#543	1	1	yes	characteristic, complemented, Sylow
73	#543	—	71	16	no	—
74	C₂×C₂²⋊Q₈	—	71	4	no	—
75	C₂×C₂²⋊Q₈	—	71	8	no	—
76	C₂²×C₂²⋊C₄	—	71	2	no	—
77	C₂²×C₂²⋊C₄	D₇₁	1	1	yes	characteristic
78	C₂⁵	C₂×D₇₁	1	1	yes	characteristic
79	C₂²×Q₈	—	71	4	no	—
80	C₂³×C₄	—	71	4	no	—
81	C₂³×C₄	C₂×D₇₁	1	2	yes	—
82	C₂²⋊Q₈	—	71	32	no	—
83	C₂²⋊Q₈	—	71	64	no	—
84	C₂×C₄⋊C₄	—	71	8	no	—
85	C₂×C₄⋊C₄	—	71	4	no	—
86	C₂×C₂²⋊C₄	—	71	4	no	—
87	C₂×C₂²⋊C₄	—	71	4	no	—
88	C₂×C₂²⋊C₄	—	71	16	no	—
89	C₂×C₂²⋊C₄	C₂×D₇₁	1	4	yes	—
90	C₂×C₂²⋊C₄	C₂×D₇₁	1	8	yes	—
91	C₂⁴	—	4	4	no	—
92	C₂⁴	Dic₁₄₂	1	8	yes	complemented
93	C₂⁴	C₂²×D₇₁	1	1	yes	characteristic
94	C₂⁴	C₂²×D₇₁	1	2	yes	—
95	C₂⁴	C₂²×D₇₁	1	4	yes	—
96	C₂×Q₈	—	284	8	no	—
97	C₂×Q₈	—	71	16	no	—
98	C₂²×C₄	—	4	4	no	—
99	C₂²×C₄	—	284	8	no	—
100	C₂²×C₄	—	71	16	no	—
101	C₂²×C₄	—	71	8	no	—
102	C₂²×C₄	C₂²×D₇₁	1	4	yes	—
103	C₂²×C₄	C₂²×D₇₁	1	8	yes	—
104	C₄⋊C₄	—	71	32	no	—
105	C₄⋊C₄	—	71	16	no	—
106	C₂²⋊C₄	—	71	16	no	—
107	C₂²⋊C₄	—	71	16	no	—
108	C₂²⋊C₄	C₂²×D₇₁	1	16	yes	—
109	C₂³	—	2	4	no	—
110	C₂³	—	2	8	no	—
111	C₂³	—	4	16	no	—
112	C₂³	—	4	8	no	—
113	C₂³	C₂×Dic₁₄₂	1	4	yes	—
114	C₂³	C₂×Dic₁₄₂	1	8	yes	—
115	C₂³	C₂×Dic₁₄₂	1	16	yes	complemented
116	C₂³	C₂³×D₇₁	1	2	yes	—
117	C₂³	C₂³×D₇₁	1	4	yes	—
118	C₂³	C₂³×D₇₁	1	1	yes	characteristic
119	Q₈	—	284	16	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	284	16	no	—
123	C₂×C₄	—	71	16	no	—
124	C₂×C₄	—	142	16	no	—
125	C₂×C₄	C₂³×D₇₁	1	8	yes	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	8	no	—
128	C₂²	—	2	8	no	—
129	C₂²	—	2	16	no	—
130	C₂²	—	4	16	no	—
131	C₂²	C₂²×Dic₁₄₂	1	4	yes	—
132	C₂²	C₂²×Dic₁₄₂	1	8	yes	—
133	C₂²	C₂²×Dic₁₄₂	1	2	yes	—
134	C₂²	#14556173	1	4	yes	—
135	C₂²	C₂⁴×D₇₁	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	—	142	16	no	—
138	C₂	—	2	4	no	—
139	C₂	—	2	8	no	—
140	C₂	#17986625	1	4	yes	direct product factor
141	C₂	C₂³×Dic₁₄₂	1	1	yes	characteristic
142	C₂	C₂×#14556173	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986625

C₂

yes

maximal, direct product factor

C₂×#14556107

C₂

yes

maximal, complemented

C₂×#14556105

C₂

yes

maximal, complemented

C₂²×C₂²⋊Dic₇₁

C₂

yes

maximal

C₂²×C₂²⋊C₄×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

complemented

C₂³×Dic₇₁

C₂²

yes

—

#14556107

C₂²

yes

complemented

#14556105

C₂²

yes

complemented

C₂×C₄⋊Dic₇₁

C₂²

yes

complemented

C₂×Dic₇₁⋊C₄

C₂²

yes

complemented

C₂×C₂²⋊Dic₇₁

C₂²

yes

—

C₂×C₂²⋊Dic₇₁

C₂²

yes

—

C₂×C₂²⋊Dic₇₁

C₂²

yes

—

C₂⁵×C₇₁

C₂²

yes

characteristic

C₂³×C₄×C₇₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂³

yes

complemented

C₂²×Dic₇₁

—

C₂²×Dic₇₁

C₂³

yes

—

C₂²×Dic₇₁

C₂³

yes

—

C₄⋊Dic₇₁

C₂³

yes

complemented

Dic₇₁⋊C₄

C₂³

yes

complemented

C₂²⋊Dic₇₁

C₂³

yes

—

C₂²⋊Dic₇₁

C₂³

yes

—

C₂⁴×C₇₁

—

C₂⁴×C₇₁

Q₈

yes

complemented

C₂⁴×C₇₁

C₂³

yes

—

C₂⁴×C₇₁

C₂³

yes

characteristic

C₂⁴×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂²⋊C₄×C₇₁

C₂³

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂×Q₈

yes

—

C₂³×C₇₁

C₂×Q₈

yes

—

C₂³×C₇₁

C₂×Q₈

yes

complemented

C₂³×C₇₁

C₂⁴

yes

—

C₂³×C₇₁

C₂⁴

yes

—

C₂³×C₇₁

C₂⁴

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂⁴

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂⁴

yes

—

Dic₁₄₂

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×Q₈

yes

—

C₂²×C₇₁

C₂²×Q₈

yes

—

C₂²×C₇₁

C₂²×Q₈

yes

—

C₂²×C₇₁

H(5,𝔽₂)

yes

—

C₂²×C₇₁

C₂⁵

yes

characteristic

C₄×C₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#543

yes

complemented

C₂×C₇₁

C₂³×Q₈

yes

characteristic

C₂×C₇₁

C₂×H(5,𝔽₂)

yes

—

C₂×#543

—

maximal, Sylow

C₇₁

C₂×#543

yes

characteristic, complemented, Sylow

#543

—

C₂×C₂²⋊Q₈

—

C₂×C₂²⋊Q₈

—

C₂²×C₂²⋊C₄

—

C₂²×C₂²⋊C₄

D₇₁

yes

characteristic

C₂⁵

C₂×D₇₁

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

C₂×D₇₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇₁

yes

—

C₂×C₂²⋊C₄

C₂×D₇₁

yes

—

C₂⁴

—

C₂⁴

Dic₁₄₂

yes

complemented

C₂⁴

C₂²×D₇₁

yes

characteristic

C₂⁴

C₂²×D₇₁

yes

—

C₂⁴

C₂²×D₇₁

yes

—

C₂×Q₈

—

284

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

284

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₇₁

yes

—

103

C₂²×C₄

C₂²×D₇₁

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

C₂²×D₇₁

yes

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×Dic₁₄₂

yes

—

114

C₂³

C₂×Dic₁₄₂

yes

—

115

C₂³

C₂×Dic₁₄₂

yes

complemented

116

C₂³

C₂³×D₇₁

yes

—

117

C₂³

C₂³×D₇₁

yes

—

118

C₂³

C₂³×D₇₁

yes

characteristic

119

Q₈

—

284

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

284

—

123

C₂×C₄

—

124

C₂×C₄

—

142

—

125

C₂×C₄

C₂³×D₇₁

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂²×Dic₁₄₂

yes

—

132

C₂²

C₂²×Dic₁₄₂

yes

—

133

C₂²

C₂²×Dic₁₄₂

yes

—

134

C₂²

#14556173

yes

—

135

C₂²

C₂⁴×D₇₁

yes

characteristic

136

C₄

—

137

C₄

—

142

—

138

C₂

—

139

C₂

—

140

C₂

#17986625

yes

direct product factor

141

C₂

C₂³×Dic₁₄₂

yes

characteristic

142

C₂

C₂×#14556173

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₇₁
commutator	C₂²×C₇₁	C₂⁵
Frattini	C₂²	C₂⁴×D₇₁
Fitting	C₂²×C₂²⋊C₄×C₇₁	C₂
socle	C₂³×C₇₁	C₂⁴
hypercenter	C₂²×C₂²⋊C₄	D₇₁
hypocenter	C₇₁	C₂×#543
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

subgroup

G/H

center

C₂³

C₂³×D₇₁

commutator

C₂²×C₇₁

C₂⁵

Frattini

C₂²

C₂⁴×D₇₁

Fitting

C₂²×C₂²⋊C₄×C₇₁

C₂

socle

C₂³×C₇₁

C₂⁴

hypercenter

C₂²×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

N	H ≅ G/N	type
1. #17986625	C₂	direct (N × H)
2. C₂×#14556107	C₂	semidirect
3. C₂×#14556105	C₂	semidirect
6. C₂²×Dic₁₄₂	C₂²	semidirect
8. #14556107	C₂²	semidirect
9. #14556105	C₂²	semidirect
10. C₂×C₄⋊Dic₇₁	C₂²	semidirect
11. C₂×Dic₇₁⋊C₄	C₂²	semidirect
20. C₂×Dic₁₄₂	C₂³	semidirect
24. C₄⋊Dic₇₁	C₂³	semidirect
25. Dic₇₁⋊C₄	C₂³	semidirect
29. C₂⁴×C₇₁	Q₈	semidirect
43. C₂³×C₇₁	C₂×Q₈	semidirect
68. C₂×C₇₁	#543	semidirect
72. C₇₁	C₂×#543	semidirect
92. C₂⁴	Dic₁₄₂	semidirect
115. C₂³	C₂×Dic₁₄₂	semidirect
140. C₂	#17986625	direct (N × H)

H ≅ G/N

type

1. #17986625

C₂

direct (N × H)

2. C₂×#14556107

C₂

semidirect

3. C₂×#14556105

C₂

semidirect

6. C₂²×Dic₁₄₂

C₂²

semidirect

8. #14556107

C₂²

semidirect

9. #14556105

C₂²

semidirect

10. C₂×C₄⋊Dic₇₁

C₂²

semidirect

11. C₂×Dic₇₁⋊C₄

C₂²

semidirect

20. C₂×Dic₁₄₂

C₂³

semidirect

24. C₄⋊Dic₇₁

C₂³

semidirect

25. Dic₇₁⋊C₄

C₂³

semidirect

29. C₂⁴×C₇₁

Q₈

semidirect

43. C₂³×C₇₁

C₂×Q₈

semidirect

68. C₂×C₇₁

#543

semidirect

72. C₇₁

C₂×#543

semidirect

92. C₂⁴

Dic₁₄₂

semidirect

115. C₂³

C₂×Dic₁₄₂

semidirect

140. C₂

#17986625

direct (N × H)

Subgroups — C₂×#17986625 #22601735

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17986625 #22601735

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17986625 #22601735