Group #22598847 — subgroups

Subgroups — C₂×#17986403 #22598847

Order 9088.

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

normal subgroups only

This group has 424 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	C₂×#14556127	C₂	1	2	yes	maximal
2	#17986403	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14556062	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556061	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₄₂	—	2	2	no	—
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#14556127	—	2	4	no	—
10	#14556127	C₂²	1	8	yes	—
11	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
12	C₂×Dic₇₁⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
14	#14556062	C₂²	1	4	yes	—
15	#14556061	C₂²	1	8	yes	—
16	C₈×Dic₇₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
19	#107×C₇₁	C₂²	1	4	yes	—
20	C₂×Dic₁₄₂	—	2	8	no	—
21	C₂×Dic₁₄₂	—	4	8	no	—
22	C₂²×Dic₇₁	—	2	2	no	—
23	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
24	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₈	Q₈	1	4	yes	complemented
26	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
27	C₄⋊Dic₇₁	—	2	2	no	—
28	C₄⋊Dic₇₁	C₂³	1	4	yes	—
29	Dic₇₁⋊C₄	—	2	8	no	—
30	C₄×Dic₇₁	D₄	1	2	yes	—
31	C₄×Dic₇₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₇₁	—	2	2	no	—
33	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₇₁	Q₈	1	4	yes	complemented
35	C₂×C₈×C₇₁	C₂³	1	2	yes	—
36	C₄⋊C₄×C₇₁	—	2	2	no	—
37	C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
38	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	—	2	4	no	—
40	C₂×C₄×C₇₁	—	4	4	no	—
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
45	C₂×Dic₇₁	—	2	4	no	—
46	C₂×Dic₇₁	—	4	4	no	—
47	C₂×Dic₇₁	D₈	1	4	yes	—
48	C₂×Dic₇₁	Q₁₆	1	4	yes	—
49	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
50	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
51	Dic₁₄₂	—	4	16	no	—
52	C₈×C₇₁	C₂×Q₈	1	4	yes	complemented
53	C₇₁⋊C₈	C₂×Q₈	1	4	yes	complemented
54	C₂²×C₇₁	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₇₁	C₂×D₈	1	2	yes	—
56	C₂²×C₇₁	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₇₁	—	4	4	no	—
59	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
60	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
61	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
62	Dic₇₁	—	4	4	no	—
63	Dic₇₁	C₂×D₈	1	4	yes	—
64	Dic₇₁	C₂×Q₁₆	1	4	yes	—
65	C₂×C₇₁	#500	1	4	yes	complemented
66	C₂×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₇₁	C₂²×D₈	1	1	yes	characteristic
68	C₂×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×#500	—	71	1	no	maximal, Sylow
70	C₇₁	C₂×#500	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	71	2	no	—
72	#500	—	71	8	no	—
73	C₂×#107	—	71	1	no	—
74	C₂×#107	—	71	2	no	—
75	C₂×#107	D₇₁	1	1	yes	characteristic
76	C₂×C₄×C₈	—	71	1	no	—
77	C₂²×Q₈	—	142	2	no	—
78	C₂²×C₈	—	71	1	no	—
79	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
80	C₄⋊Q₈	—	71	8	no	—
81	C₄⋊Q₈	—	142	4	no	—
82	C₂×C₄⋊C₄	—	71	2	no	—
83	C₂×C₄⋊C₄	—	142	2	no	—
84	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
85	C₂×C₄²	—	71	1	no	—
86	#107	—	71	4	no	—
87	#107	—	71	8	no	—
88	#107	C₂×D₇₁	1	4	yes	—
89	C₄×C₈	—	71	4	no	—
90	C₂×Q₈	—	284	8	no	—
91	C₂×Q₈	—	142	8	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	71	1	no	—
94	C₂²×C₄	—	71	1	no	—
95	C₂²×C₄	—	142	2	no	—
96	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
97	C₂×C₈	—	71	4	no	—
98	C₂×C₈	—	71	2	no	—
99	C₂×C₈	Dic₁₄₂	1	4	yes	complemented
100	C₂×C₈	C₂²×D₇₁	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	71	4	no	—
103	C₄⋊C₄	—	142	8	no	—
104	C₄⋊C₄	—	142	2	no	—
105	C₄⋊C₄	C₂²×D₇₁	1	4	yes	—
106	C₄²	—	71	2	no	—
107	C₄²	—	71	2	no	—
108	C₂³	D₄×D₇₁	1	1	yes	characteristic
109	Q₈	—	284	16	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	284	4	no	—
113	C₂×C₄	—	71	2	no	—
114	C₂×C₄	—	71	2	no	—
115	C₂×C₄	—	71	4	no	—
116	C₂×C₄	—	71	4	no	—
117	C₂×C₄	—	142	4	no	—
118	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₇₁	1	2	yes	—
120	C₂×C₄	C₂×Dic₁₄₂	1	2	yes	—
121	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
122	C₈	—	71	4	no	—
123	C₈	C₂×Dic₁₄₂	1	4	yes	complemented
124	C₂²	#14556127	1	2	yes	—
125	C₂²	D₈×D₇₁	1	2	yes	—
126	C₂²	Q₁₆×D₇₁	1	2	yes	—
127	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	284	4	no	—
130	C₄	—	71	4	no	—
131	C₄	—	71	4	no	—
132	C₄	#14556127	1	2	yes	—
133	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
134	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
135	C₂	#17986403	1	4	yes	direct product factor
136	C₂	C₂×#14556127	1	1	yes	characteristic
137	C₂	C₂×D₈×D₇₁	1	1	yes	characteristic
138	C₂	C₂×Q₁₆×D₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556127

C₂

yes

maximal

#17986403

C₂

yes

maximal, direct product factor

C₂×#14556062

C₂

yes

maximal, characteristic

C₂×#14556061

C₂

yes

maximal

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

C₂×#107×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556127

—

#14556127

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556062

C₂²

yes

—

#14556061

C₂²

yes

—

C₈×Dic₇₁

C₂²

yes

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

#107×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

characteristic

C₂²×Dic₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

Q₈

yes

complemented

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

—

C₂×Dic₇₁

Q₁₆

yes

—

C₂×Dic₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

C₂×D₄

yes

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₈

yes

complemented

C₇₁⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₇₁

C₄⋊Q₈

yes

—

C₂²×C₇₁

C₂×D₈

yes

—

C₂²×C₇₁

C₂×Q₁₆

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₄⋊Q₈

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

C₂²×Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

C₂×D₈

yes

—

Dic₇₁

C₂×Q₁₆

yes

—

C₂×C₇₁

#500

yes

complemented

C₂×C₇₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₇₁

C₂²×D₈

yes

characteristic

C₂×C₇₁

C₂²×Q₁₆

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₇₁

yes

characteristic

C₂×C₄×C₈

—

C₂²×Q₈

—

142

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₇₁

yes

—

C₄×C₈

—

C₂×Q₈

—

284

—

C₂×Q₈

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₄₂

yes

complemented

100

C₂×C₈

C₂²×D₇₁

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

142

—

104

C₄⋊C₄

—

142

—

105

C₄⋊C₄

C₂²×D₇₁

yes

—

106

C₄²

—

107

C₄²

—

108

C₂³

D₄×D₇₁

yes

characteristic

109

Q₈

—

284

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

284

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

142

—

118

C₂×C₄

D₄×D₇₁

yes

characteristic

119

C₂×C₄

Q₈×D₇₁

yes

—

120

C₂×C₄

C₂×Dic₁₄₂

yes

—

121

C₂×C₄

C₂³×D₇₁

yes

characteristic

122

C₈

—

123

C₈

C₂×Dic₁₄₂

yes

complemented

124

C₂²

#14556127

yes

—

125

C₂²

D₈×D₇₁

yes

—

126

C₂²

Q₁₆×D₇₁

yes

—

127

C₂²

C₂×D₄×D₇₁

yes

characteristic

128

C₄

—

129

C₄

—

284

—

130

C₄

—

131

C₄

—

132

C₄

#14556127

yes

—

133

C₄

C₂×Q₈×D₇₁

yes

characteristic

134

C₄

C₂²×Dic₁₄₂

yes

characteristic

135

C₂

#17986403

yes

direct product factor

136

C₂

C₂×#14556127

yes

characteristic

137

C₂

C₂×D₈×D₇₁

yes

characteristic

138

C₂

C₂×Q₁₆×D₇₁

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₂×#107×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#107	D₇₁
hypocenter	C₇₁	C₂×#500
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₂×#107×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #17986403	C₂	direct (N × H)
25. C₂×C₇₁⋊C₈	Q₈	semidirect
34. C₂×C₈×C₇₁	Q₈	semidirect
52. C₈×C₇₁	C₂×Q₈	semidirect
53. C₇₁⋊C₈	C₂×Q₈	semidirect
65. C₂×C₇₁	#500	semidirect
70. C₇₁	C₂×#500	semidirect
99. C₂×C₈	Dic₁₄₂	semidirect
123. C₈	C₂×Dic₁₄₂	semidirect
135. C₂	#17986403	direct (N × H)

H ≅ G/N

type

2. #17986403

C₂

direct (N × H)

25. C₂×C₇₁⋊C₈

Q₈

semidirect

34. C₂×C₈×C₇₁

Q₈

semidirect

52. C₈×C₇₁

C₂×Q₈

semidirect

53. C₇₁⋊C₈

C₂×Q₈

semidirect

65. C₂×C₇₁

#500

semidirect

70. C₇₁

C₂×#500

semidirect

99. C₂×C₈

Dic₁₄₂

semidirect

123. C₈

C₂×Dic₁₄₂

semidirect

135. C₂

#17986403

direct (N × H)

Subgroups — C₂×#17986403 #22598847

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17986403 #22598847

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17986403 #22598847