Group #26030759 — subgroups

Subgroups — C₂×#19152002 #26030759

Order 11392.

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×#15668011	C₂	1	2	yes	maximal
2	#19152002	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15667946	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667945	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	—	2	2	no	—
8	C₂²×#10623494	C₂²	1	1	yes	characteristic
9	#15668011	—	2	4	no	—
10	#15668011	C₂²	1	8	yes	—
11	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
12	C₂×#13083553	—	2	2	no	—
13	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
14	#15667946	C₂²	1	4	yes	—
15	#15667945	C₂²	1	8	yes	—
16	C₈×Dic₈₉	C₂²	1	4	yes	—
17	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
19	#107×C₈₉	C₂²	1	4	yes	—
20	C₂×Dic₁₇₈	—	2	8	no	—
21	C₂×Dic₁₇₈	—	4	8	no	—
22	C₂²×Dic₈₉	—	2	2	no	—
23	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
24	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
25	C₂×#10623494	Q₈	1	4	yes	complemented
26	C₂×#10623494	C₂³	1	2	yes	—
27	C₄⋊Dic₈₉	—	2	2	no	—
28	C₄⋊Dic₈₉	C₂³	1	4	yes	—
29	#13083553	—	2	8	no	—
30	C₄×Dic₈₉	D₄	1	2	yes	—
31	C₄×Dic₈₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₈₉	—	2	2	no	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₈₉	Q₈	1	4	yes	complemented
35	C₂×C₈×C₈₉	C₂³	1	2	yes	—
36	C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
38	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	2	4	no	—
40	C₂×C₄×C₈₉	—	4	4	no	—
41	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
45	C₂×Dic₈₉	—	2	4	no	—
46	C₂×Dic₈₉	—	4	4	no	—
47	C₂×Dic₈₉	D₈	1	4	yes	—
48	C₂×Dic₈₉	Q₁₆	1	4	yes	—
49	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
50	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
51	Dic₁₇₈	—	4	16	no	—
52	C₈×C₈₉	C₂×Q₈	1	4	yes	complemented
53	#10623494	C₂×Q₈	1	4	yes	complemented
54	C₂²×C₈₉	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₈₉	C₂×D₈	1	2	yes	—
56	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₈₉	—	4	4	no	—
59	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
60	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
61	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
62	Dic₈₉	—	4	4	no	—
63	Dic₈₉	C₂×D₈	1	4	yes	—
64	Dic₈₉	C₂×Q₁₆	1	4	yes	—
65	C₂×C₈₉	#500	1	4	yes	complemented
66	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₈₉	C₂²×D₈	1	1	yes	characteristic
68	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×#500	—	89	1	no	maximal, Sylow
70	C₈₉	C₂×#500	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	89	2	no	—
72	#500	—	89	8	no	—
73	C₂×#107	—	89	1	no	—
74	C₂×#107	—	89	2	no	—
75	C₂×#107	D₈₉	1	1	yes	characteristic
76	C₂×C₄×C₈	—	89	1	no	—
77	C₂²×Q₈	—	178	2	no	—
78	C₂²×C₈	—	89	1	no	—
79	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
80	C₄⋊Q₈	—	89	8	no	—
81	C₄⋊Q₈	—	178	4	no	—
82	C₂×C₄⋊C₄	—	89	2	no	—
83	C₂×C₄⋊C₄	—	178	2	no	—
84	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
85	C₂×C₄²	—	89	1	no	—
86	#107	—	89	4	no	—
87	#107	—	89	8	no	—
88	#107	C₂×D₈₉	1	4	yes	—
89	C₄×C₈	—	89	4	no	—
90	C₂×Q₈	—	356	8	no	—
91	C₂×Q₈	—	178	8	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	89	1	no	—
94	C₂²×C₄	—	89	1	no	—
95	C₂²×C₄	—	178	2	no	—
96	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
97	C₂×C₈	—	89	4	no	—
98	C₂×C₈	—	89	2	no	—
99	C₂×C₈	Dic₁₇₈	1	4	yes	complemented
100	C₂×C₈	C₂²×D₈₉	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	89	4	no	—
103	C₄⋊C₄	—	178	8	no	—
104	C₄⋊C₄	—	178	2	no	—
105	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
106	C₄²	—	89	2	no	—
107	C₄²	—	89	2	no	—
108	C₂³	D₄×D₈₉	1	1	yes	characteristic
109	Q₈	—	356	16	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	89	4	no	—
113	C₂×C₄	—	89	4	no	—
114	C₂×C₄	—	89	2	no	—
115	C₂×C₄	—	89	2	no	—
116	C₂×C₄	—	356	4	no	—
117	C₂×C₄	—	178	4	no	—
118	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₈₉	1	2	yes	—
120	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
121	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
122	C₈	—	89	4	no	—
123	C₈	C₂×Dic₁₇₈	1	4	yes	complemented
124	C₂²	#15668011	1	2	yes	—
125	C₂²	D₈×D₈₉	1	2	yes	—
126	C₂²	Q₁₆×D₈₉	1	2	yes	—
127	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	89	4	no	—
130	C₄	—	89	4	no	—
131	C₄	—	356	4	no	—
132	C₄	#15668011	1	2	yes	—
133	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
134	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
135	C₂	#19152002	1	4	yes	direct product factor
136	C₂	C₂×#15668011	1	1	yes	characteristic
137	C₂	C₂×D₈×D₈₉	1	1	yes	characteristic
138	C₂	C₂×Q₁₆×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668011

C₂

yes

maximal

#19152002

C₂

yes

maximal, direct product factor

C₂×#15667946

C₂

yes

maximal, characteristic

C₂×#15667945

C₂

yes

maximal

C₂×C₈×Dic₈₉

C₂

yes

maximal, characteristic

C₂×#107×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

—

C₂²×#10623494

C₂²

yes

characteristic

#15668011

—

#15668011

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667946

C₂²

yes

—

#15667945

C₂²

yes

—

C₈×Dic₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

D₄

yes

characteristic

C₂²×Dic₈₉

D₄

yes

characteristic

C₂×#10623494

Q₈

yes

complemented

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

Q₈

yes

complemented

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

D₈

yes

—

C₂×Dic₈₉

Q₁₆

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

C₈×C₈₉

C₂×Q₈

yes

complemented

#10623494

C₂×Q₈

yes

complemented

C₂²×C₈₉

C₄⋊Q₈

yes

—

C₂²×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₄⋊Q₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂²×Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

C₂×D₈

yes

—

Dic₈₉

C₂×Q₁₆

yes

—

C₂×C₈₉

#500

yes

complemented

C₂×C₈₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₈₉

C₂²×D₈

yes

characteristic

C₂×C₈₉

C₂²×Q₁₆

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₈₉

yes

characteristic

C₂×C₄×C₈

—

C₂²×Q₈

—

178

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₈₉

yes

—

C₄×C₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₇₈

yes

complemented

100

C₂×C₈

C₂²×D₈₉

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

178

—

104

C₄⋊C₄

—

178

—

105

C₄⋊C₄

C₂²×D₈₉

yes

—

106

C₄²

—

107

C₄²

—

108

C₂³

D₄×D₈₉

yes

characteristic

109

Q₈

—

356

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

178

—

118

C₂×C₄

D₄×D₈₉

yes

characteristic

119

C₂×C₄

Q₈×D₈₉

yes

—

120

C₂×C₄

C₂×Dic₁₇₈

yes

—

121

C₂×C₄

C₂³×D₈₉

yes

characteristic

122

C₈

—

123

C₈

C₂×Dic₁₇₈

yes

complemented

124

C₂²

#15668011

yes

—

125

C₂²

D₈×D₈₉

yes

—

126

C₂²

Q₁₆×D₈₉

yes

—

127

C₂²

C₂×D₄×D₈₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

356

—

132

C₄

#15668011

yes

—

133

C₄

C₂×Q₈×D₈₉

yes

characteristic

134

C₄

C₂²×Dic₁₇₈

yes

characteristic

135

C₂

#19152002

yes

direct product factor

136

C₂

C₂×#15668011

yes

characteristic

137

C₂

C₂×D₈×D₈₉

yes

characteristic

138

C₂

C₂×Q₁₆×D₈₉

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#107×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×#107	D₈₉
hypocenter	C₈₉	C₂×#500
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×#107×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #19152002	C₂	direct (N × H)
25. C₂×#10623494	Q₈	semidirect
34. C₂×C₈×C₈₉	Q₈	semidirect
52. C₈×C₈₉	C₂×Q₈	semidirect
53. #10623494	C₂×Q₈	semidirect
65. C₂×C₈₉	#500	semidirect
70. C₈₉	C₂×#500	semidirect
99. C₂×C₈	Dic₁₇₈	semidirect
123. C₈	C₂×Dic₁₇₈	semidirect
135. C₂	#19152002	direct (N × H)

H ≅ G/N

type

2. #19152002

C₂

direct (N × H)

25. C₂×#10623494

Q₈

semidirect

34. C₂×C₈×C₈₉

Q₈

semidirect

52. C₈×C₈₉

C₂×Q₈

semidirect

53. #10623494

C₂×Q₈

semidirect

65. C₂×C₈₉

#500

semidirect

70. C₈₉

C₂×#500

semidirect

99. C₂×C₈

Dic₁₇₈

semidirect

123. C₈

C₂×Dic₁₇₈

semidirect

135. C₂

#19152002

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15668011	2
#19152002	2
C₂×#15667946	2
C₂×#15667945	2
C₂×C₈×Dic₈₉	2
C₂×#107×C₈₉	2
C₂×#500	89

subgroup

index

C₂×C₈×Dic₈₉

C₂×#107×C₈₉

C₂×#500

Subgroups — C₂×#19152002 #26030759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19152002 #26030759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19152002 #26030759