Group #26033653 — subgroups

Subgroups — C₂×#19152229 #26033653

Order 11392.

1080 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1080 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	#19152229	C₂	1	16	yes	maximal, direct product factor
2	C₂×#15668011	C₂	1	4	yes	maximal, complemented
3	C₂×#15668003	C₂	1	4	yes	maximal, complemented
4	C₂×#15667991	C₂	1	4	yes	maximal
5	C₂×#15667971	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	C₂²	1	4	yes	—
8	#15668011	C₂²	1	32	yes	complemented
9	#15668003	C₂²	1	32	yes	complemented
10	#15667991	C₂²	1	32	yes	—
11	#15667971	C₂²	1	16	yes	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
15	C₂×#13083553	C₂²	1	8	yes	—
16	C₂×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
17	C₂×C₄×Dic₈₉	C₂²	1	4	yes	—
18	C₂³×C₄×C₈₉	C₂²	1	2	yes	—
19	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₈₉	C₂²	1	8	yes	—
22	C₂×Dic₁₇₈	—	4	8	no	—
23	C₂×Dic₁₇₈	C₂³	1	16	yes	—
24	C₂²×Dic₈₉	C₂³	1	4	yes	—
25	C₂²×Dic₈₉	C₂³	1	4	yes	—
26	C₄⋊Dic₈₉	C₂³	1	8	yes	—
27	C₄⋊Dic₈₉	C₂³	1	8	yes	—
28	C₄⋊Dic₈₉	C₂³	1	16	yes	—
29	#13083553	C₂³	1	32	yes	—
30	C₂²⋊Dic₈₉	C₂³	1	16	yes	—
31	C₄×Dic₈₉	C₂³	1	16	yes	—
32	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	—	4	4	no	—
34	C₂²×C₄×C₈₉	Q₈	1	8	yes	complemented
35	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
36	C₂²×C₄×C₈₉	C₂³	1	8	yes	—
37	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
39	C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
40	C₂³×C₈₉	—	2	2	no	—
41	C₂³×C₈₉	—	2	2	no	—
42	C₂³×C₈₉	C₂×Q₈	1	4	yes	—
43	C₂³×C₈₉	C₂⁴	1	2	yes	—
44	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
45	C₂×C₄×C₈₉	—	2	8	no	—
46	C₂×C₄×C₈₉	—	4	8	no	—
47	C₂×C₄×C₈₉	C₂×Q₈	1	8	yes	—
48	C₂×C₄×C₈₉	C₂×Q₈	1	16	yes	complemented
49	C₂×C₄×C₈₉	C₂⁴	1	8	yes	—
50	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
51	C₂×Dic₈₉	—	2	8	no	—
52	C₂×Dic₈₉	—	2	8	no	—
53	C₂×Dic₈₉	C₂⁴	1	8	yes	—
54	C₂×Dic₈₉	C₂⁴	1	8	yes	—
55	Dic₁₇₈	—	4	16	no	—
56	C₂²×C₈₉	—	2	8	no	—
57	C₂²×C₈₉	—	2	2	no	—
58	C₂²×C₈₉	—	2	2	no	—
59	C₂²×C₈₉	C₂²×Q₈	1	4	yes	—
60	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
61	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
63	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
64	C₄×C₈₉	—	2	8	no	—
65	C₄×C₈₉	C₂²×Q₈	1	8	yes	—
66	Dic₈₉	—	2	8	no	—
67	Dic₈₉	—	2	8	no	—
68	C₂×C₈₉	—	2	4	no	—
69	C₂×C₈₉	#544	1	4	yes	complemented
70	C₂×C₈₉	C₂³×Q₈	1	1	yes	characteristic
71	C₂×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
73	C₂×#544	—	89	1	no	maximal, Sylow
74	C₈₉	C₂×#544	1	1	yes	characteristic, complemented, Sylow
75	#544	—	89	16	no	—
76	C₂×C₄⋊Q₈	—	89	4	no	—
77	C₂×#125	—	89	4	no	—
78	C₂×C₂²⋊Q₈	—	89	4	no	—
79	C₂×#117	—	89	2	no	—
80	C₂²×C₄⋊C₄	D₈₉	1	1	yes	characteristic
81	C₂²×Q₈	—	89	4	no	—
82	C₂³×C₄	C₂×D₈₉	1	2	yes	—
83	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
84	C₄⋊Q₈	—	89	32	no	—
85	#125	—	89	32	no	—
86	C₂²⋊Q₈	—	89	32	no	—
87	#117	—	89	16	no	—
88	C₂×C₄⋊C₄	—	89	2	no	—
89	C₂×C₄⋊C₄	—	89	2	no	—
90	C₂×C₄⋊C₄	—	89	4	no	—
91	C₂×C₄⋊C₄	—	89	8	no	—
92	C₂×C₄⋊C₄	C₂×D₈₉	1	4	yes	—
93	C₂×C₄⋊C₄	C₂×D₈₉	1	8	yes	—
94	C₂×C₂²⋊C₄	—	89	4	no	—
95	C₂×C₄²	—	89	4	no	—
96	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
97	C₂×Q₈	—	89	16	no	—
98	C₂×Q₈	—	356	8	no	—
99	C₂²×C₄	—	4	4	no	—
100	C₂²×C₄	—	89	4	no	—
101	C₂²×C₄	—	89	4	no	—
102	C₂²×C₄	Dic₁₇₈	1	8	yes	complemented
103	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
104	C₂²×C₄	C₂²×D₈₉	1	8	yes	—
105	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
106	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
107	C₄⋊C₄	—	89	8	no	—
108	C₄⋊C₄	—	89	8	no	—
109	C₄⋊C₄	—	89	16	no	—
110	C₄⋊C₄	—	89	32	no	—
111	C₄⋊C₄	C₂²×D₈₉	1	16	yes	—
112	C₂²⋊C₄	—	89	16	no	—
113	C₄²	—	89	16	no	—
114	C₂³	—	2	2	no	—
115	C₂³	—	2	2	no	—
116	C₂³	C₂×Dic₁₇₈	1	4	yes	—
117	C₂³	C₂³×D₈₉	1	2	yes	—
118	C₂³	C₂³×D₈₉	1	1	yes	characteristic
119	Q₈	—	356	16	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	4	8	no	—
122	C₂×C₄	—	89	8	no	—
123	C₂×C₄	—	89	8	no	—
124	C₂×C₄	—	178	8	no	—
125	C₂×C₄	—	178	8	no	—
126	C₂×C₄	C₂×Dic₁₇₈	1	8	yes	—
127	C₂×C₄	C₂×Dic₁₇₈	1	16	yes	complemented
128	C₂×C₄	C₂³×D₈₉	1	8	yes	—
129	C₂×C₄	C₂³×D₈₉	1	4	yes	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	8	no	—
133	C₂²	C₂²×Dic₁₇₈	1	4	yes	—
134	C₂²	C₂²×Dic₁₇₈	1	2	yes	—
135	C₂²	#15668057	1	2	yes	—
136	C₂²	#15668060	1	2	yes	—
137	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
138	C₄	—	2	8	no	—
139	C₄	—	178	8	no	—
140	C₄	—	178	8	no	—
141	C₄	C₂²×Dic₁₇₈	1	8	yes	—
142	C₂	—	2	4	no	—
143	C₂	#19152229	1	4	yes	direct product factor
144	C₂	C₂³×Dic₁₇₈	1	1	yes	characteristic
145	C₂	C₂×#15668057	1	1	yes	characteristic
146	C₂	C₂×#15668060	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152229

C₂

yes

maximal, direct product factor

C₂×#15668011

C₂

yes

maximal, complemented

C₂×#15668003

C₂

yes

maximal, complemented

C₂×#15667991

C₂

yes

maximal

C₂×#15667971

C₂

yes

maximal

C₂²×C₄⋊C₄×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668011

C₂²

yes

complemented

#15668003

C₂²

yes

complemented

#15667991

C₂²

yes

—

#15667971

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

C₂³

yes

—

C₂²⋊Dic₈₉

C₂³

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

Q₈

yes

complemented

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

C₂⁴

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

complemented

C₂×C₄×C₈₉

C₂⁴

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂⁴

yes

—

C₂×Dic₈₉

C₂⁴

yes

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂²×C₈₉

C₂⁵

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂²×Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#544

yes

complemented

C₂×C₈₉

C₂³×Q₈

yes

characteristic

C₂×C₈₉

C₂×H(5,𝔽₂)

yes

characteristic

C₂×C₈₉

C₂×2₋¹⁺⁴

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#544

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×C₂²⋊Q₈

—

C₂×#117

—

C₂²×C₄⋊C₄

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

C₂×D₈₉

yes

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

Dic₁₇₈

yes

complemented

103

C₂²×C₄

C₂²×D₈₉

yes

—

104

C₂²×C₄

C₂²×D₈₉

yes

—

105

C₂²×C₄

C₂²×D₈₉

yes

—

106

C₂²×C₄

C₂²×D₈₉

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₈₉

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₂³

—

115

C₂³

—

116

C₂³

C₂×Dic₁₇₈

yes

—

117

C₂³

C₂³×D₈₉

yes

—

118

C₂³

C₂³×D₈₉

yes

characteristic

119

Q₈

—

356

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

178

—

125

C₂×C₄

—

178

—

126

C₂×C₄

C₂×Dic₁₇₈

yes

—

127

C₂×C₄

C₂×Dic₁₇₈

yes

complemented

128

C₂×C₄

C₂³×D₈₉

yes

—

129

C₂×C₄

C₂³×D₈₉

yes

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

C₂²×Dic₁₇₈

yes

—

134

C₂²

C₂²×Dic₁₇₈

yes

—

135

C₂²

#15668057

yes

—

136

C₂²

#15668060

yes

—

137

C₂²

C₂⁴×D₈₉

yes

characteristic

138

C₄

—

139

C₄

—

178

—

140

C₄

—

178

—

141

C₄

C₂²×Dic₁₇₈

yes

—

142

C₂

—

143

C₂

#19152229

yes

direct product factor

144

C₂

C₂³×Dic₁₇₈

yes

characteristic

145

C₂

C₂×#15668057

yes

characteristic

146

C₂

C₂×#15668060

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₈₉
commutator, cosocle	C₂²×C₈₉	C₂⁵
Frattini	C₂²	C₂⁴×D₈₉
Fitting	C₂²×C₄⋊C₄×C₈₉	C₂
socle	C₂³×C₈₉	C₂⁴
prime radical	C₂⁴×C₈₉	C₂³
hypercenter	C₂²×C₄⋊C₄	D₈₉
hypocenter	C₈₉	C₂×#544
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂³×D₈₉

commutator, cosocle

C₂²×C₈₉

C₂⁵

Frattini

C₂²

C₂⁴×D₈₉

Fitting

C₂²×C₄⋊C₄×C₈₉

C₂

socle

C₂³×C₈₉

C₂⁴

prime radical

C₂⁴×C₈₉

C₂³

hypercenter

C₂²×C₄⋊C₄

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
1. #19152229	C₂	direct (N × H)
2. C₂×#15668011	C₂	semidirect
3. C₂×#15668003	C₂	semidirect
8. #15668011	C₂²	semidirect
9. #15668003	C₂²	semidirect
34. C₂²×C₄×C₈₉	Q₈	semidirect
48. C₂×C₄×C₈₉	C₂×Q₈	semidirect
69. C₂×C₈₉	#544	semidirect
74. C₈₉	C₂×#544	semidirect
102. C₂²×C₄	Dic₁₇₈	semidirect
127. C₂×C₄	C₂×Dic₁₇₈	semidirect
143. C₂	#19152229	direct (N × H)

H ≅ G/N

type

1. #19152229

C₂

direct (N × H)

2. C₂×#15668011

C₂

semidirect

3. C₂×#15668003

C₂

semidirect

8. #15668011

C₂²

semidirect

9. #15668003

C₂²

semidirect

34. C₂²×C₄×C₈₉

Q₈

semidirect

48. C₂×C₄×C₈₉

C₂×Q₈

semidirect

69. C₂×C₈₉

#544

semidirect

74. C₈₉

C₂×#544

semidirect

102. C₂²×C₄

Dic₁₇₈

semidirect

127. C₂×C₄

C₂×Dic₁₇₈

semidirect

143. C₂

#19152229

direct (N × H)

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

subgroup	index
#19152229	2
C₂×#15668011	2
C₂×#15668003	2
C₂×#15667991	2
C₂×#15667971	2
C₂²×C₄⋊C₄×C₈₉	2
C₂×#544	89

subgroup

index

C₂²×C₄⋊C₄×C₈₉

Subgroups — C₂×#19152229 #26033653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19152229 #26033653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19152229 #26033653