Group #26028810 — subgroups

Subgroups — C₂×#19151624 #26028810

Order 11392.

512 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 269 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	C₂²×#13083558	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667971	C₂	1	1	yes	maximal, characteristic
3	#19151624	C₂	1	8	yes	maximal, direct product factor
4	C₂×#15667941	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	—	2	4	no	—
7	C₂×#13083558	C₂²	1	4	yes	—
8	C₂²×#10623494	C₂²	1	2	yes	—
9	#15667971	C₄	1	4	yes	—
10	#15667971	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₈₉	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₈₉	C₄	1	2	yes	—
13	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
14	#15667941	C₂²	1	16	yes	complemented
15	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₈₉	—	2	4	no	—
17	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
18	C₂²×C₈×C₈₉	C₂²	1	4	yes	—
19	C₂²×Dic₈₉	C₂×C₄	1	4	yes	—
20	#13083558	—	2	8	no	—
21	C₂×#10623494	—	2	4	no	—
22	C₂×#10623494	C₂³	1	4	yes	—
23	C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
24	C₂²⋊Dic₈₉	C₂×C₄	1	8	yes	—
25	C₄×Dic₈₉	C₂×C₄	1	4	yes	—
26	C₄×Dic₈₉	C₂³	1	4	yes	—
27	C₂⁴×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	D₄	1	4	yes	—
31	C₂²×C₄×C₈₉	Q₈	1	4	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₈₉	—	2	4	no	—
35	C₂×C₈×C₈₉	—	2	8	no	—
36	C₂×C₈×C₈₉	C₂³	1	4	yes	—
37	C₂³×C₈₉	—	2	4	no	—
38	C₂³×C₈₉	C₄⋊C₄	1	4	yes	—
39	C₂³×C₈₉	C₂²×C₄	1	2	yes	—
40	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₄⋊C₄	1	8	yes	—
42	C₂×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
48	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₈₉	—	2	8	no	—
51	C₂×Dic₈₉	C₂²×C₄	1	8	yes	—
52	C₈×C₈₉	—	2	4	no	—
53	#10623494	—	2	4	no	—
54	C₂²×C₈₉	—	2	4	no	—
55	C₂²×C₈₉	—	2	8	no	—
56	C₂²×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₈₉	#131	1	4	yes	—
59	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
61	C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
62	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
64	Dic₈₉	—	2	8	no	—
65	C₂×C₈₉	—	2	4	no	—
66	C₂×C₈₉	#424	1	4	yes	complemented
67	C₂×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈₉	C₂×#131	1	2	yes	—
69	C₂×#424	—	89	1	no	maximal, Sylow
70	C₈₉	C₂×#424	1	1	yes	characteristic, complemented, Sylow
71	C₂²×M₁₆	—	89	1	no	—
72	C₂³×C₈	D₈₉	1	1	yes	characteristic
73	C₂×#117	—	89	1	no	—
74	#424	—	89	8	no	—
75	C₂×C₄⋊C₈	—	89	4	no	—
76	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
77	C₂×M₁₆	—	89	4	no	—
78	C₂×M₁₆	—	178	4	no	—
79	C₂²×C₈	—	2	4	no	—
80	C₂²×C₈	—	89	2	no	—
81	C₂²×C₈	C₂×D₈₉	1	4	yes	—
82	C₂²×C₈	C₂×D₈₉	1	2	yes	—
83	#117	—	89	4	no	—
84	#117	—	89	4	no	—
85	C₂×C₄⋊C₄	—	89	2	no	—
86	C₂×C₂²⋊C₄	—	89	2	no	—
87	C₂×C₄²	—	89	2	no	—
88	C₄⋊C₈	—	89	16	no	—
89	C₂⁴	C₄×D₈₉	1	1	yes	characteristic
90	C₂²×C₄	—	89	4	no	—
91	C₂²×C₄	Dic₁₇₈	1	4	yes	—
92	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₈₉	1	2	yes	—
94	C₂²×C₄	C₂²⋊D₈₉	1	4	yes	—
95	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
97	M₁₆	—	178	8	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	8	no	—
100	C₂×C₈	—	89	4	no	—
101	C₂×C₈	—	178	4	no	—
102	C₂×C₈	C₂²×D₈₉	1	4	yes	—
103	C₄⋊C₄	—	89	8	no	—
104	C₂²⋊C₄	—	89	8	no	—
105	C₄²	—	89	4	no	—
106	C₄²	—	89	4	no	—
107	C₂³	—	2	4	no	—
108	C₂³	#13083553	1	4	yes	—
109	C₂³	C₂×C₄×D₈₉	1	2	yes	—
110	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
111	C₂×C₄	—	89	8	no	—
112	C₂×C₄	—	178	8	no	—
113	C₂×C₄	#13083553	1	8	yes	—
114	C₂×C₄	#13083553	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
119	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
120	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
121	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	178	4	no	—
124	C₂²	—	2	8	no	—
125	C₂²	—	2	4	no	—
126	C₂²	C₂×#13083553	1	4	yes	—
127	C₂²	C₂×#13083553	1	2	yes	—
128	C₂²	#15668018	1	4	yes	—
129	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
130	C₄	—	178	8	no	—
131	C₄	C₂×#13083553	1	2	yes	—
132	C₄	C₂×#13083553	1	4	yes	—
133	C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
134	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#19151624	1	4	yes	direct product factor
137	C₂	C₂²×#13083553	1	1	yes	characteristic
138	C₂	C₂×#15668018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13083558

C₂

yes

maximal, characteristic

C₂×#15667971

C₂

yes

maximal, characteristic

#19151624

C₂

yes

maximal, direct product factor

C₂×#15667941

C₂

yes

maximal, complemented

C₂³×C₈×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

—

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

#15667971

C₄

yes

—

#15667971

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₄

yes

—

C₂×C₂²⋊Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667941

C₂²

yes

complemented

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

#13083558

—

C₂×#10623494

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂²⋊Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂⁴×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂²×C₄

yes

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

—

C₂²×C₈₉

#131

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂²×Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#424

yes

complemented

C₂×C₈₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₈₉

C₂×#131

yes

—

C₂×#424

—

maximal, Sylow

C₈₉

C₂×#424

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂³×C₈

D₈₉

yes

characteristic

C₂×#117

—

#424

—

C₂×C₄⋊C₈

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

—

C₂²×C₈

C₂×D₈₉

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂⁴

C₄×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Dic₁₇₈

yes

—

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

178

—

102

C₂×C₈

C₂²×D₈₉

yes

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₂³

—

108

C₂³

#13083553

yes

—

109

C₂³

C₂×C₄×D₈₉

yes

—

110

C₂³

C₂×C₄×D₈₉

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

178

—

113

C₂×C₄

#13083553

yes

—

114

C₂×C₄

#13083553

yes

—

115

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

116

C₂×C₄

C₂×C₄×D₈₉

yes

—

117

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

118

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

119

C₂×C₄

C₂×Dic₁₇₈

yes

—

120

C₂×C₄

C₂×Dic₁₇₈

yes

—

121

C₂×C₄

C₂³×D₈₉

yes

characteristic

122

C₈

—

123

C₈

—

178

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×#13083553

yes

—

127

C₂²

C₂×#13083553

yes

—

128

C₂²

#15668018

yes

—

129

C₂²

C₂²×C₄×D₈₉

yes

characteristic

130

C₄

—

178

—

131

C₄

C₂×#13083553

yes

—

132

C₄

C₂×#13083553

yes

—

133

C₄

C₂²×C₂²⋊D₈₉

yes

characteristic

134

C₄

C₂²×Dic₁₇₈

yes

characteristic

135

C₂

—

136

C₂

#19151624

yes

direct product factor

137

C₂

C₂²×#13083553

yes

characteristic

138

C₂

C₂×#15668018

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂³×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×C₄
prime radical	C₂⁴×C₈₉	C₂×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	C₂³×C₈	D₈₉
hypocenter	C₈₉	C₂×#424
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₈₉

commutator

C₂²×C₈₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂³×C₈×C₈₉

C₂

socle

C₂³×C₈₉

C₂²×C₄

prime radical

C₂⁴×C₈₉

C₂×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

hypercenter

C₂³×C₈

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
3. #19151624	C₂	direct (N × H)
4. C₂×#15667941	C₂	semidirect
14. #15667941	C₂²	semidirect
66. C₂×C₈₉	#424	semidirect
70. C₈₉	C₂×#424	semidirect
136. C₂	#19151624	direct (N × H)

H ≅ G/N

type

3. #19151624

C₂

direct (N × H)

4. C₂×#15667941

C₂

semidirect

14. #15667941

C₂²

semidirect

66. C₂×C₈₉

#424

semidirect

70. C₈₉

C₂×#424

semidirect

136. C₂

#19151624

direct (N × H)

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

subgroup	index
C₂²×#13083558	2
C₂×#15667971	2
#19151624	2
C₂×#15667941	2
C₂³×C₈×C₈₉	2
C₂×#424	89

subgroup

index

C₂³×C₈×C₈₉

Subgroups — C₂×#19151624 #26028810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19151624 #26028810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151624 #26028810