Group #26030588 — subgroups

Subgroups — C₂×#19151964 #26030588

Order 11392.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15668009	C₂	1	1	yes	maximal, characteristic
2	#19151964	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15667932	C₂	1	4	yes	maximal, complemented
4	C₂×#15667923	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	—	2	2	no	—
7	C₂²×D₃₅₆	C₂²	1	2	yes	—
8	C₂²×#10623494	C₂²	1	2	yes	—
9	#15668009	—	2	2	no	—
10	#15668009	C₂²	1	4	yes	—
11	#15667932	C₂²	1	16	yes	complemented
12	#15667923	C₂²	1	4	yes	complemented
13	#125×C₈₉	—	2	2	no	—
14	#125×C₈₉	C₂²	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
16	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	C₂³×D₈₉	—	2	2	no	—
19	C₂×D₃₅₆	—	2	8	no	—
20	C₂×D₃₅₆	—	2	2	no	—
21	C₂×D₃₅₆	—	2	8	no	—
22	C₂×D₃₅₆	—	4	8	no	—
23	C₂×D₃₅₆	C₂³	1	4	yes	—
24	C₂×#10623494	—	2	4	no	—
25	C₂×#10623494	C₂³	1	4	yes	—
26	C₂²×C₄×C₈₉	—	2	2	no	—
27	C₂²×C₄×C₈₉	D₄	1	2	yes	—
28	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	—	2	8	no	—
30	C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
32	C₄²×C₈₉	D₄	1	2	yes	—
33	C₄²×C₈₉	C₂³	1	2	yes	—
34	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₈₉	—	2	4	no	—
36	C₂²×D₈₉	—	4	4	no	—
37	C₂²×D₈₉	—	4	8	no	—
38	C₂×C₄×C₈₉	—	2	4	no	—
39	C₂×C₄×C₈₉	—	2	4	no	—
40	C₂×C₄×C₈₉	—	4	4	no	—
41	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	#41	1	4	yes	—
44	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
45	D₃₅₆	—	2	8	no	—
46	D₃₅₆	—	4	16	no	—
47	D₃₅₆	—	4	4	no	—
48	#10623494	—	2	4	no	—
49	C₂²×C₈₉	#124	1	2	yes	—
50	C₂²×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
51	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₈₉	—	4	4	no	—
53	C₂×D₈₉	—	4	8	no	—
54	C₂×D₈₉	—	8	8	no	—
55	C₄×C₈₉	—	2	4	no	—
56	C₄×C₈₉	—	4	4	no	—
57	C₄×C₈₉	#124	1	2	yes	—
58	C₄×C₈₉	C₂×#41	1	2	yes	—
59	C₂×C₈₉	#490	1	4	yes	complemented
60	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
61	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
62	D₈₉	—	8	4	no	—
63	C₂×#490	—	89	1	no	maximal, Sylow
64	C₈₉	C₂×#490	1	1	yes	characteristic, complemented, Sylow
65	C₂×#127	—	89	1	no	—
66	C₂×#125	D₈₉	1	1	yes	characteristic, complemented
67	#490	—	89	8	no	—
68	C₂×#102	—	89	4	no	—
69	C₂×#97	—	89	1	no	—
70	C₂²×D₄	—	89	2	no	—
71	C₂²×D₄	—	178	2	no	—
72	C₂²×C₈	—	89	2	no	—
73	#127	—	89	4	no	—
74	#127	—	178	2	no	—
75	#125	—	2	2	no	—
76	#125	C₂×D₈₉	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
79	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
80	#102	—	89	16	no	—
81	#97	—	89	4	no	—
82	C₂⁴	—	178	2	no	—
83	C₂×D₄	—	89	4	no	—
84	C₂×D₄	—	356	8	no	—
85	C₂×D₄	—	178	8	no	—
86	C₂×D₄	—	178	2	no	—
87	C₂×D₄	—	178	8	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
90	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
91	C₂×C₈	—	89	4	no	—
92	C₂×C₈	—	178	4	no	—
93	C₄⋊C₄	—	2	8	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
96	C₄²	C₂²⋊D₈₉	1	2	yes	—
97	C₄²	C₂²×D₈₉	1	2	yes	—
98	C₂³	—	356	4	no	—
99	C₂³	—	356	8	no	—
100	C₂³	—	178	4	no	—
101	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
102	D₄	—	356	16	no	—
103	D₄	—	356	4	no	—
104	D₄	—	178	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
110	C₂×C₄	#13083575	1	4	yes	—
111	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
112	C₈	—	178	4	no	—
113	C₂²	—	356	8	no	—
114	C₂²	—	356	4	no	—
115	C₂²	—	712	8	no	—
116	C₂²	#15668001	1	2	yes	—
117	C₂²	#15668037	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	4	4	no	—
121	C₄	#15668001	1	2	yes	—
122	C₄	C₂×#13083575	1	2	yes	—
123	C₂	—	712	4	no	—
124	C₂	#19151964	1	4	yes	direct product factor
125	C₂	C₂×#15668001	1	1	yes	characteristic
126	C₂	C₂×#15668037	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668009

C₂

yes

maximal, characteristic

#19151964

C₂

yes

maximal, direct product factor

C₂×#15667932

C₂

yes

maximal, complemented

C₂×#15667923

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

—

C₂²×D₃₅₆

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

#15668009

—

#15668009

C₂²

yes

—

#15667932

C₂²

yes

complemented

#15667923

C₂²

yes

complemented

#125×C₈₉

—

#125×C₈₉

C₂²

yes

complemented

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

C₂³

yes

—

C₂×#10623494

—

C₂×#10623494

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

#10623494

—

C₂²×C₈₉

#124

yes

—

C₂²×C₈₉

AGL(1,ℤ/8)

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

C₄×C₈₉

C₂×#41

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

C₂×#490

—

maximal, Sylow

C₈₉

C₂×#490

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#125

D₈₉

yes

characteristic, complemented

#490

—

C₂×#102

—

C₂×#97

—

C₂²×D₄

—

C₂²×D₄

—

178

—

C₂²×C₈

—

#127

—

#127

—

178

—

#125

—

#125

C₂×D₈₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#102

—

#97

—

C₂⁴

—

178

—

C₂×D₄

—

C₂×D₄

—

356

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₈₉

yes

—

C₄²

C₂²⋊D₈₉

yes

—

C₄²

C₂²×D₈₉

yes

—

C₂³

—

356

—

C₂³

—

356

—

100

C₂³

—

178

—

101

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

102

D₄

—

356

—

103

D₄

—

356

—

104

D₄

—

178

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

109

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

110

C₂×C₄

#13083575

yes

—

111

C₂×C₄

C₂³×D₈₉

yes

characteristic

112

C₈

—

178

—

113

C₂²

—

356

—

114

C₂²

—

356

—

115

C₂²

—

712

—

116

C₂²

#15668001

yes

—

117

C₂²

#15668037

yes

—

118

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#15668001

yes

—

122

C₄

C₂×#13083575

yes

—

123

C₂

—

712

—

124

C₂

#19151964

yes

direct product factor

125

C₂

C₂×#15668001

yes

characteristic

126

C₂

C₂×#15668037

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#125×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂×#15668009	C₂
hypercenter	C₂×#125	D₈₉
hypocenter	C₈₉	C₂×#490
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×#125×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

C₂

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #19151964	C₂	direct (N × H)
3. C₂×#15667932	C₂	semidirect
4. C₂×#15667923	C₂	semidirect
5. C₂×#125×C₈₉	C₂	semidirect
11. #15667932	C₂²	semidirect
12. #15667923	C₂²	semidirect
14. #125×C₈₉	C₂²	semidirect
59. C₂×C₈₉	#490	semidirect
64. C₈₉	C₂×#490	semidirect
66. C₂×#125	D₈₉	semidirect
76. #125	C₂×D₈₉	semidirect
124. C₂	#19151964	direct (N × H)

H ≅ G/N

type

2. #19151964

C₂

direct (N × H)

3. C₂×#15667932

C₂

semidirect

4. C₂×#15667923

C₂

semidirect

5. C₂×#125×C₈₉

C₂

semidirect

11. #15667932

C₂²

semidirect

12. #15667923

C₂²

semidirect

14. #125×C₈₉

C₂²

semidirect

59. C₂×C₈₉

#490

semidirect

64. C₈₉

C₂×#490

semidirect

66. C₂×#125

D₈₉

semidirect

76. #125

C₂×D₈₉

semidirect

124. C₂

#19151964

direct (N × H)

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

subgroup	index
C₂×#15668009	2
#19151964	2
C₂×#15667932	2
C₂×#15667923	2
C₂×#125×C₈₉	2
C₂×#490	89

subgroup

index

C₂×#125×C₈₉

Subgroups — C₂×#19151964 #26030588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19151964 #26030588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151964 #26030588