Group #26028598 — subgroups

Subgroups — #26028598

Order 11392.

924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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, complemented
2	C₂×#15667987	C₂	1	6	yes	maximal, complemented
3	C₂×C₄²×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151499	C₂	1	6	yes	maximal, complemented
5	#384×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	6	yes	—
7	C₂²×D₃₅₆	C₂²	1	4	yes	—
8	C₂²×C₄×D₈₉	C₂²	1	3	yes	—
9	#15668009	—	2	4	no	—
10	#15667987	—	2	24	no	—
11	C₂×#13083551	C₂²	1	12	yes	—
12	C₂×C₄×Dic₈₉	C₂²	1	3	yes	—
13	C₄²×D₈₉	—	2	4	no	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	6	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×D₈₉	C₂³	1	4	yes	—
17	C₂³×D₈₉	C₂³	1	1	yes	characteristic
18	C₂×D₃₅₆	—	2	24	no	—
19	C₂×D₃₅₆	—	2	12	no	—
20	C₂×D₃₅₆	—	2	12	no	—
21	C₂×D₃₅₆	—	2	12	no	—
22	C₂²×Dic₈₉	C₂³	1	3	yes	—
23	C₂×C₄×D₈₉	—	2	6	no	—
24	C₂×C₄×D₈₉	—	2	6	no	—
25	C₂×C₄×D₈₉	D₄	1	12	yes	complemented
26	#13083551	—	2	24	no	—
27	C₄×Dic₈₉	—	2	6	no	—
28	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
30	C₄⋊C₄×C₈₉	—	2	12	no	—
31	C₄²×C₈₉	D₄	1	4	yes	complemented
32	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
33	C₂²×D₈₉	—	2	1	no	—
34	C₂²×D₈₉	—	2	3	no	—
35	C₂²×D₈₉	—	2	12	no	—
36	C₂²×D₈₉	—	2	4	no	—
37	C₂²×D₈₉	—	2	12	no	—
38	C₂²×D₈₉	C₂×D₄	1	6	yes	—
39	C₂×C₄×C₈₉	—	2	12	no	—
40	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
41	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
42	C₂×C₄×C₈₉	#41	1	6	yes	—
43	C₂×Dic₈₉	—	2	3	no	—
44	C₂×Dic₈₉	—	2	3	no	—
45	C₂×Dic₈₉	C₂×D₄	1	6	yes	—
46	D₃₅₆	—	4	16	no	—
47	D₃₅₆	—	4	24	no	—
48	C₄×D₈₉	—	2	24	no	—
49	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₈₉	C₂²×D₄	1	3	yes	—
51	C₂²×C₈₉	C₂×#41	1	3	yes	—
52	C₂×D₈₉	—	2	6	no	—
53	C₂×D₈₉	—	2	6	no	—
54	C₂×D₈₉	—	4	12	no	—
55	C₂×D₈₉	—	4	4	no	—
56	C₂×D₈₉	—	4	12	no	—
57	C₂×D₈₉	#127	1	4	yes	complemented
58	C₄×C₈₉	—	4	4	no	—
59	C₄×C₈₉	#121	1	12	yes	complemented
60	Dic₈₉	—	2	6	no	—
61	C₂×C₈₉	C₂×#121	1	3	yes	—
62	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
63	C₂×C₈₉	#532	1	3	yes	—
64	D₈₉	—	2	4	no	—
65	D₈₉	—	8	4	no	—
66	#2856	—	89	1	no	maximal, Sylow
67	C₈₉	#2856	1	1	yes	characteristic, complemented, Sylow
68	C₂×#127	—	89	1	no	—
69	C₂×#121	—	89	6	no	—
70	C₂²×C₄²	—	89	1	no	—
71	#390	—	89	6	no	—
72	#384	D₈₉	1	1	yes	characteristic, complemented
73	C₂²×D₄	—	89	4	no	—
74	C₂²×D₄	—	89	6	no	—
75	C₂³×C₄	—	89	3	no	—
76	#127	—	178	4	no	—
77	#121	—	178	24	no	—
78	C₂×C₄⋊C₄	C₂×D₈₉	1	6	yes	—
79	C₂×C₂²⋊C₄	—	89	12	no	—
80	C₂×C₄²	—	89	3	no	—
81	C₂×C₄²	—	178	4	no	—
82	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
83	C₂⁴	—	89	1	no	—
84	C₂⁴	—	89	4	no	—
85	C₂×D₄	—	178	24	no	—
86	C₂×D₄	—	178	12	no	—
87	C₂×D₄	—	178	12	no	—
88	C₂×D₄	—	178	12	no	—
89	C₂²×C₄	—	89	3	no	—
90	C₂²×C₄	—	89	12	no	—
91	C₂²×C₄	—	178	6	no	—
92	C₂²×C₄	—	178	6	no	—
93	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
94	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
95	C₄⋊C₄	—	2	12	no	—
96	C₂²⋊C₄	—	178	24	no	—
97	C₄²	—	178	6	no	—
98	C₄²	D₃₅₆	1	4	yes	complemented
99	C₂³	—	89	6	no	—
100	C₂³	—	178	12	no	—
101	C₂³	—	178	4	no	—
102	C₂³	—	178	12	no	—
103	C₂³	—	178	1	no	—
104	C₂³	—	178	3	no	—
105	C₂³	C₂³×D₈₉	1	1	yes	characteristic
106	D₄	—	356	16	no	—
107	D₄	—	356	24	no	—
108	C₂×C₄	—	2	12	no	—
109	C₂×C₄	—	89	6	no	—
110	C₂×C₄	—	178	3	no	—
111	C₂×C₄	—	178	3	no	—
112	C₂×C₄	—	178	24	no	—
113	C₂×C₄	D₄×D₈₉	1	6	yes	—
114	C₂×C₄	C₂×D₃₅₆	1	6	yes	—
115	C₂×C₄	#13083575	1	6	yes	—
116	C₂²	—	89	4	no	—
117	C₂²	—	356	4	no	—
118	C₂²	—	356	12	no	—
119	C₂²	—	356	12	no	—
120	C₂²	—	178	6	no	—
121	C₂²	—	178	6	no	—
122	C₂²	C₂×D₄×D₈₉	1	3	yes	—
123	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
124	C₂²	C₂×#13083575	1	3	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	178	6	no	—
127	C₄	#15667987	1	12	yes	complemented
128	C₂	—	178	4	no	—
129	C₂	—	712	4	no	—
130	C₂	C₂×#15667987	1	3	yes	—
131	C₂	#127×D₈₉	1	1	yes	characteristic
132	C₂	#19152149	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668009

C₂

yes

maximal, characteristic, complemented

C₂×#15667987

C₂

yes

maximal, complemented

C₂×C₄²×D₈₉

C₂

yes

maximal, characteristic, complemented

#19151499

C₂

yes

maximal, complemented

#384×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

—

C₂²×D₃₅₆

C₂²

yes

—

C₂²×C₄×D₈₉

C₂²

yes

—

#15668009

—

#15667987

—

C₂×#13083551

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₄²×D₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

C₂³

yes

—

C₂³×D₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

C₂³

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

D₄

yes

complemented

#13083551

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

complemented

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×D₄

yes

—

C₂²×C₈₉

C₂×#41

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

#127

yes

complemented

C₄×C₈₉

—

C₄×C₈₉

#121

yes

complemented

Dic₈₉

—

C₂×C₈₉

C₂×#121

yes

—

C₂×C₈₉

C₂×#127

yes

characteristic

C₂×C₈₉

#532

yes

—

D₈₉

—

D₈₉

—

#2856

—

maximal, Sylow

C₈₉

#2856

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#121

—

C₂²×C₄²

—

#390

—

#384

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

#127

—

178

—

#121

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₂⁴

—

C₂⁴

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

178

—

C₄²

—

178

—

C₄²

D₃₅₆

yes

complemented

C₂³

—

100

C₂³

—

178

—

101

C₂³

—

178

—

102

C₂³

—

178

—

103

C₂³

—

178

—

104

C₂³

—

178

—

105

C₂³

C₂³×D₈₉

yes

characteristic

106

D₄

—

356

—

107

D₄

—

356

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

178

—

111

C₂×C₄

—

178

—

112

C₂×C₄

—

178

—

113

C₂×C₄

D₄×D₈₉

yes

—

114

C₂×C₄

C₂×D₃₅₆

yes

—

115

C₂×C₄

#13083575

yes

—

116

C₂²

—

117

C₂²

—

356

—

118

C₂²

—

356

—

119

C₂²

—

356

—

120

C₂²

—

178

—

121

C₂²

—

178

—

122

C₂²

C₂×D₄×D₈₉

yes

—

123

C₂²

C₂²×D₃₅₆

yes

characteristic

124

C₂²

C₂×#13083575

yes

—

125

C₄

—

126

C₄

—

178

—

127

C₄

#15667987

yes

complemented

128

C₂

—

178

—

129

C₂

—

712

—

130

C₂

C₂×#15667987

yes

—

131

C₂

#127×D₈₉

yes

characteristic

132

C₂

#19152149

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#384×C₈₉	C₂
hypercenter	#384	D₈₉
hypocenter	C₈₉	#2856
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#384×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#15668009	C₂	semidirect
2. C₂×#15667987	C₂	semidirect
3. C₂×C₄²×D₈₉	C₂	semidirect
4. #19151499	C₂	semidirect
5. #384×C₈₉	C₂	semidirect
25. C₂×C₄×D₈₉	D₄	semidirect
31. C₄²×C₈₉	D₄	semidirect
57. C₂×D₈₉	#127	semidirect
59. C₄×C₈₉	#121	semidirect
67. C₈₉	#2856	semidirect
72. #384	D₈₉	semidirect
98. C₄²	D₃₅₆	semidirect
127. C₄	#15667987	semidirect

H ≅ G/N

type

1. C₂×#15668009

C₂

semidirect

2. C₂×#15667987

C₂

semidirect

3. C₂×C₄²×D₈₉

C₂

semidirect

4. #19151499

C₂

semidirect

5. #384×C₈₉

C₂

semidirect

25. C₂×C₄×D₈₉

D₄

semidirect

31. C₄²×C₈₉

D₄

semidirect

57. C₂×D₈₉

#127

semidirect

59. C₄×C₈₉

#121

semidirect

67. C₈₉

#2856

semidirect

72. #384

D₈₉

semidirect

98. C₄²

D₃₅₆

semidirect

127. C₄

#15667987

semidirect

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

subgroup	index
C₂×#15668009	2
C₂×#15667987	2
C₂×C₄²×D₈₉	2
#19151499	2
#384×C₈₉	2
#2856	89

subgroup

index

C₂×C₄²×D₈₉

#384×C₈₉

Subgroups — #26028598

Conjugacy classes of subgroups

Complements

Maximal subgroups