Group #26023432 — subgroups

Subgroups — #26023432

Order 11392.

424 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15668012	C₂	1	1	yes	maximal, characteristic
2	#19151502	C₂	1	4	yes	maximal
3	#19151483	C₂	1	4	yes	maximal
4	#19151481	C₂	1	4	yes	maximal
5	#19151466	C₂	1	1	yes	maximal, characteristic
6	#383×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
8	#15668012	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
11	C₂×#13083553	C₂²	1	4	yes	—
12	C₂×C₄×Dic₈₉	C₂²	1	4	yes	—
13	#15667919	C₂²	1	2	yes	—
14	#15667919	C₂²	1	2	yes	—
15	#15667919	C₂²	1	2	yes	—
16	#15667919	C₂²	1	8	yes	—
17	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#95×C₈₉	C₂²	1	2	yes	—
19	#95×C₈₉	C₂²	1	2	yes	—
20	#95×C₈₉	C₂²	1	2	yes	—
21	C₂×Dic₁₇₈	—	4	2	no	—
22	C₂×Dic₁₇₈	—	4	2	no	—
23	C₂×Dic₁₇₈	C₂×C₄	1	8	yes	—
24	C₂²×Dic₈₉	C₂³	1	4	yes	—
25	C₂²×Dic₈₉	C₂³	1	4	yes	—
26	C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
27	C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
28	#13083553	—	4	4	no	—
29	C₄×Dic₈₉	—	4	4	no	—
30	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₄²×C₈₉	C₂×C₄	1	4	yes	—
35	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₈₉	—	2	2	no	—
37	C₂×C₄×C₈₉	—	2	2	no	—
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	2	4	no	—
40	C₂×C₄×C₈₉	—	2	4	no	—
41	C₂×C₄×C₈₉	—	2	4	no	—
42	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
45	C₂×Dic₈₉	—	2	4	no	—
46	C₂×Dic₈₉	—	2	4	no	—
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	—	2	4	no	—
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	C₂²×C₄	1	8	yes	—
51	Dic₁₇₈	—	4	8	no	—
52	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
55	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
56	C₄×C₈₉	—	2	2	no	—
57	C₄×C₈₉	—	2	2	no	—
58	C₄×C₈₉	—	2	2	no	—
59	C₄×C₈₉	—	4	4	no	—
60	Dic₈₉	—	2	8	no	—
61	Dic₈₉	—	4	4	no	—
62	C₂×C₈₉	#518	1	1	yes	characteristic
63	C₂×C₈₉	#519	1	1	yes	characteristic
64	C₂×C₈₉	#519	1	1	yes	characteristic
65	C₂×C₈₉	#563	1	4	yes	—
66	#2381	—	89	1	no	maximal, Sylow
67	C₈₉	#2381	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄⋊Q₈	—	89	1	no	—
69	#391	—	89	4	no	—
70	#387	—	89	4	no	—
71	#387	—	89	4	no	—
72	#383	—	89	1	no	—
73	#383	D₈₉	1	1	yes	characteristic
74	C₂²×Q₈	—	89	2	no	—
75	C₄⋊Q₈	—	89	8	no	—
76	C₂×C₄⋊C₄	—	89	2	no	—
77	C₂×C₄⋊C₄	—	89	2	no	—
78	C₂×C₄⋊C₄	—	89	4	no	—
79	C₂×C₄²	—	89	4	no	—
80	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
81	#95	—	89	2	no	—
82	#95	—	89	2	no	—
83	#95	—	89	2	no	—
84	#95	—	89	8	no	—
85	#95	C₂×D₈₉	1	2	yes	—
86	#95	C₂×D₈₉	1	2	yes	—
87	#95	C₂×D₈₉	1	2	yes	—
88	C₂×Q₈	—	89	8	no	—
89	C₂×Q₈	—	356	2	no	—
90	C₂×Q₈	—	356	2	no	—
91	C₂²×C₄	—	89	4	no	—
92	C₂²×C₄	—	89	4	no	—
93	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
94	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
97	C₄⋊C₄	—	89	8	no	—
98	C₄⋊C₄	—	89	8	no	—
99	C₄⋊C₄	—	356	4	no	—
100	C₄²	—	356	4	no	—
101	C₄²	C₄×D₈₉	1	4	yes	—
102	C₂³	C₂³×D₈₉	1	1	yes	characteristic
103	Q₈	—	356	8	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	89	8	no	—
111	C₂×C₄	—	178	4	no	—
112	C₂×C₄	—	178	4	no	—
113	C₂×C₄	—	178	4	no	—
114	C₂×C₄	—	178	4	no	—
115	C₂×C₄	—	178	4	no	—
116	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
118	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
119	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
120	C₂²	#15668058	1	2	yes	—
121	C₂²	#15668059	1	2	yes	—
122	C₂²	#15668059	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	356	4	no	—
128	C₄	—	178	8	no	—
129	C₂	#19152072	1	1	yes	characteristic
130	C₂	#19152075	1	1	yes	characteristic
131	C₂	#19152075	1	1	yes	characteristic
132	C₂	#19152350	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668012

C₂

yes

maximal, characteristic

#19151502

C₂

yes

maximal

#19151483

C₂

yes

maximal

#19151481

C₂

yes

maximal

#19151466

C₂

yes

maximal, characteristic

#383×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668012

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083553

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂²×C₄

yes

—

Dic₁₇₈

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

#2381

—

maximal, Sylow

C₈₉

#2381

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

—

C₂×D₈₉

yes

—

#95

C₂×D₈₉

yes

—

#95

C₂×D₈₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

100

C₄²

—

356

—

101

C₄²

C₄×D₈₉

yes

—

102

C₂³

C₂³×D₈₉

yes

characteristic

103

Q₈

—

356

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

178

—

112

C₂×C₄

—

178

—

113

C₂×C₄

—

178

—

114

C₂×C₄

—

178

—

115

C₂×C₄

—

178

—

116

C₂×C₄

C₂×C₄×D₈₉

yes

—

117

C₂×C₄

C₂×C₄×D₈₉

yes

—

118

C₂×C₄

C₂×C₄×D₈₉

yes

—

119

C₂²

C₂²×C₄×D₈₉

yes

characteristic

120

C₂²

#15668058

yes

—

121

C₂²

#15668059

yes

—

122

C₂²

#15668059

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

356

—

128

C₄

—

178

—

129

C₂

#19152072

yes

characteristic

130

C₂

#19152075

yes

characteristic

131

C₂

#19152075

yes

characteristic

132

C₂

#19152350

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator	C₂²×C₈₉	C₂³×C₄
Fitting	#383×C₈₉	C₂
socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
hypercenter	#383	D₈₉
hypocenter	C₈₉	#2381
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator

C₂²×C₈₉

C₂³×C₄

Fitting

#383×C₈₉

C₂

socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
8. #15668012	C₄	semidirect
67. C₈₉	#2381	semidirect

H ≅ G/N

type

8. #15668012

C₄

semidirect

67. C₈₉

#2381

semidirect

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

subgroup	index
C₂×#15668012	2
#19151502	2
#19151483	2
#19151481	2
#19151466	2
#383×C₈₉	2
#2381	89

subgroup

index

#383×C₈₉

Subgroups — #26023432

Conjugacy classes of subgroups

Complements

Maximal subgroups