Group #26025922 — subgroups

Subgroups — #26025922

Order 11392.

428 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 141 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	#19151521	C₂	1	4	yes	maximal
2	#19151495	C₂	1	4	yes	maximal, complemented
3	#19151481	C₂	1	4	yes	maximal, complemented
4	C₄×C₂²⋊Dic₈₉	C₂	1	2	yes	maximal
5	#381×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
7	C₂×#13083553	C₂²	1	4	yes	—
8	C₂×#13083553	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₈₉	C₂²	1	4	yes	—
10	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
12	#15667919	C₂²	1	4	yes	—
13	#15667919	C₂²	1	4	yes	—
14	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
16	#95×C₈₉	C₂²	1	4	yes	—
17	C₂²×Dic₈₉	C₂³	1	4	yes	—
18	C₂²×Dic₈₉	C₂³	1	4	yes	—
19	C₄⋊Dic₈₉	—	2	8	no	—
20	#13083553	—	2	8	no	—
21	#13083553	—	4	4	no	—
22	C₂²⋊Dic₈₉	—	2	8	no	—
23	C₄×Dic₈₉	—	4	2	no	—
24	C₄×Dic₈₉	—	4	2	no	—
25	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	—	4	2	no	—
27	C₂²×C₄×C₈₉	Q₈	1	4	yes	—
28	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
30	C₂²⋊C₄×C₈₉	—	4	2	no	—
31	C₂³×C₈₉	—	2	4	no	—
32	C₂³×C₈₉	—	2	2	no	—
33	C₂³×C₈₉	C₂×Q₈	1	2	yes	—
34	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	2	2	no	—
36	C₂×C₄×C₈₉	—	2	2	no	—
37	C₂×C₄×C₈₉	—	2	4	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×Q₈	1	4	yes	—
42	C₂×Dic₈₉	—	2	4	no	—
43	C₂×Dic₈₉	—	2	4	no	—
44	C₂×Dic₈₉	—	2	4	no	—
45	C₂×Dic₈₉	—	2	4	no	—
46	C₂×Dic₈₉	—	2	4	no	—
47	C₂×Dic₈₉	#41	1	8	yes	—
48	C₂²×C₈₉	—	2	4	no	—
49	C₂²×C₈₉	—	2	2	no	—
50	C₂²×C₈₉	—	4	4	no	—
51	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₈₉	C₂×#41	1	4	yes	—
53	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
54	C₄×C₈₉	—	2	4	no	—
55	C₄×C₈₉	—	4	4	no	—
56	Dic₈₉	—	2	8	no	—
57	Dic₈₉	—	4	4	no	—
58	C₂×C₈₉	—	4	2	no	—
59	C₂×C₈₉	#533	1	2	yes	—
60	C₂×C₈₉	#543	1	1	yes	characteristic
61	C₂×C₈₉	#553	1	4	yes	—
62	#2608	—	89	1	no	maximal, Sylow
63	C₈₉	#2608	1	1	yes	characteristic, complemented, Sylow
64	#396	—	89	4	no	—
65	#389	—	89	4	no	—
66	#387	—	89	4	no	—
67	#381	D₈₉	1	1	yes	characteristic
68	C₄×C₂²⋊C₄	—	89	2	no	—
69	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	89	4	no	—
71	C₂×C₄⋊C₄	—	89	4	no	—
72	C₂×C₄⋊C₄	—	89	4	no	—
73	C₂×C₂²⋊C₄	—	89	4	no	—
74	C₂×C₂²⋊C₄	C₂×D₈₉	1	2	yes	—
75	C₂×C₄²	—	89	2	no	—
76	C₂×C₄²	—	89	2	no	—
77	#95	—	89	4	no	—
78	#95	—	89	4	no	—
79	#95	C₂×D₈₉	1	4	yes	—
80	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
81	C₂²×C₄	—	4	2	no	—
82	C₂²×C₄	—	89	4	no	—
83	C₂²×C₄	—	89	4	no	—
84	C₂²×C₄	Dic₁₇₈	1	4	yes	—
85	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
86	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
87	C₄⋊C₄	—	356	4	no	—
88	C₄⋊C₄	—	178	8	no	—
89	C₄⋊C₄	—	178	8	no	—
90	C₂²⋊C₄	—	4	2	no	—
91	C₂²⋊C₄	—	178	8	no	—
92	C₄²	—	356	2	no	—
93	C₄²	—	356	2	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	2	2	no	—
96	C₂³	C₂×Dic₁₇₈	1	2	yes	—
97	C₂³	C₂³×D₈₉	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	89	8	no	—
105	C₂×C₄	—	178	4	no	—
106	C₂×C₄	—	178	4	no	—
107	C₂×C₄	—	178	4	no	—
108	C₂×C₄	—	178	4	no	—
109	C₂×C₄	—	178	4	no	—
110	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
111	C₂²	—	2	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
115	C₂²	#41×D₈₉	1	4	yes	—
116	C₂²	#15668057	1	2	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	356	4	no	—
120	C₄	—	178	8	no	—
121	C₂	—	4	2	no	—
122	C₂	#19152151	1	2	yes	—
123	C₂	#19152224	1	1	yes	characteristic
124	C₂	#19152293	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151521

C₂

yes

maximal

#19151495

C₂

yes

maximal, complemented

#19151481

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₈₉

C₂

yes

maximal

#381×C₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×Q₈

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

C₂×Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#41

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#533

yes

—

C₂×C₈₉

#543

yes

characteristic

C₂×C₈₉

#553

yes

—

#2608

—

maximal, Sylow

C₈₉

#2608

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

C₄×C₂²⋊C₄

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₈₉

yes

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₇₈

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

C₂²⋊C₄

—

178

—

C₄²

—

356

—

C₄²

—

356

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₁₇₈

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

—

178

—

108

C₂×C₄

—

178

—

109

C₂×C₄

—

178

—

110

C₂×C₄

C₂×Dic₁₇₈

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂²×Dic₁₇₈

yes

characteristic

115

C₂²

#41×D₈₉

yes

—

116

C₂²

#15668057

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

356

—

120

C₄

—

178

—

121

C₂

—

122

C₂

#19152151

yes

—

123

C₂

#19152224

yes

characteristic

124

C₂

#19152293

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#381×C₈₉	C₂
prime radical	C₂⁴×C₈₉	C₂³
hypercenter	#381	D₈₉
hypocenter	C₈₉	#2608
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #381

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#381×C₈₉

C₂

prime radical

C₂⁴×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #381

N	H ≅ G/N	type
2. #19151495	C₂	semidirect
3. #19151481	C₂	semidirect
63. C₈₉	#2608	semidirect

H ≅ G/N

type

2. #19151495

C₂

semidirect

3. #19151481

C₂

semidirect

63. C₈₉

#2608

semidirect

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

subgroup	index
#19151521	2
#19151495	2
#19151481	2
C₄×C₂²⋊Dic₈₉	2
#381×C₈₉	2
#2608	89

subgroup

index

#19151521

#19151495

#19151481

C₄×C₂²⋊Dic₈₉

#381×C₈₉

#2608

Subgroups — #26025922

Conjugacy classes of subgroups

Complements

Maximal subgroups