Group #26027631 — subgroups

Subgroups — #26027631

Order 11392.

520 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₈₉⋊#401	C₂	1	2	yes	maximal, complemented
2	#19151510	C₂	1	3	yes	maximal, complemented
3	#19151478	C₂	1	6	yes	maximal, complemented
4	#19151459	C₂	1	3	yes	maximal, complemented
5	#392×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₂³×Dic₈₉	C₂²	1	3	yes	—
8	C₂×C₂²⋊Dic₈₉	C₂²	1	3	yes	—
9	C₂×C₂²⋊Dic₈₉	C₂²	1	6	yes	—
10	#15667919	C₂²	1	3	yes	—
11	#15667919	C₂²	1	6	yes	—
12	#15667919	C₂²	1	6	yes	—
13	C₂²×D₄×C₈₉	C₂²	1	3	yes	—
14	C₂×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
15	#95×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₈	—	4	3	no	—
17	C₂²×Dic₈₉	—	2	6	no	—
18	C₂²×Dic₈₉	—	4	3	no	—
19	C₂²×Dic₈₉	—	4	3	no	—
20	C₂²×Dic₈₉	C₂³	1	6	yes	—
21	C₂²×Dic₈₉	C₂³	1	2	yes	—
22	C₂²⋊Dic₈₉	—	4	3	no	—
23	C₂²⋊Dic₈₉	—	4	6	no	—
24	C₂⁴×C₈₉	C₂³	1	3	yes	—
25	C₂×D₄×C₈₉	—	2	12	no	—
26	C₂×D₄×C₈₉	—	4	3	no	—
27	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
29	C₂²⋊C₄×C₈₉	—	4	3	no	—
30	C₂³×C₈₉	—	2	6	no	—
31	C₂³×C₈₉	—	2	3	no	—
32	C₂³×C₈₉	—	2	6	no	—
33	C₂³×C₈₉	—	2	3	no	—
34	C₂³×C₈₉	#41	1	6	yes	—
35	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
36	D₄×C₈₉	—	4	12	no	—
37	C₂×C₄×C₈₉	—	2	6	no	—
38	C₂×C₄×C₈₉	—	2	3	no	—
39	C₂×C₄×C₈₉	—	2	3	no	—
40	C₂×Dic₈₉	—	2	6	no	—
41	C₂×Dic₈₉	—	2	6	no	—
42	C₂×Dic₈₉	—	2	6	no	—
43	C₂×Dic₈₉	—	2	6	no	—
44	C₂×Dic₈₉	—	4	12	no	—
45	Dic₁₇₈	—	4	4	no	—
46	C₂²×C₈₉	—	2	6	no	—
47	C₂²×C₈₉	—	2	12	no	—
48	C₂²×C₈₉	—	4	6	no	—
49	C₂²×C₈₉	—	4	3	no	—
50	C₂²×C₈₉	—	4	3	no	—
51	C₂²×C₈₉	C₂×#41	1	3	yes	—
52	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
54	C₄×C₈₉	—	4	3	no	—
55	C₄×C₈₉	—	4	1	no	—
56	Dic₈₉	—	4	6	no	—
57	Dic₈₉	—	4	2	no	—
58	C₂×C₈₉	—	4	6	no	—
59	C₂×C₈₉	#538	1	3	yes	—
60	C₂×C₈₉	#551	1	3	yes	—
61	C₂×C₈₉	#560	1	1	yes	characteristic
62	#2760	—	89	1	no	maximal, Sylow
63	C₈₉	#2760	1	1	yes	characteristic, complemented, Sylow
64	#401	—	89	2	no	—
65	#393	—	89	3	no	—
66	#392	D₈₉	1	1	yes	characteristic
67	#386	—	89	6	no	—
68	#381	—	89	3	no	—
69	C₂²×Q₈	—	89	1	no	—
70	C₂²×D₄	C₂×D₈₉	1	3	yes	—
71	C₂³×C₄	—	89	3	no	—
72	C₂×C₂²⋊C₄	—	89	3	no	—
73	C₂×C₂²⋊C₄	—	89	6	no	—
74	C₂×C₂²⋊C₄	C₂×D₈₉	1	3	yes	—
75	#95	—	89	3	no	—
76	#95	—	89	6	no	—
77	#95	—	89	6	no	—
78	#95	C₂×D₈₉	1	1	yes	characteristic
79	C₂⁴	C₂²×D₈₉	1	3	yes	—
80	C₂×Q₈	—	356	3	no	—
81	C₂×D₄	—	2	12	no	—
82	C₂×D₄	—	4	3	no	—
83	C₂²×C₄	—	89	6	no	—
84	C₂²×C₄	—	89	2	no	—
85	C₂²×C₄	—	356	3	no	—
86	C₂²×C₄	—	356	3	no	—
87	C₂²×C₄	—	178	6	no	—
88	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
89	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
90	C₂²⋊C₄	—	4	3	no	—
91	C₂²⋊C₄	—	356	6	no	—
92	C₂²⋊C₄	—	356	3	no	—
93	C₂³	—	2	6	no	—
94	C₂³	—	2	3	no	—
95	C₂³	—	2	6	no	—
96	C₂³	—	2	3	no	—
97	C₂³	#13083573	1	6	yes	—
98	C₂³	C₂³×D₈₉	1	1	yes	characteristic
99	Q₈	—	356	4	no	—
100	D₄	—	4	12	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	356	12	no	—
105	C₂×C₄	—	178	6	no	—
106	C₂×C₄	—	178	6	no	—
107	C₂×C₄	—	178	6	no	—
108	C₂×C₄	—	178	6	no	—
109	C₂²	—	2	6	no	—
110	C₂²	—	2	12	no	—
111	C₂²	—	4	6	no	—
112	C₂²	—	4	3	no	—
113	C₂²	—	4	3	no	—
114	C₂²	C₂×#13083573	1	3	yes	—
115	C₂²	#15668057	1	3	yes	—
116	C₂²	#15668058	1	1	yes	characteristic
117	C₄	—	4	3	no	—
118	C₄	—	4	1	no	—
119	C₄	—	356	6	no	—
120	C₄	—	356	2	no	—
121	C₂	—	4	6	no	—
122	C₂	#19152185	1	3	yes	—
123	C₂	#19152279	1	3	yes	—
124	C₂	#19152335	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉⋊#401

C₂

yes

maximal, complemented

#19151510

C₂

yes

maximal, complemented

#19151478

C₂

yes

maximal, complemented

#19151459

C₂

yes

maximal, complemented

#392×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂³×Dic₈₉

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×D₄×C₈₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

#41

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#538

yes

—

C₂×C₈₉

#551

yes

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#401

—

#393

—

#392

D₈₉

yes

characteristic

#386

—

#381

—

C₂²×Q₈

—

C₂²×D₄

C₂×D₈₉

yes

—

C₂³×C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

—

—

—

C₂×D₈₉

yes

characteristic

C₂⁴

C₂²×D₈₉

yes

—

C₂×Q₈

—

356

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

356

—

C₂²×C₄

—

356

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

356

—

C₂²⋊C₄

—

356

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#13083573

yes

—

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

356

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

356

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

—

178

—

108

C₂×C₄

—

178

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×#13083573

yes

—

115

C₂²

#15668057

yes

—

116

C₂²

#15668058

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

356

—

120

C₄

—

356

—

121

C₂

—

122

C₂

#19152185

yes

—

123

C₂

#19152279

yes

—

124

C₂

#19152335

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting, prime radical	#392×C₈₉	C₂
hypercenter	#392	D₈₉
hypocenter	C₈₉	#2760
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #392

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting, prime radical

#392×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #392

N	H ≅ G/N	type
1. C₈₉⋊#401	C₂	semidirect
2. #19151510	C₂	semidirect
3. #19151478	C₂	semidirect
4. #19151459	C₂	semidirect
63. C₈₉	#2760	semidirect

H ≅ G/N

type

1. C₈₉⋊#401

C₂

semidirect

2. #19151510

C₂

semidirect

3. #19151478

C₂

semidirect

4. #19151459

C₂

semidirect

63. C₈₉

#2760

semidirect

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

subgroup	index
C₈₉⋊#401	2
#19151510	2
#19151478	2
#19151459	2
#392×C₈₉	2
#2760	89

subgroup

index

C₈₉⋊#401

#392×C₈₉

Subgroups — #26027631

Conjugacy classes of subgroups

Complements

Maximal subgroups