Group #26026219 — subgroups

Subgroups — #26026219

Order 11392.

352 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 123 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	1	yes	maximal, characteristic
2	#19151540	C₂	1	3	yes	maximal
3	#19151538	C₂	1	3	yes	maximal
4	#19151502	C₂	1	3	yes	maximal
5	#19151495	C₂	1	3	yes	maximal
6	#19151443	C₂	1	1	yes	maximal, characteristic
7	#401×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
10	C₂×#13083553	C₂²	1	3	yes	—
11	C₂×#13083553	C₂²	1	3	yes	—
12	C₂×C₄×Dic₈₉	C₂²	1	3	yes	—
13	#15667919	C₂²	1	3	yes	—
14	#15667919	C₂²	1	3	yes	—
15	#15667919	C₂²	1	3	yes	—
16	#15667919	C₂²	1	3	yes	—
17	#15667919	C₂²	1	3	yes	—
18	#95×C₈₉	C₂²	1	1	yes	characteristic
19	#95×C₈₉	C₂²	1	3	yes	—
20	#95×C₈₉	C₂²	1	3	yes	—
21	C₂×Dic₁₇₈	—	2	6	no	—
22	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₈₉	C₂³	1	3	yes	—
24	C₂²×Dic₈₉	C₂³	1	3	yes	—
25	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₈₉	—	4	3	no	—
27	#13083553	—	2	6	no	—
28	#13083553	—	4	3	no	—
29	C₄×Dic₈₉	—	2	6	no	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
33	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
34	C₂×C₄×C₈₉	—	2	3	no	—
35	C₂×C₄×C₈₉	—	2	3	no	—
36	C₂×C₄×C₈₉	—	2	3	no	—
37	C₂×C₄×C₈₉	—	2	3	no	—
38	C₂×C₄×C₈₉	—	2	6	no	—
39	C₂×C₄×C₈₉	—	2	3	no	—
40	C₂×Dic₈₉	—	2	3	no	—
41	C₂×Dic₈₉	—	2	3	no	—
42	C₂×Dic₈₉	—	2	3	no	—
43	C₂×Dic₈₉	—	2	3	no	—
44	C₂×Dic₈₉	—	2	3	no	—
45	C₂×Dic₈₉	—	2	3	no	—
46	C₂×Dic₈₉	—	2	3	no	—
47	C₂×Dic₈₉	#41	1	6	yes	—
48	Dic₁₇₈	—	4	4	no	—
49	C₂²×C₈₉	C₂×#41	1	3	yes	—
50	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₈₉	2₋¹⁺⁴	1	3	yes	—
52	C₄×C₈₉	—	4	1	no	—
53	C₄×C₈₉	—	4	3	no	—
54	C₄×C₈₉	—	4	3	no	—
55	Dic₈₉	—	4	1	no	—
56	Dic₈₉	—	4	3	no	—
57	Dic₈₉	—	4	3	no	—
58	Dic₈₉	#126	1	4	yes	—
59	C₂×C₈₉	C₂×#126	1	1	yes	characteristic
60	C₂×C₈₉	#541	1	3	yes	—
61	C₂×C₈₉	#542	1	3	yes	—
62	#2645	—	89	1	no	maximal, Sylow
63	C₈₉	#2645	1	1	yes	characteristic, complemented, Sylow
64	#401	—	89	1	no	—
65	#401	D₈₉	1	1	yes	characteristic
66	#400	—	89	3	no	—
67	#400	—	89	3	no	—
68	#391	—	89	3	no	—
69	#389	—	89	3	no	—
70	#376	—	89	1	no	—
71	C₂²×Q₈	—	89	1	no	—
72	C₂×C₄⋊C₄	—	89	3	no	—
73	C₂×C₄⋊C₄	—	89	3	no	—
74	C₂×C₄⋊C₄	—	89	3	no	—
75	C₂×C₄²	—	89	3	no	—
76	#95	—	89	3	no	—
77	#95	—	89	3	no	—
78	#95	—	89	3	no	—
79	#95	—	89	3	no	—
80	#95	—	89	3	no	—
81	#95	C₂×D₈₉	1	1	yes	characteristic
82	#95	C₂×D₈₉	1	3	yes	—
83	#95	C₂×D₈₉	1	3	yes	—
84	C₂×Q₈	—	178	6	no	—
85	C₂²×C₄	—	89	1	no	—
86	C₂²×C₄	—	89	1	no	—
87	C₂²×C₄	—	89	3	no	—
88	C₂²×C₄	—	89	3	no	—
89	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
90	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
91	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
92	C₄⋊C₄	—	356	3	no	—
93	C₄⋊C₄	—	356	3	no	—
94	C₄⋊C₄	—	178	6	no	—
95	C₄²	—	178	6	no	—
96	C₂³	C₂³×D₈₉	1	1	yes	characteristic
97	Q₈	—	356	4	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	89	6	no	—
105	C₂×C₄	—	178	3	no	—
106	C₂×C₄	—	178	3	no	—
107	C₂×C₄	—	178	3	no	—
108	C₂×C₄	—	178	3	no	—
109	C₂×C₄	—	178	3	no	—
110	C₂×C₄	—	178	3	no	—
111	C₂×C₄	—	178	3	no	—
112	C₂²	#41×D₈₉	1	3	yes	—
113	C₂²	#15668058	1	1	yes	characteristic
114	C₂²	#15668059	1	3	yes	—
115	C₄	—	4	3	no	—
116	C₄	—	4	3	no	—
117	C₄	—	4	1	no	—
118	C₄	—	89	4	no	—
119	C₄	—	356	3	no	—
120	C₄	—	356	3	no	—
121	C₄	—	356	1	no	—
122	C₂	#126×D₈₉	1	1	yes	characteristic
123	C₂	#19152205	1	3	yes	—
124	C₂	#19152214	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉⋊#401

C₂

yes

maximal, characteristic

#19151540

C₂

yes

maximal

#19151538

C₂

yes

maximal

#19151502

C₂

yes

maximal

#19151495

C₂

yes

maximal

#19151443

C₂

yes

maximal, characteristic

#401×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#95×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

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×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#41

yes

—

Dic₁₇₈

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#126

yes

—

C₂×C₈₉

C₂×#126

yes

characteristic

C₂×C₈₉

#541

yes

—

C₂×C₈₉

#542

yes

—

#2645

—

maximal, Sylow

C₈₉

#2645

yes

characteristic, complemented, Sylow

#401

—

#401

D₈₉

yes

—

—

—

—

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

—

—

—

—

—

C₂×D₈₉

yes

characteristic

#95

C₂×D₈₉

yes

—

#95

C₂×D₈₉

yes

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

356

—

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

—

178

—

111

C₂×C₄

—

178

—

112

C₂²

#41×D₈₉

yes

—

113

C₂²

#15668058

yes

characteristic

114

C₂²

#15668059

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

356

—

120

C₄

—

356

—

121

C₄

—

356

—

122

C₂

#126×D₈₉

yes

characteristic

123

C₂

#19152205

yes

—

124

C₂

#19152214

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#401×C₈₉	C₂
hypercenter	#401	D₈₉
hypocenter	C₈₉	#2645
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #401

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#401×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #401

N	H ≅ G/N	type
63. C₈₉	#2645	semidirect

H ≅ G/N

type

63. C₈₉

#2645

semidirect

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

subgroup	index
C₈₉⋊#401	2
#19151540	2
#19151538	2
#19151502	2
#19151495	2
#19151443	2
#401×C₈₉	2
#2645	89

subgroup

index

C₈₉⋊#401

#401×C₈₉

Subgroups — #26026219

Conjugacy classes of subgroups

Complements

Maximal subgroups