Group #26028519 — subgroups

Subgroups — #26028519

Order 11392.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#19151540	C₂	1	3	yes	maximal
2	#19151532	C₂	1	3	yes	maximal
3	#19151530	C₂	1	3	yes	maximal
4	#19151519	C₂	1	3	yes	maximal
5	#19151468	C₂	1	1	yes	maximal, characteristic
6	#19151466	C₂	1	1	yes	maximal, characteristic
7	#383×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×C₄⋊Dic₈₉	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
12	C₂×#13083553	C₂²	1	3	yes	—
13	C₂×#13083553	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	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	#95×C₈₉	C₂²	1	3	yes	—
20	#95×C₈₉	C₂²	1	3	yes	—
21	C₂×Dic₁₇₈	—	4	3	no	—
22	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
23	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₈₉	C₂³	1	3	yes	—
25	C₂²×Dic₈₉	C₂³	1	3	yes	—
26	C₄⋊Dic₈₉	—	4	3	no	—
27	C₄⋊Dic₈₉	—	4	3	no	—
28	C₄⋊Dic₈₉	—	4	3	no	—
29	#13083553	—	4	3	no	—
30	#13083553	—	4	3	no	—
31	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
34	C₄²×C₈₉	Q₈	1	4	yes	complemented
35	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₈₉	—	2	3	no	—
37	C₂×C₄×C₈₉	—	2	3	no	—
38	C₂×C₄×C₈₉	—	2	3	no	—
39	C₂×C₄×C₈₉	—	2	3	no	—
40	C₂×C₄×C₈₉	—	2	6	no	—
41	C₂×C₄×C₈₉	C₂×Q₈	1	6	yes	—
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	—	2	3	no	—
48	C₂×Dic₈₉	—	2	6	no	—
49	Dic₁₇₈	—	4	4	no	—
50	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
52	C₂²×C₈₉	2₋¹⁺⁴	1	3	yes	—
53	C₄×C₈₉	—	2	6	no	—
54	C₄×C₈₉	—	4	1	no	—
55	C₄×C₈₉	—	4	3	no	—
56	Dic₈₉	—	4	1	no	—
57	Dic₈₉	—	4	1	no	—
58	Dic₈₉	—	4	3	no	—
59	Dic₈₉	—	4	3	no	—
60	C₂×C₈₉	#544	1	3	yes	—
61	C₂×C₈₉	#560	1	1	yes	characteristic
62	C₂×C₈₉	#563	1	3	yes	—
63	#2843	—	89	1	no	maximal, Sylow
64	C₈₉	#2843	1	1	yes	characteristic, complemented, Sylow
65	#400	—	89	3	no	—
66	#398	—	89	3	no	—
67	#398	—	89	3	no	—
68	#395	—	89	3	no	—
69	#384	—	89	1	no	—
70	#383	—	89	1	no	—
71	#383	D₈₉	1	1	yes	characteristic
72	C₂²×Q₈	—	89	1	no	—
73	C₂×C₄⋊C₄	—	89	3	no	—
74	C₂×C₄⋊C₄	—	89	3	no	—
75	C₂×C₄⋊C₄	—	89	3	no	—
76	C₂×C₄⋊C₄	—	89	3	no	—
77	C₂×C₄⋊C₄	—	89	3	no	—
78	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
79	#95	—	89	3	no	—
80	#95	—	89	3	no	—
81	#95	—	89	3	no	—
82	#95	—	89	3	no	—
83	#95	C₂×D₈₉	1	3	yes	—
84	#95	C₂×D₈₉	1	3	yes	—
85	C₂×Q₈	—	356	3	no	—
86	C₂²×C₄	—	89	3	no	—
87	C₂²×C₄	—	89	1	no	—
88	C₂²×C₄	—	89	1	no	—
89	C₂²×C₄	—	89	3	no	—
90	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
91	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
93	C₄⋊C₄	—	356	3	no	—
94	C₄⋊C₄	—	356	3	no	—
95	C₄⋊C₄	—	356	3	no	—
96	C₄⋊C₄	—	356	3	no	—
97	C₄⋊C₄	—	356	3	no	—
98	C₄²	Dic₁₇₈	1	4	yes	complemented
99	C₂³	C₂³×D₈₉	1	1	yes	characteristic
100	Q₈	—	356	4	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	6	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	C₂×C₄	—	178	6	no	—
113	C₂×C₄	C₂×Dic₁₇₈	1	6	yes	—
114	C₂²	C₂²×Dic₁₇₈	1	1	yes	characteristic
115	C₂²	#15668058	1	3	yes	—
116	C₂²	#15668059	1	3	yes	—
117	C₄	—	2	6	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	3	no	—
120	C₄	—	356	1	no	—
121	C₄	—	356	1	no	—
122	C₄	—	356	3	no	—
123	C₄	—	356	3	no	—
124	C₂	#19152226	1	3	yes	—
125	C₂	#19152339	1	1	yes	characteristic
126	C₂	#19152350	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151540

C₂

yes

maximal

#19151532

C₂

yes

maximal

#19151530

C₂

yes

maximal

#19151519

C₂

yes

maximal

#19151468

C₂

yes

maximal, characteristic

#19151466

C₂

yes

maximal, characteristic

#383×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

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

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

Q₈

yes

complemented

C₂³×C₈₉

C₂⁴

yes

characteristic

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

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#544

yes

—

C₂×C₈₉

#560

yes

characteristic

C₂×C₈₉

#563

yes

—

#2843

—

maximal, Sylow

C₈₉

#2843

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

—

C₂×D₈₉

yes

—

#95

C₂×D₈₉

yes

—

C₂×Q₈

—

356

—

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

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

Dic₁₇₈

yes

complemented

C₂³

C₂³×D₈₉

yes

characteristic

100

Q₈

—

356

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

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

C₂×C₄

—

178

—

113

C₂×C₄

C₂×Dic₁₇₈

yes

—

114

C₂²

C₂²×Dic₁₇₈

yes

characteristic

115

C₂²

#15668058

yes

—

116

C₂²

#15668059

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

356

—

121

C₄

—

356

—

122

C₄

—

356

—

123

C₄

—

356

—

124

C₂

#19152226

yes

—

125

C₂

#19152339

yes

characteristic

126

C₂

#19152350

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#383×C₈₉	C₂
hypercenter	#383	D₈₉
hypocenter	C₈₉	#2843
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#383×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
34. C₄²×C₈₉	Q₈	semidirect
64. C₈₉	#2843	semidirect
98. C₄²	Dic₁₇₈	semidirect

H ≅ G/N

type

34. C₄²×C₈₉

Q₈

semidirect

64. C₈₉

#2843

semidirect

98. C₄²

Dic₁₇₈

semidirect

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

subgroup	index
#19151540	2
#19151532	2
#19151530	2
#19151519	2
#19151468	2
#19151466	2
#383×C₈₉	2
#2843	89

subgroup

index

#383×C₈₉

Subgroups — #26028519

Conjugacy classes of subgroups

Complements

Maximal subgroups