Group #26030801 — subgroups

Subgroups — #26030801

Order 11392.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#19152155	C₂	1	1	yes	maximal, characteristic
2	#19152008	C₂	1	4	yes	maximal, complemented
3	#19151997	C₂	1	4	yes	maximal, complemented
4	#19151638	C₂	1	4	yes	maximal
5	C₄×#13083558	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#13083558	C₂²	1	2	yes	—
8	#15668012	C₂²	1	2	yes	—
9	#15668004	C₂²	1	2	yes	—
10	#15667991	—	2	2	no	—
11	C₄×Dic₁₇₈	—	2	2	no	—
12	#15667971	C₂²	1	2	yes	—
13	#15667945	C₂²	1	8	yes	—
14	#15667943	C₂²	1	8	yes	—
15	#15667923	C₂²	1	2	yes	—
16	C₄×#10623494	C₂²	1	2	yes	—
17	#125×C₈₉	—	2	2	no	—
18	#125×C₈₉	C₂²	1	2	yes	—
19	#125×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	2	no	—
24	#13083558	Q₈	1	8	yes	complemented
25	C₂×#10623494	C₂³	1	4	yes	—
26	C₄⋊Dic₈₉	—	2	2	no	—
27	C₄⋊Dic₈₉	C₂³	1	4	yes	—
28	#13083553	—	2	4	no	—
29	C₂²⋊Dic₈₉	—	2	2	no	—
30	C₄×Dic₈₉	—	2	2	no	—
31	C₂²×C₄×C₈₉	—	2	2	no	—
32	C₂²×C₄×C₈₉	D₄	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₈₉	—	2	4	no	—
35	C₄⋊C₄×C₈₉	—	2	4	no	—
36	C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
38	C₄²×C₈₉	D₄	1	2	yes	—
39	C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄²×C₈₉	C₂³	1	1	yes	characteristic
41	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	—	2	4	no	—
43	C₂×C₄×C₈₉	—	4	2	no	—
44	C₂×C₄×C₈₉	—	4	4	no	—
45	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₈₉	—	2	4	no	—
51	Dic₁₇₈	—	4	4	no	—
52	#10623494	C₂×Q₈	1	8	yes	—
53	C₂²×C₈₉	—	2	2	no	—
54	C₂²×C₈₉	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₈₉	—	2	2	no	—
57	C₄×C₈₉	—	2	2	no	—
58	C₄×C₈₉	—	4	4	no	—
59	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
60	C₄×C₈₉	C₂²×Q₈	1	2	yes	—
61	Dic₈₉	—	4	4	no	—
62	C₂×C₈₉	—	2	2	no	—
63	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₈₉	#575	1	2	yes	—
65	#3166	—	89	1	no	maximal, Sylow
66	C₈₉	#3166	1	1	yes	characteristic, complemented, Sylow
67	#533	—	89	1	no	—
68	C₂×#125	D₈₉	1	1	yes	characteristic
69	#501	—	89	4	no	—
70	#499	—	89	4	no	—
71	#428	—	89	4	no	—
72	C₄×M₁₆	—	89	1	no	—
73	C₂×M₁₆	—	89	2	no	—
74	C₄⋊Q₈	—	89	2	no	—
75	#125	—	2	2	no	—
76	#125	—	89	2	no	—
77	#125	C₂×D₈₉	1	2	yes	—
78	#125	C₂×D₈₉	1	2	yes	—
79	C₂²⋊Q₈	—	178	2	no	—
80	C₄×Q₈	—	178	2	no	—
81	#117	—	89	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
84	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
85	#107	—	89	8	no	—
86	#106	—	89	8	no	—
87	#97	—	89	2	no	—
88	C₄×C₈	—	89	2	no	—
89	C₂×Q₈	—	178	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
92	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
93	M₁₆	—	89	8	no	—
94	C₂×C₈	—	89	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	89	4	no	—
99	C₄⋊C₄	—	178	4	no	—
100	C₄⋊C₄	—	178	2	no	—
101	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
102	C₂²⋊C₄	—	178	2	no	—
103	C₄²	—	178	2	no	—
104	C₄²	C₂²⋊D₈₉	1	2	yes	—
105	C₄²	C₂²×D₈₉	1	1	yes	characteristic
106	C₄²	C₂²×D₈₉	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
108	Q₈	—	356	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	178	4	no	—
113	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₈₉	1	4	yes	—
117	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
118	C₈	—	89	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#15668010	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	356	4	no	—
126	C₄	#15668010	1	2	yes	—
127	C₄	C₂×Q₈×D₈₉	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#15668010	1	1	yes	characteristic
130	C₂	#19152396	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152155

C₂

yes

maximal, characteristic

#19152008

C₂

yes

maximal, complemented

#19151997

C₂

yes

maximal, complemented

#19151638

C₂

yes

maximal

C₄×#13083558

C₂

yes

maximal, characteristic

C₂×#125×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

#15668012

C₂²

yes

—

#15668004

C₂²

yes

—

#15667991

—

C₄×Dic₁₇₈

—

#15667971

C₂²

yes

—

#15667945

C₂²

yes

—

#15667943

C₂²

yes

—

#15667923

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

#125×C₈₉

—

#125×C₈₉

C₂²

yes

—

#125×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

#13083558

Q₈

yes

complemented

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

#10623494

C₂×Q₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄⋊Q₈

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄⋊Q₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₈₉

#575

yes

—

#3166

—

maximal, Sylow

C₈₉

#3166

yes

characteristic, complemented, Sylow

#533

—

C₂×#125

D₈₉

yes

—

—

—

C₄×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

—

#125

C₂×D₈₉

yes

—

#125

C₂×D₈₉

yes

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

178

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

—

—

—

C₄×C₈

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

100

C₄⋊C₄

—

178

—

101

C₄⋊C₄

C₂²×D₈₉

yes

—

102

C₂²⋊C₄

—

178

—

103

C₄²

—

178

—

104

C₄²

C₂²⋊D₈₉

yes

—

105

C₄²

C₂²×D₈₉

yes

characteristic

106

C₄²

C₂²×D₈₉

yes

characteristic

107

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

108

Q₈

—

356

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

178

—

113

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

114

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

115

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

116

C₂×C₄

Q₈×D₈₉

yes

—

117

C₂×C₄

C₂³×D₈₉

yes

characteristic

118

C₈

—

119

C₂²

—

120

C₂²

#15668010

yes

—

121

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

356

—

126

C₄

#15668010

yes

—

127

C₄

C₂×Q₈×D₈₉

yes

—

128

C₂

—

129

C₂

C₂×#15668010

yes

characteristic

130

C₂

#19152396

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#125×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×#125	D₈₉
hypocenter	C₈₉	#3166
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×#125×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #19152008	C₂	semidirect
3. #19151997	C₂	semidirect
24. #13083558	Q₈	semidirect
66. C₈₉	#3166	semidirect

H ≅ G/N

type

2. #19152008

C₂

semidirect

3. #19151997

C₂

semidirect

24. #13083558

Q₈

semidirect

66. C₈₉

#3166

semidirect

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

subgroup	index
#19152155	2
#19152008	2
#19151997	2
#19151638	2
C₄×#13083558	2
C₂×#125×C₈₉	2
#3166	89

subgroup

index

C₂×#125×C₈₉

Subgroups — #26030801

Conjugacy classes of subgroups

Complements

Maximal subgroups