Group #26021671 — subgroups

Subgroups — #26021671

Order 11392.

222 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#15667935	C₂	1	2	yes	maximal
2	#19151504	C₂	1	1	yes	maximal, characteristic
3	#19151360	C₂	1	2	yes	maximal
4	#19151353	C₂	1	1	yes	maximal, characteristic
5	#382×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₂²×#10623494	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
9	#15667935	—	2	4	no	—
10	#15667919	—	2	2	no	—
11	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	#95×C₈₉	—	2	2	no	—
14	C₂×Dic₁₇₈	—	2	2	no	—
15	C₂×Dic₁₇₈	—	2	2	no	—
16	C₂×Dic₁₇₈	—	2	1	no	—
17	C₂×Dic₁₇₈	D₄	1	2	yes	—
18	C₂²×Dic₈₉	—	2	1	no	—
19	C₂²×Dic₈₉	—	2	1	no	—
20	C₂×#10623494	—	2	2	no	—
21	C₂×#10623494	—	2	2	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₄⋊Dic₈₉	—	2	2	no	—
24	C₂²×C₄×C₈₉	—	2	2	no	—
25	C₂²×C₄×C₈₉	D₄	1	2	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₄²×C₈₉	—	2	2	no	—
30	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₉	—	2	2	no	—
32	C₂×C₄×C₈₉	—	2	2	no	—
33	C₂×C₄×C₈₉	—	2	2	no	—
34	C₂×C₄×C₈₉	—	4	2	no	—
35	C₂×C₄×C₈₉	—	4	2	no	—
36	C₂×C₄×C₈₉	—	4	2	no	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#41	1	2	yes	—
40	C₂×C₄×C₈₉	#41	1	2	yes	—
41	C₂×Dic₈₉	—	2	2	no	—
42	C₂×Dic₈₉	—	4	1	no	—
43	C₂×Dic₈₉	—	4	1	no	—
44	C₂×Dic₈₉	—	4	1	no	—
45	C₂×Dic₈₉	—	4	2	no	—
46	Dic₁₇₈	—	2	4	no	—
47	Dic₁₇₈	—	4	2	no	—
48	#10623494	—	4	2	no	—
49	C₂²×C₈₉	#121	1	1	yes	characteristic
50	C₂²×C₈₉	#123	1	1	yes	characteristic
51	C₂²×C₈₉	#124	1	1	yes	characteristic
52	C₂²×C₈₉	#135	1	2	yes	—
53	C₂²×C₈₉	#137	1	2	yes	—
54	C₄×C₈₉	—	4	2	no	—
55	C₄×C₈₉	—	8	2	no	—
56	C₄×C₈₉	#123	1	2	yes	—
57	C₄×C₈₉	#126	1	2	yes	—
58	Dic₈₉	—	4	2	no	—
59	Dic₈₉	—	8	1	no	—
60	C₂×C₈₉	#397	1	1	yes	characteristic
61	C₂×C₈₉	#464	1	2	yes	—
62	C₂×C₈₉	#485	1	2	yes	—
63	C₂×C₈₉	#488	1	1	yes	characteristic
64	C₂×C₈₉	#491	1	1	yes	characteristic
65	#2074	—	89	1	no	maximal, Sylow
66	C₈₉	#2074	1	1	yes	characteristic, complemented, Sylow
67	C₂×#103	—	89	2	no	—
68	#391	—	89	1	no	—
69	#382	D₈₉	1	1	yes	characteristic
70	#340	—	89	2	no	—
71	#336	—	89	1	no	—
72	C₂²×Q₈	—	89	1	no	—
73	C₂²×C₈	—	89	2	no	—
74	C₂×C₄⋊C₄	—	89	1	no	—
75	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
76	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
77	#103	—	178	4	no	—
78	#95	—	2	2	no	—
79	#95	—	178	2	no	—
80	C₂×Q₈	—	89	2	no	—
81	C₂×Q₈	—	178	2	no	—
82	C₂×Q₈	—	178	2	no	—
83	C₂×Q₈	—	178	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	178	1	no	—
86	C₂²×C₄	—	178	1	no	—
87	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
88	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
89	C₂×C₈	—	178	2	no	—
90	C₂×C₈	—	178	2	no	—
91	C₂×C₈	—	178	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	178	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
97	Q₈	—	356	2	no	—
98	Q₈	—	178	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	356	1	no	—
106	C₂×C₄	—	356	1	no	—
107	C₂×C₄	—	356	1	no	—
108	C₂×C₄	—	356	2	no	—
109	C₂×C₄	—	178	2	no	—
110	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
111	C₂×C₄	#13083573	1	1	yes	characteristic
112	C₂×C₄	#13083574	1	2	yes	—
113	C₂×C₄	#13083575	1	2	yes	—
114	C₈	—	356	2	no	—
115	C₂²	#15667983	1	1	yes	characteristic
116	C₂²	#15667995	1	1	yes	characteristic
117	C₂²	#15668001	1	1	yes	characteristic
118	C₂²	#15668032	1	2	yes	—
119	C₂²	#15668042	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	8	2	no	—
122	C₄	—	356	2	no	—
123	C₄	—	712	1	no	—
124	C₄	#15667997	1	2	yes	—
125	C₄	#15668007	1	2	yes	—
126	C₂	#19151524	1	1	yes	characteristic
127	C₂	#19151817	1	2	yes	—
128	C₂	#19151939	1	2	yes	—
129	C₂	#19151955	1	1	yes	characteristic
130	C₂	#19151968	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667935

C₂

yes

maximal

#19151504

C₂

yes

maximal, characteristic

#19151360

C₂

yes

maximal

#19151353

C₂

yes

maximal, characteristic

#382×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

#15667935

—

#15667919

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#137

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#126

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#397

yes

characteristic

C₂×C₈₉

#464

yes

—

C₂×C₈₉

#485

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×#103

—

#391

—

#382

D₈₉

yes

characteristic

#340

—

#336

—

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#103

—

178

—

#95

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄²

—

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

Q₈

—

356

—

Q₈

—

178

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

356

—

106

C₂×C₄

—

356

—

107

C₂×C₄

—

356

—

108

C₂×C₄

—

356

—

109

C₂×C₄

—

178

—

110

C₂×C₄

D₄×D₈₉

yes

characteristic

111

C₂×C₄

#13083573

yes

characteristic

112

C₂×C₄

#13083574

yes

—

113

C₂×C₄

#13083575

yes

—

114

C₈

—

356

—

115

C₂²

#15667983

yes

characteristic

116

C₂²

#15667995

yes

characteristic

117

C₂²

#15668001

yes

characteristic

118

C₂²

#15668032

yes

—

119

C₂²

#15668042

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

356

—

123

C₄

—

712

—

124

C₄

#15667997

yes

—

125

C₄

#15668007

yes

—

126

C₂

#19151524

yes

characteristic

127

C₂

#19151817

yes

—

128

C₂

#19151939

yes

—

129

C₂

#19151955

yes

characteristic

130

C₂

#19151968

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#382×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#382	D₈₉
hypocenter	C₈₉	#2074
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#382×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
66. C₈₉	#2074	semidirect

H ≅ G/N

type

66. C₈₉

#2074

semidirect

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

subgroup	index
C₂×#15667935	2
#19151504	2
#19151360	2
#19151353	2
#382×C₈₉	2
#2074	89

subgroup

index

#382×C₈₉

Subgroups — #26021671

Conjugacy classes of subgroups

Complements

Maximal subgroups