Group #26019708 — subgroups

Subgroups — #26019708

Order 11392.

144 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 55 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	#19151948	C₂	1	1	yes	maximal, characteristic
2	#19151893	C₂	1	1	yes	maximal, characteristic
3	Q₈×#10623494	C₂	1	1	yes	maximal, characteristic
4	#19151350	C₂	1	1	yes	maximal, characteristic
5	#19151346	C₂	1	1	yes	maximal, characteristic
6	#19151344	C₂	1	1	yes	maximal, characteristic
7	#475×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#15668012	C₂²	1	1	yes	characteristic
9	#15667945	—	2	1	no	—
10	C₄⋊#10623494	—	2	1	no	—
11	C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	#15667936	—	2	1	no	—
14	#15667936	—	2	1	no	—
15	#15667936	—	2	1	no	—
16	C₄×#10623494	—	2	1	no	—
17	C₄×#10623494	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	#106×C₈₉	—	2	1	no	—
21	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#103×C₈₉	—	2	1	no	—
23	C₂×Dic₁₇₈	—	2	1	no	—
24	C₂×#10623494	—	2	1	no	—
25	C₂×#10623494	—	2	1	no	—
26	C₂×#10623494	D₄	1	1	yes	characteristic
27	C₂×#10623494	D₄	1	1	yes	characteristic
28	C₄⋊Dic₈₉	—	2	1	no	—
29	C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×Q₈×C₈₉	—	2	1	no	—
31	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	—	2	1	no	—
33	C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₄⋊C₄×C₈₉	—	2	1	no	—
35	C₄⋊C₄×C₈₉	—	2	1	no	—
36	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
37	C₄²×C₈₉	—	2	1	no	—
38	C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	Q₈×C₈₉	—	2	1	no	—
40	Q₈×C₈₉	—	4	1	no	—
41	Q₈×C₈₉	—	4	1	no	—
42	Q₈×C₈₉	Q₁₆	1	2	yes	—
43	C₂×C₄×C₈₉	—	2	1	no	—
44	C₂×C₄×C₈₉	—	2	1	no	—
45	C₂×C₄×C₈₉	—	4	1	no	—
46	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
49	C₂×Dic₈₉	—	4	1	no	—
50	Dic₁₇₈	—	8	1	no	—
51	C₈×C₈₉	—	4	1	no	—
52	#10623494	—	2	1	no	—
53	#10623494	—	2	1	no	—
54	#10623494	—	4	1	no	—
55	#10623494	—	4	1	no	—
56	C₂²×C₈₉	#121	1	1	yes	characteristic
57	C₄×C₈₉	—	2	2	no	—
58	C₄×C₈₉	—	2	1	no	—
59	C₄×C₈₉	—	4	1	no	—
60	C₄×C₈₉	—	8	1	no	—
61	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₈₉	#135	1	1	yes	characteristic
63	C₄×C₈₉	#135	1	1	yes	characteristic
64	C₄×C₈₉	#137	1	1	yes	characteristic
65	Dic₈₉	—	8	1	no	—
66	C₂×C₈₉	#464	1	1	yes	characteristic
67	C₂×C₈₉	#467	1	1	yes	characteristic
68	C₂×C₈₉	#471	1	1	yes	characteristic
69	#1684	—	89	1	no	maximal, Sylow
70	C₈₉	#1684	1	1	yes	characteristic, complemented, Sylow
71	#487	—	89	1	no	—
72	#477	—	89	1	no	—
73	#475	D₈₉	1	1	yes	characteristic
74	C₈×Q₈	—	89	1	no	—
75	#335	—	89	1	no	—
76	#333	—	89	1	no	—
77	#332	—	89	1	no	—
78	C₄⋊Q₈	—	89	1	no	—
79	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
80	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
81	#107	—	178	1	no	—
82	#106	—	2	1	no	—
83	C₄⋊C₈	—	89	1	no	—
84	C₄⋊C₈	—	89	1	no	—
85	C₄⋊C₈	—	178	1	no	—
86	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
87	#103	—	2	1	no	—
88	#103	—	178	1	no	—
89	#103	—	178	1	no	—
90	#103	—	178	1	no	—
91	C₄×C₈	—	89	1	no	—
92	C₄×C₈	—	178	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	178	1	no	—
95	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	89	1	no	—
98	C₂×C₈	—	89	1	no	—
99	C₂×C₈	—	178	1	no	—
100	C₂×C₈	—	178	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	178	1	no	—
105	C₄⋊C₄	—	178	1	no	—
106	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
107	C₄²	—	2	1	no	—
108	C₄²	C₂²×D₈₉	1	1	yes	characteristic
109	Q₈	—	2	1	no	—
110	Q₈	—	4	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	712	1	no	—
113	Q₈	#13083564	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	356	1	no	—
118	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
120	C₂×C₄	#13083573	1	1	yes	characteristic
121	C₈	—	4	1	no	—
122	C₈	—	356	1	no	—
123	C₈	—	356	1	no	—
124	C₈	—	178	1	no	—
125	C₈	—	178	1	no	—
126	C₂²	#15667984	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	712	1	no	—
132	C₄	C₂×#13083564	1	1	yes	characteristic
133	C₄	#15668032	1	1	yes	characteristic
134	C₄	#15668033	1	1	yes	characteristic
135	C₄	#15668041	1	1	yes	characteristic
136	C₂	#19151818	1	1	yes	characteristic
137	C₂	#19151837	1	1	yes	characteristic
138	C₂	#19151863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151948

C₂

yes

maximal, characteristic

#19151893

C₂

yes

maximal, characteristic

Q₈×#10623494

C₂

yes

maximal, characteristic

#19151350

C₂

yes

maximal, characteristic

#19151346

C₂

yes

maximal, characteristic

#19151344

C₂

yes

maximal, characteristic

#475×C₈₉

C₂

yes

maximal, characteristic

#15668012

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄×Q₈×C₈₉

C₂²

yes

characteristic

#106×C₈₉

—

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

#103×C₈₉

—

C₂×Dic₁₇₈

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

D₄

yes

characteristic

C₂×#10623494

D₄

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

Q₁₆

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

—

—

—

—

C₂²×C₈₉

#121

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#487

—

#477

—

#475

D₈₉

yes

characteristic

C₈×Q₈

—

#335

—

#333

—

#332

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

C₄×Q₈

C₂×D₈₉

yes

characteristic

#107

—

178

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

characteristic

#103

—

#103

—

178

—

#103

—

178

—

#103

—

178

—

C₄×C₈

—

C₄×C₈

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

100

C₂×C₈

—

178

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

178

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

C₂²⋊D₈₉

yes

characteristic

107

C₄²

—

108

C₄²

C₂²×D₈₉

yes

characteristic

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

712

—

113

Q₈

#13083564

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

D₄×D₈₉

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

120

C₂×C₄

#13083573

yes

characteristic

121

C₈

—

122

C₈

—

356

—

123

C₈

—

356

—

124

C₈

—

178

—

125

C₈

—

178

—

126

C₂²

#15667984

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

712

—

132

C₄

C₂×#13083564

yes

characteristic

133

C₄

#15668032

yes

characteristic

134

C₄

#15668033

yes

characteristic

135

C₄

#15668041

yes

characteristic

136

C₂

#19151818

yes

characteristic

137

C₂

#19151837

yes

characteristic

138

C₂

#19151863

yes

characteristic

	subgroup	G/H
center	C₂²	#15667984
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#475×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#121
hypercenter	#475	D₈₉
hypocenter	C₈₉	#1684
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#15667984

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#475×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
70. C₈₉	#1684	semidirect

H ≅ G/N

type

70. C₈₉

#1684

semidirect

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

subgroup	index
#19151948	2
#19151893	2
Q₈×#10623494	2
#19151350	2
#19151346	2
#19151344	2
#475×C₈₉	2
#1684	89

subgroup

index

#475×C₈₉

Subgroups — #26019708

Conjugacy classes of subgroups

Complements

Maximal subgroups