Group #26019865 — subgroups

Subgroups — #26019865

Order 11392.

178 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19151950	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151948	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151946	C₂	1	1	yes	maximal, characteristic
4	#19151946	C₂	1	1	yes	maximal, characteristic
5	#19151945	C₂	1	1	yes	maximal, characteristic
6	C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
7	#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15668012	C₂²	1	1	yes	characteristic
9	#15668009	C₂²	1	1	yes	characteristic
10	#15667936	—	2	2	no	—
11	#15667935	—	2	2	no	—
12	#15667934	—	2	2	no	—
13	#15667932	—	2	2	no	—
14	#15667932	—	2	2	no	—
15	C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	#103×C₈₉	—	2	2	no	—
20	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₇₈	—	2	1	no	—
22	C₂×D₃₅₆	—	2	1	no	—
23	C₂×D₃₅₆	—	2	1	no	—
24	C₂×D₃₅₆	—	2	1	no	—
25	C₂×#10623494	D₄	1	2	yes	—
26	C₂×#10623494	D₄	1	2	yes	—
27	C₄⋊Dic₈₉	—	2	1	no	—
28	C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×Q₈×C₈₉	—	2	1	no	—
30	C₂×Q₈×C₈₉	—	2	1	no	—
31	C₂×C₈×C₈₉	D₄	1	2	yes	—
32	C₄⋊C₄×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	C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	C₂²×D₈₉	—	4	1	no	—
38	Q₈×C₈₉	—	4	2	no	—
39	Q₈×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	—	4	1	no	—
41	C₂×C₄×C₈₉	—	4	1	no	—
42	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₈₉	—	4	1	no	—
46	D₃₅₆	—	4	2	no	—
47	D₃₅₆	—	4	2	no	—
48	D₃₅₆	—	4	2	no	—
49	Dic₁₇₈	—	4	2	no	—
50	C₈×C₈₉	—	2	2	no	—
51	#10623494	—	2	2	no	—
52	#10623494	—	2	2	no	—
53	C₂²×C₈₉	#127	1	1	yes	characteristic
54	C₂×D₈₉	—	8	1	no	—
55	C₂×D₈₉	—	8	1	no	—
56	C₂×D₈₉	—	8	1	no	—
57	C₄×C₈₉	—	8	1	no	—
58	C₄×C₈₉	—	8	1	no	—
59	C₄×C₈₉	#135	1	2	yes	—
60	C₄×C₈₉	#135	1	2	yes	—
61	C₄×C₈₉	#135	1	2	yes	—
62	Dic₈₉	—	8	1	no	—
63	C₂×C₈₉	#495	1	1	yes	characteristic
64	C₂×C₈₉	#495	1	1	yes	characteristic
65	C₂×C₈₉	#495	1	1	yes	characteristic
66	D₈₉	—	16	1	no	—
67	#1711	—	89	1	no	maximal, Sylow
68	C₈₉	#1711	1	1	yes	characteristic, complemented, Sylow
69	#487	—	89	1	no	—
70	#487	—	89	1	no	—
71	#487	D₈₉	1	1	yes	characteristic, complemented
72	#486	—	89	1	no	—
73	#486	—	89	1	no	—
74	#486	—	89	1	no	—
75	C₈²	—	89	1	no	—
76	C₄⋊Q₈	—	89	1	no	—
77	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
78	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
79	#127	—	89	1	no	—
80	#103	—	2	2	no	—
81	#103	—	178	2	no	—
82	#103	—	178	2	no	—
83	#102	—	178	2	no	—
84	#102	—	178	2	no	—
85	#102	—	178	2	no	—
86	C₄×C₈	—	89	1	no	—
87	C₄×C₈	—	89	1	no	—
88	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	178	1	no	—
92	C₂×D₄	—	178	1	no	—
93	C₂×D₄	—	178	1	no	—
94	C₂×D₄	—	178	1	no	—
95	C₂×C₈	—	89	2	no	—
96	C₂×C₈	—	89	2	no	—
97	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	178	1	no	—
103	C₄⋊C₄	—	178	1	no	—
104	C₄²	C₂²×D₈₉	1	1	yes	characteristic
105	C₂³	—	356	1	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	356	2	no	—
109	D₄	—	356	2	no	—
110	D₄	—	356	2	no	—
111	D₄	—	356	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	356	1	no	—
115	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
116	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	178	2	no	—
120	C₈	—	178	2	no	—
121	C₂²	—	712	1	no	—
122	C₂²	—	712	1	no	—
123	C₂²	—	712	1	no	—
124	C₂²	#15668008	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	712	1	no	—
128	C₄	#15668032	1	2	yes	—
129	C₄	#15668033	1	2	yes	—
130	C₄	#15668035	1	2	yes	—
131	C₂	—	1424	1	no	—
132	C₂	#19151979	1	1	yes	characteristic
133	C₂	#19151980	1	1	yes	characteristic
134	C₂	#19151983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151950

C₂

yes

maximal, characteristic, complemented

#19151948

C₂

yes

maximal, characteristic, complemented

#19151946

C₂

yes

maximal, characteristic

#19151946

C₂

yes

maximal, characteristic

#19151945

C₂

yes

maximal, characteristic

C₈×#10623494

C₂

yes

maximal, characteristic, complemented

#487×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668012

C₂²

yes

characteristic

#15668009

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#103×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×#10623494

D₄

yes

—

C₂×#10623494

D₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

#127

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

#135

yes

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#1711

—

maximal, Sylow

C₈₉

#1711

yes

characteristic, complemented, Sylow

#487

—

#487

—

#487

D₈₉

yes

characteristic, complemented

#486

—

#486

—

#486

—

C₈²

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

C₂×D₈₉

yes

—

—

—

178

—

#103

—

178

—

#102

—

178

—

#102

—

178

—

#102

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

178

—

103

C₄⋊C₄

—

178

—

104

C₄²

C₂²×D₈₉

yes

characteristic

105

C₂³

—

356

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

356

—

109

D₄

—

356

—

110

D₄

—

356

—

111

D₄

—

356

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

356

—

115

C₂×C₄

D₄×D₈₉

yes

characteristic

116

C₂×C₄

D₄×D₈₉

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

118

C₈

—

119

C₈

—

178

—

120

C₈

—

178

—

121

C₂²

—

712

—

122

C₂²

—

712

—

123

C₂²

—

712

—

124

C₂²

#15668008

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

712

—

128

C₄

#15668032

yes

—

129

C₄

#15668033

yes

—

130

C₄

#15668035

yes

—

131

C₂

—

1424

—

132

C₂

#19151979

yes

characteristic

133

C₂

#19151980

yes

characteristic

134

C₂

#19151983

yes

characteristic

	subgroup	G/H
center	C₂²	#15668008
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#487×C₈₉	C₂
socle	C₂²×C₈₉	#127
prime radical	#15668009	C₂²
hypercenter	#487	D₈₉
hypocenter	C₈₉	#1711
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#15668008

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#487×C₈₉

C₂

C₂²×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
1. #19151950	C₂	semidirect
2. #19151948	C₂	semidirect
6. C₈×#10623494	C₂	semidirect
7. #487×C₈₉	C₂	semidirect
68. C₈₉	#1711	semidirect
71. #487	D₈₉	semidirect

H ≅ G/N

type

1. #19151950

C₂

semidirect

2. #19151948

C₂

semidirect

6. C₈×#10623494

C₂

semidirect

7. #487×C₈₉

C₂

semidirect

68. C₈₉

#1711

semidirect

71. #487

D₈₉

semidirect

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

subgroup	index
#19151950	2
#19151948	2
#19151946	2
#19151945	2
C₈×#10623494	2
#487×C₈₉	2
#1711	89

subgroup

index

#487×C₈₉

Subgroups — #26019865

Conjugacy classes of subgroups

Complements

Maximal subgroups