Group #36469156 — subgroups

Subgroups — #36469156

Order 22784.

360 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 99 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	#26022508	C₂	1	2	yes	maximal, complemented
2	#26022502	C₂	1	2	yes	maximal, complemented
3	#26019867	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019858	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15667957	—	2	4	no	—
7	C₂×D₁₄₂₄	—	2	4	no	—
8	#19151978	C₂²	1	1	yes	characteristic
9	#19151975	—	2	2	no	—
10	#19151975	C₂²	1	1	yes	characteristic
11	#19151975	C₂²	1	2	yes	—
12	#19151945	—	2	2	no	—
13	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₈²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₅₆	—	2	2	no	—
16	C₂×D₇₁₂	—	2	2	no	—
17	C₂×D₇₁₂	—	2	2	no	—
18	C₂×D₇₁₂	—	2	2	no	—
19	C₂×D₇₁₂	—	4	4	no	—
20	#15668012	—	2	1	no	—
21	#15668009	—	2	1	no	—
22	#15668009	—	2	2	no	—
23	#15667957	—	4	8	no	—
24	D₁₄₂₄	—	4	8	no	—
25	#15667934	—	4	4	no	—
26	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
27	C₄×C₈×C₈₉	D₄	1	2	yes	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₇₈	—	4	1	no	—
30	C₂×D₃₅₆	—	4	1	no	—
31	C₂×D₃₅₆	—	4	2	no	—
32	C₂×D₃₅₆	—	4	4	no	—
33	C₂×D₃₅₆	—	4	2	no	—
34	Dic₃₅₆	—	4	4	no	—
35	D₇₁₂	—	4	4	no	—
36	D₇₁₂	—	4	4	no	—
37	D₇₁₂	—	4	4	no	—
38	D₇₁₂	—	8	8	no	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₂×C₈×C₈₉	D₈	1	4	yes	—
41	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
42	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₈₉	D₈	1	8	yes	complemented
44	C₂²×D₈₉	—	8	2	no	—
45	C₂²×D₈₉	—	8	1	no	—
46	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
48	C₂×Dic₈₉	—	8	1	no	—
49	D₃₅₆	—	8	2	no	—
50	D₃₅₆	—	8	2	no	—
51	D₃₅₆	—	8	2	no	—
52	D₃₅₆	—	8	8	no	—
53	D₃₅₆	—	8	4	no	—
54	Dic₁₇₈	—	8	2	no	—
55	C₈×C₈₉	—	2	4	no	—
56	C₈×C₈₉	C₂×D₈	1	4	yes	—
57	C₂²×C₈₉	#493	1	1	yes	characteristic
58	C₂×D₈₉	—	16	1	no	—
59	C₂×D₈₉	—	16	2	no	—
60	C₂×D₈₉	—	16	4	no	—
61	C₂×D₈₉	—	16	2	no	—
62	C₄×C₈₉	#493	1	2	yes	—
63	C₄×C₈₉	#509	1	4	yes	—
64	Dic₈₉	—	16	1	no	—
65	#12519	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	#1712	1	1	yes	characteristic
67	C₂×C₈₉	#2250	1	2	yes	—
68	D₈₉	—	32	2	no	—
69	D₈₉	—	32	1	no	—
70	#2248	—	89	2	no	—
71	#2246	—	89	2	no	—
72	#1712	—	89	1	no	—
73	#1709	—	89	1	no	—
74	#1311	D₈₉	1	1	yes	characteristic, complemented
75	C₈₉	#12519	1	1	yes	characteristic, complemented, Sylow
76	C₂×SD₃₂	—	178	4	no	—
77	C₂×D₁₆	—	178	4	no	—
78	#494	—	89	1	no	—
79	#493	—	89	1	no	—
80	#493	—	89	2	no	—
81	#493	—	178	2	no	—
82	#486	—	178	2	no	—
83	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
84	C₈²	C₂×D₈₉	1	1	yes	characteristic
85	C₂×Q₁₆	—	178	2	no	—
86	C₂×D₈	—	356	4	no	—
87	C₂×D₈	—	178	2	no	—
88	C₂×D₈	—	178	2	no	—
89	C₂×D₈	—	178	2	no	—
90	C₄⋊Q₈	—	178	1	no	—
91	#127	—	178	1	no	—
92	#127	—	178	2	no	—
93	SD₃₂	—	356	8	no	—
94	D₁₆	—	356	8	no	—
95	C₂×C₁₆	D₃₅₆	1	4	yes	—
96	#102	—	356	4	no	—
97	C₄×C₈	D₃₅₆	1	2	yes	—
98	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×Q₈	—	356	1	no	—
100	C₂×D₄	—	356	1	no	—
101	C₂×D₄	—	356	2	no	—
102	C₂×D₄	—	356	2	no	—
103	C₂×D₄	—	356	4	no	—
104	Q₁₆	—	356	4	no	—
105	D₈	—	356	4	no	—
106	D₈	—	356	4	no	—
107	D₈	—	356	4	no	—
108	D₈	—	712	8	no	—
109	C₂×C₈	D₇₁₂	1	4	yes	—
110	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
111	C₄⋊C₄	—	356	2	no	—
112	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
113	C₁₆	D₇₁₂	1	8	yes	complemented
114	C₂³	—	712	1	no	—
115	C₂³	—	712	2	no	—
116	Q₈	—	712	2	no	—
117	D₄	—	712	2	no	—
118	D₄	—	712	2	no	—
119	D₄	—	712	2	no	—
120	D₄	—	712	8	no	—
121	D₄	—	712	4	no	—
122	C₂×C₄	—	712	1	no	—
123	C₂×C₄	#15668009	1	1	yes	characteristic
124	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	C₂×D₇₁₂	1	4	yes	—
127	C₂²	—	1424	1	no	—
128	C₂²	—	1424	2	no	—
129	C₂²	—	1424	4	no	—
130	C₂²	—	1424	2	no	—
131	C₂²	#19151975	1	1	yes	characteristic
132	C₄	—	1424	1	no	—
133	C₄	#19151975	1	2	yes	—
134	C₄	#19152036	1	4	yes	—
135	C₂	—	2848	2	no	—
136	C₂	—	2848	1	no	—
137	C₂	#26019867	1	1	yes	characteristic
138	C₂	#26022518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022508

C₂

yes

maximal, complemented

#26022502

C₂

yes

maximal, complemented

#26019867

C₂

yes

maximal, characteristic, complemented

#26019858

C₂

yes

maximal, characteristic, complemented

C₈₉×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#15667957

—

C₂×D₁₄₂₄

—

#19151978

C₂²

yes

characteristic

#19151975

—

#19151975

C₂²

yes

characteristic

#19151975

C₂²

yes

—

#19151945

—

C₄×C₁₆×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

—

—

—

—

D₁₄₂₄

—

#15667934

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

Dic₃₅₆

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

D₈

yes

complemented

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#493

yes

—

C₄×C₈₉

#509

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

D₈₉

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#12519

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

178

—

C₂×D₁₆

—

178

—

—

—

—

—

178

—

#486

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₂×D₈

—

356

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

178

—

#127

—

178

—

#127

—

178

—

SD₃₂

—

356

—

D₁₆

—

356

—

C₂×C₁₆

D₃₅₆

yes

—

#102

—

356

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

100

C₂×D₄

—

356

—

101

C₂×D₄

—

356

—

102

C₂×D₄

—

356

—

103

C₂×D₄

—

356

—

104

Q₁₆

—

356

—

105

D₈

—

356

—

106

D₈

—

356

—

107

D₈

—

356

—

108

D₈

—

712

—

109

C₂×C₈

D₇₁₂

yes

—

110

C₂×C₈

C₂×D₃₅₆

yes

—

111

C₄⋊C₄

—

356

—

112

C₄²

C₂×D₃₅₆

yes

characteristic

113

C₁₆

D₇₁₂

yes

complemented

114

C₂³

—

712

—

115

C₂³

—

712

—

116

Q₈

—

712

—

117

D₄

—

712

—

118

D₄

—

712

—

119

D₄

—

712

—

120

D₄

—

712

—

121

D₄

—

712

—

122

C₂×C₄

—

712

—

123

C₂×C₄

#15668009

yes

characteristic

124

C₂×C₄

C₂×D₇₁₂

yes

—

125

C₈

—

126

C₈

C₂×D₇₁₂

yes

—

127

C₂²

—

1424

—

128

C₂²

—

1424

—

129

C₂²

—

1424

—

130

C₂²

—

1424

—

131

C₂²

#19151975

yes

characteristic

132

C₄

—

1424

—

133

C₄

#19151975

yes

—

134

C₄

#19152036

yes

—

135

C₂

—

2848

—

136

C₂

—

2848

—

137

C₂

#26019867

yes

characteristic

138

C₂

#26022518

yes

—

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1311	C₂
socle	C₂²×C₈₉	#493
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#12519
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1311

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #26022508	C₂	semidirect
2. #26022502	C₂	semidirect
3. #26019867	C₂	semidirect
4. #26019858	C₂	semidirect
5. C₈₉×#1311	C₂	semidirect
43. C₁₆×C₈₉	D₈	semidirect
74. #1311	D₈₉	semidirect
75. C₈₉	#12519	semidirect
113. C₁₆	D₇₁₂	semidirect

H ≅ G/N

type

1. #26022508

C₂

semidirect

2. #26022502

C₂

semidirect

3. #26019867

C₂

semidirect

4. #26019858

C₂

semidirect

5. C₈₉×#1311

C₂

semidirect

43. C₁₆×C₈₉

D₈

semidirect

74. #1311

D₈₉

semidirect

75. C₈₉

#12519

semidirect

113. C₁₆

D₇₁₂

semidirect

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

subgroup	index
#26022508	2
#26022502	2
#26019867	2
#26019858	2
C₈₉×#1311	2
#12519	89

subgroup

index

C₈₉×#1311

Subgroups — #36469156

Conjugacy classes of subgroups

Complements

Maximal subgroups