Group #36469151 — subgroups

Subgroups — #36469151

Order 22784.

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

normal subgroups only

This group has 330 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	#26022505	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022502	C₂	1	1	yes	maximal, characteristic
4	#26019858	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₇₁₂	—	2	2	no	—
7	C₂×#15667957	—	2	4	no	—
8	C₂×D₁₄₂₄	—	2	2	no	—
9	#19151978	C₂²	1	2	yes	—
10	#19151975	C₂²	1	2	yes	—
11	#19151945	—	2	4	no	—
12	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₈²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₅₆	—	2	2	no	—
16	C₂×Dic₃₅₆	—	2	2	no	—
17	C₂×D₇₁₂	—	2	2	no	—
18	C₂×D₇₁₂	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	#15668009	—	2	2	no	—
21	Dic₇₁₂	—	4	4	no	—
22	#15667957	—	4	8	no	—
23	D₁₄₂₄	—	4	4	no	—
24	#15667934	—	4	8	no	—
25	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
27	C₄×C₈×C₈₉	D₄	1	2	yes	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₇₈	—	4	2	no	—
30	C₂×D₃₅₆	—	4	2	no	—
31	C₂×D₃₅₆	—	4	4	no	—
32	Dic₃₅₆	—	4	4	no	—
33	Dic₃₅₆	—	4	4	no	—
34	D₇₁₂	—	4	4	no	—
35	D₇₁₂	—	4	4	no	—
36	C₄⋊Dic₈₉	—	4	4	no	—
37	C₂×C₈×C₈₉	D₈	1	4	yes	—
38	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₈₉	D₈	1	4	yes	—
42	C₁₆×C₈₉	D₈	1	4	yes	—
43	C₂²×D₈₉	—	8	2	no	—
44	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
46	C₂×Dic₈₉	—	8	2	no	—
47	D₃₅₆	—	8	2	no	—
48	D₃₅₆	—	8	2	no	—
49	D₃₅₆	—	8	8	no	—
50	Dic₁₇₈	—	8	2	no	—
51	Dic₁₇₈	—	8	2	no	—
52	C₈×C₈₉	—	2	4	no	—
53	C₈×C₈₉	C₂×D₈	1	2	yes	—
54	C₈×C₈₉	C₂×D₈	1	2	yes	—
55	C₂²×C₈₉	#493	1	1	yes	characteristic
56	C₂×D₈₉	—	16	2	no	—
57	C₂×D₈₉	—	16	4	no	—
58	C₄×C₈₉	#493	1	1	yes	characteristic
59	C₄×C₈₉	#493	1	1	yes	characteristic
60	C₄×C₈₉	#508	1	4	yes	—
61	Dic₈₉	—	16	2	no	—
62	#12518	—	89	1	no	maximal, Sylow
63	C₂×C₈₉	#1712	1	1	yes	characteristic
64	C₂×C₈₉	#2249	1	2	yes	—
65	D₈₉	—	32	2	no	—
66	#2248	—	89	2	no	—
67	#2247	—	89	1	no	—
68	#2246	—	89	1	no	—
69	#1709	—	89	2	no	—
70	C₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
71	C₈₉	#12518	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	178	2	no	—
73	C₂×SD₃₂	—	178	4	no	—
74	C₂×D₁₆	—	178	2	no	—
75	#494	—	89	2	no	—
76	#493	—	89	2	no	—
77	#486	—	178	4	no	—
78	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
79	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
80	C₈²	C₂×D₈₉	1	1	yes	characteristic
81	C₂×Q₁₆	—	178	2	no	—
82	C₂×Q₁₆	—	178	2	no	—
83	C₂×D₈	—	178	2	no	—
84	C₂×D₈	—	178	2	no	—
85	C₄⋊Q₈	—	178	2	no	—
86	#127	—	178	2	no	—
87	Q₃₂	—	356	4	no	—
88	SD₃₂	—	356	8	no	—
89	D₁₆	—	356	4	no	—
90	C₂×C₁₆	D₃₅₆	1	2	yes	—
91	C₂×C₁₆	D₃₅₆	1	2	yes	—
92	#102	—	356	8	no	—
93	C₄×C₈	D₃₅₆	1	2	yes	—
94	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
95	C₂×Q₈	—	356	2	no	—
96	C₂×D₄	—	356	2	no	—
97	C₂×D₄	—	356	4	no	—
98	Q₁₆	—	356	4	no	—
99	Q₁₆	—	356	4	no	—
100	D₈	—	356	4	no	—
101	D₈	—	356	4	no	—
102	C₂×C₈	D₇₁₂	1	4	yes	—
103	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
104	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
105	C₄⋊C₄	—	356	4	no	—
106	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
107	C₁₆	D₇₁₂	1	4	yes	—
108	C₁₆	D₇₁₂	1	4	yes	—
109	C₂³	—	712	2	no	—
110	Q₈	—	712	2	no	—
111	Q₈	—	712	2	no	—
112	D₄	—	712	2	no	—
113	D₄	—	712	2	no	—
114	D₄	—	712	8	no	—
115	C₂×C₄	—	712	2	no	—
116	C₂×C₄	#15668009	1	1	yes	characteristic
117	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
118	C₈	—	2	4	no	—
119	C₈	C₂×D₇₁₂	1	2	yes	—
120	C₈	C₂×D₇₁₂	1	2	yes	—
121	C₂²	—	1424	2	no	—
122	C₂²	—	1424	4	no	—
123	C₂²	#19151975	1	1	yes	characteristic
124	C₄	—	1424	2	no	—
125	C₄	#19151975	1	1	yes	characteristic
126	C₄	#19151975	1	1	yes	characteristic
127	C₄	#19152030	1	4	yes	—
128	C₂	—	2848	2	no	—
129	C₂	#26019867	1	1	yes	characteristic
130	C₂	#26022513	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022508

C₂

yes

maximal, complemented

#26022505

C₂

yes

maximal, characteristic, complemented

#26022502

C₂

yes

maximal, characteristic

#26019858

C₂

yes

maximal, complemented

C₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₁₂

—

C₂×#15667957

—

C₂×D₁₄₂₄

—

#19151978

C₂²

yes

—

#19151975

C₂²

yes

—

#19151945

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668012

—

#15668009

—

Dic₇₁₂

—

#15667957

—

D₁₄₂₄

—

#15667934

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

D₈

yes

—

C₁₆×C₈₉

D₈

yes

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

—

—

—

—

C₈×C₁₆

D₈₉

yes

characteristic, complemented

C₈₉

#12518

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

178

—

C₂×SD₃₂

—

178

—

C₂×D₁₆

—

178

—

#494

—

#493

—

#486

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

178

—

#127

—

178

—

Q₃₂

—

356

—

SD₃₂

—

356

—

D₁₆

—

356

—

C₂×C₁₆

D₃₅₆

yes

—

C₂×C₁₆

D₃₅₆

yes

—

#102

—

356

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

Q₁₆

—

356

—

Q₁₆

—

356

—

100

D₈

—

356

—

101

D₈

—

356

—

102

C₂×C₈

D₇₁₂

yes

—

103

C₂×C₈

C₂×D₃₅₆

yes

characteristic

104

C₂×C₈

C₂×D₃₅₆

yes

characteristic

105

C₄⋊C₄

—

356

—

106

C₄²

C₂×D₃₅₆

yes

characteristic

107

C₁₆

D₇₁₂

yes

—

108

C₁₆

D₇₁₂

yes

—

109

C₂³

—

712

—

110

Q₈

—

712

—

111

Q₈

—

712

—

112

D₄

—

712

—

113

D₄

—

712

—

114

D₄

—

712

—

115

C₂×C₄

—

712

—

116

C₂×C₄

#15668009

yes

characteristic

117

C₂×C₄

C₂×D₇₁₂

yes

—

118

C₈

—

119

C₈

C₂×D₇₁₂

yes

—

120

C₈

C₂×D₇₁₂

yes

—

121

C₂²

—

1424

—

122

C₂²

—

1424

—

123

C₂²

#19151975

yes

characteristic

124

C₄

—

1424

—

125

C₄

#19151975

yes

characteristic

126

C₄

#19151975

yes

characteristic

127

C₄

#19152030

yes

—

128

C₂

—

2848

—

129

C₂

#26019867

yes

characteristic

130

C₂

#26022513

yes

—

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	#493
prime radical	#26022502	C₂
hypercenter	C₈×C₁₆	D₈₉
hypocenter	C₈₉	#12518
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈×C₁₆×C₈₉

C₂

C₂²×C₈₉

C₂

C₈×C₁₆

D₈₉

hypocenter

C₈₉

#12518

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #26022508	C₂	semidirect
2. #26022505	C₂	semidirect
4. #26019858	C₂	semidirect
5. C₈×C₁₆×C₈₉	C₂	semidirect
70. C₈×C₁₆	D₈₉	semidirect
71. C₈₉	#12518	semidirect

H ≅ G/N

type

1. #26022508

C₂

semidirect

2. #26022505

C₂

semidirect

4. #26019858

C₂

semidirect

5. C₈×C₁₆×C₈₉

C₂

semidirect

70. C₈×C₁₆

D₈₉

semidirect

71. C₈₉

#12518

semidirect

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

subgroup	index
#26022508	2
#26022505	2
#26022502	2
#26019858	2
C₈×C₁₆×C₈₉	2
#12518	89

subgroup

index

C₈×C₁₆×C₈₉

Subgroups — #36469151

Conjugacy classes of subgroups

Complements

Maximal subgroups