Group #36469114 — subgroups

Subgroups — #36469114

Order 22784.

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

normal subgroups only

This group has 306 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	#26022476	C₂	1	4	yes	maximal, complemented
2	#26019870	C₂	1	1	yes	maximal, characteristic, complemented
3	#26019867	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#19152003	—	2	1	no	—
6	#19152003	C₂²	1	2	yes	—
7	#19151978	—	2	1	no	—
8	#19151975	—	2	1	no	—
9	#19151975	—	2	1	no	—
10	#19151975	C₂²	1	2	yes	—
11	#19151400	—	2	8	no	—
12	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
13	C₈²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₅₆	—	4	2	no	—
15	C₂×D₇₁₂	—	2	4	no	—
16	C₂×D₇₁₂	—	4	2	no	—
17	C₂×D₇₁₂	—	4	2	no	—
18	#15668012	—	2	2	no	—
19	#15668009	—	2	2	no	—
20	#15667946	—	2	4	no	—
21	#15667946	—	4	2	no	—
22	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
23	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₇₈	—	4	2	no	—
27	C₂×D₃₅₆	—	4	2	no	—
28	C₂×D₃₅₆	—	4	2	no	—
29	C₂×D₃₅₆	—	4	2	no	—
30	Dic₃₅₆	—	8	4	no	—
31	D₇₁₂	—	4	8	no	—
32	D₇₁₂	—	8	4	no	—
33	D₇₁₂	—	8	4	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₂×C₈×C₈₉	D₈	1	2	yes	—
37	C₂×C₈×C₈₉	D₈	1	2	yes	—
38	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
39	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₈₉	—	2	4	no	—
41	C₂²×D₈₉	—	8	2	no	—
42	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
45	C₂×Dic₈₉	—	8	2	no	—
46	D₃₅₆	—	8	4	no	—
47	D₃₅₆	—	8	4	no	—
48	D₃₅₆	—	8	4	no	—
49	Dic₁₇₈	—	8	4	no	—
50	C₈×C₈₉	D₁₆	1	4	yes	—
51	C₈×C₈₉	SD₃₂	1	4	yes	—
52	C₈×C₈₉	#135	1	4	yes	—
53	C₂²×C₈₉	#493	1	1	yes	characteristic
54	C₂×D₈₉	—	16	2	no	—
55	C₂×D₈₉	—	16	2	no	—
56	C₂×D₈₉	—	16	2	no	—
57	C₄×C₈₉	#495	1	2	yes	—
58	C₄×C₈₉	C₂×D₁₆	1	2	yes	—
59	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
60	Dic₈₉	—	16	2	no	—
61	#12509	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	#1709	1	1	yes	characteristic
63	C₂×C₈₉	#2246	1	1	yes	characteristic
64	C₂×C₈₉	#2248	1	1	yes	characteristic
65	D₈₉	—	32	2	no	—
66	#2240	—	89	4	no	—
67	#1713	—	89	1	no	—
68	#1712	—	89	1	no	—
69	C₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
70	C₈₉	#12509	1	1	yes	characteristic, complemented, Sylow
71	#500	—	89	2	no	—
72	#500	—	178	1	no	—
73	#494	—	178	1	no	—
74	#493	—	89	2	no	—
75	#493	—	178	1	no	—
76	#493	—	178	1	no	—
77	#357	—	178	8	no	—
78	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
79	C₈²	C₂×D₈₉	1	1	yes	characteristic
80	C₂×Q₁₆	—	356	2	no	—
81	C₂×D₈	—	356	2	no	—
82	C₂×D₈	—	356	2	no	—
83	C₂×D₈	—	178	4	no	—
84	C₄⋊Q₈	—	178	2	no	—
85	#127	—	178	2	no	—
86	C₂×C₁₆	D₃₅₆	1	4	yes	—
87	#107	—	356	2	no	—
88	#107	—	178	4	no	—
89	C₄×C₈	D₃₅₆	1	1	yes	characteristic
90	C₄×C₈	D₃₅₆	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
92	C₂×Q₈	—	356	2	no	—
93	C₂×D₄	—	356	2	no	—
94	C₂×D₄	—	356	2	no	—
95	C₂×D₄	—	356	2	no	—
96	Q₁₆	—	712	4	no	—
97	D₈	—	356	8	no	—
98	D₈	—	712	4	no	—
99	D₈	—	712	4	no	—
100	C₂×C₈	D₇₁₂	1	2	yes	—
101	C₂×C₈	D₇₁₂	1	2	yes	—
102	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
103	C₄⋊C₄	—	356	2	no	—
104	C₄⋊C₄	—	356	2	no	—
105	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
106	C₁₆	—	2	4	no	—
107	C₂³	—	712	2	no	—
108	Q₈	—	712	4	no	—
109	D₄	—	712	4	no	—
110	D₄	—	712	4	no	—
111	D₄	—	712	4	no	—
112	C₂×C₄	—	712	2	no	—
113	C₂×C₄	#15668009	1	1	yes	characteristic
114	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
115	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
116	C₈	D₁₄₂₄	1	4	yes	—
117	C₈	#15667957	1	4	yes	—
118	C₈	#15668034	1	4	yes	—
119	C₂²	—	1424	2	no	—
120	C₂²	—	1424	2	no	—
121	C₂²	—	1424	2	no	—
122	C₂²	#19151975	1	1	yes	characteristic
123	C₄	—	1424	2	no	—
124	C₄	#19151982	1	2	yes	—
125	C₄	C₂×D₁₄₂₄	1	2	yes	—
126	C₄	C₂×#15667957	1	2	yes	—
127	C₂	—	2848	2	no	—
128	C₂	#26019858	1	1	yes	characteristic
129	C₂	#26022502	1	1	yes	characteristic
130	C₂	#26022508	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022476

C₂

yes

maximal, complemented

#26019870

C₂

yes

maximal, characteristic, complemented

#26019867

C₂

yes

maximal, characteristic

C₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

#19152003

—

#19152003

C₂²

yes

—

—

—

—

C₂²

yes

—

#19151400

—

C₄×C₁₆×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

—

—

—

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

Dic₃₅₆

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

D₁₆

yes

—

C₈×C₈₉

SD₃₂

yes

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#495

yes

—

C₄×C₈₉

C₂×D₁₆

yes

—

C₄×C₈₉

C₂×SD₃₂

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2240

—

#1713

—

#1712

—

C₈×C₁₆

D₈₉

yes

characteristic, complemented

C₈₉

#12509

yes

characteristic, complemented, Sylow

#500

—

#500

—

178

—

#494

—

178

—

#493

—

#493

—

178

—

#493

—

178

—

#357

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

356

—

C₂×D₈

—

356

—

C₂×D₈

—

356

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

178

—

#127

—

178

—

C₂×C₁₆

D₃₅₆

yes

—

#107

—

356

—

#107

—

178

—

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

Q₁₆

—

712

—

D₈

—

356

—

D₈

—

712

—

D₈

—

712

—

100

C₂×C₈

D₇₁₂

yes

—

101

C₂×C₈

D₇₁₂

yes

—

102

C₂×C₈

C₂×D₃₅₆

yes

—

103

C₄⋊C₄

—

356

—

104

C₄⋊C₄

—

356

—

105

C₄²

C₂×D₃₅₆

yes

characteristic

106

C₁₆

—

107

C₂³

—

712

—

108

Q₈

—

712

—

109

D₄

—

712

—

110

D₄

—

712

—

111

D₄

—

712

—

112

C₂×C₄

—

712

—

113

C₂×C₄

#15668009

yes

characteristic

114

C₂×C₄

C₂×D₇₁₂

yes

characteristic

115

C₂×C₄

C₂×D₇₁₂

yes

characteristic

116

C₈

D₁₄₂₄

yes

—

117

C₈

#15667957

yes

—

118

C₈

#15668034

yes

—

119

C₂²

—

1424

—

120

C₂²

—

1424

—

121

C₂²

—

1424

—

122

C₂²

#19151975

yes

characteristic

123

C₄

—

1424

—

124

C₄

#19151982

yes

—

125

C₄

C₂×D₁₄₂₄

yes

—

126

C₄

C₂×#15667957

yes

—

127

C₂

—

2848

—

128

C₂

#26019858

yes

characteristic

129

C₂

#26022502

yes

characteristic

130

C₂

#26022508

yes

characteristic

	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	#26019867	C₂
hypercenter	C₈×C₁₆	D₈₉
hypocenter	C₈₉	#12509
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

#12509

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #26022476	C₂	semidirect
2. #26019870	C₂	semidirect
4. C₈×C₁₆×C₈₉	C₂	semidirect
69. C₈×C₁₆	D₈₉	semidirect
70. C₈₉	#12509	semidirect

H ≅ G/N

type

1. #26022476

C₂

semidirect

2. #26019870

C₂

semidirect

4. C₈×C₁₆×C₈₉

C₂

semidirect

69. C₈×C₁₆

D₈₉

semidirect

70. C₈₉

#12509

semidirect

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

subgroup	index
#26022476	2
#26019870	2
#26019867	2
C₈×C₁₆×C₈₉	2
#12509	89

subgroup

index

C₈×C₁₆×C₈₉

Subgroups — #36469114

Conjugacy classes of subgroups

Complements

Maximal subgroups