Group #36470880 — subgroups

Subgroups — #36470880

Order 22784.

306 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₂×#19151428	C₂	1	4	yes	maximal
2	#26020424	C₂	1	2	yes	maximal
3	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667948	—	2	2	no	—
5	C₂×#15667948	C₂²	1	4	yes	—
6	#19151622	—	2	2	no	—
7	#19151428	—	2	8	no	—
8	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	#346×C₈₉	C₄	1	4	yes	—
11	C₂×#13083558	—	2	4	no	—
12	#15667948	—	2	8	no	—
13	#15667948	—	4	4	no	—
14	C₄⋊#10623494	—	4	4	no	—
15	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
20	C₂×C₁₆×C₈₉	Q₈	1	4	yes	—
21	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₈₉	Q₈	1	2	yes	—
23	#13083558	—	4	4	no	—
24	#13083558	—	4	4	no	—
25	C₂×#10623494	—	4	4	no	—
26	C₂²×C₄×C₈₉	Q₁₆	1	2	yes	—
27	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₈₉	D₈	1	2	yes	—
30	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
31	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
33	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
34	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
35	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
36	C₄²×C₈₉	D₈	1	2	yes	—
37	C₁₆×C₈₉	C₄⋊C₄	1	8	yes	—
38	C₂³×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	#107	1	4	yes	—
41	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
47	C₈×C₈₉	#107	1	4	yes	—
48	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
49	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
50	#10623494	—	8	4	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
53	C₂²×C₈₉	#494	1	1	yes	characteristic
54	C₂²×C₈₉	#500	1	1	yes	characteristic
55	C₄×C₈₉	—	2	4	no	—
56	C₄×C₈₉	#384	1	1	yes	characteristic
57	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
58	C₄×C₈₉	#493	1	1	yes	characteristic
59	C₄×C₈₉	#500	1	1	yes	characteristic
60	#12910	—	89	1	no	maximal, Sylow
61	C₂×C₈₉	—	2	1	no	—
62	C₂×C₈₉	—	2	1	no	—
63	C₂×C₈₉	#1845	1	1	yes	characteristic
64	C₂×C₈₉	#2161	1	2	yes	—
65	C₂×#368	—	89	4	no	—
66	C₂×#346	D₈₉	1	1	yes	characteristic
67	#1848	—	89	2	no	—
68	C₈₉	#12910	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
70	C₂×#108	—	89	4	no	—
71	C₂×#108	—	178	2	no	—
72	#423	—	178	2	no	—
73	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
74	#368	—	178	8	no	—
75	#346	Dic₈₉	1	4	yes	—
76	C₂×M₁₆	—	178	4	no	—
77	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
79	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
80	C₂×C₁₆	Dic₁₇₈	1	4	yes	—
81	C₂×C₁₆	D₃₅₆	1	4	yes	—
82	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	—
83	#108	—	356	4	no	—
84	#108	—	178	8	no	—
85	C₄⋊C₈	—	356	4	no	—
86	C₄×C₈	Dic₁₇₈	1	2	yes	—
87	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
88	C₂²×C₄	Dic₃₅₆	1	2	yes	—
89	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
90	M₁₆	—	356	4	no	—
91	M₁₆	—	356	4	no	—
92	C₂×C₈	—	356	4	no	—
93	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
94	C₂×C₈	D₇₁₂	1	2	yes	—
95	C₂×C₈	Dic₃₅₆	1	2	yes	—
96	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
97	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
98	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
99	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
100	C₄²	C₄⋊Dic₈₉	1	2	yes	—
101	C₄²	D₇₁₂	1	2	yes	—
102	C₁₆	C₄⋊Dic₈₉	1	8	yes	—
103	C₂³	C₂×Dic₃₅₆	1	1	yes	characteristic
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	#15667946	1	4	yes	—
106	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
107	C₂×C₄	#15668009	1	1	yes	characteristic
108	C₂×C₄	#15668012	1	1	yes	characteristic
109	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
110	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
112	C₈	—	712	4	no	—
113	C₈	#15667946	1	4	yes	—
114	C₈	C₂×C₄⋊Dic₈₉	1	2	yes	—
115	C₈	#15668012	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#15667946	1	1	yes	characteristic
118	C₂²	#19151978	1	1	yes	characteristic
119	C₂²	#19152003	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	#19151468	1	1	yes	characteristic
122	C₄	C₂×#15667946	1	1	yes	characteristic
123	C₄	#19151975	1	1	yes	characteristic
124	C₄	#19152003	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	#26020412	1	1	yes	characteristic
128	C₂	#26022047	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151428

C₂

yes

maximal

#26020424

C₂

yes

maximal

C₂×#346×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667948

—

C₂×#15667948

C₂²

yes

—

#19151622

—

#19151428

—

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₄

yes

—

—

—

—

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

Q₈

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₂²×C₄×C₈₉

Q₁₆

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

D₈

yes

—

C₁₆×C₈₉

C₄⋊C₄

yes

—

C₂³×C₈₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#107

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₈×C₈₉

#107

yes

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

—

C₈×C₈₉

C₄⋊Q₈

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1845

yes

characteristic

C₂×C₈₉

#2161

yes

—

C₂×#368

—

C₂×#346

D₈₉

yes

characteristic

#1848

—

C₈₉

#12910

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₈₉

yes

—

C₂×#108

—

C₂×#108

—

178

—

#423

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#368

—

178

—

#346

Dic₈₉

yes

—

C₂×M₁₆

—

178

—

C₂²×C₈

D₃₅₆

yes

characteristic

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₁₆

Dic₁₇₈

yes

—

C₂×C₁₆

D₃₅₆

yes

—

C₂×C₁₆

C₂×Dic₈₉

yes

—

#108

—

356

—

#108

—

178

—

C₄⋊C₈

—

356

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

C₂×Dic₈₉

yes

—

C₂²×C₄

Dic₃₅₆

yes

—

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

M₁₆

—

356

—

M₁₆

—

356

—

C₂×C₈

—

356

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

D₇₁₂

yes

—

C₂×C₈

Dic₃₅₆

yes

—

C₂×C₈

C₂²×Dic₈₉

yes

characteristic

C₂×C₈

C₂×D₃₅₆

yes

—

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

C₂×C₈

C₂×Dic₁₇₈

yes

—

100

C₄²

C₄⋊Dic₈₉

yes

—

101

C₄²

D₇₁₂

yes

—

102

C₁₆

C₄⋊Dic₈₉

yes

—

103

C₂³

C₂×Dic₃₅₆

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#15667946

yes

—

106

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

107

C₂×C₄

#15668009

yes

characteristic

108

C₂×C₄

#15668012

yes

characteristic

109

C₂×C₄

C₂×D₇₁₂

yes

characteristic

110

C₂×C₄

C₂×D₇₁₂

yes

characteristic

111

C₂×C₄

C₂×Dic₃₅₆

yes

characteristic

112

C₈

—

712

—

113

C₈

#15667946

yes

—

114

C₈

C₂×C₄⋊Dic₈₉

yes

—

115

C₈

#15668012

yes

—

116

C₂²

—

117

C₂²

C₂×#15667946

yes

characteristic

118

C₂²

#19151978

yes

characteristic

119

C₂²

#19152003

yes

characteristic

120

C₄

—

121

C₄

#19151468

yes

characteristic

122

C₄

C₂×#15667946

yes

characteristic

123

C₄

#19151975

yes

characteristic

124

C₄

#19152003

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

#26020412

yes

characteristic

128

C₂

#26022047

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	#12910
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₇₁₂

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₂×#346×C₈₉

C₂

socle

C₂²×C₈₉

C₂×#107

prime radical

C₂³×C₈₉

C₂×Q₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
68. C₈₉	#12910	semidirect

H ≅ G/N

type

68. C₈₉

#12910

semidirect

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

subgroup	index
C₂×#19151428	2
#26020424	2
C₂×#346×C₈₉	2
#12910	89

subgroup

index

C₂×#19151428

#26020424

C₂×#346×C₈₉

#12910

Subgroups — #36470880

Conjugacy classes of subgroups

Complements

Maximal subgroups