Group #36448232 — subgroups

Subgroups — #36448232

Order 22784.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	#26020424	C₂	1	2	yes	maximal
2	#26019011	C₂	1	4	yes	maximal
3	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667948	—	2	4	no	—
5	#19151622	C₂²	1	4	yes	—
6	#19151351	—	2	8	no	—
7	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₈²×C₈₉	C₄	1	1	yes	characteristic
10	C₈²×C₈₉	C₄	1	2	yes	—
11	C₈²×C₈₉	C₄	1	1	yes	characteristic
12	C₂×#13083558	—	2	4	no	—
13	#15667948	—	4	8	no	—
14	C₄⋊#10623494	—	2	8	no	—
15	C₂²×C₈×C₈₉	D₄	1	2	yes	—
16	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
17	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₈₉	D₄	1	4	yes	—
23	C₄×C₈×C₈₉	Q₈	1	2	yes	—
24	#13083558	—	4	8	no	—
25	C₂×#10623494	—	4	4	no	—
26	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
28	C₂×C₈×C₈₉	—	2	8	no	—
29	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₈₉	D₈	1	4	yes	—
34	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
35	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₈₉	C₂×D₄	1	2	yes	—
37	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	2	4	no	—
40	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
44	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
46	C₈×C₈₉	—	2	4	no	—
47	C₈×C₈₉	#107	1	4	yes	—
48	C₈×C₈₉	#107	1	4	yes	—
49	C₈×C₈₉	#108	1	8	yes	complemented
50	#10623494	—	8	4	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	#384	1	1	yes	characteristic
53	C₂²×C₈₉	#500	1	2	yes	—
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	C₂×#107	1	2	yes	—
56	C₄×C₈₉	C₂×#108	1	4	yes	—
57	C₄×C₈₉	#493	1	1	yes	characteristic
58	C₄×C₈₉	#494	1	1	yes	characteristic
59	#8823	—	89	1	no	maximal, Sylow
60	C₂×C₈₉	—	2	2	no	—
61	C₂×C₈₉	#1845	1	1	yes	characteristic
62	C₂×C₈₉	#1848	1	2	yes	—
63	#1848	—	89	2	no	—
64	#1567	—	89	4	no	—
65	C₂×C₈²	D₈₉	1	1	yes	characteristic
66	C₈₉	#8823	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	178	4	no	—
68	#423	—	89	4	no	—
69	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
71	#335	—	178	8	no	—
72	C₈²	Dic₈₉	1	1	yes	characteristic
73	C₈²	Dic₈₉	1	2	yes	—
74	C₈²	Dic₈₉	1	1	yes	characteristic
75	C₂×M₁₆	—	178	4	no	—
76	C₂²×C₈	Dic₁₇₈	1	4	yes	—
77	C₂²×C₈	D₃₅₆	1	2	yes	—
78	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
79	#108	—	356	8	no	—
80	C₄⋊C₈	—	178	8	no	—
81	C₄×C₈	Dic₁₇₈	1	2	yes	—
82	C₄×C₈	D₃₅₆	1	4	yes	—
83	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
84	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
87	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
89	M₁₆	—	356	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	356	4	no	—
92	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
93	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
96	C₂×C₈	D₇₁₂	1	4	yes	—
97	C₂×C₈	Dic₃₅₆	1	4	yes	—
98	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
99	C₄²	C₂×D₃₅₆	1	2	yes	—
100	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
101	C₂³	#15668012	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
105	C₂×C₄	#15668009	1	1	yes	characteristic
106	C₂×C₄	#15668012	1	2	yes	—
107	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
108	C₂×C₄	C₂×Dic₃₅₆	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	712	4	no	—
111	C₈	#15667946	1	4	yes	—
112	C₈	#15667946	1	4	yes	—
113	C₈	#15667948	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#19151468	1	1	yes	characteristic
116	C₂²	#19152003	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#15667946	1	2	yes	—
119	C₄	C₂×#15667948	1	4	yes	—
120	C₄	#19151975	1	1	yes	characteristic
121	C₄	#19151978	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#26020412	1	1	yes	characteristic
124	C₂	#26020424	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020424

C₂

yes

maximal

#26019011

C₂

yes

maximal

C₂×C₈²×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667948

—

#19151622

C₂²

yes

—

#19151351

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₄

yes

characteristic

C₈²×C₈₉

C₄

yes

—

C₈²×C₈₉

C₄

yes

—

—

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#13083558

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

#107

yes

—

C₈×C₈₉

#107

yes

—

C₈×C₈₉

#108

yes

complemented

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#384

yes

characteristic

C₂²×C₈₉

#500

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#107

yes

—

C₄×C₈₉

C₂×#108

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

#1845

yes

characteristic

C₂×C₈₉

#1848

yes

—

#1848

—

#1567

—

C₂×C₈²

D₈₉

yes

characteristic

C₈₉

#8823

yes

characteristic, complemented, Sylow

C₂×#108

—

178

—

#423

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄×C₈

C₂×D₈₉

yes

—

#335

—

178

—

C₈²

Dic₈₉

yes

characteristic

C₈²

Dic₈₉

yes

—

C₈²

Dic₈₉

yes

characteristic

C₂×M₁₆

—

178

—

C₂²×C₈

Dic₁₇₈

yes

—

C₂²×C₈

D₃₅₆

yes

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#108

—

356

—

C₄⋊C₈

—

178

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂×Dic₈₉

yes

—

C₄×C₈

C₂×Dic₈₉

yes

characteristic

C₄×C₈

C₂×Dic₈₉

yes

characteristic

C₄×C₈

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₈

yes

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

D₇₁₂

yes

—

C₂×C₈

Dic₃₅₆

yes

—

C₄²

C₂²×Dic₈₉

yes

characteristic

C₄²

C₂×D₃₅₆

yes

—

100

C₄²

C₂×Dic₁₇₈

yes

characteristic

101

C₂³

#15668012

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

104

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

105

C₂×C₄

#15668009

yes

characteristic

106

C₂×C₄

#15668012

yes

—

107

C₂×C₄

C₂×D₇₁₂

yes

—

108

C₂×C₄

C₂×Dic₃₅₆

yes

—

109

C₈

—

110

C₈

—

712

—

111

C₈

#15667946

yes

—

112

C₈

#15667946

yes

—

113

C₈

#15667948

yes

complemented

114

C₂²

—

115

C₂²

#19151468

yes

characteristic

116

C₂²

#19152003

yes

—

117

C₄

—

118

C₄

C₂×#15667946

yes

—

119

C₄

C₂×#15667948

yes

—

120

C₄

#19151975

yes

characteristic

121

C₄

#19151978

yes

characteristic

122

C₂

—

123

C₂

#26020412

yes

characteristic

124

C₂

#26020424

yes

—

	subgroup	G/H
center	C₂×C₄	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂²×C₈₉	#384
prime radical	C₂³×C₈₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#8823
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#15668009

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×C₈²×C₈₉

C₂

socle

C₂²×C₈₉

#384

prime radical

C₂³×C₈₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₉

C₂³

hypercenter

C₂×C₈²

D₈₉

hypocenter

C₈₉

#8823

LCS

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
49. C₈×C₈₉	#108	semidirect
66. C₈₉	#8823	semidirect
113. C₈	#15667948	semidirect

H ≅ G/N

type

49. C₈×C₈₉

#108

semidirect

66. C₈₉

#8823

semidirect

113. C₈

#15667948

semidirect

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

subgroup	index
#26020424	2
#26019011	2
C₂×C₈²×C₈₉	2
#8823	89

subgroup

index

#26020424

#26019011

C₂×C₈²×C₈₉

#8823

Subgroups — #36448232

Conjugacy classes of subgroups

Complements

Maximal subgroups