Group #36470792 — subgroups

Subgroups — #36470792

Order 22784.

302 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 155 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	#26022008	C₂	1	2	yes	maximal
2	#26018334	C₂	1	4	yes	maximal
3	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	C₂²	1	4	yes	—
5	C₄⋊#13083549	—	2	4	no	—
6	C₂×#97×C₈₉	C₄	1	2	yes	—
7	C₂×#97×C₈₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	#376×C₈₉	C₄	1	1	yes	characteristic
11	#15667952	—	2	8	no	—
12	C₂×#13083549	—	2	4	no	—
13	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
18	#97×C₈₉	—	2	4	no	—
19	#97×C₈₉	C₈	1	4	yes	—
20	C₄×C₈×C₈₉	—	2	4	no	—
21	C₄×C₈×C₈₉	D₄	1	2	yes	—
22	C₄×C₈×C₈₉	Q₈	1	2	yes	—
23	#95×C₈₉	—	2	2	no	—
24	#13083549	—	4	4	no	—
25	C₂²×C₄×C₈₉	—	2	2	no	—
26	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
27	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	#41	1	4	yes	—
36	C₄²×C₈₉	—	2	2	no	—
37	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
38	C₄²×C₈₉	C₂×C₈	1	2	yes	—
39	C₄²×C₈₉	M₁₆	1	4	yes	—
40	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₄×C₈₉	—	2	4	no	—
42	C₂×C₄×C₈₉	—	4	2	no	—
43	C₂×C₄×C₈₉	—	4	4	no	—
44	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#117	1	2	yes	—
47	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
49	C₈×C₈₉	—	2	2	no	—
50	C₈×C₈₉	—	2	4	no	—
51	C₈×C₈₉	#124	1	2	yes	—
52	C₈×C₈₉	#125	1	2	yes	—
53	C₂²×C₈₉	—	2	2	no	—
54	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂²×C₈₉	#432	1	2	yes	—
56	C₄×C₈₉	—	2	4	no	—
57	C₄×C₈₉	—	4	4	no	—
58	C₄×C₈₉	#382	1	1	yes	characteristic
59	C₄×C₈₉	#423	1	1	yes	characteristic
60	C₄×C₈₉	#431	1	2	yes	—
61	#12884	—	89	1	no	maximal, Sylow
62	C₂×C₈₉	—	2	2	no	—
63	C₂×C₈₉	#1831	1	1	yes	characteristic
64	C₂×C₈₉	#2153	1	2	yes	—
65	#2150	—	89	2	no	—
66	#1726	D₈₉	1	1	yes	characteristic
67	#1376	—	89	4	no	—
68	C₈₉	#12884	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₃₂	—	89	4	no	—
70	C₂×#97	Dic₈₉	1	2	yes	—
71	C₂×#97	Dic₈₉	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
73	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
74	#376	Dic₈₉	1	1	yes	characteristic
75	C₄⋊C₁₆	—	178	4	no	—
76	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
78	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
79	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
80	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
81	M₃₂	—	178	8	no	—
82	C₂×C₁₆	—	178	4	no	—
83	#97	—	2	4	no	—
84	#97	#10623494	1	4	yes	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	Dic₁₇₈	1	2	yes	—
87	C₄×C₈	D₃₅₆	1	2	yes	—
88	#95	—	2	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	#13083558	1	2	yes	—
91	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
96	C₂×C₈	C₂×#10623494	1	4	yes	—
97	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
98	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
99	C₂×C₈	#13083574	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	C₄⋊Dic₈₉	1	2	yes	—
102	C₄²	C₂×#10623494	1	2	yes	—
103	C₄²	#13083558	1	4	yes	—
104	C₁₆	—	356	4	no	—
105	C₂³	C₂×#13083558	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₄⋊#10623494	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
111	C₂×C₄	#15667971	1	2	yes	—
112	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
113	C₂×C₄	C₂×#13083558	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	#15668000	1	2	yes	—
117	C₈	#15668004	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
120	C₂²	#19151655	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#19151464	1	1	yes	characteristic
124	C₄	#19151622	1	1	yes	characteristic
125	C₄	#19151651	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#26020369	1	1	yes	characteristic
128	C₂	#26022018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022008

C₂

yes

maximal

#26018334

C₂

yes

maximal

C₈₉×#1726

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

C₄⋊#13083549

—

C₂×#97×C₈₉

C₄

yes

—

C₂×#97×C₈₉

C₄

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#376×C₈₉

C₄

yes

characteristic

#15667952

—

C₂×#13083549

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

#97×C₈₉

—

#97×C₈₉

C₈

yes

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#95×C₈₉

—

#13083549

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

M₁₆

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

yes

—

C₄²×C₈₉

M₁₆

yes

—

C₂³×C₈₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₈

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#124

yes

—

C₈×C₈₉

#125

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₈₉

#432

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#12884

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1831

yes

characteristic

C₂×C₈₉

#2153

yes

—

#2150

—

#1726

D₈₉

yes

characteristic

#1376

—

C₈₉

#12884

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#97

Dic₈₉

yes

—

C₂×#97

Dic₈₉

yes

characteristic

C₂×C₄×C₈

C₂×D₈₉

yes

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#376

Dic₈₉

yes

characteristic

C₄⋊C₁₆

—

178

—

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

C₂²×C₈

C₂×Dic₈₉

yes

—

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

M₃₂

—

178

—

C₂×C₁₆

—

178

—

#97

—

#97

#10623494

yes

—

C₄×C₈

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

D₃₅₆

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#13083558

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

C₂×#10623494

yes

—

C₂×C₈

C₂×D₃₅₆

yes

characteristic

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

C₂×C₈

#13083574

yes

—

100

C₄²

—

101

C₄²

C₄⋊Dic₈₉

yes

—

102

C₄²

C₂×#10623494

yes

—

103

C₄²

#13083558

yes

—

104

C₁₆

—

356

—

105

C₂³

C₂×#13083558

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊#10623494

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

111

C₂×C₄

#15667971

yes

—

112

C₂×C₄

C₂²×#10623494

yes

characteristic

113

C₂×C₄

C₂×#13083558

yes

—

114

C₈

—

115

C₈

—

116

C₈

#15668000

yes

—

117

C₈

#15668004

yes

—

118

C₂²

—

119

C₂²

C₂×C₄⋊#10623494

yes

characteristic

120

C₂²

#19151655

yes

—

121

C₄

—

122

C₄

—

123

C₄

#19151464

yes

characteristic

124

C₄

#19151622

yes

characteristic

125

C₄

#19151651

yes

—

126

C₂

—

127

C₂

#26020369

yes

characteristic

128

C₂

#26022018

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1726	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₈
prime radical	C₂³×C₈₉	C₂×M₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#12884
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₈₉

Fitting

C₈₉×#1726

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₈

prime radical

C₂³×C₈₉

C₂×M₁₆

cosocle

C₂²×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
68. C₈₉	#12884	semidirect

H ≅ G/N

type

68. C₈₉

#12884

semidirect

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

subgroup	index
#26022008	2
#26018334	2
C₈₉×#1726	2
#12884	89

subgroup

index

#26022008

#26018334

C₈₉×#1726

#12884

Subgroups — #36470792

Conjugacy classes of subgroups

Complements

Maximal subgroups