Group #36470848 — subgroups

Subgroups — #36470848

Order 22784.

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

normal subgroups only

This group has 342 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	C₂×#19151419	C₂	1	4	yes	maximal
2	#26022008	C₂	1	2	yes	maximal
3	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667952	C₂²	1	4	yes	—
5	#19151419	—	2	8	no	—
6	C₄⋊#13083549	—	2	4	no	—
7	C₂×#97×C₈₉	C₄	1	2	yes	—
8	C₂×#97×C₈₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
11	#376×C₈₉	C₄	1	1	yes	characteristic
12	#15667952	—	2	8	no	—
13	C₂×#13083549	—	2	4	no	—
14	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
19	#97×C₈₉	—	2	4	no	—
20	#97×C₈₉	C₈	1	4	yes	—
21	C₄×C₈×C₈₉	D₄	1	2	yes	—
22	C₄×C₈×C₈₉	D₄	1	4	yes	—
23	C₄×C₈×C₈₉	Q₈	1	4	yes	—
24	C₄×C₈×C₈₉	Q₈	1	2	yes	—
25	#95×C₈₉	—	2	2	no	—
26	#13083549	—	4	4	no	—
27	C₂²×C₄×C₈₉	—	2	2	no	—
28	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
29	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
35	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
36	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
38	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₈₉	—	2	2	no	—
40	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
41	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
42	C₄²×C₈₉	C₂×C₈	1	2	yes	—
43	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₈₉	—	2	4	no	—
45	C₂×C₄×C₈₉	—	4	2	no	—
46	C₂×C₄×C₈₉	—	4	4	no	—
47	C₂×C₄×C₈₉	C₄⋊C₈	1	4	yes	—
48	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
52	C₈×C₈₉	—	2	2	no	—
53	C₈×C₈₉	C₄⋊C₈	1	8	yes	—
54	C₈×C₈₉	#127	1	1	yes	characteristic
55	C₈×C₈₉	C₄⋊Q₈	1	2	yes	—
56	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
57	C₂²×C₈₉	—	2	2	no	—
58	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₈₉	#423	1	2	yes	—
60	C₄×C₈₉	—	2	4	no	—
61	C₄×C₈₉	—	4	4	no	—
62	C₄×C₈₉	#384	1	1	yes	characteristic
63	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
64	C₄×C₈₉	#423	1	1	yes	characteristic
65	#12901	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	—	2	2	no	—
67	C₂×C₈₉	#1842	1	1	yes	characteristic
68	C₂×C₈₉	#2153	1	2	yes	—
69	C₂×#364	—	89	4	no	—
70	#2150	—	89	2	no	—
71	#1726	D₈₉	1	1	yes	characteristic
72	C₈₉	#12901	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₃₂	—	89	4	no	—
74	C₂×#97	Dic₈₉	1	2	yes	—
75	C₂×#97	Dic₈₉	1	1	yes	characteristic
76	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
77	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
78	#376	Dic₈₉	1	1	yes	characteristic
79	#364	—	178	8	no	—
80	C₄⋊C₁₆	—	178	4	no	—
81	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
83	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
84	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
85	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
86	M₃₂	—	178	8	no	—
87	C₂×C₁₆	—	178	4	no	—
88	#97	—	2	4	no	—
89	#97	#10623494	1	4	yes	—
90	C₄×C₈	Dic₁₇₈	1	4	yes	—
91	C₄×C₈	Dic₁₇₈	1	2	yes	—
92	C₄×C₈	D₃₅₆	1	4	yes	—
93	C₄×C₈	D₃₅₆	1	2	yes	—
94	#95	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#13083558	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
100	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
101	C₂×C₈	C₂×#10623494	1	4	yes	—
102	C₂×C₈	#13083558	1	4	yes	—
103	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
104	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
105	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
106	C₂×C₈	C₂×Dic₁₇₈	1	2	yes	—
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊Dic₈₉	1	4	yes	—
109	C₄²	C₄⋊Dic₈₉	1	2	yes	—
110	C₄²	C₂×#10623494	1	2	yes	—
111	C₁₆	—	356	4	no	—
112	C₂³	C₂×#13083558	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	C₄⋊#10623494	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
119	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
120	C₂×C₄	C₂×#13083558	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	C₄⋊#10623494	1	8	yes	—
123	C₈	#15668009	1	1	yes	characteristic
124	C₈	#15668012	1	2	yes	—
125	C₈	#15668012	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
128	C₂²	#19151622	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#19151468	1	1	yes	characteristic
132	C₄	C₂×C₄⋊#10623494	1	2	yes	—
133	C₄	#19151622	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#26020404	1	1	yes	characteristic
136	C₂	#26022018	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151419

C₂

yes

maximal

#26022008

C₂

yes

maximal

C₈₉×#1726

C₂

yes

maximal, characteristic

C₂×#15667952

C₂²

yes

—

#19151419

—

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

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

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

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₂×C₈

yes

—

C₂×C₈×C₈₉

M₁₆

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

—

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

C₂×C₈

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

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₄⋊C₈

yes

—

C₈×C₈₉

#127

yes

characteristic

C₈×C₈₉

C₄⋊Q₈

yes

—

C₈×C₈₉

C₄⋊Q₈

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₈₉

#423

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#384

yes

characteristic

C₄×C₈₉

C₂×C₄⋊C₈

yes

—

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

#1842

yes

characteristic

C₂×C₈₉

#2153

yes

—

C₂×#364

—

#2150

—

#1726

D₈₉

yes

characteristic

C₈₉

#12901

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

#364

—

178

—

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

Dic₁₇₈

yes

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

D₃₅₆

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#13083558

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

100

C₂×C₈

C₄⋊Dic₈₉

yes

—

101

C₂×C₈

C₂×#10623494

yes

—

102

C₂×C₈

#13083558

yes

—

103

C₂×C₈

C₂×D₃₅₆

yes

—

104

C₂×C₈

C₂×D₃₅₆

yes

characteristic

105

C₂×C₈

C₂×Dic₁₇₈

yes

characteristic

106

C₂×C₈

C₂×Dic₁₇₈

yes

—

107

C₄²

—

108

C₄²

C₄⋊Dic₈₉

yes

—

109

C₄²

C₄⋊Dic₈₉

yes

—

110

C₄²

C₂×#10623494

yes

—

111

C₁₆

—

356

—

112

C₂³

C₂×#13083558

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₄⋊#10623494

yes

—

117

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

118

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

119

C₂×C₄

C₂²×#10623494

yes

characteristic

120

C₂×C₄

C₂×#13083558

yes

—

121

C₈

—

122

C₈

C₄⋊#10623494

yes

—

123

C₈

#15668009

yes

characteristic

124

C₈

#15668012

yes

—

125

C₈

#15668012

yes

characteristic

126

C₂²

—

127

C₂²

C₂×C₄⋊#10623494

yes

characteristic

128

C₂²

#19151622

yes

—

129

C₄

—

130

C₄

—

131

C₄

#19151468

yes

characteristic

132

C₄

C₂×C₄⋊#10623494

yes

—

133

C₄

#19151622

yes

characteristic

134

C₂

—

135

C₂

#26020404

yes

characteristic

136

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	#12901
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
72. C₈₉	#12901	semidirect

H ≅ G/N

type

72. C₈₉

#12901

semidirect

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

subgroup	index
C₂×#19151419	2
#26022008	2
C₈₉×#1726	2
#12901	89

subgroup

index

C₂×#19151419

#26022008

C₈₉×#1726

#12901

Subgroups — #36470848

Conjugacy classes of subgroups

Complements

Maximal subgroups