Group #36453929 — subgroups

Subgroups — #36453929

Order 22784.

454 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 127 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	#26020390	C₂	1	1	yes	maximal, characteristic
2	#26018924	C₂	1	4	yes	maximal
3	#26018544	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	2	yes	—
6	#19151622	C₂²	1	2	yes	—
7	#19151363	—	2	2	no	—
8	#19151344	—	2	8	no	—
9	#19151320	—	2	2	no	—
10	C₂²×Q₁₆×C₈₉	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
12	#494×C₈₉	—	2	2	no	—
13	#494×C₈₉	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	2	no	—
16	C₄⋊#10623494	—	2	4	no	—
17	C₄⋊#10623494	D₄	1	4	yes	—
18	C₂²⋊#10623494	—	2	2	no	—
19	C₄×#10623494	—	2	2	no	—
20	C₂²×Q₈×C₈₉	—	2	2	no	—
21	C₂×Q₁₆×C₈₉	—	2	8	no	—
22	C₂×Q₁₆×C₈₉	—	4	8	no	—
23	C₂²×C₈×C₈₉	D₄	1	2	yes	—
24	C₄⋊Q₈×C₈₉	—	2	4	no	—
25	C₄⋊Q₈×C₈₉	—	2	2	no	—
26	C₄⋊Q₈×C₈₉	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
28	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₈₉	D₄	1	2	yes	—
31	#13083558	—	4	4	no	—
32	C₂×#10623494	—	2	4	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₂×Q₈×C₈₉	—	2	4	no	—
35	C₂×Q₈×C₈₉	—	4	2	no	—
36	C₂×Q₈×C₈₉	—	4	8	no	—
37	C₂²×C₄×C₈₉	—	4	2	no	—
38	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	#41	1	2	yes	—
40	Q₁₆×C₈₉	—	4	16	no	—
41	C₂×C₈×C₈₉	—	2	4	no	—
42	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
43	C₄⋊C₄×C₈₉	—	2	8	no	—
44	C₄⋊C₄×C₈₉	—	4	4	no	—
45	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₈₉	C₂×D₄	1	2	yes	—
48	C₂³×C₈₉	#124	1	1	yes	characteristic
49	Q₈×C₈₉	—	4	8	no	—
50	Q₈×C₈₉	—	8	4	no	—
51	C₂×C₄×C₈₉	—	2	4	no	—
52	C₂×C₄×C₈₉	—	4	4	no	—
53	C₂×C₄×C₈₉	—	8	4	no	—
54	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₈₉	#121	1	2	yes	—
57	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
58	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
59	C₈×C₈₉	—	2	4	no	—
60	#10623494	—	4	4	no	—
61	#10623494	—	8	2	no	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	#390	1	1	yes	characteristic
64	C₂²×C₈₉	#490	1	2	yes	—
65	C₄×C₈₉	—	2	2	no	—
66	C₄×C₈₉	—	8	4	no	—
67	C₄×C₈₉	#418	1	2	yes	—
68	C₄×C₈₉	#473	1	4	yes	—
69	C₄×C₈₉	#496	1	2	yes	—
70	#9779	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	—	2	2	no	—
72	C₂×C₈₉	#1961	1	1	yes	characteristic
73	C₂×C₈₉	#1979	1	2	yes	—
74	C₂×#494	D₈₉	1	1	yes	characteristic
75	#1837	—	89	1	no	—
76	#1546	—	89	4	no	—
77	#1452	—	89	1	no	—
78	C₈₉	#9779	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	2	2	no	—
80	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
81	#494	—	2	2	no	—
82	#494	Dic₈₉	1	4	yes	—
83	#431	—	89	2	no	—
84	#423	—	89	2	no	—
85	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
86	#341	—	178	2	no	—
87	#332	—	178	8	no	—
88	#322	—	178	2	no	—
89	C₂²×Q₈	—	2	2	no	—
90	C₂×Q₁₆	—	2	8	no	—
91	C₂×Q₁₆	—	4	8	no	—
92	C₂×M₁₆	—	178	2	no	—
93	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
94	C₄⋊Q₈	—	2	4	no	—
95	C₄⋊Q₈	—	2	2	no	—
96	C₄⋊Q₈	C₂×Dic₈₉	1	4	yes	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
99	C₄⋊C₈	—	89	4	no	—
100	C₄⋊C₈	—	178	4	no	—
101	C₂²⋊C₈	—	178	2	no	—
102	C₄×C₈	—	178	2	no	—
103	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
104	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	8	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
110	C₂²×C₄	#13083573	1	2	yes	—
111	Q₁₆	—	4	16	no	—
112	M₁₆	—	356	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	356	2	no	—
115	C₂×C₈	—	178	4	no	—
116	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
120	C₄²	D₄×D₈₉	1	2	yes	—
121	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
122	C₂³	#15667999	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	8	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
129	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
130	C₂×C₄	#15667983	1	2	yes	—
131	C₂×C₄	#15668008	1	1	yes	characteristic
132	C₂×C₄	#15668038	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	356	4	no	—
135	C₈	—	712	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#19151501	1	1	yes	characteristic
138	C₂²	#19151962	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	4	no	—
141	C₄	#19151603	1	2	yes	—
142	C₄	#19151870	1	4	yes	—
143	C₄	#19151985	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#26021077	1	1	yes	characteristic
146	C₂	#26021168	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020390

C₂

yes

maximal, characteristic

#26018924

C₂

yes

maximal

#26018544

C₂

yes

maximal, characteristic

C₂×#494×C₈₉

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

—

#19151622

C₂²

yes

—

#19151363

—

#19151344

—

#19151320

—

C₂²×Q₁₆×C₈₉

—

C₂×C₄⋊Q₈×C₈₉

C₂²

yes

—

#494×C₈₉

—

#494×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

—

D₄

yes

—

C₂²⋊#10623494

—

C₄×#10623494

—

C₂²×Q₈×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

#13083558

—

C₂×#10623494

—

C₂×#10623494

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

#124

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

#121

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#490

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#418

yes

—

C₄×C₈₉

#473

yes

—

C₄×C₈₉

#496

yes

—

#9779

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1961

yes

characteristic

C₂×C₈₉

#1979

yes

—

C₂×#494

D₈₉

yes

—

—

—

C₈₉

#9779

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₈₉

yes

—

#494

—

#494

Dic₈₉

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#341

—

178

—

#332

—

178

—

#322

—

178

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

178

—

C₂²×C₈

C₂²⋊D₈₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₈₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

178

—

101

C₂²⋊C₈

—

178

—

102

C₄×C₈

—

178

—

103

C₄×C₈

C₂×Dic₈₉

yes

—

104

C₄×C₈

C₂²⋊D₈₉

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

110

C₂²×C₄

#13083573

yes

—

111

Q₁₆

—

112

M₁₆

—

356

—

113

C₂×C₈

—

114

C₂×C₈

—

356

—

115

C₂×C₈

—

178

—

116

C₂×C₈

C₂²⋊Dic₈₉

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂²×Dic₈₉

yes

characteristic

120

C₄²

D₄×D₈₉

yes

—

121

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

122

C₂³

#15667999

yes

characteristic

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊Dic₈₉

yes

characteristic

129

C₂×C₄

D₄×Dic₈₉

yes

—

130

C₂×C₄

#15667983

yes

—

131

C₂×C₄

#15668008

yes

characteristic

132

C₂×C₄

#15668038

yes

—

133

C₈

—

134

C₈

—

356

—

135

C₈

—

712

—

136

C₂²

—

137

C₂²

#19151501

yes

characteristic

138

C₂²

#19151962

yes

—

139

C₄

—

140

C₄

—

141

C₄

#19151603

yes

—

142

C₄

#19151870

yes

—

143

C₄

#19151985

yes

—

144

C₂

—

145

C₂

#26021077

yes

characteristic

146

C₂

#26021168

yes

—

	subgroup	G/H
center	C₂²	#19151501
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#494×C₈₉	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#494	D₈₉
hypocenter	C₈₉	#9779
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#19151501

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×#494×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
78. C₈₉	#9779	semidirect

H ≅ G/N

type

78. C₈₉

#9779

semidirect

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

subgroup	index
#26020390	2
#26018924	2
#26018544	2
C₂×#494×C₈₉	2
#9779	89

subgroup

index

C₂×#494×C₈₉

Subgroups — #36453929

Conjugacy classes of subgroups

Complements

Maximal subgroups