Group #36453928 — subgroups

Subgroups — #36453928

Order 22784.

454 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 123 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×#19151978	C₂	1	1	yes	maximal, characteristic
2	#26020390	C₂	1	1	yes	maximal, characteristic
3	#26018923	C₂	1	4	yes	maximal
4	C₈₉×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₆	—	2	2	no	—
6	C₂×#15668012	C₂²	1	2	yes	—
7	#19151978	—	2	2	no	—
8	#19151978	C₄	1	4	yes	—
9	#19151622	C₂²	1	2	yes	—
10	#19151363	—	2	2	no	—
11	#19151343	—	2	8	no	—
12	#431×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#322×C₈₉	—	2	2	no	—
15	C₂²×Dic₁₇₈	—	2	2	no	—
16	C₂×Dic₃₅₆	—	2	8	no	—
17	C₂×Dic₃₅₆	—	4	8	no	—
18	C₂×#13083558	—	2	2	no	—
19	#15668012	—	2	4	no	—
20	#15668012	—	2	2	no	—
21	#15668012	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₈₉	—	2	4	no	—
23	C₄⋊#10623494	—	2	4	no	—
24	C₂²×C₈×C₈₉	D₄	1	2	yes	—
25	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₈₉	D₄	1	4	yes	—
27	C₂²⋊C₈×C₈₉	—	2	2	no	—
28	C₄×C₈×C₈₉	—	2	2	no	—
29	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₈₉	D₄	1	2	yes	—
31	C₂×Dic₁₇₈	—	2	4	no	—
32	C₂×Dic₁₇₈	—	4	8	no	—
33	C₂×Dic₁₇₈	—	4	2	no	—
34	C₂²×Dic₈₉	—	4	2	no	—
35	Dic₃₅₆	—	4	16	no	—
36	#13083558	—	4	4	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₄⋊Dic₈₉	—	2	8	no	—
39	C₄⋊Dic₈₉	—	4	4	no	—
40	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	#41	1	2	yes	—
42	C₂×C₈×C₈₉	—	2	4	no	—
43	C₂×C₈×C₈₉	—	2	4	no	—
44	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
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	C₂×C₄×C₈₉	—	2	4	no	—
50	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
52	C₂×C₄×C₈₉	#121	1	2	yes	—
53	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
54	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
55	C₂×Dic₈₉	—	4	4	no	—
56	C₂×Dic₈₉	—	8	4	no	—
57	Dic₁₇₈	—	4	8	no	—
58	Dic₁₇₈	—	8	4	no	—
59	C₈×C₈₉	—	2	4	no	—
60	C₈×C₈₉	—	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	#418	1	2	yes	—
67	C₄×C₈₉	#473	1	4	yes	—
68	C₄×C₈₉	#496	1	2	yes	—
69	Dic₈₉	—	8	4	no	—
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	—	89	1	no	—
75	#1837	—	89	1	no	—
76	#1546	—	89	4	no	—
77	#1452	D₈₉	1	1	yes	characteristic
78	C₈₉	#9779	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	178	2	no	—
80	C₂×C₄⋊Q₈	—	89	2	no	—
81	#494	—	89	4	no	—
82	#494	—	178	2	no	—
83	#431	C₂×D₈₉	1	2	yes	—
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	—	2	2	no	—
89	C₂²×Q₈	—	178	2	no	—
90	C₂×Q₁₆	—	356	8	no	—
91	C₂×Q₁₆	—	178	8	no	—
92	C₂×M₁₆	—	178	2	no	—
93	C₂²×C₈	D₃₅₆	1	2	yes	—
94	C₄⋊Q₈	—	89	4	no	—
95	C₄⋊Q₈	—	178	2	no	—
96	C₄⋊Q₈	—	178	4	no	—
97	C₂×C₄⋊C₄	—	178	4	no	—
98	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
99	C₄⋊C₈	—	178	4	no	—
100	C₄⋊C₈	D₃₅₆	1	4	yes	—
101	C₂²⋊C₈	—	2	2	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	C₄×D₈₉	1	2	yes	—
104	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
105	C₂×Q₈	—	356	2	no	—
106	C₂×Q₈	—	356	8	no	—
107	C₂×Q₈	—	178	4	no	—
108	C₂²×C₄	—	356	2	no	—
109	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
110	C₂²×C₄	#13083574	1	2	yes	—
111	Q₁₆	—	356	16	no	—
112	M₁₆	—	356	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	356	2	no	—
116	C₂×C₈	#13083551	1	4	yes	—
117	C₄⋊C₄	—	356	4	no	—
118	C₄⋊C₄	—	178	8	no	—
119	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
120	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
121	C₄²	C₂×D₃₅₆	1	2	yes	—
122	C₂³	#15668000	1	1	yes	characteristic
123	Q₈	—	356	8	no	—
124	Q₈	—	712	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	356	4	no	—
127	C₂×C₄	—	712	4	no	—
128	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
129	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
130	C₂×C₄	#15667986	1	2	yes	—
131	C₂×C₄	#15668009	1	1	yes	characteristic
132	C₂×C₄	#15668040	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	4	4	no	—
135	C₈	—	712	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#19151500	1	1	yes	characteristic
138	C₂²	#19151963	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	712	4	no	—
141	C₄	#19151604	1	2	yes	—
142	C₄	#19151873	1	4	yes	—
143	C₄	#19151987	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#26021076	1	1	yes	characteristic
146	C₂	#26021167	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151978

C₂

yes

maximal, characteristic

#26020390

C₂

yes

maximal, characteristic

#26018923

C₂

yes

maximal

C₈₉×#1452

C₂

yes

maximal, characteristic

C₂²×Dic₃₅₆

—

C₂×#15668012

C₂²

yes

—

#19151978

—

#19151978

C₄

yes

—

#19151622

C₂²

yes

—

#19151363

—

#19151343

—

#431×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#322×C₈₉

—

C₂²×Dic₁₇₈

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₄⋊#10623494

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

D₄

yes

—

C₂²⋊C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

#124

yes

characteristic

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

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#390

yes

characteristic

C₂²×C₈₉

#490

yes

—

C₄×C₈₉

—

C₄×C₈₉

#418

yes

—

C₄×C₈₉

#473

yes

—

C₄×C₈₉

#496

yes

—

Dic₈₉

—

#9779

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1961

yes

characteristic

C₂×C₈₉

#1979

yes

—

—

—

—

D₈₉

yes

characteristic

C₈₉

#9779

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

178

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

178

—

#431

C₂×D₈₉

yes

—

#423

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#341

—

178

—

#332

—

178

—

#322

—

C₂²×Q₈

—

178

—

C₂×Q₁₆

—

356

—

C₂×Q₁₆

—

178

—

C₂×M₁₆

—

178

—

C₂²×C₈

D₃₅₆

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

178

—

100

C₄⋊C₈

D₃₅₆

yes

—

101

C₂²⋊C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₄×D₈₉

yes

—

104

C₄×C₈

C₂²⋊D₈₉

yes

—

105

C₂×Q₈

—

356

—

106

C₂×Q₈

—

356

—

107

C₂×Q₈

—

178

—

108

C₂²×C₄

—

356

—

109

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

110

C₂²×C₄

#13083574

yes

—

111

Q₁₆

—

356

—

112

M₁₆

—

356

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

356

—

116

C₂×C₈

#13083551

yes

—

117

C₄⋊C₄

—

356

—

118

C₄⋊C₄

—

178

—

119

C₄²

C₂×C₄×D₈₉

yes

characteristic

120

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

121

C₄²

C₂×D₃₅₆

yes

—

122

C₂³

#15668000

yes

characteristic

123

Q₈

—

356

—

124

Q₈

—

712

—

125

C₂×C₄

—

126

C₂×C₄

—

356

—

127

C₂×C₄

—

712

—

128

C₂×C₄

C₂×#13083551

yes

characteristic

129

C₂×C₄

C₄×D₃₅₆

yes

—

130

C₂×C₄

#15667986

yes

—

131

C₂×C₄

#15668009

yes

characteristic

132

C₂×C₄

#15668040

yes

—

133

C₈

—

134

C₈

—

135

C₈

—

712

—

136

C₂²

—

137

C₂²

#19151500

yes

characteristic

138

C₂²

#19151963

yes

—

139

C₄

—

140

C₄

—

712

—

141

C₄

#19151604

yes

—

142

C₄

#19151873

yes

—

143

C₄

#19151987

yes

—

144

C₂

—

145

C₂

#26021076

yes

characteristic

146

C₂

#26021167

yes

—

	subgroup	G/H
center	C₂²	#19151500
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1452	C₂
socle	C₂²×C₈₉	#390
prime radical	C₂³×C₈₉	#124
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1452	D₈₉
hypocenter	C₈₉	#9779
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#19151500

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1452

C₂

C₂²×C₈₉

C₂³×C₈₉

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

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
C₂×#19151978	2
#26020390	2
#26018923	2
C₈₉×#1452	2
#9779	89

subgroup

index

C₈₉×#1452

Subgroups — #36453928

Conjugacy classes of subgroups

Complements

Maximal subgroups