Group #36562833 — subgroups

Subgroups — #36562833

Order 22784.

1154 conjugacy classes of subgroups; only the 299 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 299 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×#19151677	C₂	1	6	yes	maximal, complemented
2	#26023533	C₂	1	1	yes	maximal, characteristic
3	#26021050	C₂	1	6	yes	maximal, complemented
4	#26020406	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13083557	C₂²	1	4	yes	—
7	C₂×C₄×D₃₅₆	C₂²	1	6	yes	—
8	#19151677	—	2	24	no	—
9	C₂×C₄⋊#10623494	C₂²	1	6	yes	—
10	C₂×#15667924	C₂²	1	12	yes	—
11	#19151535	C₄	1	4	yes	—
12	#19151508	C₄	1	4	yes	—
13	C₂×#97×C₈₉	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
15	#376×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×D₃₅₆	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₈₉	C₂³	1	4	yes	—
18	C₂×#13083557	—	2	24	no	—
19	C₂²×#10623494	C₂³	1	4	yes	—
20	C₄×D₃₅₆	—	2	12	no	—
21	C₄×D₃₅₆	—	2	12	no	—
22	C₂×C₄⋊Dic₈₉	C₂×C₄	1	6	yes	—
23	C₂×#13083551	C₂×C₄	1	12	yes	—
24	C₄⋊#10623494	—	2	12	no	—
25	#15667924	—	2	24	no	—
26	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
27	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
29	#97×C₈₉	—	2	6	no	—
30	C₄×C₈×C₈₉	D₄	1	12	yes	complemented
31	#95×C₈₉	C₂×C₄	1	4	yes	—
32	C₂³×D₈₉	C₂²×C₄	1	4	yes	—
33	C₂×D₃₅₆	—	2	12	no	—
34	C₂×D₃₅₆	—	4	12	no	—
35	C₂²×Dic₈₉	C₂²×C₄	1	4	yes	—
36	C₂×C₄×D₈₉	—	2	12	no	—
37	C₂×C₄×D₈₉	—	2	12	no	—
38	#13083557	—	4	16	no	—
39	C₂×#10623494	—	2	12	no	—
40	C₄⋊Dic₈₉	—	2	12	no	—
41	#13083551	—	2	24	no	—
42	C₂²×C₄×C₈₉	C₂²×C₄	1	6	yes	—
43	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
44	C₂×C₈×C₈₉	—	2	6	no	—
45	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
46	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
47	C₄²×C₈₉	C₂×D₄	1	6	yes	—
48	C₄²×C₈₉	#41	1	6	yes	—
49	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
50	C₂²×D₈₉	—	2	12	no	—
51	C₂²×D₈₉	—	2	4	no	—
52	C₂²×D₈₉	—	2	12	no	—
53	C₂×C₄×C₈₉	—	2	12	no	—
54	C₂×C₄×C₈₉	C₄×D₄	1	12	yes	—
55	C₂×C₄×C₈₉	C₂²×D₄	1	3	yes	—
56	C₂×C₄×C₈₉	C₂×#41	1	3	yes	—
57	C₂×Dic₈₉	—	2	12	no	—
58	D₃₅₆	—	4	24	no	—
59	C₄×D₈₉	—	4	4	no	—
60	C₄×D₈₉	—	4	12	no	—
61	C₈×C₈₉	—	2	6	no	—
62	C₈×C₈₉	#127	1	4	yes	—
63	#10623494	—	4	4	no	—
64	C₂²×C₈₉	C₂×C₄×D₄	1	3	yes	—
65	C₂²×C₈₉	#568	1	4	yes	—
66	C₂×D₈₉	—	4	4	no	—
67	C₂×D₈₉	—	4	12	no	—
68	C₂×D₈₉	—	4	12	no	—
69	C₄×C₈₉	—	2	12	no	—
70	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
71	C₄×C₈₉	#532	1	3	yes	—
72	Dic₈₉	—	4	4	no	—
73	#21684	—	89	1	no	maximal, Sylow
74	C₂×C₈₉	C₄×#127	1	1	yes	characteristic
75	C₂×C₈₉	#2930	1	6	yes	—
76	D₈₉	—	8	4	no	—
77	C₂×#436	—	89	6	no	—
78	#2397	—	89	1	no	—
79	#1955	—	89	6	no	—
80	#1843	—	89	1	no	—
81	#1726	D₈₉	1	1	yes	characteristic, complemented
82	C₈₉	#21684	1	1	yes	characteristic, complemented, Sylow
83	C₂²×M₁₆	—	89	4	no	—
84	C₂×C₄×D₄	—	89	6	no	—
85	#436	—	178	24	no	—
86	C₂×C₄⋊C₈	—	89	6	no	—
87	C₂×C₂²⋊C₈	—	89	12	no	—
88	C₂×#97	C₂×D₈₉	1	3	yes	—
89	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
90	#399	—	89	4	no	—
91	#392	—	89	4	no	—
92	#376	C₂×D₈₉	1	1	yes	characteristic
93	C₂²×D₄	—	89	6	no	—
94	C₂³×C₄	—	89	4	no	—
95	C₂×M₁₆	—	178	24	no	—
96	C₂²×C₈	—	89	4	no	—
97	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
98	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
99	C₄×D₄	—	178	12	no	—
100	C₄×D₄	—	178	12	no	—
101	C₂×C₄⋊C₄	—	89	6	no	—
102	C₂×C₂²⋊C₄	—	89	12	no	—
103	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
104	C₄⋊C₈	—	178	12	no	—
105	C₂²⋊C₈	—	178	24	no	—
106	#97	—	2	6	no	—
107	C₄×C₈	D₃₅₆	1	12	yes	complemented
108	#95	C₄×D₈₉	1	4	yes	—
109	C₂⁴	—	89	4	no	—
110	C₂×D₄	—	356	12	no	—
111	C₂×D₄	—	178	12	no	—
112	C₂²×C₄	—	89	4	no	—
113	C₂²×C₄	—	178	12	no	—
114	C₂²×C₄	—	178	12	no	—
115	C₂²×C₄	C₂×C₄×D₈₉	1	6	yes	—
116	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
117	M₁₆	—	356	16	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	—	178	12	no	—
120	C₂×C₈	C₂×D₃₅₆	1	6	yes	—
121	C₂×C₈	C₂×D₃₅₆	1	6	yes	—
122	C₄⋊C₄	—	178	12	no	—
123	C₂²⋊C₄	—	178	24	no	—
124	C₄²	C₂×D₃₅₆	1	6	yes	—
125	C₄²	#13083574	1	6	yes	—
126	C₂³	—	178	12	no	—
127	C₂³	—	178	4	no	—
128	C₂³	—	178	12	no	—
129	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
130	D₄	—	356	24	no	—
131	C₂×C₄	—	2	12	no	—
132	C₂×C₄	—	356	4	no	—
133	C₂×C₄	—	356	12	no	—
134	C₂×C₄	—	178	12	no	—
135	C₂×C₄	C₄×D₃₅₆	1	12	yes	—
136	C₂×C₄	C₂²×D₃₅₆	1	3	yes	—
137	C₂×C₄	C₂×#13083574	1	3	yes	—
138	C₈	—	2	6	no	—
139	C₈	—	356	4	no	—
140	C₈	#15668009	1	4	yes	—
141	C₂²	—	356	12	no	—
142	C₂²	—	356	12	no	—
143	C₂²	—	356	4	no	—
144	C₂²	C₂×C₄×D₃₅₆	1	3	yes	—
145	C₂²	#19152363	1	4	yes	—
146	C₄	—	2	12	no	—
147	C₄	—	356	4	no	—
148	C₄	C₂×#15668009	1	1	yes	characteristic
149	C₄	#19152148	1	3	yes	—
150	C₂	—	712	4	no	—
151	C₂	C₄×#15668009	1	1	yes	characteristic
152	C₂	#26028957	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19151677

C₂

yes

maximal, complemented

#26023533

C₂

yes

maximal, characteristic

#26021050

C₂

yes

maximal, complemented

#26020406

C₂

yes

maximal, characteristic, complemented

C₈₉×#1726

C₂

yes

maximal, characteristic, complemented

C₂²×#13083557

C₂²

yes

—

C₂×C₄×D₃₅₆

C₂²

yes

—

#19151677

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×#15667924

C₂²

yes

—

#19151535

C₄

yes

—

#19151508

C₄

yes

—

C₂×#97×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

#376×C₈₉

C₂²

yes

characteristic

C₂²×D₃₅₆

C₂×C₄

yes

—

C₂²×C₄×D₈₉

C₂³

yes

—

C₂×#13083557

—

C₂²×#10623494

C₂³

yes

—

C₄×D₃₅₆

—

C₄×D₃₅₆

—

C₂×C₄⋊Dic₈₉

C₂×C₄

yes

—

C₂×#13083551

C₂×C₄

yes

—

C₄⋊#10623494

—

#15667924

—

C₂²×C₈×C₈₉

C₂³

yes

—

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄²×C₈₉

C₂³

yes

—

#97×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

complemented

#95×C₈₉

C₂×C₄

yes

—

C₂³×D₈₉

C₂²×C₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

C₂²×C₄

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

#13083557

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂³×C₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

—

C₂×C₄×C₈₉

C₂×#41

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#127

yes

—

#10623494

—

C₂²×C₈₉

C₂×C₄×D₄

yes

—

C₂²×C₈₉

#568

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#127

yes

characteristic

C₄×C₈₉

#532

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#21684

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

C₂×C₄×D₄

—

#436

—

178

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

C₂×D₈₉

yes

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

#399

—

#392

—

#376

C₂×D₈₉

yes

characteristic

C₂²×D₄

—

C₂³×C₄

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₈₉

yes

—

C₂²×C₈

C₂²×D₈₉

yes

characteristic

C₄×D₄

—

178

—

100

C₄×D₄

—

178

—

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

C₂²×D₈₉

yes

—

104

C₄⋊C₈

—

178

—

105

C₂²⋊C₈

—

178

—

106

#97

—

107

C₄×C₈

D₃₅₆

yes

complemented

108

#95

C₄×D₈₉

yes

—

109

C₂⁴

—

110

C₂×D₄

—

356

—

111

C₂×D₄

—

178

—

112

C₂²×C₄

—

113

C₂²×C₄

—

178

—

114

C₂²×C₄

—

178

—

115

C₂²×C₄

C₂×C₄×D₈₉

yes

—

116

C₂²×C₄

C₂³×D₈₉

yes

characteristic

117

M₁₆

—

356

—

118

C₂×C₈

—

119

C₂×C₈

—

178

—

120

C₂×C₈

C₂×D₃₅₆

yes

—

121

C₂×C₈

C₂×D₃₅₆

yes

—

122

C₄⋊C₄

—

178

—

123

C₂²⋊C₄

—

178

—

124

C₄²

C₂×D₃₅₆

yes

—

125

C₄²

#13083574

yes

—

126

C₂³

—

178

—

127

C₂³

—

178

—

128

C₂³

—

178

—

129

C₂³

C₂²×C₄×D₈₉

yes

characteristic

130

D₄

—

356

—

131

C₂×C₄

—

132

C₂×C₄

—

356

—

133

C₂×C₄

—

356

—

134

C₂×C₄

—

178

—

135

C₂×C₄

C₄×D₃₅₆

yes

—

136

C₂×C₄

C₂²×D₃₅₆

yes

—

137

C₂×C₄

C₂×#13083574

yes

—

138

C₈

—

139

C₈

—

356

—

140

C₈

#15668009

yes

—

141

C₂²

—

356

—

142

C₂²

—

356

—

143

C₂²

—

356

—

144

C₂²

C₂×C₄×D₃₅₆

yes

—

145

C₂²

#19152363

yes

—

146

C₄

—

147

C₄

—

356

—

148

C₄

C₂×#15668009

yes

characteristic

149

C₄

#19152148

yes

—

150

C₂

—

712

—

151

C₂

C₄×#15668009

yes

characteristic

152

C₂

#26028957

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₉
commutator, socle	C₂³×C₈₉	C₂³×C₄
Fitting	C₈₉×#1726	C₂
prime radical	#26023533	C₂
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#21684
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #1726

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₈₉

commutator, socle

C₂³×C₈₉

C₂³×C₄

Fitting

C₈₉×#1726

C₂

prime radical

#26023533

C₂

cosocle

C₂²×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ #1726

N	H ≅ G/N	type
1. C₂×#19151677	C₂	semidirect
3. #26021050	C₂	semidirect
4. #26020406	C₂	semidirect
5. C₈₉×#1726	C₂	semidirect
30. C₄×C₈×C₈₉	D₄	semidirect
81. #1726	D₈₉	semidirect
82. C₈₉	#21684	semidirect
107. C₄×C₈	D₃₅₆	semidirect

H ≅ G/N

type

1. C₂×#19151677

C₂

semidirect

3. #26021050

C₂

semidirect

4. #26020406

C₂

semidirect

5. C₈₉×#1726

C₂

semidirect

30. C₄×C₈×C₈₉

D₄

semidirect

81. #1726

D₈₉

semidirect

82. C₈₉

#21684

semidirect

107. C₄×C₈

D₃₅₆

semidirect

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

subgroup	index
C₂×#19151677	2
#26023533	2
#26021050	2
#26020406	2
C₈₉×#1726	2
#21684	89

subgroup

index

C₈₉×#1726

Subgroups — #36562833

Conjugacy classes of subgroups

Complements

Maximal subgroups