Group #36441813 — subgroups

Subgroups — #36441813

Order 22784.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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	#26018198	C₂	1	2	yes	maximal
2	C₈₉×#1956	C₂	1	1	yes	maximal, characteristic
3	#19151324	—	2	2	no	—
4	C₂×C₄⋊Q₈×C₈₉	C₄	1	1	yes	characteristic
5	C₂×#103×C₈₉	—	2	2	no	—
6	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
7	#384×C₈₉	C₂²	1	1	yes	characteristic
8	C₂²⋊#10623494	—	4	2	no	—
9	C₂²×Q₈×C₈₉	—	2	1	no	—
10	C₂²×C₈×C₈₉	—	2	1	no	—
11	C₄⋊Q₈×C₈₉	—	2	1	no	—
12	C₄⋊Q₈×C₈₉	—	4	1	no	—
13	C₄⋊Q₈×C₈₉	C₈	1	2	yes	complemented
14	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
15	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
16	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
17	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
18	#103×C₈₉	—	4	4	no	—
19	C₄×C₈×C₈₉	—	4	1	no	—
20	C₂×#10623494	—	8	2	no	—
21	C₂×Q₈×C₈₉	—	2	2	no	—
22	C₂×Q₈×C₈₉	—	4	2	no	—
23	C₂×Q₈×C₈₉	—	4	2	no	—
24	C₂×Q₈×C₈₉	—	4	1	no	—
25	C₂²×C₄×C₈₉	—	2	1	no	—
26	C₂²×C₄×C₈₉	—	2	2	no	—
27	C₂²×C₄×C₈₉	—	4	1	no	—
28	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	—	4	1	no	—
31	C₂×C₈×C₈₉	—	4	1	no	—
32	C₄⋊C₄×C₈₉	—	4	2	no	—
33	C₄⋊C₄×C₈₉	—	4	2	no	—
34	C₄⋊C₄×C₈₉	—	4	2	no	—
35	C₄⋊C₄×C₈₉	—	8	2	no	—
36	C₄²×C₈₉	—	2	1	no	—
37	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
38	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
39	C₂³×C₈₉	#99	1	1	yes	characteristic
40	Q₈×C₈₉	—	4	4	no	—
41	Q₈×C₈₉	—	8	2	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	1	no	—
45	C₂×C₄×C₈₉	—	2	1	no	—
46	C₂×C₄×C₈₉	—	2	1	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	—	4	2	no	—
49	C₂×C₄×C₈₉	—	4	2	no	—
50	C₂×C₄×C₈₉	—	8	1	no	—
51	C₂×C₄×C₈₉	—	8	1	no	—
52	C₂×C₄×C₈₉	—	8	2	no	—
53	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
55	C₈×C₈₉	—	4	2	no	—
56	#10623494	—	16	2	no	—
57	C₂²×C₈₉	—	2	1	no	—
58	C₂²×C₈₉	—	2	1	no	—
59	C₂²×C₈₉	#323	1	1	yes	characteristic
60	C₂²×C₈₉	#353	1	1	yes	characteristic
61	C₂²×C₈₉	#355	1	1	yes	characteristic
62	C₄×C₈₉	—	2	2	no	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	—	2	1	no	—
65	C₄×C₈₉	—	2	1	no	—
66	C₄×C₈₉	—	8	2	no	—
67	C₄×C₈₉	—	8	2	no	—
68	#7355	—	89	1	no	maximal, Sylow
69	C₂×C₈₉	—	2	1	no	—
70	C₂×C₈₉	—	2	1	no	—
71	C₂×C₈₉	#1318	1	1	yes	characteristic
72	C₂×C₈₉	#1409	1	1	yes	characteristic
73	C₂×C₈₉	#1411	1	1	yes	characteristic
74	#1956	D₈₉	1	1	yes	characteristic
75	#1325	—	89	2	no	—
76	C₈₉	#7355	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₄⋊Q₈	Dic₈₉	1	1	yes	characteristic
78	C₂×#103	—	2	2	no	—
79	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
80	#384	C₂×D₈₉	1	1	yes	characteristic
81	#324	—	178	2	no	—
82	C₂²×Q₈	—	2	1	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	4	1	no	—
86	C₄⋊Q₈	#10623494	1	2	yes	complemented
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄⋊C₄	—	4	1	no	—
89	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
90	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
91	#103	—	4	4	no	—
92	C₂²⋊C₈	—	356	2	no	—
93	C₄×C₈	—	4	1	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	4	1	no	—
101	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	712	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₄²	—	2	1	no	—
111	C₄²	C₂×#10623494	1	1	yes	characteristic
112	C₄²	#13083558	1	1	yes	characteristic
113	C₂³	#15667927	1	1	yes	characteristic
114	Q₈	—	4	4	no	—
115	Q₈	—	8	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	—	8	1	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	C₂²⋊#10623494	1	1	yes	characteristic
128	C₂×C₄	#15667929	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₈	—	1424	2	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#19151322	1	1	yes	characteristic
134	C₂²	#19151391	1	1	yes	characteristic
135	C₂²	#19151395	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	8	2	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#26018184	1	1	yes	characteristic
145	C₂	#26018417	1	1	yes	characteristic
146	C₂	#26018421	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26018198

C₂

yes

maximal

C₈₉×#1956

C₂

yes

maximal, characteristic

#19151324

—

C₂×C₄⋊Q₈×C₈₉

C₄

yes

characteristic

C₂×#103×C₈₉

—

C₂×C₄×C₈×C₈₉

C₄

yes

characteristic

#384×C₈₉

C₂²

yes

characteristic

C₂²⋊#10623494

—

C₂²×Q₈×C₈₉

—

C₂²×C₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₈

yes

complemented

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

#103×C₈₉

—

C₄×C₈×C₈₉

—

C₂×#10623494

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

characteristic

C₂³×C₈₉

#99

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₈₉

#100

yes

characteristic

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

#7355

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

#1325

—

C₈₉

#7355

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

Dic₈₉

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

Dic₈₉

yes

characteristic

#384

C₂×D₈₉

yes

characteristic

#324

—

178

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#10623494

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₈₉

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

characteristic

#103

—

C₂²⋊C₈

—

356

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²⋊Dic₈₉

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

712

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

—

111

C₄²

C₂×#10623494

yes

characteristic

112

C₄²

#13083558

yes

characteristic

113

C₂³

#15667927

yes

characteristic

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂²⋊#10623494

yes

characteristic

128

C₂×C₄

#15667929

yes

characteristic

129

C₈

—

130

C₈

—

1424

—

131

C₂²

—

132

C₂²

—

133

C₂²

#19151322

yes

characteristic

134

C₂²

#19151391

yes

characteristic

135

C₂²

#19151395

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₂

—

143

C₂

—

144

C₂

#26018184

yes

characteristic

145

C₂

#26018417

yes

characteristic

146

C₂

#26018421

yes

characteristic

	subgroup	G/H
center	C₂²	#19151391
commutator	C₄²×C₈₉	C₂×C₈
Frattini	#384	C₂×D₈₉
Fitting	C₈₉×#1956	C₂
socle	C₂²×C₈₉	#353
prime radical	C₂³×C₈₉	#99
cosocle	#384×C₈₉	C₂²
hypercenter	#1956	D₈₉
hypocenter	C₈₉	#7355
LCS	G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1956

subgroup

G/H

center

C₂²

#19151391

commutator

C₄²×C₈₉

C₂×C₈

Frattini

#384

C₂×D₈₉

Fitting

C₈₉×#1956

C₂

C₂²×C₈₉

C₂³×C₈₉

#384×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1956

N	H ≅ G/N	type
13. C₄⋊Q₈×C₈₉	C₈	semidirect
76. C₈₉	#7355	semidirect
86. C₄⋊Q₈	#10623494	semidirect

H ≅ G/N

type

13. C₄⋊Q₈×C₈₉

C₈

semidirect

76. C₈₉

#7355

semidirect

86. C₄⋊Q₈

#10623494

semidirect

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

subgroup	index
#26018198	2
C₈₉×#1956	2
#7355	89

subgroup

index

#26018198

C₈₉×#1956

#7355

Subgroups — #36441813

Conjugacy classes of subgroups

Complements

Maximal subgroups