Group #36468787 — subgroups

Subgroups — #36468787

Order 22784.

204 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 75 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	#26022527	C₂	1	1	yes	maximal, characteristic
2	#26019646	C₂	1	2	yes	maximal
3	#26018324	C₂	1	1	yes	maximal, characteristic
4	#26018306	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19152003	C₂²	1	2	yes	—
7	#19151808	—	2	2	no	—
8	#19151424	—	2	1	no	—
9	#19151424	—	2	1	no	—
10	#19151351	C₂²	1	2	yes	—
11	#19151343	—	2	2	no	—
12	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₈₉	—	2	1	no	—
14	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₈₉	—	2	1	no	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#13083564	—	4	2	no	—
18	#15668012	—	2	2	no	—
19	#15667946	—	2	2	no	—
20	#15667946	—	2	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	#15667935	—	4	2	no	—
23	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	—	2	1	no	—
25	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	1	no	—
28	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	—	2	1	no	—
30	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₇₈	—	4	2	no	—
32	#13083564	—	8	4	no	—
33	C₂×#10623494	—	4	2	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	—	2	1	no	—
41	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
42	C₄²×C₈₉	—	2	1	no	—
43	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₈₉	—	2	1	no	—
45	C₁₆×C₈₉	—	4	1	no	—
46	C₁₆×C₈₉	Q₁₆	1	2	yes	complemented
47	Q₈×C₈₉	—	2	1	no	—
48	Q₈×C₈₉	D₁₆	1	2	yes	—
49	C₂×C₄×C₈₉	—	2	1	no	—
50	C₂×C₄×C₈₉	—	2	1	no	—
51	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
54	C₂×Dic₈₉	—	8	2	no	—
55	Dic₁₇₈	—	8	2	no	—
56	Dic₁₇₈	—	8	2	no	—
57	C₈×C₈₉	—	2	1	no	—
58	C₈×C₈₉	—	4	1	no	—
59	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
60	C₈×C₈₉	#137	1	1	yes	characteristic
61	#10623494	—	8	2	no	—
62	C₂²×C₈₉	#466	1	1	yes	characteristic
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	—	2	1	no	—
65	C₄×C₈₉	—	4	1	no	—
66	C₄×C₈₉	#462	1	1	yes	characteristic
67	C₄×C₈₉	#473	1	1	yes	characteristic
68	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₈₉	#508	1	1	yes	characteristic
70	Dic₈₉	—	16	2	no	—
71	#12457	—	89	1	no	maximal, Sylow
72	C₂×C₈₉	#1674	1	1	yes	characteristic
73	C₂×C₈₉	#2215	1	1	yes	characteristic
74	C₂×C₈₉	#2222	1	1	yes	characteristic
75	#2252	—	89	1	no	—
76	Q₈×C₁₆	D₈₉	1	1	yes	characteristic
77	#1674	—	89	2	no	—
78	#1371	—	89	1	no	—
79	#1364	—	89	2	no	—
80	C₈₉	#12457	1	1	yes	characteristic, complemented, Sylow
81	#500	—	89	2	no	—
82	#462	—	178	2	no	—
83	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
84	#366	—	178	1	no	—
85	#366	—	178	1	no	—
86	C₄⋊C₁₆	—	2	1	no	—
87	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
88	C₄×C₁₆	—	2	1	no	—
89	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
90	#335	—	89	2	no	—
91	#332	—	178	2	no	—
92	C₂×Q₁₆	—	356	2	no	—
93	C₄⋊Q₈	—	178	2	no	—
94	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
98	#107	—	178	2	no	—
99	#107	—	178	2	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	178	2	no	—
102	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
103	#103	—	356	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
106	C₂×Q₈	—	356	2	no	—
107	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
108	Q₁₆	—	712	4	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	356	2	no	—
111	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
112	C₂×C₈	#13083574	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	356	2	no	—
115	C₄⋊C₄	—	356	2	no	—
116	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
117	C₄²	—	2	1	no	—
118	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	#13083564	1	2	yes	complemented
122	Q₈	—	2	1	no	—
123	Q₈	—	712	2	no	—
124	Q₈	—	712	2	no	—
125	Q₈	D₁₄₂₄	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	712	2	no	—
129	C₂×C₄	#15667986	1	1	yes	characteristic
130	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
131	C₂×C₄	#15668034	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	712	2	no	—
135	C₈	C₂×#13083564	1	1	yes	characteristic
136	C₈	#15668042	1	1	yes	characteristic
137	C₂²	#19151834	1	1	yes	characteristic
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	1	no	—
141	C₄	—	1424	2	no	—
142	C₄	#19151808	1	1	yes	characteristic
143	C₄	#19151873	1	1	yes	characteristic
144	C₄	C₂×D₁₄₂₄	1	1	yes	characteristic
145	C₄	#19152030	1	1	yes	characteristic
146	C₂	#26019646	1	1	yes	characteristic
147	C₂	#26022338	1	1	yes	characteristic
148	C₂	#26022377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022527

C₂

yes

maximal, characteristic

#26019646

C₂

yes

maximal

#26018324

C₂

yes

maximal, characteristic

#26018306

C₂

yes

maximal

Q₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic

#19152003

C₂²

yes

—

—

—

—

C₂²

yes

—

#19151343

—

C₈×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₄⋊C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

—

—

—

—

—

—

C₄×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

#13083564

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₈

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

Q₁₆

yes

complemented

Q₈×C₈₉

—

Q₈×C₈₉

D₁₆

yes

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#466

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

#508

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

Q₈×C₁₆

D₈₉

yes

—

—

—

C₈₉

#12457

yes

characteristic, complemented, Sylow

#500

—

#462

—

178

—

C₈×Q₈

C₂×D₈₉

yes

characteristic

#366

—

178

—

#366

—

178

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

#332

—

178

—

C₂×Q₁₆

—

356

—

C₄⋊Q₈

—

178

—

C₄×Q₈

D₃₅₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

#107

—

178

—

#107

—

178

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

178

—

102

C₄⋊C₈

D₃₅₆

yes

characteristic

103

#103

—

356

—

104

C₄×C₈

—

105

C₄×C₈

C₂²×D₈₉

yes

characteristic

106

C₂×Q₈

—

356

—

107

C₂×Q₈

D₇₁₂

yes

characteristic

108

Q₁₆

—

712

—

109

C₂×C₈

—

110

C₂×C₈

—

356

—

111

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

112

C₂×C₈

#13083574

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

356

—

115

C₄⋊C₄

—

356

—

116

C₄⋊C₄

D₇₁₂

yes

characteristic

117

C₄²

—

118

C₄²

C₂×D₃₅₆

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#13083564

yes

complemented

122

Q₈

—

123

Q₈

—

712

—

124

Q₈

—

712

—

125

Q₈

D₁₄₂₄

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

712

—

129

C₂×C₄

#15667986

yes

characteristic

130

C₂×C₄

C₂×D₇₁₂

yes

characteristic

131

C₂×C₄

#15668034

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

712

—

135

C₈

C₂×#13083564

yes

characteristic

136

C₈

#15668042

yes

characteristic

137

C₂²

#19151834

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

1424

—

142

C₄

#19151808

yes

characteristic

143

C₄

#19151873

yes

characteristic

144

C₄

C₂×D₁₄₂₄

yes

characteristic

145

C₄

#19152030

yes

characteristic

146

C₂

#26019646

yes

characteristic

147

C₂

#26022338

yes

characteristic

148

C₂

#26022377

yes

characteristic

	subgroup	G/H
center	C₂²	#19151834
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	Q₈×C₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#466
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#12457
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

#19151834

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

Q₈×C₁₆×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

#466

hypercenter

Q₈×C₁₆

D₈₉

hypocenter

C₈₉

#12457

LCS

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
46. C₁₆×C₈₉	Q₁₆	semidirect
80. C₈₉	#12457	semidirect
121. C₁₆	#13083564	semidirect

H ≅ G/N

type

46. C₁₆×C₈₉

Q₁₆

semidirect

80. C₈₉

#12457

semidirect

121. C₁₆

#13083564

semidirect

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

subgroup	index
#26022527	2
#26019646	2
#26018324	2
#26018306	2
Q₈×C₁₆×C₈₉	2
#12457	89

subgroup

index

Q₈×C₁₆×C₈₉

Subgroups — #36468787

Conjugacy classes of subgroups

Complements

Maximal subgroups