Group #36468690 — subgroups

Subgroups — #36468690

Order 22784.

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

normal subgroups only

This group has 216 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	#26022505	C₂	1	1	yes	maximal, characteristic
2	#26019622	C₂	1	2	yes	maximal
3	#26018324	C₂	1	1	yes	maximal, characteristic
4	#26018302	C₂	1	2	yes	maximal
5	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	—	2	1	no	—
7	C₂×Dic₇₁₂	—	2	1	no	—
8	#19151978	C₂²	1	2	yes	—
9	#19151883	—	2	2	no	—
10	#19151351	C₂²	1	2	yes	—
11	#19151345	—	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×Dic₃₅₆	—	2	2	no	—
18	C₂×Dic₃₅₆	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	Dic₇₁₂	—	4	2	no	—
21	Dic₇₁₂	—	4	2	no	—
22	#15667943	—	4	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	#15667936	—	4	2	no	—
25	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	—	2	1	no	—
27	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	—	2	1	no	—
30	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	—	2	1	no	—
32	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₇₈	—	4	2	no	—
34	Dic₃₅₆	—	4	2	no	—
35	Dic₃₅₆	—	4	2	no	—
36	Dic₃₅₆	—	8	2	no	—
37	C₂×#10623494	—	4	2	no	—
38	C₄⋊Dic₈₉	—	4	2	no	—
39	C₄⋊Dic₈₉	—	4	2	no	—
40	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
41	C₂×C₈×C₈₉	—	2	1	no	—
42	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
46	C₄²×C₈₉	—	2	1	no	—
47	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₈₉	—	2	1	no	—
49	C₁₆×C₈₉	—	4	1	no	—
50	C₁₆×C₈₉	SD₁₆	1	2	yes	—
51	Q₈×C₈₉	—	2	1	no	—
52	Q₈×C₈₉	Q₃₂	1	2	yes	complemented
53	C₂×C₄×C₈₉	—	2	1	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
58	C₂×Dic₈₉	—	8	2	no	—
59	Dic₁₇₈	—	8	2	no	—
60	Dic₁₇₈	—	8	2	no	—
61	C₈×C₈₉	—	2	1	no	—
62	C₈×C₈₉	—	4	1	no	—
63	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
64	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	#10623494	—	8	2	no	—
66	C₂²×C₈₉	#466	1	1	yes	characteristic
67	C₄×C₈₉	—	2	2	no	—
68	C₄×C₈₉	—	2	1	no	—
69	C₄×C₈₉	—	4	1	no	—
70	C₄×C₈₉	#460	1	1	yes	characteristic
71	C₄×C₈₉	#472	1	1	yes	characteristic
72	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
73	C₄×C₈₉	#508	1	1	yes	characteristic
74	Dic₈₉	—	16	2	no	—
75	#12440	—	89	1	no	maximal, Sylow
76	C₂×C₈₉	#1668	1	1	yes	characteristic
77	C₂×C₈₉	#2216	1	1	yes	characteristic
78	C₂×C₈₉	#2222	1	1	yes	characteristic
79	#2247	—	89	1	no	—
80	Q₈×C₁₆	D₈₉	1	1	yes	characteristic
81	#1670	—	89	2	no	—
82	#1371	—	89	1	no	—
83	#1362	—	89	2	no	—
84	C₈₉	#12440	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₃₂	—	178	1	no	—
86	C₂×Q₃₂	—	178	1	no	—
87	#494	—	89	2	no	—
88	#475	—	178	2	no	—
89	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
90	C₄⋊C₁₆	—	2	1	no	—
91	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
92	C₄×C₁₆	—	2	1	no	—
93	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
94	#335	—	89	2	no	—
95	#332	—	178	2	no	—
96	C₂×Q₁₆	—	178	2	no	—
97	C₂×Q₁₆	—	178	2	no	—
98	C₄⋊Q₈	—	178	2	no	—
99	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
100	Q₃₂	—	356	2	no	—
101	Q₃₂	—	356	2	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
105	#106	—	356	2	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	178	2	no	—
108	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
109	#103	—	356	2	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
112	C₂×Q₈	—	356	2	no	—
113	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
114	Q₁₆	—	356	2	no	—
115	Q₁₆	—	356	2	no	—
116	Q₁₆	—	712	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	356	2	no	—
119	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
120	C₂×C₈	#13083574	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	356	2	no	—
123	C₄⋊C₄	—	356	2	no	—
124	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
125	C₄²	—	2	1	no	—
126	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	4	1	no	—
129	C₁₆	#13083561	1	2	yes	—
130	Q₈	—	2	1	no	—
131	Q₈	—	712	2	no	—
132	Q₈	—	712	2	no	—
133	Q₈	Dic₇₁₂	1	2	yes	complemented
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	712	2	no	—
137	C₂×C₄	#15667986	1	1	yes	characteristic
138	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
139	C₂×C₄	#15668034	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	4	1	no	—
142	C₈	—	712	2	no	—
143	C₈	C₂×#13083561	1	1	yes	characteristic
144	C₈	#15668037	1	1	yes	characteristic
145	C₂²	#19151834	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	1424	2	no	—
150	C₄	#19151798	1	1	yes	characteristic
151	C₄	#19151868	1	1	yes	characteristic
152	C₄	C₂×Dic₇₁₂	1	1	yes	characteristic
153	C₄	#19152030	1	1	yes	characteristic
154	C₂	#26019612	1	1	yes	characteristic
155	C₂	#26022343	1	1	yes	characteristic
156	C₂	#26022377	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022505

C₂

yes

maximal, characteristic

#26019622

C₂

yes

maximal

#26018324

C₂

yes

maximal, characteristic

#26018302

C₂

yes

maximal

Q₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

C₂×Dic₇₁₂

—

#19151978

C₂²

yes

—

#19151883

—

#19151351

C₂²

yes

—

#19151345

—

C₈×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₄⋊C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

Dic₇₁₂

—

Dic₇₁₂

—

#15667943

—

C₄⋊#10623494

—

#15667936

—

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

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

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

SD₁₆

yes

—

Q₈×C₈₉

—

Q₈×C₈₉

Q₃₂

yes

complemented

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×SD₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#466

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×Q₃₂

yes

characteristic

C₄×C₈₉

#508

yes

characteristic

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

Q₈×C₁₆

D₈₉

yes

—

—

—

C₈₉

#12440

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

178

—

C₂×Q₃₂

—

178

—

#494

—

#475

—

178

—

C₈×Q₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

#332

—

178

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

178

—

C₄×Q₈

D₃₅₆

yes

characteristic

100

Q₃₂

—

356

—

101

Q₃₂

—

356

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

105

#106

—

356

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

178

—

108

C₄⋊C₈

D₃₅₆

yes

characteristic

109

#103

—

356

—

110

C₄×C₈

—

111

C₄×C₈

C₂²×D₈₉

yes

characteristic

112

C₂×Q₈

—

356

—

113

C₂×Q₈

D₇₁₂

yes

characteristic

114

Q₁₆

—

356

—

115

Q₁₆

—

356

—

116

Q₁₆

—

712

—

117

C₂×C₈

—

118

C₂×C₈

—

356

—

119

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

120

C₂×C₈

#13083574

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

356

—

123

C₄⋊C₄

—

356

—

124

C₄⋊C₄

D₇₁₂

yes

characteristic

125

C₄²

—

126

C₄²

C₂×D₃₅₆

yes

characteristic

127

C₁₆

—

128

C₁₆

—

129

C₁₆

#13083561

yes

—

130

Q₈

—

131

Q₈

—

712

—

132

Q₈

—

712

—

133

Q₈

Dic₇₁₂

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

712

—

137

C₂×C₄

#15667986

yes

characteristic

138

C₂×C₄

C₂×D₇₁₂

yes

characteristic

139

C₂×C₄

#15668034

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

712

—

143

C₈

C₂×#13083561

yes

characteristic

144

C₈

#15668037

yes

characteristic

145

C₂²

#19151834

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

1424

—

150

C₄

#19151798

yes

characteristic

151

C₄

#19151868

yes

characteristic

152

C₄

C₂×Dic₇₁₂

yes

characteristic

153

C₄

#19152030

yes

characteristic

154

C₂

#26019612

yes

characteristic

155

C₂

#26022343

yes

characteristic

156

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	#12440
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

#12440

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
52. Q₈×C₈₉	Q₃₂	semidirect
84. C₈₉	#12440	semidirect
133. Q₈	Dic₇₁₂	semidirect

H ≅ G/N

type

52. Q₈×C₈₉

Q₃₂

semidirect

84. C₈₉

#12440

semidirect

133. Q₈

Dic₇₁₂

semidirect

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

subgroup	index
#26022505	2
#26019622	2
#26018324	2
#26018302	2
Q₈×C₁₆×C₈₉	2
#12440	89

subgroup

index

Q₈×C₁₆×C₈₉

Subgroups — #36468690

Conjugacy classes of subgroups

Complements

Maximal subgroups