Group #36468685 — subgroups

Subgroups — #36468685

Order 22784.

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

normal subgroups only

This group has 248 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	#26022502	C₂	1	1	yes	maximal, characteristic
2	#26019612	C₂	1	2	yes	maximal, complemented
3	#26018324	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018299	C₂	1	2	yes	maximal, complemented
5	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₂₄	—	2	1	no	—
7	C₂×D₁₄₂₄	—	2	1	no	—
8	#19151975	C₂²	1	2	yes	—
9	#19151798	—	2	2	no	—
10	#19151351	C₂²	1	2	yes	—
11	#19151341	—	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×#13083561	—	4	2	no	—
18	C₂×D₇₁₂	—	2	2	no	—
19	C₂×D₇₁₂	—	2	2	no	—
20	#15668009	—	2	2	no	—
21	D₁₄₂₄	—	4	2	no	—
22	D₁₄₂₄	—	4	2	no	—
23	C₄⋊#10623494	—	2	2	no	—
24	#15667932	—	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×D₃₅₆	—	4	2	no	—
34	C₂×D₃₅₆	—	4	2	no	—
35	C₂×D₃₅₆	—	4	2	no	—
36	#13083561	—	8	4	no	—
37	D₇₁₂	—	4	2	no	—
38	D₇₁₂	—	4	2	no	—
39	D₇₁₂	—	8	2	no	—
40	C₂×#10623494	—	4	2	no	—
41	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
42	C₂×C₈×C₈₉	—	2	1	no	—
43	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₈₉	—	2	1	no	—
46	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
47	C₄²×C₈₉	—	2	1	no	—
48	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₈₉	—	2	1	no	—
50	C₁₆×C₈₉	—	4	1	no	—
51	C₁₆×C₈₉	SD₁₆	1	2	yes	complemented
52	C₂²×D₈₉	—	8	2	no	—
53	Q₈×C₈₉	—	2	1	no	—
54	Q₈×C₈₉	D₁₆	1	2	yes	complemented
55	C₂×C₄×C₈₉	—	2	1	no	—
56	C₂×C₄×C₈₉	—	2	1	no	—
57	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
58	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
60	D₃₅₆	—	8	2	no	—
61	D₃₅₆	—	8	2	no	—
62	D₃₅₆	—	8	2	no	—
63	D₃₅₆	—	8	2	no	—
64	D₃₅₆	—	16	2	no	—
65	C₈×C₈₉	—	2	1	no	—
66	C₈×C₈₉	—	4	1	no	—
67	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	#10623494	—	8	2	no	—
70	C₂²×C₈₉	#466	1	1	yes	characteristic
71	C₂×D₈₉	—	16	2	no	—
72	C₂×D₈₉	—	16	2	no	—
73	C₂×D₈₉	—	16	2	no	—
74	C₄×C₈₉	—	2	2	no	—
75	C₄×C₈₉	—	2	1	no	—
76	C₄×C₈₉	—	4	1	no	—
77	C₄×C₈₉	#460	1	1	yes	characteristic
78	C₄×C₈₉	#472	1	1	yes	characteristic
79	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
80	C₄×C₈₉	#508	1	1	yes	characteristic
81	#12439	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	#1668	1	1	yes	characteristic
83	C₂×C₈₉	#2215	1	1	yes	characteristic
84	C₂×C₈₉	#2221	1	1	yes	characteristic
85	D₈₉	—	32	2	no	—
86	#2246	—	89	1	no	—
87	Q₈×C₁₆	D₈₉	1	1	yes	characteristic, complemented
88	#1668	—	89	2	no	—
89	#1371	—	89	1	no	—
90	#1361	—	89	2	no	—
91	C₈₉	#12439	1	1	yes	characteristic, complemented, Sylow
92	C₂×D₁₆	—	178	1	no	—
93	C₂×D₁₆	—	178	1	no	—
94	#493	—	89	2	no	—
95	#460	—	178	2	no	—
96	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
97	C₄⋊C₁₆	—	2	1	no	—
98	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
99	C₄×C₁₆	—	2	1	no	—
100	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
101	#335	—	89	2	no	—
102	#331	—	178	2	no	—
103	C₂×SD₁₆	—	356	2	no	—
104	C₂×D₈	—	178	2	no	—
105	C₂×D₈	—	178	2	no	—
106	#127	—	178	2	no	—
107	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
108	D₁₆	—	356	2	no	—
109	D₁₆	—	356	2	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
112	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	178	2	no	—
115	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
116	#102	—	356	2	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
119	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
120	C₂×D₄	—	356	2	no	—
121	C₂×D₄	—	356	2	no	—
122	C₂×D₄	—	356	2	no	—
123	SD₁₆	—	712	4	no	—
124	D₈	—	356	2	no	—
125	D₈	—	356	2	no	—
126	D₈	—	712	2	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	356	2	no	—
129	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
130	C₂×C₈	#13083574	1	1	yes	characteristic
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
133	C₄²	—	2	1	no	—
134	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	4	1	no	—
137	C₁₆	#13083561	1	2	yes	complemented
138	C₂³	—	712	2	no	—
139	Q₈	—	2	1	no	—
140	Q₈	D₁₄₂₄	1	2	yes	complemented
141	D₄	—	1424	2	no	—
142	D₄	—	712	2	no	—
143	D₄	—	712	2	no	—
144	D₄	—	712	2	no	—
145	D₄	—	712	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	#15667986	1	1	yes	characteristic
149	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
150	C₂×C₄	#15668034	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	712	2	no	—
154	C₈	C₂×#13083561	1	1	yes	characteristic
155	C₈	#15668037	1	1	yes	characteristic
156	C₂²	—	1424	2	no	—
157	C₂²	—	1424	2	no	—
158	C₂²	—	1424	2	no	—
159	C₂²	#19151834	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	1	no	—
162	C₄	—	4	1	no	—
163	C₄	#19151798	1	1	yes	characteristic
164	C₄	#19151868	1	1	yes	characteristic
165	C₄	C₂×D₁₄₂₄	1	1	yes	characteristic
166	C₄	#19152030	1	1	yes	characteristic
167	C₂	—	2848	2	no	—
168	C₂	#26019612	1	1	yes	characteristic
169	C₂	#26022338	1	1	yes	characteristic
170	C₂	#26022372	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022502

C₂

yes

maximal, characteristic

#26019612

C₂

yes

maximal, complemented

#26018324

C₂

yes

maximal, characteristic, complemented

#26018299

C₂

yes

maximal, complemented

Q₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₂₄

—

C₂×D₁₄₂₄

—

#19151975

C₂²

yes

—

#19151798

—

#19151351

C₂²

yes

—

#19151341

—

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×#13083561

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

D₁₄₂₄

—

D₁₄₂₄

—

C₄⋊#10623494

—

#15667932

—

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×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083561

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

C₂×#10623494

—

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

complemented

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

D₁₆

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

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×SD₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#466

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×D₁₆

yes

C₄×C₈₉

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2246

—

Q₈×C₁₆

D₈₉

yes

characteristic, complemented

#1668

—

#1371

—

#1361

—

C₈₉

#12439

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

178

—

C₂×D₁₆

—

178

—

#493

—

#460

—

178

—

C₈×Q₈

C₂×D₈₉

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

—

100

C₄×C₁₆

C₂×D₈₉

yes

characteristic

101

#335

—

102

#331

—

178

—

103

C₂×SD₁₆

—

356

—

104

C₂×D₈

—

178

—

105

C₂×D₈

—

178

—

106

#127

—

178

—

107

C₄×Q₈

D₃₅₆

yes

characteristic

108

D₁₆

—

356

—

109

D₁₆

—

356

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

112

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

178

—

115

C₄⋊C₈

D₃₅₆

yes

characteristic

116

#102

—

356

—

117

C₄×C₈

—

118

C₄×C₈

C₂²×D₈₉

yes

characteristic

119

C₂×Q₈

D₇₁₂

yes

characteristic

120

C₂×D₄

—

356

—

121

C₂×D₄

—

356

—

122

C₂×D₄

—

356

—

123

SD₁₆

—

712

—

124

D₈

—

356

—

125

D₈

—

356

—

126

D₈

—

712

—

127

C₂×C₈

—

128

C₂×C₈

—

356

—

129

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

130

C₂×C₈

#13083574

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

D₇₁₂

yes

characteristic

133

C₄²

—

134

C₄²

C₂×D₃₅₆

yes

characteristic

135

C₁₆

—

136

C₁₆

—

137

C₁₆

#13083561

yes

complemented

138

C₂³

—

712

—

139

Q₈

—

140

Q₈

D₁₄₂₄

yes

complemented

141

D₄

—

1424

—

142

D₄

—

712

—

143

D₄

—

712

—

144

D₄

—

712

—

145

D₄

—

712

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#15667986

yes

characteristic

149

C₂×C₄

C₂×D₇₁₂

yes

characteristic

150

C₂×C₄

#15668034

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

712

—

154

C₈

C₂×#13083561

yes

characteristic

155

C₈

#15668037

yes

characteristic

156

C₂²

—

1424

—

157

C₂²

—

1424

—

158

C₂²

—

1424

—

159

C₂²

#19151834

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#19151798

yes

characteristic

164

C₄

#19151868

yes

characteristic

165

C₄

C₂×D₁₄₂₄

yes

characteristic

166

C₄

#19152030

yes

characteristic

167

C₂

—

2848

—

168

C₂

#26019612

yes

characteristic

169

C₂

#26022338

yes

characteristic

170

C₂

#26022372

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	C₂²×C₈₉	#466
prime radical	#26022502	C₂
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	#12439
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

C₂²×C₈₉

C₂

Q₈×C₁₆

D₈₉

hypocenter

C₈₉

#12439

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
2. #26019612	C₂	semidirect
3. #26018324	C₂	semidirect
4. #26018299	C₂	semidirect
5. Q₈×C₁₆×C₈₉	C₂	semidirect
51. C₁₆×C₈₉	SD₁₆	semidirect
54. Q₈×C₈₉	D₁₆	semidirect
87. Q₈×C₁₆	D₈₉	semidirect
91. C₈₉	#12439	semidirect
137. C₁₆	#13083561	semidirect
140. Q₈	D₁₄₂₄	semidirect

H ≅ G/N

type

2. #26019612

C₂

semidirect

3. #26018324

C₂

semidirect

4. #26018299

C₂

semidirect

5. Q₈×C₁₆×C₈₉

C₂

semidirect

51. C₁₆×C₈₉

SD₁₆

semidirect

54. Q₈×C₈₉

D₁₆

semidirect

87. Q₈×C₁₆

D₈₉

semidirect

91. C₈₉

#12439

semidirect

137. C₁₆

#13083561

semidirect

140. Q₈

D₁₄₂₄

semidirect

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

subgroup	index
#26022502	2
#26019612	2
#26018324	2
#26018299	2
Q₈×C₁₆×C₈₉	2
#12439	89

subgroup

index

Q₈×C₁₆×C₈₉

Subgroups — #36468685

Conjugacy classes of subgroups

Complements

Maximal subgroups