Group #36468687 — subgroups

Subgroups — #36468687

Order 22784.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	#26019619	C₂	1	2	yes	maximal
3	#26018324	C₂	1	1	yes	maximal, characteristic
4	#26018301	C₂	1	2	yes	maximal
5	C₈₉×#2247	C₂	1	1	yes	maximal, characteristic
6	#19151880	—	2	2	no	—
7	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
8	C₄⋊#13083549	C₂²	1	1	yes	characteristic
9	#19151416	—	2	1	no	—
10	C₁₆×Dic₈₉	—	2	1	no	—
11	#19151351	C₂²	1	2	yes	—
12	#19151344	—	2	2	no	—
13	C₂×Q₃₂×C₈₉	—	2	1	no	—
14	C₂×Q₃₂×C₈₉	—	2	1	no	—
15	#494×C₈₉	C₂²	1	2	yes	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
18	C₂×#13083549	—	2	1	no	—
19	#15667943	—	4	2	no	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	D₄	1	1	yes	characteristic
22	#15667941	—	2	1	no	—
23	#15667935	—	4	2	no	—
24	C₈×Dic₈₉	—	2	1	no	—
25	C₂×Q₁₆×C₈₉	—	2	2	no	—
26	C₂×Q₁₆×C₈₉	—	2	2	no	—
27	C₄⋊Q₈×C₈₉	—	2	2	no	—
28	Q₃₂×C₈₉	—	4	2	no	—
29	Q₃₂×C₈₉	—	4	2	no	—
30	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
34	C₂×#10623494	—	2	1	no	—
35	C₂×#10623494	—	4	2	no	—
36	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
37	#13083553	—	2	1	no	—
38	C₄×Dic₈₉	—	2	1	no	—
39	#13083549	—	4	1	no	—
40	C₂×Q₈×C₈₉	—	4	2	no	—
41	Q₁₆×C₈₉	—	4	2	no	—
42	Q₁₆×C₈₉	—	4	2	no	—
43	Q₁₆×C₈₉	—	8	2	no	—
44	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₈₉	—	4	2	no	—
47	C₄⋊C₄×C₈₉	—	4	2	no	—
48	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₈₉	—	2	1	no	—
50	C₁₆×C₈₉	SD₁₆	1	2	yes	—
51	Q₈×C₈₉	—	8	2	no	—
52	Q₈×C₈₉	—	8	2	no	—
53	C₂×C₄×C₈₉	—	8	2	no	—
54	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
57	C₂×Dic₈₉	—	2	1	no	—
58	C₂×Dic₈₉	—	2	1	no	—
59	Dic₁₇₈	—	2	1	no	—
60	Dic₁₇₈	Q₃₂	1	2	yes	complemented
61	C₈×C₈₉	—	2	1	no	—
62	C₈×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
64	#10623494	—	4	1	no	—
65	#10623494	—	8	2	no	—
66	C₂²×C₈₉	#466	1	1	yes	characteristic
67	C₄×C₈₉	—	2	1	no	—
68	C₄×C₈₉	—	16	2	no	—
69	C₄×C₈₉	#460	1	1	yes	characteristic
70	C₄×C₈₉	#472	1	1	yes	characteristic
71	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
72	C₄×C₈₉	#508	1	1	yes	characteristic
73	Dic₈₉	—	2	2	no	—
74	Dic₈₉	—	4	1	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	D₈₉	1	1	yes	characteristic
80	Q₈×C₁₆	—	89	1	no	—
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	—	2	1	no	—
86	C₂×Q₃₂	—	2	1	no	—
87	#494	C₂×D₈₉	1	2	yes	—
88	#475	—	178	2	no	—
89	C₈×Q₈	—	89	1	no	—
90	C₄⋊C₁₆	—	89	1	no	—
91	C₄⋊C₁₆	—	178	1	no	—
92	C₄×C₁₆	—	178	1	no	—
93	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
94	#335	—	89	2	no	—
95	#332	—	178	2	no	—
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	2	2	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄×Q₈	—	89	1	no	—
100	Q₃₂	—	4	2	no	—
101	Q₃₂	—	4	2	no	—
102	C₂×C₁₆	—	178	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	—	89	1	no	—
107	C₄⋊C₈	—	178	1	no	—
108	C₄⋊C₈	—	178	2	no	—
109	#103	—	356	2	no	—
110	C₄×C₈	—	178	1	no	—
111	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	89	1	no	—
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	8	2	no	—
117	C₂×C₈	—	356	2	no	—
118	C₂×C₈	—	178	1	no	—
119	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
120	C₂×C₈	#13083573	1	1	yes	characteristic
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	89	1	no	—
124	C₄⋊C₄	—	178	1	no	—
125	C₄²	—	178	1	no	—
126	C₄²	D₄×D₈₉	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	356	1	no	—
129	C₁₆	#13083562	1	2	yes	—
130	Q₈	—	8	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	89	2	no	—
133	Q₈	—	178	1	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	178	1	no	—
136	C₂×C₄	—	178	1	no	—
137	C₂×C₄	#15667983	1	1	yes	characteristic
138	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
139	C₂×C₄	#15668031	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	356	1	no	—
142	C₈	—	712	2	no	—
143	C₈	C₂×#13083562	1	1	yes	characteristic
144	C₈	#15668038	1	1	yes	characteristic
145	C₂²	#19151831	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	16	2	no	—
148	C₄	—	356	1	no	—
149	C₄	—	178	2	no	—
150	C₄	#19151795	1	1	yes	characteristic
151	C₄	#19151865	1	1	yes	characteristic
152	C₄	Q₃₂×D₈₉	1	1	yes	characteristic
153	C₄	#19152031	1	1	yes	characteristic
154	C₂	#26019609	1	1	yes	characteristic
155	C₂	#26022340	1	1	yes	characteristic
156	C₂	#26022374	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₇₈

C₂

yes

maximal, characteristic

#26019619

C₂

yes

maximal

#26018324

C₂

yes

maximal, characteristic

#26018301

C₂

yes

maximal

C₈₉×#2247

C₂

yes

maximal, characteristic

#19151880

—

C₈×Dic₁₇₈

C₂²

yes

characteristic

C₄⋊#13083549

C₂²

yes

characteristic

#19151416

—

C₁₆×Dic₈₉

—

#19151351

C₂²

yes

—

#19151344

—

C₂×Q₃₂×C₈₉

—

C₂×Q₃₂×C₈₉

—

#494×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×Dic₁₇₈

D₄

yes

—

—

—

D₄

yes

characteristic

#15667941

—

#15667935

—

C₈×Dic₈₉

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

Q₃₂×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

D₈

yes

characteristic

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₈

yes

characteristic

#13083553

—

C₄×Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

SD₁₆

yes

—

Q₈×C₈₉

—

Q₈×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

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

Q₃₂

yes

complemented

C₈×C₈₉

—

C₈×C₈₉

C₂×SD₁₆

yes

C₈×C₈₉

yes

—

—

C₂²×C₈₉

#466

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×Q₃₂

yes

characteristic

C₄×C₈₉

#508

yes

characteristic

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

yes

characteristic

Q₈×C₁₆

—

#1670

—

#1371

—

#1362

—

C₈₉

#12440

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂×D₈₉

yes

—

#475

—

178

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

C₄×C₁₆

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

#332

—

178

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

100

Q₃₂

—

101

Q₃₂

—

102

C₂×C₁₆

—

178

—

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

—

178

—

109

#103

—

356

—

110

C₄×C₈

—

178

—

111

C₄×C₈

C₂²×D₈₉

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

Q₁₆

—

115

Q₁₆

—

116

Q₁₆

—

117

C₂×C₈

—

356

—

118

C₂×C₈

—

178

—

119

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

120

C₂×C₈

#13083573

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

178

—

125

C₄²

—

178

—

126

C₄²

D₄×D₈₉

yes

characteristic

127

C₁₆

—

128

C₁₆

—

356

—

129

C₁₆

#13083562

yes

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

178

—

134

C₂×C₄

—

135

C₂×C₄

—

178

—

136

C₂×C₄

—

178

—

137

C₂×C₄

#15667983

yes

characteristic

138

C₂×C₄

D₈×D₈₉

yes

characteristic

139

C₂×C₄

#15668031

yes

characteristic

140

C₈

—

141

C₈

—

356

—

142

C₈

—

712

—

143

C₈

C₂×#13083562

yes

characteristic

144

C₈

#15668038

yes

characteristic

145

C₂²

#19151831

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

356

—

149

C₄

—

178

—

150

C₄

#19151795

yes

characteristic

151

C₄

#19151865

yes

characteristic

152

C₄

Q₃₂×D₈₉

yes

characteristic

153

C₄

#19152031

yes

characteristic

154

C₂

#26019609

yes

characteristic

155

C₂

#26022340

yes

characteristic

156

C₂

#26022374

yes

characteristic

	subgroup	G/H
center	C₂²	#19151831
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2247	C₂
socle, prime radical	C₂²×C₈₉	#466
hypercenter	#2247	D₈₉
hypocenter	C₈₉	#12440
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#19151831

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2247

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
60. Dic₁₇₈	Q₃₂	semidirect
84. C₈₉	#12440	semidirect

H ≅ G/N

type

60. Dic₁₇₈

Q₃₂

semidirect

84. C₈₉

#12440

semidirect

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

subgroup	index
C₁₆×Dic₁₇₈	2
#26019619	2
#26018324	2
#26018301	2
C₈₉×#2247	2
#12440	89

subgroup

index

C₁₆×Dic₁₇₈

C₈₉×#2247

Subgroups — #36468687

Conjugacy classes of subgroups

Complements

Maximal subgroups