Group #36466175 — subgroups

Subgroups — #36466175

Order 22784.

222 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₃₅₆	C₂	1	2	yes	maximal
3	#26018326	C₂	1	1	yes	maximal, characteristic
4	#26018217	C₂	1	2	yes	maximal
5	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	C₄	1	1	yes	characteristic
7	C₈×Dic₁₇₈	—	2	2	no	—
8	C₄×Dic₃₅₆	C₂²	1	2	yes	—
9	#19151424	C₄	1	1	yes	characteristic
10	#19151403	C₄	1	2	yes	—
11	#19151351	C₂²	1	2	yes	—
12	#19151330	—	2	2	no	—
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₈²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₇₈	—	2	2	no	—
18	Dic₇₁₂	C₈	1	4	yes	complemented
19	#15667946	C₂×C₄	1	2	yes	—
20	C₄⋊#10623494	—	2	2	no	—
21	#15667941	—	4	2	no	—
22	#15667937	—	2	2	no	—
23	C₈×Dic₈₉	—	4	2	no	—
24	C₂×C₁₆×C₈₉	—	2	1	no	—
25	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₇₈	—	2	2	no	—
31	Dic₃₅₆	—	2	2	no	—
32	Dic₃₅₆	C₂×C₈	1	4	yes	—
33	C₂×#10623494	—	4	2	no	—
34	C₄⋊Dic₈₉	—	2	2	no	—
35	#13083553	—	4	2	no	—
36	C₄×Dic₈₉	—	4	2	no	—
37	C₂×C₈×C₈₉	—	2	1	no	—
38	C₂×C₈×C₈₉	D₈	1	2	yes	—
39	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
41	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₉	—	2	2	no	—
43	C₁₆×C₈₉	—	2	1	no	—
44	C₁₆×C₈₉	C₂×C₈	1	2	yes	—
45	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
48	C₂×Dic₈₉	—	4	2	no	—
49	C₂×Dic₈₉	—	4	2	no	—
50	Dic₁₇₈	—	2	4	no	—
51	Dic₁₇₈	—	4	2	no	—
52	C₈×C₈₉	—	2	1	no	—
53	C₈×C₈₉	—	2	2	no	—
54	C₈×C₈₉	—	4	1	no	—
55	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₈₉	#131	1	1	yes	characteristic
57	#10623494	—	8	2	no	—
58	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
59	C₄×C₈₉	—	2	1	no	—
60	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
61	C₄×C₈₉	#443	1	1	yes	characteristic
62	C₄×C₈₉	#510	1	2	yes	—
63	Dic₈₉	—	4	4	no	—
64	Dic₈₉	—	8	2	no	—
65	#12044	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	C₈×D₈	1	1	yes	characteristic
67	C₂×C₈₉	#2176	1	1	yes	characteristic
68	C₂×C₈₉	#2179	1	1	yes	characteristic
69	C₄×Q₃₂	—	89	1	no	—
70	C₈×Q₁₆	—	89	2	no	—
71	#1372	—	89	1	no	—
72	#1332	—	89	2	no	—
73	#1311	D₈₉	1	1	yes	characteristic
74	C₈₉	#12044	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	—	89	1	no	—
76	C₈×Q₈	—	178	2	no	—
77	C₄×Q₁₆	—	89	2	no	—
78	#366	—	89	1	no	—
79	#358	—	89	2	no	—
80	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
82	#335	—	89	2	no	—
83	Q₈⋊C₈	—	178	2	no	—
84	C₈²	C₂×D₈₉	1	1	yes	characteristic
85	C₂×Q₁₆	—	89	2	no	—
86	C₄×Q₈	—	178	2	no	—
87	Q₃₂	—	89	4	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
91	#107	—	89	2	no	—
92	C₄⋊C₈	—	356	2	no	—
93	C₄⋊C₈	—	178	2	no	—
94	#103	—	178	2	no	—
95	C₄×C₈	—	356	2	no	—
96	C₄×C₈	D₃₅₆	1	1	yes	characteristic
97	C₄×C₈	D₃₅₆	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×Q₈	—	178	2	no	—
100	Q₁₆	—	89	4	no	—
101	Q₁₆	—	178	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	356	2	no	—
104	C₂×C₈	D₇₁₂	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
106	C₂×C₈	#13083574	1	1	yes	characteristic
107	C₄⋊C₄	—	356	2	no	—
108	C₄⋊C₄	—	178	2	no	—
109	C₄²	—	356	2	no	—
110	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	C₈×D₈₉	1	2	yes	—
114	Q₈	—	356	2	no	—
115	Q₈	—	178	4	no	—
116	C₂×C₄	—	356	2	no	—
117	C₂×C₄	—	356	2	no	—
118	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
119	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
120	C₂×C₄	#15668034	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	2	no	—
123	C₈	—	4	1	no	—
124	C₈	—	712	2	no	—
125	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
126	C₈	#15668018	1	1	yes	characteristic
127	C₂²	C₄×D₇₁₂	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	356	4	no	—
130	C₄	—	712	2	no	—
131	C₄	C₈×D₃₅₆	1	1	yes	characteristic
132	C₄	#19151716	1	1	yes	characteristic
133	C₄	#19152041	1	2	yes	—
134	C₂	C₈×D₇₁₂	1	1	yes	characteristic
135	C₂	#26022124	1	1	yes	characteristic
136	C₂	#26022139	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₇₁₂

C₂

yes

maximal, characteristic

C₈×Dic₃₅₆

C₂

yes

maximal

#26018326

C₂

yes

maximal, characteristic

#26018217

C₂

yes

maximal

C₈₉×#1311

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₄

yes

characteristic

C₈×Dic₁₇₈

—

C₄×Dic₃₅₆

C₂²

yes

—

#19151424

C₄

yes

characteristic

#19151403

C₄

yes

—

#19151351

C₂²

yes

—

#19151330

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

C₂×C₄

yes

—

C₄×Dic₁₇₈

—

Dic₇₁₂

C₈

yes

complemented

#15667946

C₂×C₄

yes

—

C₄⋊#10623494

—

#15667941

—

#15667937

—

C₈×Dic₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

C₂×C₈

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×C₈

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂²×C₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₄×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

#12044

—

maximal, Sylow

C₂×C₈₉

C₈×D₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

D₈₉

yes

characteristic

C₈₉

#12044

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₈×Q₈

—

178

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

Q₈⋊C₈

—

178

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

178

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

characteristic

#107

—

C₄⋊C₈

—

356

—

C₄⋊C₈

—

178

—

#103

—

178

—

C₄×C₈

—

356

—

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

178

—

100

Q₁₆

—

101

Q₁₆

—

178

—

102

C₂×C₈

—

103

C₂×C₈

—

356

—

104

C₂×C₈

D₇₁₂

yes

—

105

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

106

C₂×C₈

#13083574

yes

characteristic

107

C₄⋊C₄

—

356

—

108

C₄⋊C₄

—

178

—

109

C₄²

—

356

—

110

C₄²

C₂×D₃₅₆

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₈×D₈₉

yes

—

114

Q₈

—

356

—

115

Q₈

—

178

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

356

—

118

C₂×C₄

C₄×D₃₅₆

yes

characteristic

119

C₂×C₄

C₂×D₇₁₂

yes

characteristic

120

C₂×C₄

#15668034

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

712

—

125

C₈

C₂×C₈×D₈₉

yes

characteristic

126

C₈

#15668018

yes

characteristic

127

C₂²

C₄×D₇₁₂

yes

characteristic

128

C₄

—

129

C₄

—

356

—

130

C₄

—

712

—

131

C₄

C₈×D₃₅₆

yes

characteristic

132

C₄

#19151716

yes

characteristic

133

C₄

#19152041

yes

—

134

C₂

C₈×D₇₁₂

yes

characteristic

135

C₂

#26022124

yes

characteristic

136

C₂

#26022139

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1311	C₂
socle, prime radical	C₂²×C₈₉	C₄×D₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#12044
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

C₂×D₇₁₂

commutator

C₈×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1311

C₂

socle, prime radical

C₂²×C₈₉

C₄×D₈

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
18. Dic₇₁₂	C₈	semidirect
74. C₈₉	#12044	semidirect

H ≅ G/N

type

18. Dic₇₁₂

C₈

semidirect

74. C₈₉

#12044

semidirect

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

subgroup	index
C₄×Dic₇₁₂	2
C₈×Dic₃₅₆	2
#26018326	2
#26018217	2
C₈₉×#1311	2
#12044	89

subgroup

index

C₄×Dic₇₁₂

C₈×Dic₃₅₆

#26018326

#26018217

C₈₉×#1311

#12044

Subgroups — #36466175

Conjugacy classes of subgroups

Complements

Maximal subgroups