Group #32997976 — subgroups

Subgroups — #32997976

Order 18688.

210 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 83 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×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic
2	#22608829	C₂	1	2	yes	maximal
3	#22608003	C₂	1	1	yes	maximal, characteristic
4	#22607896	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₈₄	C₄	1	1	yes	characteristic
7	#17988717	—	2	2	no	—
8	C₄×Dic₂₉₂	C₂²	1	2	yes	—
9	#17988413	C₄	1	1	yes	characteristic
10	#17988392	C₄	1	2	yes	—
11	#17988340	C₂²	1	2	yes	—
12	#17988319	—	2	2	no	—
13	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₈²×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₄₆	—	2	2	no	—
18	Dic₅₈₄	—	2	2	no	—
19	#14556525	C₂×C₄	1	2	yes	—
20	C₄⋊#10596831	—	2	2	no	—
21	#14556520	—	4	2	no	—
22	#14556516	—	2	2	no	—
23	#14556501	—	4	2	no	—
24	C₂×C₁₆×C₇₃	—	2	1	no	—
25	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₄₆	—	2	2	no	—
31	Dic₂₉₂	—	2	2	no	—
32	Dic₂₉₂	—	2	2	no	—
33	C₂×#10596831	—	4	2	no	—
34	C₄⋊Dic₇₃	—	2	2	no	—
35	#11924007	—	4	2	no	—
36	C₄×Dic₇₃	—	4	2	no	—
37	C₂×C₈×C₇₃	—	2	1	no	—
38	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
42	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₃	—	2	2	no	—
44	C₁₆×C₇₃	—	2	1	no	—
45	C₁₆×C₇₃	M₁₆	1	2	yes	—
46	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
49	C₂×Dic₇₃	—	4	2	no	—
50	C₂×Dic₇₃	—	4	2	no	—
51	Dic₁₄₆	—	4	2	no	—
52	Dic₁₄₆	—	4	2	no	—
53	C₈×C₇₃	—	2	2	no	—
54	C₈×C₇₃	—	2	1	no	—
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	Q₃₂	1	2	yes	complemented
57	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
58	C₈×C₇₃	#131	1	1	yes	characteristic
59	#10596831	—	8	2	no	—
60	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
61	C₄×C₇₃	—	2	1	no	—
62	C₄×C₇₃	#436	1	1	yes	characteristic
63	C₄×C₇₃	#443	1	1	yes	characteristic
64	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
65	C₄×C₇₃	#508	1	1	yes	characteristic
66	Dic₇₃	—	8	2	no	—
67	Dic₇₃	—	8	2	no	—
68	#12083	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	#1589	1	1	yes	characteristic
70	C₂×C₇₃	C₄×Q₃₂	1	1	yes	characteristic
71	C₂×C₇₃	#2178	1	1	yes	characteristic
72	C₄×Q₃₂	—	73	1	no	—
73	#1591	—	73	2	no	—
74	#1371	—	73	1	no	—
75	#1332	—	73	2	no	—
76	C₈×C₁₆	D₇₃	1	1	yes	characteristic
77	C₇₃	#12083	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₃₂	—	73	1	no	—
79	#446	—	146	2	no	—
80	C₄×Q₁₆	—	73	2	no	—
81	#366	—	73	1	no	—
82	#358	—	73	2	no	—
83	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
84	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
85	#335	—	73	2	no	—
86	Q₈⋊C₈	—	146	2	no	—
87	C₈²	C₂×D₇₃	1	1	yes	characteristic
88	C₂×Q₁₆	—	73	2	no	—
89	C₄×Q₈	—	146	2	no	—
90	Q₃₂	—	146	2	no	—
91	C₂×C₁₆	—	2	1	no	—
92	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
94	#107	—	73	2	no	—
95	C₄⋊C₈	—	292	2	no	—
96	C₄⋊C₈	—	146	2	no	—
97	#103	—	146	2	no	—
98	#97	—	292	2	no	—
99	C₄×C₈	D₂₉₂	1	1	yes	characteristic
100	C₄×C₈	D₂₉₂	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
102	C₂×Q₈	—	146	2	no	—
103	Q₁₆	—	146	2	no	—
104	Q₁₆	—	146	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	292	2	no	—
107	C₂×C₈	D₅₈₄	1	1	yes	characteristic
108	C₂×C₈	D₅₈₄	1	1	yes	characteristic
109	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
110	C₂×C₈	#11924028	1	1	yes	characteristic
111	C₄⋊C₄	—	292	2	no	—
112	C₄⋊C₄	—	146	2	no	—
113	C₄²	—	292	2	no	—
114	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
115	C₁₆	—	2	2	no	—
116	C₁₆	—	2	1	no	—
117	C₁₆	#11924011	1	2	yes	—
118	Q₈	—	292	2	no	—
119	Q₈	—	292	2	no	—
120	C₂×C₄	—	292	2	no	—
121	C₂×C₄	—	292	2	no	—
122	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
123	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
124	C₂×C₄	#14556613	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	1	no	—
128	C₈	—	584	2	no	—
129	C₈	Dic₅₈₄	1	2	yes	complemented
130	C₈	C₂×#11924011	1	1	yes	characteristic
131	C₈	#14556597	1	1	yes	characteristic
132	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	584	2	no	—
135	C₄	—	584	2	no	—
136	C₄	#17988666	1	1	yes	characteristic
137	C₄	#17988705	1	1	yes	characteristic
138	C₄	C₂×Dic₅₈₄	1	1	yes	characteristic
139	C₄	#17989019	1	1	yes	characteristic
140	C₂	#22608817	1	1	yes	characteristic
141	C₂	C₄×Dic₅₈₄	1	1	yes	characteristic
142	C₂	#22611813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₅₈₄

C₂

yes

maximal, characteristic

#22608829

C₂

yes

maximal

#22608003

C₂

yes

maximal, characteristic

#22607896

C₂

yes

maximal

C₈×C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

C₄

yes

characteristic

#17988717

—

C₄×Dic₂₉₂

C₂²

yes

—

#17988413

C₄

yes

characteristic

#17988392

C₄

yes

—

#17988340

C₂²

yes

—

#17988319

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

C₂×C₄

yes

—

C₄×Dic₁₄₆

—

Dic₅₈₄

—

#14556525

C₂×C₄

yes

—

—

—

—

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

Q₃₂

yes

complemented

C₈×C₇₃

C₂×M₁₆

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×Q₃₂

yes

characteristic

C₄×C₇₃

#508

yes

characteristic

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₄×Q₃₂

yes

characteristic

C₂×C₇₃

#2178

yes

characteristic

C₄×Q₃₂

—

#1591

—

#1371

—

#1332

—

C₈×C₁₆

D₇₃

yes

characteristic

C₇₃

#12083

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#446

—

146

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

—

Q₈⋊C₈

—

146

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

146

—

Q₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₄×D₇₃

yes

characteristic

#107

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

#103

—

146

—

#97

—

292

—

C₄×C₈

D₂₉₂

yes

characteristic

100

C₄×C₈

D₂₉₂

yes

characteristic

101

C₄×C₈

C₂²×D₇₃

yes

characteristic

102

C₂×Q₈

—

146

—

103

Q₁₆

—

146

—

104

Q₁₆

—

146

—

105

C₂×C₈

—

106

C₂×C₈

—

292

—

107

C₂×C₈

D₅₈₄

yes

characteristic

108

C₂×C₈

D₅₈₄

yes

characteristic

109

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

110

C₂×C₈

#11924028

yes

characteristic

111

C₄⋊C₄

—

292

—

112

C₄⋊C₄

—

146

—

113

C₄²

—

292

—

114

C₄²

C₂×D₂₉₂

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

#11924011

yes

—

118

Q₈

—

292

—

119

Q₈

—

292

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

292

—

122

C₂×C₄

C₄×D₂₉₂

yes

characteristic

123

C₂×C₄

C₂×D₅₈₄

yes

characteristic

124

C₂×C₄

#14556613

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

584

—

129

C₈

Dic₅₈₄

yes

complemented

130

C₈

C₂×#11924011

yes

characteristic

131

C₈

#14556597

yes

characteristic

132

C₂²

C₄×D₅₈₄

yes

characteristic

133

C₄

—

134

C₄

—

584

—

135

C₄

—

584

—

136

C₄

#17988666

yes

characteristic

137

C₄

#17988705

yes

characteristic

138

C₄

C₂×Dic₅₈₄

yes

characteristic

139

C₄

#17989019

yes

characteristic

140

C₂

#22608817

yes

characteristic

141

C₂

C₄×Dic₅₈₄

yes

characteristic

142

C₂

#22611813

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₈×C₁₆×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	C₄×D₈
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₈×C₁₆	D₇₃
hypocenter	C₇₃	#12083
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₅₈₄

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₈×C₁₆×C₇₃

C₂

socle, prime radical

C₂²×C₇₃

C₄×D₈

cosocle

C₄×C₈×C₇₃

C₂³

hypercenter

C₈×C₁₆

D₇₃

hypocenter

C₇₃

#12083

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
56. C₈×C₇₃	Q₃₂	semidirect
77. C₇₃	#12083	semidirect
129. C₈	Dic₅₈₄	semidirect

H ≅ G/N

type

56. C₈×C₇₃

Q₃₂

semidirect

77. C₇₃

#12083

semidirect

129. C₈

Dic₅₈₄

semidirect

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

subgroup	index
C₄×Dic₅₈₄	2
#22608829	2
#22608003	2
#22607896	2
C₈×C₁₆×C₇₃	2
#12083	73

subgroup

index

C₄×Dic₅₈₄

C₈×C₁₆×C₇₃

Subgroups — #32997976

Conjugacy classes of subgroups

Complements

Maximal subgroups