Group #32997992 — subgroups

Subgroups — #32997992

Order 18688.

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

normal subgroups only

This group has 234 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	#22608816	C₂	1	2	yes	maximal, complemented
2	#22608005	C₂	1	1	yes	maximal, characteristic, complemented
3	#22607892	C₂	1	2	yes	maximal, complemented
4	#22607847	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988665	—	2	2	no	—
7	C₄×#11924003	C₂²	1	1	yes	characteristic
8	#17988340	C₂²	1	2	yes	—
9	D₄⋊#10596831	—	2	2	no	—
10	C₈×#10596831	C₂²	1	1	yes	characteristic
11	C₂×D₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₄×D₈×C₇₃	C₂²	1	2	yes	—
13	#366×C₇₃	C₄	1	1	yes	characteristic
14	#357×C₇₃	C₄	1	2	yes	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924012	—	4	2	no	—
17	C₂×#11924003	—	2	1	no	—
18	C₄⋊#10596831	—	2	2	no	—
19	C₂²⋊#10596831	—	4	2	no	—
20	C₄×#10596831	D₄	1	1	yes	characteristic
21	C₄×#10596831	D₄	1	1	yes	characteristic
22	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	—
23	C₄×D₄×C₇₃	—	2	2	no	—
24	D₁₆×C₇₃	—	2	2	no	—
25	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
27	#107×C₇₃	C₂×C₄	1	2	yes	—
28	#102×C₇₃	—	2	2	no	—
29	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	#11924012	—	8	4	no	—
31	C₂×#10596831	—	2	1	no	—
32	C₂×#10596831	—	4	2	no	—
33	C₂×#10596831	D₈	1	2	yes	—
34	#11924003	—	2	2	no	—
35	C₂×D₄×C₇₃	—	2	2	no	—
36	C₂²×C₄×C₇₃	—	4	2	no	—
37	D₈×C₇₃	—	2	2	no	—
38	D₈×C₇₃	—	2	2	no	—
39	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₇₃	—	2	2	no	—
42	C₂²⋊C₄×C₇₃	—	4	2	no	—
43	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₇₃	—	2	1	no	—
45	C₁₆×C₇₃	M₁₆	1	2	yes	complemented
46	C₂³×C₇₃	—	4	2	no	—
47	D₄×C₇₃	—	4	2	no	—
48	D₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	—	8	2	no	—
51	C₂×C₄×C₇₃	—	8	2	no	—
52	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
57	C₈×C₇₃	#131	1	1	yes	characteristic
58	#10596831	—	2	2	no	—
59	#10596831	—	4	1	no	—
60	#10596831	—	8	2	no	—
61	C₂²×C₇₃	—	8	2	no	—
62	C₂²×C₇₃	—	8	2	no	—
63	C₂²×C₇₃	—	8	2	no	—
64	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	8	2	no	—
67	C₄×C₇₃	#436	1	1	yes	characteristic
68	C₄×C₇₃	#443	1	1	yes	characteristic
69	C₄×C₇₃	#509	1	2	yes	—
70	#12086	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	—	16	2	no	—
72	C₂×C₇₃	#1589	1	1	yes	characteristic
73	C₂×C₇₃	#2177	1	1	yes	characteristic
74	C₂×C₇₃	#2179	1	1	yes	characteristic
75	C₄×D₁₆	D₇₃	1	1	yes	characteristic
76	#1589	—	73	2	no	—
77	#1372	—	73	1	no	—
78	#1331	—	73	2	no	—
79	#1311	—	73	1	no	—
80	C₇₃	#12086	1	1	yes	characteristic, complemented, Sylow
81	C₂×D₁₆	Dic₇₃	1	1	yes	characteristic
82	C₄×D₈	C₂×D₇₃	1	2	yes	—
83	#436	—	146	2	no	—
84	#366	Dic₇₃	1	1	yes	characteristic
85	#357	Dic₇₃	1	2	yes	—
86	C₄×C₁₆	—	73	1	no	—
87	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
88	#335	—	73	2	no	—
89	D₄⋊C₈	—	146	2	no	—
90	C₈²	—	73	1	no	—
91	C₂×D₈	C₂×Dic₇₃	1	2	yes	—
92	C₂×M₁₆	—	292	2	no	—
93	C₄×D₄	—	2	2	no	—
94	D₁₆	—	2	2	no	—
95	C₂×C₁₆	—	146	1	no	—
96	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
97	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
98	#107	C₂×Dic₇₃	1	2	yes	—
99	C₄⋊C₈	—	146	2	no	—
100	#102	—	2	2	no	—
101	C₂²⋊C₈	—	292	2	no	—
102	C₄×C₈	—	73	1	no	—
103	C₄×C₈	—	73	1	no	—
104	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
105	C₂×D₄	—	2	2	no	—
106	C₂²×C₄	—	4	2	no	—
107	D₈	—	2	2	no	—
108	D₈	—	2	2	no	—
109	M₁₆	—	584	4	no	—
110	C₂×C₈	—	292	2	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	146	1	no	—
113	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
114	C₂×C₈	#11924027	1	1	yes	characteristic
115	C₄⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₄²	D₄×D₇₃	1	1	yes	characteristic
118	C₁₆	—	2	1	no	—
119	C₁₆	—	146	2	no	—
120	C₁₆	#11924012	1	2	yes	complemented
121	C₂³	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
128	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
129	C₂×C₄	#14556610	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	292	1	no	—
132	C₈	—	584	2	no	—
133	C₈	—	146	2	no	—
134	C₈	C₂×#11924012	1	1	yes	characteristic
135	C₈	#14556598	1	1	yes	characteristic
136	C₂²	—	8	2	no	—
137	C₂²	—	8	2	no	—
138	C₂²	—	8	2	no	—
139	C₂²	D₈×Dic₇₃	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	8	2	no	—
142	C₄	#17988665	1	1	yes	characteristic
143	C₄	#17988704	1	1	yes	characteristic
144	C₄	#17989021	1	2	yes	—
145	C₂	—	16	2	no	—
146	C₂	#22608816	1	1	yes	characteristic
147	C₂	#22611807	1	1	yes	characteristic
148	C₂	#22611817	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22608816

C₂

yes

maximal, complemented

#22608005

C₂

yes

maximal, characteristic, complemented

#22607892

C₂

yes

maximal, complemented

#22607847

C₂

yes

maximal, characteristic, complemented

C₄×D₁₆×C₇₃

C₂

yes

maximal, characteristic

#17988665

—

C₄×#11924003

C₂²

yes

characteristic

#17988340

C₂²

yes

—

D₄⋊#10596831

—

C₈×#10596831

C₂²

yes

characteristic

C₂×D₁₆×C₇₃

C₄

yes

characteristic

C₄×D₈×C₇₃

C₂²

yes

—

#366×C₇₃

C₄

yes

characteristic

#357×C₇₃

C₄

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

—

—

D₄

yes

characteristic

C₄×#10596831

D₄

yes

characteristic

C₂×D₈×C₇₃

C₂×C₄

yes

—

C₄×D₄×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#107×C₇₃

C₂×C₄

yes

—

#102×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

—

—

—

D₈

yes

—

#11924003

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

complemented

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

C₈×C₇₃

yes

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

—

#12086

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×D₁₆

D₇₃

yes

—

—

—

—

C₇₃

#12086

yes

characteristic, complemented, Sylow

C₂×D₁₆

Dic₇₃

yes

characteristic

C₄×D₈

C₂×D₇₃

yes

—

#436

—

146

—

#366

Dic₇₃

yes

characteristic

#357

Dic₇₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

#335

—

D₄⋊C₈

—

146

—

C₈²

—

C₂×D₈

C₂×Dic₇₃

yes

—

C₂×M₁₆

—

292

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

#107

C₂×Dic₇₃

yes

—

C₄⋊C₈

—

146

—

100

#102

—

101

C₂²⋊C₈

—

292

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₂²×D₇₃

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

107

D₈

—

108

D₈

—

109

M₁₆

—

584

—

110

C₂×C₈

—

292

—

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

114

C₂×C₈

#11924027

yes

characteristic

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

D₄×D₇₃

yes

characteristic

118

C₁₆

—

119

C₁₆

—

146

—

120

C₁₆

#11924012

yes

complemented

121

C₂³

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

D₄×Dic₇₃

yes

characteristic

128

C₂×C₄

D₈×D₇₃

yes

characteristic

129

C₂×C₄

#14556610

yes

characteristic

130

C₈

—

131

C₈

—

292

—

132

C₈

—

584

—

133

C₈

—

146

—

134

C₈

C₂×#11924012

yes

characteristic

135

C₈

#14556598

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

D₈×Dic₇₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#17988665

yes

characteristic

143

C₄

#17988704

yes

characteristic

144

C₄

#17989021

yes

—

145

C₂

—

146

C₂

#22608816

yes

characteristic

147

C₂

#22611807

yes

characteristic

148

C₂

#22611817

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₄×D₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₄×D₈
prime radical	C₂×D₁₆×C₇₃	C₄
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₄×D₁₆	D₇₃
hypocenter	C₇₃	#12086
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

subgroup

G/H

center

C₂×C₄

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₄×D₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₄×D₈

prime radical

C₂×D₁₆×C₇₃

C₄

cosocle

C₄×C₈×C₇₃

C₂³

hypercenter

C₄×D₁₆

D₇₃

hypocenter

C₇₃

#12086

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

N	H ≅ G/N	type
1. #22608816	C₂	semidirect
2. #22608005	C₂	semidirect
3. #22607892	C₂	semidirect
4. #22607847	C₂	semidirect
45. C₁₆×C₇₃	M₁₆	semidirect
80. C₇₃	#12086	semidirect
120. C₁₆	#11924012	semidirect

H ≅ G/N

type

1. #22608816

C₂

semidirect

2. #22608005

C₂

semidirect

3. #22607892

C₂

semidirect

4. #22607847

C₂

semidirect

45. C₁₆×C₇₃

M₁₆

semidirect

80. C₇₃

#12086

semidirect

120. C₁₆

#11924012

semidirect

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

subgroup	index
#22608816	2
#22608005	2
#22607892	2
#22607847	2
C₄×D₁₆×C₇₃	2
#12086	73

subgroup

index

C₄×D₁₆×C₇₃

Subgroups — #32997992

Conjugacy classes of subgroups

Complements

Maximal subgroups