Group #32973417 — subgroups

Subgroups — #32973417

Order 18688.

170 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 39 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	#22610150	C₂	1	1	yes	maximal, characteristic
2	#22607869	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1315	C₂	1	1	yes	maximal, characteristic
4	#17989354	C₄	1	1	yes	characteristic
5	#17988635	—	2	1	no	—
6	#17988571	C₄	1	1	yes	characteristic
7	#17988369	—	2	1	no	—
8	#17988352	—	2	1	no	—
9	#408×C₇₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₁₆×C₇₃	—	2	1	no	—
11	#14556598	—	4	2	no	—
12	C₂×#11924012	—	2	1	no	—
13	C₂×#11924012	—	2	1	no	—
14	C₂×#11924012	—	4	1	no	—
15	#14556531	—	4	1	no	—
16	#14556526	—	4	2	no	—
17	#14556510	—	2	1	no	—
18	#14556508	C₈	1	2	yes	—
19	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₃	—	4	1	no	—
23	C₂²⋊C₈×C₇₃	—	4	1	no	—
24	#11924012	—	2	2	no	—
25	#11924012	—	2	2	no	—
26	#11924012	—	4	1	no	—
27	#11924012	—	4	2	no	—
28	#11924012	—	4	1	no	—
29	C₂×#10596831	—	4	1	no	—
30	C₂×#10596831	—	4	1	no	—
31	C₂×#10596831	—	4	2	no	—
32	#11924003	—	8	1	no	—
33	#41×C₇₃	—	4	1	no	—
34	#41×C₇₃	—	4	1	no	—
35	C₂×Q₈×C₇₃	M₁₆	1	1	yes	characteristic
36	C₂×D₄×C₇₃	—	2	1	no	—
37	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	—	2	1	no	—
39	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	M₁₆×C₇₃	—	4	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	4	1	no	—
44	C₁₆×C₇₃	—	8	1	no	—
45	C₂³×C₇₃	—	2	1	no	—
46	C₂³×C₇₃	#101	1	1	yes	characteristic
47	Q₈×C₇₃	—	4	1	no	—
48	D₄×C₇₃	—	4	1	no	—
49	D₄×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
57	C₈×C₇₃	—	4	1	no	—
58	C₈×C₇₃	—	4	1	no	—
59	#10596831	—	4	2	no	—
60	#10596831	—	4	2	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	—	4	1	no	—
63	C₂²×C₇₃	—	4	1	no	—
64	C₂²×C₇₃	—	8	1	no	—
65	C₂²×C₇₃	#324	1	1	yes	characteristic
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	4	1	no	—
68	C₄×C₇₃	AΣL(1,𝔽₁₆)	1	1	yes	characteristic
69	C₄×C₇₃	#352	1	1	yes	characteristic
70	#7377	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	—	4	1	no	—
73	C₂×C₇₃	—	8	1	no	—
74	C₂×C₇₃	#1326	1	1	yes	characteristic
75	#1858	—	73	1	no	—
76	#1321	—	73	1	no	—
77	#1315	D₇₃	1	1	yes	characteristic
78	C₇₃	#7377	1	1	yes	characteristic, complemented, Sylow
79	#568	—	73	1	no	—
80	#430	—	146	1	no	—
81	#413	—	73	1	no	—
82	#408	C₂×D₇₃	1	1	yes	characteristic
83	#349	—	146	1	no	—
84	C₂²⋊C₁₆	—	2	1	no	—
85	#341	—	146	1	no	—
86	C₂×#41	C₄×D₇₃	1	1	yes	characteristic
87	#131	—	292	2	no	—
88	C₂×M₁₆	—	292	1	no	—
89	C₂×M₁₆	—	146	1	no	—
90	C₂×M₁₆	—	146	1	no	—
91	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
92	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
93	M₃₂	—	292	1	no	—
94	C₂×C₁₆	—	4	1	no	—
95	#108	—	292	2	no	—
96	#101	—	146	1	no	—
97	#100	—	73	2	no	—
98	C₂²⋊C₈	—	4	1	no	—
99	#41	—	4	1	no	—
100	#41	—	4	1	no	—
101	C₂×Q₈	#11924011	1	1	yes	characteristic
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	C₈×D₇₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	#11924005	1	1	yes	characteristic
106	M₁₆	—	4	1	no	—
107	M₁₆	—	292	1	no	—
108	M₁₆	—	292	2	no	—
109	M₁₆	—	292	1	no	—
110	M₁₆	—	146	2	no	—
111	M₁₆	—	146	2	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	292	1	no	—
116	C₂×C₈	—	292	1	no	—
117	C₂×C₈	—	292	2	no	—
118	C₁₆	—	8	1	no	—
119	C₁₆	—	584	1	no	—
120	C₂³	—	2	1	no	—
121	C₂³	#14556509	1	1	yes	characteristic
122	Q₈	—	4	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#14556503	1	1	yes	characteristic
131	C₂×C₄	#14556505	1	1	yes	characteristic
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	292	2	no	—
135	C₈	—	292	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	#17988312	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	#17988375	1	1	yes	characteristic
144	C₄	#17988376	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	4	1	no	—
147	C₂	—	8	1	no	—
148	C₂	#22607880	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610150

C₂

yes

maximal, characteristic

#22607869

C₂

yes

maximal, characteristic

C₇₃×#1315

C₂

yes

maximal, characteristic

#17989354

C₄

yes

characteristic

#17988635

—

#17988571

C₄

yes

characteristic

#17988369

—

#17988352

—

#408×C₇₃

C₂²

yes

characteristic

C₂²⋊C₁₆×C₇₃

—

—

—

—

—

—

—

—

C₈

yes

—

C₂×#41×C₇₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂²⋊C₈×C₇₃

—

—

—

—

—

—

—

—

—

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

M₁₆

yes

characteristic

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#101

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

#99

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#324

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

AΣL(1,𝔽₁₆)

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

C₇₃

#7377

yes

characteristic, complemented, Sylow

#568

—

#430

—

146

—

#413

—

#408

C₂×D₇₃

yes

characteristic

#349

—

146

—

C₂²⋊C₁₆

—

#341

—

146

—

C₂×#41

C₄×D₇₃

yes

characteristic

#131

—

292

—

C₂×M₁₆

—

292

—

C₂×M₁₆

—

146

—

C₂×M₁₆

—

146

—

C₂×M₁₆

D₂₉₂

yes

characteristic

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

M₃₂

—

292

—

C₂×C₁₆

—

#108

—

292

—

#101

—

146

—

#100

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

#11924011

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

C₈×D₇₃

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

#11924005

yes

characteristic

106

M₁₆

—

107

M₁₆

—

292

—

108

M₁₆

—

292

—

109

M₁₆

—

292

—

110

M₁₆

—

146

—

111

M₁₆

—

146

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

292

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

292

—

118

C₁₆

—

119

C₁₆

—

584

—

120

C₂³

—

121

C₂³

#14556509

yes

characteristic

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#14556503

yes

characteristic

131

C₂×C₄

#14556505

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

292

—

135

C₈

—

292

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#17988312

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

#17988375

yes

characteristic

144

C₄

#17988376

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#22607880

yes

characteristic

	subgroup	G/H
center	C₄	#17988375
commutator	C₂×D₄×C₇₃	C₂×C₈
Frattini	#408	C₂×D₇₃
Fitting	C₇₃×#1315	C₂
socle	C₂×C₇₃	#1326
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	#408×C₇₃	C₂²
hypercenter	#1315	D₇₃
hypocenter	C₇₃	#7377
LCS	G ▷ C₂×D₄×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1315

subgroup

G/H

center

C₄

#17988375

commutator

C₂×D₄×C₇₃

C₂×C₈

Frattini

#408

C₂×D₇₃

Fitting

C₇₃×#1315

C₂

socle

C₂×C₇₃

#1326

prime radical

C₂×#41×C₇₃

C₂×C₄

cosocle

#408×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×D₄×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ #1315

N	H ≅ G/N	type
78. C₇₃	#7377	semidirect

H ≅ G/N

type

78. C₇₃

#7377

semidirect

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

subgroup	index
#22610150	2
#22607869	2
C₇₃×#1315	2
#7377	73

subgroup

index

#22610150

#22607869

C₇₃×#1315

#7377

Subgroups — #32973417

Conjugacy classes of subgroups

Complements

Maximal subgroups