Group #32973067 — subgroups

Subgroups — #32973067

Order 18688.

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

normal subgroups only

This group has 264 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	#22609969	C₂	1	2	yes	maximal
2	C₇₃×#2112	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556527	—	2	2	no	—
4	#17988551	—	2	2	no	—
5	#17988352	C₄	1	4	yes	—
6	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₇₃	C₄	1	2	yes	—
8	C₂²⋊C₁₆×C₇₃	—	2	2	no	—
9	C₂×#11924012	—	2	4	no	—
10	#14556527	—	4	4	no	—
11	C₂²⋊#10596831	—	4	2	no	—
12	C₂²⋊#10596831	—	4	2	no	—
13	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	—	4	2	no	—
15	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₇₃	D₄	1	2	yes	—
18	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
19	M₃₂×C₇₃	—	4	2	no	—
20	C₂×C₁₆×C₇₃	—	2	2	no	—
21	#11924012	—	4	4	no	—
22	#11924012	—	4	4	no	—
23	C₂×#10596831	—	4	4	no	—
24	C₂⁴×C₇₃	Q₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₇₃	—	4	1	no	—
26	C₂²×C₄×C₇₃	—	4	1	no	—
27	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
28	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
31	C₂×C₈×C₇₃	—	2	2	no	—
32	C₂×C₈×C₇₃	—	2	4	no	—
33	C₂×C₈×C₇₃	—	4	2	no	—
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₇₃	—	4	2	no	—
39	C₂³×C₇₃	—	4	1	no	—
40	C₂³×C₇₃	—	4	2	no	—
41	C₂³×C₇₃	#103	1	2	yes	—
42	C₂³×C₇₃	#107	1	1	yes	characteristic
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#102	1	2	yes	—
51	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
52	C₈×C₇₃	—	2	1	no	—
53	C₈×C₇₃	—	2	2	no	—
54	C₈×C₇₃	#99	1	2	yes	—
55	#10596831	—	8	4	no	—
56	C₂²×C₇₃	—	2	2	no	—
57	C₂²×C₇₃	—	2	1	no	—
58	C₂²×C₇₃	—	2	4	no	—
59	C₂²×C₇₃	—	2	2	no	—
60	C₂²×C₇₃	—	4	4	no	—
61	C₂²×C₇₃	#340	1	1	yes	characteristic
62	C₄×C₇₃	—	2	2	no	—
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
65	C₄×C₇₃	#342	1	1	yes	characteristic
66	#7234	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	—	4	2	no	—
68	C₂×C₇₃	—	4	1	no	—
69	C₂×C₇₃	#1289	1	1	yes	characteristic
70	C₂×C₇₃	#1388	1	2	yes	—
71	#2112	D₇₃	1	1	yes	characteristic
72	#1810	—	73	2	no	—
73	C₇₃	#7234	1	1	yes	characteristic, complemented, Sylow
74	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
75	C₂×M₃₂	Dic₇₃	1	2	yes	—
76	C₂×#108	—	146	2	no	—
77	#407	—	146	2	no	—
78	C₂²⋊C₁₆	—	2	2	no	—
79	#341	—	73	4	no	—
80	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
81	C₂×M₁₆	—	146	4	no	—
82	C₂²×C₈	—	4	2	no	—
83	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
84	C₂²×C₈	C₄×D₇₃	1	2	yes	—
85	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
86	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
87	M₃₂	—	4	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	#108	—	292	4	no	—
90	C₂²⋊C₈	—	292	2	no	—
91	C₂²⋊C₈	—	292	2	no	—
92	C₂⁴	Dic₂₉₂	1	1	yes	characteristic
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	#11924005	1	2	yes	—
96	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
97	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
98	C₂²×C₄	#11924017	1	2	yes	—
99	M₁₆	—	292	4	no	—
100	M₁₆	—	292	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	292	4	no	—
106	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
108	C₂×C₈	#11924007	1	2	yes	—
109	C₁₆	—	4	2	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	4	2	no	—
112	C₂³	#14556516	1	2	yes	—
113	C₂³	#14556525	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	#14556498	1	1	yes	characteristic
121	C₂×C₄	#14556513	1	2	yes	—
122	C₂×C₄	#14556523	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	2	2	no	—
125	C₈	—	584	4	no	—
126	C₈	#14556506	1	2	yes	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	4	4	no	—
132	C₂²	#17988350	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	#17988344	1	1	yes	characteristic
136	C₄	#17988354	1	1	yes	characteristic
137	C₂	—	4	2	no	—
138	C₂	—	4	1	no	—
139	C₂	#22607799	1	1	yes	characteristic
140	C₂	#22608043	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22609969

C₂

yes

maximal

C₇₃×#2112

C₂

yes

maximal, characteristic

C₂×#14556527

—

#17988551

—

#17988352

C₄

yes

—

C₂³×C₈×C₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₄

yes

—

C₂²⋊C₁₆×C₇₃

—

—

—

—

—

C₂³×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂²×C₈×C₇₃

Q₈

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

C₂⁴×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂²×C₄×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#103

yes

—

C₂³×C₇₃

#107

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#95

yes

characteristic

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

#106

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

#99

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1289

yes

characteristic

C₂×C₇₃

#1388

yes

—

#2112

D₇₃

yes

characteristic

#1810

—

C₇₃

#7234

yes

characteristic, complemented, Sylow

C₂³×C₈

C₂×D₇₃

yes

characteristic

C₂×M₃₂

Dic₇₃

yes

—

C₂×#108

—

146

—

#407

—

146

—

C₂²⋊C₁₆

—

#341

—

C₂³×C₄

D₂₉₂

yes

characteristic

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

Dic₁₄₆

yes

characteristic

C₂²×C₈

C₄×D₇₃

yes

—

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

C₂²×C₈

C₂²⋊D₇₃

yes

—

M₃₂

—

C₂×C₁₆

—

#108

—

292

—

C₂²⋊C₈

—

292

—

C₂²⋊C₈

—

292

—

C₂⁴

Dic₂₉₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#11924005

yes

—

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

C₂²×C₄

D₅₈₄

yes

characteristic

C₂²×C₄

#11924017

yes

—

M₁₆

—

292

—

100

M₁₆

—

292

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

292

—

106

C₂×C₈

C₄×Dic₇₃

yes

characteristic

107

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

108

C₂×C₈

#11924007

yes

—

109

C₁₆

—

110

C₂³

—

111

C₂³

—

112

C₂³

#14556516

yes

—

113

C₂³

#14556525

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#14556498

yes

characteristic

121

C₂×C₄

#14556513

yes

—

122

C₂×C₄

#14556523

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

584

—

126

C₈

#14556506

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#17988350

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

#17988344

yes

characteristic

136

C₄

#17988354

yes

characteristic

137

C₂

—

138

C₂

—

139

C₂

#22607799

yes

characteristic

140

C₂

#22608043

yes

—

	subgroup	G/H
center	C₂²	#17988350
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂³×C₈	C₂×D₇₃
Fitting	C₇₃×#2112	C₂
socle	C₂²×C₇₃	#340
prime radical	C₂⁴×C₇₃	Q₁₆
cosocle	C₂³×C₈×C₇₃	C₂²
hypercenter	#2112	D₇₃
hypocenter	C₇₃	#7234
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

subgroup

G/H

center

C₂²

#17988350

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂³×C₈

C₂×D₇₃

Fitting

C₇₃×#2112

C₂

socle

C₂²×C₇₃

#340

prime radical

C₂⁴×C₇₃

Q₁₆

cosocle

C₂³×C₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #2112

N	H ≅ G/N	type
73. C₇₃	#7234	semidirect

H ≅ G/N

type

73. C₇₃

#7234

semidirect

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

subgroup	index
#22609969	2
C₇₃×#2112	2
#7234	73

subgroup

index

#22609969

C₇₃×#2112

#7234

Subgroups — #32973067

Conjugacy classes of subgroups

Complements

Maximal subgroups