Group #32973056 — subgroups

Subgroups — #32973056

Order 18688.

244 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 79 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	#22609981	C₂	1	2	yes	maximal
2	C₇₃×#2113	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556527	—	2	2	no	—
4	#17988571	—	2	2	no	—
5	#17988345	C₄	1	4	yes	—
6	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
9	C₂²⋊C₁₆×C₇₃	—	2	2	no	—
10	C₂×#11924012	—	2	4	no	—
11	#14556527	—	4	2	no	—
12	#14556527	—	4	2	no	—
13	#14556510	—	4	2	no	—
14	#14556508	—	4	2	no	—
15	C₂×#41×C₇₃	D₄	1	1	yes	characteristic
16	#131×C₇₃	—	4	1	no	—
17	#131×C₇₃	—	4	1	no	—
18	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₇₃	Q₈	1	1	yes	characteristic
20	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	D₄	1	2	yes	—
22	M₃₂×C₇₃	—	4	1	no	—
23	C₂×C₁₆×C₇₃	—	2	1	no	—
24	C₂×C₁₆×C₇₃	—	2	1	no	—
25	C₂×C₁₆×C₇₃	—	4	1	no	—
26	#11924012	—	4	4	no	—
27	#11924012	—	4	4	no	—
28	C₂×#10596831	—	4	4	no	—
29	#41×C₇₃	—	4	1	no	—
30	#41×C₇₃	—	4	1	no	—
31	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
32	C₂×D₄×C₇₃	SD₁₆	1	2	yes	—
33	C₂×D₄×C₇₃	Q₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
36	M₁₆×C₇₃	—	4	2	no	—
37	M₁₆×C₇₃	—	4	1	no	—
38	C₂×C₈×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	2	2	no	—
41	C₂×C₈×C₇₃	—	2	2	no	—
42	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
45	C₁₆×C₇₃	—	4	1	no	—
46	C₁₆×C₇₃	—	4	1	no	—
47	C₂³×C₇₃	#103	1	2	yes	—
48	C₂³×C₇₃	#106	1	1	yes	characteristic
49	Q₈×C₇₃	—	4	1	no	—
50	D₄×C₇₃	—	4	2	no	—
51	D₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	—	2	2	no	—
53	C₂×C₄×C₇₃	—	2	1	no	—
54	C₂×C₄×C₇₃	—	2	2	no	—
55	C₂×C₄×C₇₃	—	2	1	no	—
56	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#102	1	2	yes	—
58	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
59	C₈×C₇₃	—	2	1	no	—
60	C₈×C₇₃	—	2	2	no	—
61	C₈×C₇₃	#99	1	1	yes	characteristic
62	C₈×C₇₃	#99	1	1	yes	characteristic
63	#10596831	—	8	4	no	—
64	C₂²×C₇₃	—	2	2	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	—	4	2	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	#340	1	1	yes	characteristic
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
72	C₄×C₇₃	#342	1	1	yes	characteristic
73	#7229	—	73	1	no	maximal, Sylow
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	—	4	2	no	—
76	C₂×C₇₃	—	4	1	no	—
77	C₂×C₇₃	#1289	1	1	yes	characteristic
78	#2113	D₇₃	1	1	yes	characteristic
79	#1813	—	73	2	no	—
80	C₇₃	#7229	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
82	C₂×M₃₂	Dic₇₃	1	1	yes	characteristic
83	C₂²×C₁₆	Dic₇₃	1	1	yes	characteristic
84	C₂×#108	—	146	2	no	—
85	#413	—	146	2	no	—
86	C₂²⋊C₁₆	—	2	2	no	—
87	#338	—	73	4	no	—
88	C₂×#41	D₂₉₂	1	1	yes	characteristic
89	#131	—	4	1	no	—
90	#131	—	4	1	no	—
91	C₂×M₁₆	—	146	4	no	—
92	C₂×M₁₆	Dic₁₄₆	1	1	yes	characteristic
93	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
94	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
95	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
96	M₃₂	—	4	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	4	1	no	—
100	#108	—	292	2	no	—
101	#108	—	292	2	no	—
102	#101	—	292	2	no	—
103	#100	—	292	2	no	—
104	#41	—	4	1	no	—
105	#41	—	4	1	no	—
106	C₂×Q₈	D₅₈₄	1	1	yes	characteristic
107	C₂×D₄	#11924017	1	2	yes	—
108	C₂×D₄	Dic₂₉₂	1	1	yes	characteristic
109	C₂²×C₄	#11924005	1	2	yes	—
110	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	2	no	—
113	M₁₆	—	292	4	no	—
114	M₁₆	—	292	4	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	292	4	no	—
120	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
121	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
122	C₂×C₈	#11924007	1	2	yes	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	4	1	no	—
125	C₂³	#14556516	1	2	yes	—
126	C₂³	#14556523	1	1	yes	characteristic
127	Q₈	—	4	1	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	#14556498	1	1	yes	characteristic
135	C₂×C₄	#14556513	1	2	yes	—
136	C₂×C₄	#14556525	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	2	2	no	—
139	C₈	—	584	4	no	—
140	C₈	#14556506	1	1	yes	characteristic
141	C₈	#14556506	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	4	2	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#17988350	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	#17988344	1	1	yes	characteristic
150	C₄	#17988354	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	4	1	no	—
154	C₂	#22607799	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22609981

C₂

yes

maximal

C₇₃×#2113

C₂

yes

maximal, characteristic

C₂×#14556527

—

#17988571

—

#17988345

C₄

yes

—

C₂×#131×C₇₃

C₂²

yes

characteristic

C₂×M₃₂×C₇₃

C₄

yes

characteristic

C₂²×C₁₆×C₇₃

C₄

yes

characteristic

C₂²⋊C₁₆×C₇₃

—

—

—

—

—

—

C₂×#41×C₇₃

D₄

yes

characteristic

#131×C₇₃

—

#131×C₇₃

—

C₂×M₁₆×C₇₃

C₂×C₄

yes

—

C₂×M₁₆×C₇₃

Q₈

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₃

D₄

yes

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

#11924012

—

#11924012

—

C₂×#10596831

—

#41×C₇₃

—

#41×C₇₃

—

C₂×Q₈×C₇₃

D₈

yes

characteristic

C₂×D₄×C₇₃

SD₁₆

yes

—

C₂×D₄×C₇₃

Q₁₆

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

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

—

C₁₆×C₇₃

—

C₂³×C₇₃

#103

yes

—

C₂³×C₇₃

#106

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×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

#107

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

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

—

C₂×C₇₃

#1289

yes

characteristic

#2113

D₇₃

yes

characteristic

#1813

—

C₇₃

#7229

yes

characteristic, complemented, Sylow

C₂×#131

C₂×D₇₃

yes

characteristic

C₂×M₃₂

Dic₇₃

yes

characteristic

C₂²×C₁₆

Dic₇₃

yes

characteristic

C₂×#108

—

146

—

#413

—

146

—

C₂²⋊C₁₆

—

#338

—

C₂×#41

D₂₉₂

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

146

—

C₂×M₁₆

Dic₁₄₆

yes

characteristic

C₂×M₁₆

C₄×D₇₃

yes

—

C₂²×C₈

C₂×Dic₇₃

yes

characteristic

C₂²×C₈

C₂²⋊D₇₃

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

100

#108

—

292

—

101

#108

—

292

—

102

#101

—

292

—

103

#100

—

292

—

104

#41

—

105

#41

—

106

C₂×Q₈

D₅₈₄

yes

characteristic

107

C₂×D₄

#11924017

yes

—

108

C₂×D₄

Dic₂₉₂

yes

characteristic

109

C₂²×C₄

#11924005

yes

—

110

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

292

—

114

M₁₆

—

292

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

292

—

120

C₂×C₈

C₄×Dic₇₃

yes

characteristic

121

C₂×C₈

C₂²⋊Dic₇₃

yes

characteristic

122

C₂×C₈

#11924007

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

#14556516

yes

—

126

C₂³

#14556523

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#14556498

yes

characteristic

135

C₂×C₄

#14556513

yes

—

136

C₂×C₄

#14556525

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

584

—

140

C₈

#14556506

yes

characteristic

141

C₈

#14556506

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#17988350

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

#17988344

yes

characteristic

150

C₄

#17988354

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

#22607799

yes

characteristic

	subgroup	G/H
center	C₄	#17988344
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂×#131	C₂×D₇₃
Fitting	C₇₃×#2113	C₂
socle	C₂×C₇₃	#1289
prime radical	C₂×#41×C₇₃	D₄
cosocle	C₂×#131×C₇₃	C₂²
hypercenter	#2113	D₇₃
hypocenter	C₇₃	#7229
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

subgroup

G/H

center

C₄

#17988344

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂×#131

C₂×D₇₃

Fitting

C₇₃×#2113

C₂

socle

C₂×C₇₃

#1289

prime radical

C₂×#41×C₇₃

D₄

cosocle

C₂×#131×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2113

N	H ≅ G/N	type
80. C₇₃	#7229	semidirect

H ≅ G/N

type

80. C₇₃

#7229

semidirect

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

subgroup	index
#22609981	2
C₇₃×#2113	2
#7229	73

subgroup

index

#22609981

C₇₃×#2113

#7229

Subgroups — #32973056

Conjugacy classes of subgroups

Complements

Maximal subgroups