Group #32979799 — subgroups

Subgroups — #32979799

Order 18688.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22610069	C₂	1	1	yes	maximal, characteristic
2	#22608689	C₂	1	2	yes	maximal
3	#22608686	C₂	1	2	yes	maximal
4	C₂×#17988308	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1837	C₂	1	1	yes	maximal, characteristic
6	#17988611	C₂²	1	2	yes	—
7	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	#17988352	—	2	2	no	—
10	#17988339	—	2	4	no	—
11	#17988337	—	2	4	no	—
12	#17988308	C₄	1	4	yes	—
13	#423×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	#341×C₇₃	—	2	2	no	—
16	C₂×#11924012	—	2	2	no	—
17	C₂²×#10596831	Q₈	1	2	yes	—
18	C₂²×#10596831	Q₈	1	2	yes	—
19	C₄⋊#10596831	—	2	2	no	—
20	C₄⋊#10596831	—	2	2	no	—
21	C₄×#10596831	C₂×C₄	1	2	yes	—
22	C₄×#10596831	C₂×C₄	1	2	yes	—
23	C₄×#10596831	D₄	1	2	yes	—
24	C₄×#10596831	D₄	1	2	yes	—
25	C₂×M₁₆×C₇₃	—	2	2	no	—
26	C₂²×C₈×C₇₃	D₄	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₇₃	—	2	2	no	—
29	C₄⋊C₈×C₇₃	—	2	2	no	—
30	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₃	Q₈	1	2	yes	—
32	#11924012	—	4	4	no	—
33	C₂×#10596831	—	4	2	no	—
34	C₂×#10596831	—	4	2	no	—
35	C₂×#10596831	—	4	2	no	—
36	C₂×#10596831	C₄⋊C₄	1	4	yes	—
37	C₂×#10596831	C₄⋊C₄	1	4	yes	—
38	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
41	M₁₆×C₇₃	—	4	4	no	—
42	C₂×C₈×C₇₃	—	2	4	no	—
43	C₂×C₈×C₇₃	—	4	2	no	—
44	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
45	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
49	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₇₃	#135	1	4	yes	—
59	C₈×C₇₃	—	2	4	no	—
60	C₈×C₇₃	—	8	2	no	—
61	#10596831	—	2	4	no	—
62	#10596831	—	2	4	no	—
63	#10596831	—	8	2	no	—
64	C₂²×C₇₃	—	2	1	no	—
65	C₂²×C₇₃	—	2	1	no	—
66	C₂²×C₇₃	#384	1	1	yes	characteristic
67	C₂²×C₇₃	#499	1	2	yes	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	#427	1	2	yes	—
71	C₄×C₇₃	#430	1	2	yes	—
72	C₄×C₇₃	#430	1	2	yes	—
73	C₄×C₇₃	#495	1	2	yes	—
74	#8827	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	2	2	no	—
76	C₂×C₇₃	#1846	1	1	yes	characteristic
77	C₂×C₇₃	#1849	1	1	yes	characteristic
78	C₂×C₇₃	#1849	1	1	yes	characteristic
79	#1837	—	73	1	no	—
80	#1837	D₇₃	1	1	yes	characteristic
81	#1567	—	73	2	no	—
82	#1566	—	73	2	no	—
83	C₂×#322	—	73	1	no	—
84	C₇₃	#8827	1	1	yes	characteristic, complemented, Sylow
85	#423	—	73	2	no	—
86	#423	C₂×D₇₃	1	2	yes	—
87	C₂×C₄×C₈	—	73	1	no	—
88	C₂×C₄×C₈	—	73	1	no	—
89	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
90	#341	—	2	2	no	—
91	#341	—	146	2	no	—
92	#335	—	146	4	no	—
93	#334	—	146	4	no	—
94	#322	—	73	4	no	—
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	146	2	no	—
97	C₂²×C₈	—	73	2	no	—
98	C₂²×C₈	—	73	2	no	—
99	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
100	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	146	2	no	—
104	C₄⋊C₈	—	146	2	no	—
105	C₄×C₈	—	73	2	no	—
106	C₄×C₈	—	73	2	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	—	73	2	no	—
109	C₄×C₈	Dic₁₄₆	1	2	yes	—
110	C₄×C₈	C₄×D₇₃	1	2	yes	—
111	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
112	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
113	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
114	M₁₆	—	4	4	no	—
115	M₁₆	—	292	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	292	2	no	—
119	C₂×C₈	—	292	2	no	—
120	C₂×C₈	—	292	2	no	—
121	C₂×C₈	—	73	4	no	—
122	C₂×C₈	—	73	4	no	—
123	C₂×C₈	#11924007	1	4	yes	—
124	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
125	C₄²	D₄×D₇₃	1	1	yes	characteristic
126	C₄²	D₄×D₇₃	1	1	yes	characteristic
127	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
128	C₂³	#14556589	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
132	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
133	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
134	C₂×C₄	#14556587	1	1	yes	characteristic
135	C₂×C₄	#14556590	1	1	yes	characteristic
136	C₂×C₄	#14556590	1	1	yes	characteristic
137	C₂×C₄	#14556611	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	8	2	no	—
140	C₈	—	584	2	no	—
141	C₈	—	146	4	no	—
142	C₈	—	146	4	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#17988456	1	1	yes	characteristic
146	C₂²	#17988986	1	2	yes	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#17988623	1	2	yes	—
150	C₄	#17988634	1	2	yes	—
151	C₄	#17988634	1	2	yes	—
152	C₄	#17988969	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#22610093	1	1	yes	characteristic
155	C₂	#22610104	1	1	yes	characteristic
156	C₂	#22610104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610069

C₂

yes

maximal, characteristic

#22608689

C₂

yes

maximal

#22608686

C₂

yes

maximal

C₂×#17988308

C₂

yes

maximal, characteristic

C₇₃×#1837

C₂

yes

maximal, characteristic

#17988611

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₂×C₄×#10596831

C₂²

yes

—

—

—

C₄

yes

—

#423×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#341×C₇₃

—

C₂×#11924012

—

C₂²×#10596831

Q₈

yes

—

C₂²×#10596831

Q₈

yes

—

C₄⋊#10596831

—

C₄⋊#10596831

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

D₄

yes

—

C₄×#10596831

D₄

yes

—

C₂×M₁₆×C₇₃

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

Q₈

yes

—

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×Q₈

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₂³×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#384

yes

characteristic

C₂²×C₇₃

#499

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#427

yes

—

C₄×C₇₃

#430

yes

—

C₄×C₇₃

#430

yes

—

C₄×C₇₃

#495

yes

—

#8827

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

yes

—

—

—

C₇₃

#8827

yes

characteristic, complemented, Sylow

#423

—

#423

C₂×D₇₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#341

—

#341

—

146

—

#335

—

146

—

#334

—

146

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₇₃

yes

—

100

C₂×C₄²

C₂²×D₇₃

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

146

—

104

C₄⋊C₈

—

146

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

Dic₁₄₆

yes

—

110

C₄×C₈

C₄×D₇₃

yes

—

111

C₂²×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

112

C₂²×C₄

Q₈×D₇₃

yes

characteristic

113

C₂²×C₄

Q₈×D₇₃

yes

characteristic

114

M₁₆

—

115

M₁₆

—

292

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

292

—

119

C₂×C₈

—

292

—

120

C₂×C₈

—

292

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

#11924007

yes

—

124

C₄²

C₂×C₄×D₇₃

yes

characteristic

125

C₄²

D₄×D₇₃

yes

characteristic

126

C₄²

D₄×D₇₃

yes

characteristic

127

C₄²

C₂×Dic₁₄₆

yes

characteristic

128

C₂³

#14556589

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

132

C₂×C₄

C₄⋊C₄×D₇₃

yes

characteristic

133

C₂×C₄

C₂×#11924007

yes

characteristic

134

C₂×C₄

#14556587

yes

characteristic

135

C₂×C₄

#14556590

yes

characteristic

136

C₂×C₄

#14556590

yes

characteristic

137

C₂×C₄

#14556611

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

584

—

141

C₈

—

146

—

142

C₈

—

146

—

143

C₂²

—

144

C₂²

—

145

C₂²

#17988456

yes

characteristic

146

C₂²

#17988986

yes

—

147

C₄

—

148

C₄

—

149

C₄

#17988623

yes

—

150

C₄

#17988634

yes

—

151

C₄

#17988634

yes

—

152

C₄

#17988969

yes

—

153

C₂

—

154

C₂

#22610093

yes

characteristic

155

C₂

#22610104

yes

characteristic

156

C₂

#22610104

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556587
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1837	C₂
socle	C₂²×C₇₃	#384
prime radical	C₂³×C₇₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1837	D₇₃
hypocenter	C₇₃	#8827
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂×C₄

#14556587

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1837

C₂

socle

C₂²×C₇₃

#384

prime radical

C₂³×C₇₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

N	H ≅ G/N	type
84. C₇₃	#8827	semidirect

H ≅ G/N

type

84. C₇₃

#8827

semidirect

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

subgroup	index
#22610069	2
#22608689	2
#22608686	2
C₂×#17988308	2
C₇₃×#1837	2
#8827	73

subgroup

index

C₇₃×#1837

Subgroups — #32979799

Conjugacy classes of subgroups

Complements

Maximal subgroups