Group #32973203 — subgroups

Subgroups — #32973203

Order 18688.

250 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22608255	C₂	1	2	yes	maximal
2	C₇₃×#1985	C₂	1	1	yes	maximal, characteristic
3	#17988640	—	2	2	no	—
4	#17988313	—	2	2	no	—
5	#17988313	C₄	1	4	yes	—
6	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
8	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
9	#336×C₇₃	—	2	2	no	—
10	C₄⋊#10596831	—	4	2	no	—
11	C₂²⋊#10596831	—	2	4	no	—
12	C₄×#10596831	—	4	2	no	—
13	C₂²×Q₈×C₇₃	Q₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	—	2	1	no	—
15	C₂²×C₈×C₇₃	—	2	1	no	—
16	C₄×Q₈×C₇₃	—	4	1	no	—
17	C₄×Q₈×C₇₃	—	4	1	no	—
18	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₃	D₄	1	2	yes	—
22	C₄⋊C₈×C₇₃	—	4	1	no	—
23	#97×C₇₃	—	4	1	no	—
24	C₂×#10596831	—	4	4	no	—
25	C₂×Q₈×C₇₃	—	2	4	no	—
26	C₂×Q₈×C₇₃	—	4	1	no	—
27	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
29	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₃	—	2	2	no	—
33	C₂×C₈×C₇₃	—	4	1	no	—
34	C₂×C₈×C₇₃	—	4	1	no	—
35	C₂×C₈×C₇₃	—	4	2	no	—
36	C₂×C₈×C₇₃	—	4	1	no	—
37	C₄⋊C₄×C₇₃	—	4	2	no	—
38	C₄⋊C₄×C₇₃	—	4	1	no	—
39	C₄²×C₇₃	—	2	1	no	—
40	C₄²×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	—	2	2	no	—
42	C₄²×C₇₃	—	2	2	no	—
43	C₂³×C₇₃	#95	1	1	yes	characteristic
44	Q₈×C₇₃	—	4	4	no	—
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	2	no	—
52	C₂×C₄×C₇₃	—	2	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	4	2	no	—
55	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#108	1	4	yes	—
58	C₈×C₇₃	—	4	2	no	—
59	C₈×C₇₃	—	8	1	no	—
60	#10596831	—	8	4	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	#341	1	2	yes	—
64	C₂²×C₇₃	#342	1	1	yes	characteristic
65	C₄×C₇₃	—	2	4	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	4	1	no	—
68	C₄×C₇₃	—	4	2	no	—
69	C₄×C₇₃	—	4	1	no	—
70	C₄×C₇₃	#355	1	1	yes	characteristic
71	C₄×C₇₃	#356	1	1	yes	characteristic
72	#7286	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	—	2	2	no	—
74	C₂×C₇₃	#1298	1	1	yes	characteristic
75	C₂×C₇₃	#1391	1	1	yes	characteristic
76	C₂×C₇₃	#1394	1	1	yes	characteristic
77	#1985	D₇₃	1	1	yes	characteristic
78	#1461	—	73	2	no	—
79	C₇₃	#7286	1	1	yes	characteristic, complemented, Sylow
80	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
81	#431	—	146	2	no	—
82	C₂×C₄⋊C₈	Dic₇₃	1	1	yes	characteristic
83	C₂×#97	Dic₇₃	1	1	yes	characteristic
84	#336	—	2	2	no	—
85	#324	—	73	4	no	—
86	#324	—	146	2	no	—
87	C₂²×Q₈	Dic₁₄₆	1	1	yes	characteristic
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	—	2	1	no	—
90	C₄×Q₈	—	4	1	no	—
91	C₄×Q₈	—	4	1	no	—
92	C₂×C₄⋊C₄	C₄×D₇₃	1	2	yes	—
93	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
96	C₄⋊C₈	—	4	1	no	—
97	C₄⋊C₈	—	292	2	no	—
98	C₂²⋊C₈	—	146	4	no	—
99	#97	—	4	1	no	—
100	C₄×C₈	—	292	2	no	—
101	C₂×Q₈	—	2	4	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
104	C₂²×C₄	#11924005	1	2	yes	—
105	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
106	C₂²×C₄	#11924007	1	2	yes	—
107	C₂²×C₄	C₄⋊Dic₇₃	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	292	4	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	1	no	—
120	C₂³	#14556498	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	#14556506	1	1	yes	characteristic
133	C₂×C₄	#14556506	1	1	yes	characteristic
134	C₂×C₄	#14556526	1	4	yes	—
135	C₈	—	4	2	no	—
136	C₈	—	8	1	no	—
137	C₈	—	584	4	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#17988351	1	2	yes	—
141	C₂²	#17988354	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	4	no	—
144	C₄	—	4	2	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	#17988384	1	1	yes	characteristic
148	C₄	#17988386	1	1	yes	characteristic
149	C₂	—	2	2	no	—
150	C₂	#22607818	1	1	yes	characteristic
151	C₂	#22608052	1	1	yes	characteristic
152	C₂	#22608058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22608255

C₂

yes

maximal

C₇₃×#1985

C₂

yes

maximal, characteristic

#17988640

—

#17988313

—

#17988313

C₄

yes

—

C₂×C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₇₃

C₄

yes

characteristic

C₂×#97×C₇₃

C₄

yes

characteristic

#336×C₇₃

—

C₄⋊#10596831

—

C₂²⋊#10596831

—

C₄×#10596831

—

C₂²×Q₈×C₇₃

Q₈

yes

characteristic

C₂²×C₈×C₇₃

—

C₂²×C₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₄⋊C₈×C₇₃

—

#97×C₇₃

—

C₂×#10596831

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

C₄²

yes

characteristic

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₄⋊C₄

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

#95

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#341

yes

—

C₂²×C₇₃

#342

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

#1461

—

C₇₃

#7286

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

C₂×D₇₃

yes

characteristic

#431

—

146

—

C₂×C₄⋊C₈

Dic₇₃

yes

characteristic

C₂×#97

Dic₇₃

yes

—

—

—

146

—

C₂²×Q₈

Dic₁₄₆

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₄×D₇₃

yes

—

C₂×C₄⋊C₄

D₂₉₂

yes

characteristic

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

C₂×C₄²

C₂²⋊D₇₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

292

—

C₂²⋊C₈

—

146

—

#97

—

100

C₄×C₈

—

292

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂²×C₄

C₄×Dic₇₃

yes

characteristic

104

C₂²×C₄

#11924005

yes

—

105

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

106

C₂²×C₄

#11924007

yes

—

107

C₂²×C₄

C₄⋊Dic₇₃

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

292

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₂³

#14556498

yes

characteristic

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#14556506

yes

characteristic

133

C₂×C₄

#14556506

yes

characteristic

134

C₂×C₄

#14556526

yes

—

135

C₈

—

136

C₈

—

137

C₈

—

584

—

138

C₂²

—

139

C₂²

—

140

C₂²

#17988351

yes

—

141

C₂²

#17988354

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#17988384

yes

characteristic

148

C₄

#17988386

yes

characteristic

149

C₂

—

150

C₂

#22607818

yes

characteristic

151

C₂

#22608052

yes

characteristic

152

C₂

#22608058

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556506
commutator	C₂²×C₄×C₇₃	C₄²
Frattini	C₂×C₄×Q₈	C₂×D₇₃
Fitting	C₇₃×#1985	C₂
socle	C₂²×C₇₃	#342
prime radical	C₂³×C₇₃	#95
cosocle	C₂×C₄×Q₈×C₇₃	C₂²
hypercenter	#1985	D₇₃
hypocenter	C₇₃	#7286
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

subgroup

G/H

center

C₂×C₄

#14556506

commutator

C₂²×C₄×C₇₃

C₄²

Frattini

C₂×C₄×Q₈

C₂×D₇₃

Fitting

C₇₃×#1985

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄×Q₈×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ #1985

N	H ≅ G/N	type
79. C₇₃	#7286	semidirect

H ≅ G/N

type

79. C₇₃

#7286

semidirect

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

subgroup	index
#22608255	2
C₇₃×#1985	2
#7286	73

subgroup

index

#22608255

C₇₃×#1985

#7286

Subgroups — #32973203

Conjugacy classes of subgroups

Complements

Maximal subgroups