Group #32973380 — subgroups

Subgroups — #32973380

Order 18688.

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

normal subgroups only

This group has 184 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	#22610907	C₂	1	1	yes	maximal, characteristic
2	#22607882	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1327	C₂	1	1	yes	maximal, characteristic
4	#17989144	C₄	1	1	yes	characteristic
5	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
6	#17988369	—	2	1	no	—
7	#17988348	—	2	1	no	—
8	#17988348	—	2	1	no	—
9	#423×C₇₃	C₂²	1	1	yes	characteristic
10	#349×C₇₃	—	2	1	no	—
11	#14556591	—	2	1	no	—
12	#14556583	C₈	1	2	yes	—
13	#14556570	—	4	1	no	—
14	C₄×Dic₁₄₆	—	4	1	no	—
15	#14556550	—	2	1	no	—
16	C₂×C₄×Dic₇₃	—	2	1	no	—
17	C₂×C₄×Dic₇₃	—	4	1	no	—
18	#14556531	—	4	1	no	—
19	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
22	M₃₂×C₇₃	—	4	1	no	—
23	C₄⋊C₈×C₇₃	—	4	1	no	—
24	C₂×Dic₁₄₆	—	4	1	no	—
25	C₂²×Dic₇₃	—	4	1	no	—
26	C₄⋊Dic₇₃	—	2	2	no	—
27	C₄⋊Dic₇₃	—	4	1	no	—
28	#11924007	—	4	2	no	—
29	C₂²⋊Dic₇₃	—	4	1	no	—
30	C₄×Dic₇₃	—	2	2	no	—
31	C₄×Dic₇₃	—	4	1	no	—
32	C₄×Dic₇₃	—	4	4	no	—
33	C₄×Dic₇₃	—	4	1	no	—
34	#11924003	—	8	1	no	—
35	C₂²×C₄×C₇₃	—	2	1	no	—
36	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	M₁₆×C₇₃	—	4	1	no	—
38	M₁₆×C₇₃	—	4	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	—	2	1	no	—
42	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
44	C₁₆×C₇₃	—	8	1	no	—
45	C₂³×C₇₃	#100	1	1	yes	characteristic
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	2	1	no	—
50	C₂×C₄×C₇₃	—	4	2	no	—
51	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
53	C₂×Dic₇₃	—	4	2	no	—
54	C₂×Dic₇₃	—	4	4	no	—
55	C₂×Dic₇₃	—	4	2	no	—
56	Dic₁₄₆	—	8	2	no	—
57	C₈×C₇₃	—	4	1	no	—
58	C₈×C₇₃	—	4	1	no	—
59	C₂²×C₇₃	—	2	1	no	—
60	C₂²×C₇₃	—	4	1	no	—
61	C₂²×C₇₃	#323	1	1	yes	characteristic
62	C₄×C₇₃	—	2	2	no	—
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	—	4	1	no	—
65	C₄×C₇₃	#354	1	1	yes	characteristic
66	C₄×C₇₃	#356	1	1	yes	characteristic
67	Dic₇₃	—	4	4	no	—
68	Dic₇₃	—	8	2	no	—
69	#7361	—	73	1	no	maximal, Sylow
70	C₂×C₇₃	—	2	1	no	—
71	C₂×C₇₃	—	4	1	no	—
72	C₂×C₇₃	#1319	1	1	yes	characteristic
73	#1994	—	73	1	no	—
74	#1327	—	73	1	no	—
75	#1327	D₇₃	1	1	yes	characteristic
76	C₇₃	#7361	1	1	yes	characteristic, complemented, Sylow
77	#533	—	73	1	no	—
78	#423	C₂×D₇₃	1	1	yes	characteristic
79	C₄³	—	73	1	no	—
80	#349	—	2	1	no	—
81	#349	—	146	1	no	—
82	#339	—	146	1	no	—
83	#339	—	146	1	no	—
84	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
85	C₂×M₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
86	C₄⋊Q₈	—	146	1	no	—
87	#125	—	73	2	no	—
88	C₂²⋊Q₈	—	292	1	no	—
89	C₄×Q₈	—	292	1	no	—
90	#117	—	146	1	no	—
91	C₂×C₄²	—	292	1	no	—
92	C₂×C₄²	—	146	1	no	—
93	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
94	M₃₂	—	4	1	no	—
95	M₃₂	—	292	1	no	—
96	C₄⋊C₈	—	4	1	no	—
97	C₂×Q₈	—	292	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	292	1	no	—
100	C₂²×C₄	#11924005	1	1	yes	characteristic
101	M₁₆	—	4	1	no	—
102	M₁₆	—	4	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₄⋊C₄	—	292	2	no	—
106	C₄⋊C₄	—	292	1	no	—
107	C₄⋊C₄	—	146	2	no	—
108	C₂²⋊C₄	—	292	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	292	1	no	—
111	C₄²	—	292	4	no	—
112	C₄²	—	292	1	no	—
113	C₄²	—	146	2	no	—
114	C₄²	C₈×D₇₃	1	1	yes	characteristic
115	C₄²	#11924011	1	1	yes	characteristic
116	C₁₆	—	8	1	no	—
117	C₁₆	—	584	1	no	—
118	C₂³	#14556507	1	1	yes	characteristic
119	Q₈	—	584	2	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	292	2	no	—
126	C₂×C₄	—	292	4	no	—
127	C₂×C₄	—	292	2	no	—
128	C₂×C₄	#14556503	1	1	yes	characteristic
129	C₂×C₄	#14556505	1	1	yes	characteristic
130	C₈	—	4	1	no	—
131	C₈	—	4	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	#17988310	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	1	no	—
137	C₄	—	4	1	no	—
138	C₄	—	292	4	no	—
139	C₄	—	584	2	no	—
140	C₄	#17988381	1	1	yes	characteristic
141	C₄	#17988385	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	4	1	no	—
144	C₂	#22607864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610907

C₂

yes

maximal, characteristic

#22607882

C₂

yes

maximal, characteristic

C₇₃×#1327

C₂

yes

maximal, characteristic

#17989144

C₄

yes

characteristic

C₄²×Dic₇₃

C₄

yes

—

—

—

#423×C₇₃

C₂²

yes

characteristic

#349×C₇₃

—

#14556591

—

#14556583

C₈

yes

—

#14556570

—

C₄×Dic₁₄₆

—

#14556550

—

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

—

#14556531

—

C₂×M₁₆×C₇₃

D₄

yes

characteristic

C₂×M₁₆×C₇₃

D₄

yes

characteristic

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₄⋊C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

#11924003

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#100

yes

characteristic

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

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#323

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7361

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

C₇₃

#7361

yes

characteristic, complemented, Sylow

#533

—

#423

C₂×D₇₃

yes

characteristic

C₄³

—

#349

—

#349

—

146

—

#339

—

146

—

#339

—

146

—

C₂×M₁₆

D₂₉₂

yes

characteristic

C₂×M₁₆

C₂²⋊D₇₃

yes

characteristic

C₄⋊Q₈

—

146

—

#125

—

C₂²⋊Q₈

—

292

—

C₄×Q₈

—

292

—

#117

—

146

—

C₂×C₄²

—

292

—

C₂×C₄²

—

146

—

C₂×C₄²

C₄×D₇₃

yes

characteristic

M₃₂

—

M₃₂

—

292

—

C₄⋊C₈

—

C₂×Q₈

—

292

—

C₂²×C₄

—

C₂²×C₄

—

292

—

100

C₂²×C₄

#11924005

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

—

292

—

106

C₄⋊C₄

—

292

—

107

C₄⋊C₄

—

146

—

108

C₂²⋊C₄

—

292

—

109

C₄²

—

110

C₄²

—

292

—

111

C₄²

—

292

—

112

C₄²

—

292

—

113

C₄²

—

146

—

114

C₄²

C₈×D₇₃

yes

characteristic

115

C₄²

#11924011

yes

characteristic

116

C₁₆

—

117

C₁₆

—

584

—

118

C₂³

#14556507

yes

characteristic

119

Q₈

—

584

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

292

—

128

C₂×C₄

#14556503

yes

characteristic

129

C₂×C₄

#14556505

yes

characteristic

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

#17988310

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

292

—

139

C₄

—

584

—

140

C₄

#17988381

yes

characteristic

141

C₄

#17988385

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

#22607864

yes

characteristic

	subgroup	G/H
center	C₄	#17988381
commutator	C₄²×C₇₃	C₂×C₈
Frattini	#423	C₂×D₇₃
Fitting	C₇₃×#1327	C₂
socle	C₂×C₇₃	#1319
prime radical	C₂³×C₇₃	#100
cosocle	#423×C₇₃	C₂²
hypercenter	#1327	D₇₃
hypocenter	C₇₃	#7361
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1327

subgroup

G/H

center

C₄

#17988381

commutator

C₄²×C₇₃

C₂×C₈

Frattini

#423

C₂×D₇₃

Fitting

C₇₃×#1327

C₂

C₂×C₇₃

C₂³×C₇₃

#423×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ #1327

N	H ≅ G/N	type
76. C₇₃	#7361	semidirect

H ≅ G/N

type

76. C₇₃

#7361

semidirect

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

subgroup	index
#22610907	2
#22607882	2
C₇₃×#1327	2
#7361	73

subgroup

index

#22610907

#22607882

C₇₃×#1327

#7361

Subgroups — #32973380

Conjugacy classes of subgroups

Complements

Maximal subgroups