Group #32972808 — subgroups

Subgroups — #32972808

Order 18688.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22607764	C₂	1	2	yes	maximal
2	C₇₃×#1994	C₂	1	1	yes	maximal, characteristic
3	#17988640	—	2	2	no	—
4	#533×C₇₃	C₄	1	1	yes	characteristic
5	#423×C₇₃	C₄	1	1	yes	characteristic
6	C₄³×C₇₃	C₂²	1	1	yes	characteristic
7	#339×C₇₃	—	2	2	no	—
8	C₄⋊#10596831	—	2	4	no	—
9	C₂²⋊#10596831	—	4	2	no	—
10	C₄×#10596831	—	4	2	no	—
11	C₂×M₁₆×C₇₃	—	2	1	no	—
12	C₄⋊Q₈×C₇₃	—	2	1	no	—
13	#125×C₇₃	C₈	1	2	yes	complemented
14	C₂²⋊Q₈×C₇₃	—	4	1	no	—
15	C₄×Q₈×C₇₃	—	4	1	no	—
16	#117×C₇₃	—	2	1	no	—
17	C₂×C₄²×C₇₃	—	4	1	no	—
18	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₇₃	D₄	1	2	yes	—
20	C₄⋊C₈×C₇₃	—	4	1	no	—
21	C₂×#10596831	—	4	4	no	—
22	C₂×Q₈×C₇₃	—	4	1	no	—
23	C₂²×C₄×C₇₃	—	2	1	no	—
24	C₂²×C₄×C₇₃	—	2	2	no	—
25	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	M₁₆×C₇₃	—	4	2	no	—
27	C₂×C₈×C₇₃	—	4	1	no	—
28	C₄⋊C₄×C₇₃	—	2	2	no	—
29	C₄⋊C₄×C₇₃	—	4	2	no	—
30	C₄⋊C₄×C₇₃	—	4	1	no	—
31	C₂²⋊C₄×C₇₃	—	4	1	no	—
32	C₄²×C₇₃	—	2	1	no	—
33	C₄²×C₇₃	—	2	2	no	—
34	C₄²×C₇₃	—	4	4	no	—
35	C₄²×C₇₃	—	4	1	no	—
36	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
38	C₄²×C₇₃	SD₁₆	1	2	yes	—
39	C₄²×C₇₃	Q₁₆	1	2	yes	—
40	C₂³×C₇₃	#99	1	1	yes	characteristic
41	Q₈×C₇₃	—	8	2	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
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	—	4	4	no	—
47	C₂×C₄×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#103	1	2	yes	—
52	C₂×C₄×C₇₃	#104	1	2	yes	—
53	C₈×C₇₃	—	8	1	no	—
54	#10596831	—	8	4	no	—
55	C₂²×C₇₃	—	4	1	no	—
56	C₂²×C₇₃	#323	1	1	yes	characteristic
57	C₂²×C₇₃	#328	1	2	yes	—
58	C₄×C₇₃	—	2	4	no	—
59	C₄×C₇₃	—	2	2	no	—
60	C₄×C₇₃	—	4	1	no	—
61	C₄×C₇₃	—	4	2	no	—
62	C₄×C₇₃	—	8	2	no	—
63	C₄×C₇₃	Q₈⋊C₈	1	2	yes	—
64	C₄×C₇₃	#329	1	1	yes	characteristic
65	C₄×C₇₃	#333	1	1	yes	characteristic
66	#7134	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	—	2	1	no	—
68	C₂×C₇₃	—	2	2	no	—
69	C₂×C₇₃	#1272	1	1	yes	characteristic
70	#1994	D₇₃	1	1	yes	characteristic
71	#1276	—	73	2	no	—
72	C₇₃	#7134	1	1	yes	characteristic, complemented, Sylow
73	#533	Dic₇₃	1	1	yes	characteristic
74	#431	—	146	2	no	—
75	#423	Dic₇₃	1	1	yes	characteristic
76	C₄³	C₂×D₇₃	1	1	yes	characteristic
77	#339	—	2	2	no	—
78	C₂×M₁₆	—	2	1	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	#125	#10596831	1	2	yes	complemented
81	C₂²⋊Q₈	—	4	1	no	—
82	C₄×Q₈	—	4	1	no	—
83	#117	—	2	1	no	—
84	C₂×C₄²	—	4	1	no	—
85	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
86	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
87	C₄⋊C₈	—	4	1	no	—
88	C₄⋊C₈	—	146	4	no	—
89	C₂²⋊C₈	—	292	2	no	—
90	C₄×C₈	—	292	2	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
95	M₁₆	—	4	2	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	292	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₂²⋊C₄	—	4	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	4	4	no	—
105	C₄²	—	4	1	no	—
106	C₄²	C₂×#10596831	1	1	yes	characteristic
107	C₄²	#11924012	1	1	yes	characteristic
108	C₄²	#11924015	1	2	yes	—
109	C₄²	#11924018	1	2	yes	—
110	C₂³	#14556506	1	1	yes	characteristic
111	Q₈	—	8	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₂²⋊#10596831	1	1	yes	characteristic
120	C₂×C₄	#14556508	1	1	yes	characteristic
121	C₂×C₄	#14556515	1	2	yes	—
122	C₂×C₄	#14556518	1	2	yes	—
123	C₈	—	8	1	no	—
124	C₈	—	584	4	no	—
125	C₂²	—	4	1	no	—
126	C₂²	#17988311	1	1	yes	characteristic
127	C₂²	#17988324	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	8	2	no	—
133	C₄	Q₈⋊#10596831	1	2	yes	—
134	C₄	#17988327	1	1	yes	characteristic
135	C₄	#17988336	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	2	2	no	—
138	C₂	#22607757	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22607764

C₂

yes

maximal

C₇₃×#1994

C₂

yes

maximal, characteristic

#17988640

—

#533×C₇₃

C₄

yes

characteristic

#423×C₇₃

C₄

yes

characteristic

C₄³×C₇₃

C₂²

yes

characteristic

#339×C₇₃

—

C₄⋊#10596831

—

C₂²⋊#10596831

—

C₄×#10596831

—

C₂×M₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

—

#125×C₇₃

C₈

yes

complemented

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

D₄

yes

—

C₄⋊C₈×C₇₃

—

C₂×#10596831

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₄²×C₇₃

M₁₆

yes

characteristic

C₄²×C₇₃

SD₁₆

yes

—

C₄²×C₇₃

Q₁₆

yes

—

C₂³×C₇₃

#99

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

C₂²⋊C₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×C₄×C₇₃

#104

yes

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#323

yes

characteristic

C₂²×C₇₃

#328

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

Q₈⋊C₈

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1272

yes

characteristic

#1994

D₇₃

yes

characteristic

#1276

—

C₇₃

#7134

yes

characteristic, complemented, Sylow

#533

Dic₇₃

yes

characteristic

#431

—

146

—

#423

Dic₇₃

yes

characteristic

C₄³

C₂×D₇₃

yes

characteristic

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

#10596831

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₇₃

yes

characteristic

C₂×C₄²

C₂²⋊D₇₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

292

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₇₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

292

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×#10596831

yes

characteristic

107

C₄²

#11924012

yes

characteristic

108

C₄²

#11924015

yes

—

109

C₄²

#11924018

yes

—

110

C₂³

#14556506

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊#10596831

yes

characteristic

120

C₂×C₄

#14556508

yes

characteristic

121

C₂×C₄

#14556515

yes

—

122

C₂×C₄

#14556518

yes

—

123

C₈

—

124

C₈

—

584

—

125

C₂²

—

126

C₂²

#17988311

yes

characteristic

127

C₂²

#17988324

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

Q₈⋊#10596831

yes

—

134

C₄

#17988327

yes

characteristic

135

C₄

#17988336

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#22607757

yes

characteristic

	subgroup	G/H
center	C₄	#17988336
commutator	C₄²×C₇₃	C₂×C₈
Frattini	C₄³	C₂×D₇₃
Fitting	C₇₃×#1994	C₂
socle	C₂×C₇₃	#1272
prime radical	C₂³×C₇₃	#99
cosocle	C₄³×C₇₃	C₂²
hypercenter	#1994	D₇₃
hypocenter	C₇₃	#7134
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄³ ◁ #1994

subgroup

G/H

center

C₄

#17988336

commutator

C₄²×C₇₃

C₂×C₈

Frattini

C₄³

C₂×D₇₃

Fitting

C₇₃×#1994

C₂

C₂×C₇₃

C₂³×C₇₃

C₄³×C₇₃

C₂²

D₇₃

C₇₃

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄³ ◁ #1994

N	H ≅ G/N	type
13. #125×C₇₃	C₈	semidirect
72. C₇₃	#7134	semidirect
80. #125	#10596831	semidirect

H ≅ G/N

type

13. #125×C₇₃

C₈

semidirect

72. C₇₃

#7134

semidirect

80. #125

#10596831

semidirect

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

subgroup	index
#22607764	2
C₇₃×#1994	2
#7134	73

subgroup

index

#22607764

C₇₃×#1994

#7134

Subgroups — #32972808

Conjugacy classes of subgroups

Complements

Maximal subgroups