Group #22607402 — subgroups

Subgroups — C₇₃×#3249 #22607402

Order 9344.

348 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 188 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	#533×C₇₃	C₂	1	2	yes	maximal
2	#477×C₇₃	C₂	1	4	yes	maximal, complemented
3	#467×C₇₃	C₂	1	4	yes	maximal
4	#462×C₇₃	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₇₃	C₂²	1	4	yes	—
7	C₂²×C₈×C₇₃	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
10	#125×C₇₃	—	2	2	no	—
11	C₂²⋊Q₈×C₇₃	—	2	2	no	—
12	C₂²⋊Q₈×C₇₃	C₂²	1	4	yes	—
13	C₄×Q₈×C₇₃	—	2	2	no	—
14	C₄×Q₈×C₇₃	C₂²	1	4	yes	—
15	#117×C₇₃	—	2	2	no	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	#107×C₇₃	C₂²	1	4	yes	—
18	C₄⋊C₈×C₇₃	C₂²	1	4	yes	—
19	#103×C₇₃	C₂²	1	8	yes	—
20	C₂×Q₈×C₇₃	—	2	2	no	—
21	C₂×Q₈×C₇₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
25	Q₁₆×C₇₃	—	4	4	no	—
26	C₂×C₈×C₇₃	—	4	2	no	—
27	C₂×C₈×C₇₃	C₂³	1	4	yes	—
28	C₄⋊C₄×C₇₃	—	2	2	no	—
29	C₄⋊C₄×C₇₃	—	2	2	no	—
30	C₄⋊C₄×C₇₃	—	2	2	no	—
31	C₄⋊C₄×C₇₃	—	2	4	no	—
32	C₄⋊C₄×C₇₃	—	2	4	no	—
33	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
34	C₂²⋊C₄×C₇₃	—	2	4	no	—
35	C₄²×C₇₃	—	2	4	no	—
36	C₄²×C₇₃	D₄	1	2	yes	—
37	C₄²×C₇₃	C₂³	1	2	yes	—
38	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₇₃	—	2	4	no	—
40	Q₈×C₇₃	—	2	4	no	—
41	Q₈×C₇₃	—	4	4	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	2	4	no	—
45	C₂×C₄×C₇₃	—	2	4	no	—
46	C₂×C₄×C₇₃	—	4	1	no	—
47	C₂×C₄×C₇₃	Q₁₆	1	4	yes	complemented
48	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
52	C₈×C₇₃	—	2	4	no	—
53	C₂²×C₇₃	—	2	1	no	—
54	C₂²×C₇₃	—	2	1	no	—
55	C₂²×C₇₃	C₂×Q₁₆	1	2	yes	—
56	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	—	2	2	no	—
59	C₄×C₇₃	—	4	4	no	—
60	C₄×C₇₃	—	4	4	no	—
61	C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
62	C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
63	C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
64	C₂×C₇₃	—	2	2	no	—
65	C₂×C₇₃	#537	1	1	yes	characteristic
66	C₂×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
67	C₂×C₇₃	#575	1	1	yes	characteristic
68	#3249	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₇₃	#3249	1	1	yes	characteristic, direct product factor, Sylow
70	#533	C₂×C₇₃	1	2	yes	—
71	#477	C₂×C₇₃	1	4	yes	complemented
72	#467	C₂×C₇₃	1	4	yes	—
73	#462	C₂×C₇₃	1	4	yes	complemented
74	C₂×C₄⋊C₈	C₂×C₇₃	1	1	yes	characteristic
75	C₂×Q₁₆	C₂²×C₇₃	1	4	yes	—
76	C₂²×C₈	C₂²×C₇₃	1	2	yes	—
77	C₄⋊Q₈	C₂²×C₇₃	1	2	yes	—
78	C₄⋊Q₈	C₂²×C₇₃	1	2	yes	—
79	#125	—	2	2	no	—
80	C₂²⋊Q₈	—	2	2	no	—
81	C₂²⋊Q₈	C₂²×C₇₃	1	4	yes	—
82	C₄×Q₈	—	2	2	no	—
83	C₄×Q₈	C₂²×C₇₃	1	4	yes	—
84	#117	—	2	2	no	—
85	C₂×C₄²	C₂²×C₇₃	1	1	yes	characteristic
86	#107	C₂²×C₇₃	1	4	yes	—
87	C₄⋊C₈	C₂²×C₇₃	1	4	yes	—
88	#103	C₂²×C₇₃	1	8	yes	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	C₂³×C₇₃	1	4	yes	—
91	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
92	C₂²×C₄	D₄×C₇₃	1	1	yes	characteristic
93	C₂²×C₄	C₂³×C₇₃	1	1	yes	characteristic
94	Q₁₆	—	4	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂³×C₇₃	1	4	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
103	C₂²⋊C₄	—	2	4	no	—
104	C₄²	—	2	4	no	—
105	C₄²	D₄×C₇₃	1	2	yes	—
106	C₄²	C₂³×C₇₃	1	2	yes	—
107	C₂³	C₂×D₄×C₇₃	1	1	yes	characteristic
108	Q₈	—	2	4	no	—
109	Q₈	—	2	4	no	—
110	Q₈	—	4	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	Q₁₆×C₇₃	1	4	yes	complemented
117	C₂×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
118	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
119	C₂×C₄	C₂×D₄×C₇₃	1	2	yes	—
120	C₂×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Q₁₆×C₇₃	1	2	yes	—
125	C₂²	C₂²×D₄×C₇₃	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	4	4	no	—
130	C₄	C₂×Q₁₆×C₇₃	1	4	yes	—
131	C₄	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
132	C₄	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#537×C₇₃	1	1	yes	characteristic
135	C₂	C₂²×Q₁₆×C₇₃	1	1	yes	characteristic
136	C₂	#575×C₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#533×C₇₃

C₂

yes

maximal

#477×C₇₃

C₂

yes

maximal, complemented

#467×C₇₃

C₂

yes

maximal

#462×C₇₃

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₇₃

C₂²

yes

—

C₂²×C₈×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

C₂²

yes

—

#125×C₇₃

—

C₂²⋊Q₈×C₇₃

—

C₂²⋊Q₈×C₇₃

C₂²

yes

—

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

C₂²

yes

—

#117×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

—

C₄⋊C₈×C₇₃

C₂²

yes

—

#103×C₇₃

C₂²

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

Q₁₆

yes

complemented

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×Q₁₆

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×Q₁₆

yes

—

C₄×C₇₃

H(5,𝔽₂)

yes

characteristic

C₄×C₇₃

2₋¹⁺⁴

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

#537

yes

characteristic

C₂×C₇₃

C₂²×Q₁₆

yes

C₂×C₇₃

yes

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₇₃

#3249

yes

characteristic, direct product factor, Sylow

#533

C₂×C₇₃

yes

—

#477

C₂×C₇₃

yes

complemented

#467

C₂×C₇₃

yes

—

#462

C₂×C₇₃

yes

complemented

C₂×C₄⋊C₈

C₂×C₇₃

yes

characteristic

C₂×Q₁₆

C₂²×C₇₃

yes

—

C₂²×C₈

C₂²×C₇₃

yes

—

C₄⋊Q₈

C₂²×C₇₃

yes

—

C₄⋊Q₈

C₂²×C₇₃

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²×C₇₃

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₇₃

yes

—

#117

—

C₂×C₄²

C₂²×C₇₃

yes

characteristic

#107

C₂²×C₇₃

yes

—

C₄⋊C₈

C₂²×C₇₃

yes

—

#103

C₂²×C₇₃

yes

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₇₃

yes

—

C₂²×C₄

D₄×C₇₃

yes

characteristic

C₂²×C₄

D₄×C₇₃

yes

characteristic

C₂²×C₄

C₂³×C₇₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₂³×C₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂³×C₇₃

yes

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

D₄×C₇₃

yes

—

106

C₄²

C₂³×C₇₃

yes

—

107

C₂³

C₂×D₄×C₇₃

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Q₁₆×C₇₃

yes

complemented

117

C₂×C₄

C₂×D₄×C₇₃

yes

characteristic

118

C₂×C₄

C₂×D₄×C₇₃

yes

—

119

C₂×C₄

C₂×D₄×C₇₃

yes

—

120

C₂×C₄

C₂⁴×C₇₃

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Q₁₆×C₇₃

yes

—

125

C₂²

C₂²×D₄×C₇₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

C₂×Q₁₆×C₇₃

yes

—

131

C₄

H(5,𝔽₂)×C₇₃

yes

characteristic

132

C₄

2₋¹⁺⁴×C₇₃

yes

characteristic

133

C₂

—

134

C₂

#537×C₇₃

yes

characteristic

135

C₂

C₂²×Q₁₆×C₇₃

yes

characteristic

136

C₂

#575×C₇₃

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₇₃
prime radical	C₂³×C₇₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₃

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₇₃

prime radical

C₂³×C₇₃

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

N	H ≅ G/N	type
2. #477×C₇₃	C₂	semidirect
4. #462×C₇₃	C₂	semidirect
47. C₂×C₄×C₇₃	Q₁₆	semidirect
68. #3249	C₇₃	direct (N × H)
69. C₇₃	#3249	direct (N × H)
71. #477	C₂×C₇₃	semidirect
73. #462	C₂×C₇₃	semidirect
116. C₂×C₄	Q₁₆×C₇₃	semidirect

H ≅ G/N

type

2. #477×C₇₃

C₂

semidirect

4. #462×C₇₃

C₂

semidirect

47. C₂×C₄×C₇₃

Q₁₆

semidirect

68. #3249

C₇₃

direct (N × H)

69. C₇₃

#3249

direct (N × H)

71. #477

C₂×C₇₃

semidirect

73. #462

C₂×C₇₃

semidirect

116. C₂×C₄

Q₁₆×C₇₃

semidirect

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

subgroup	index
#533×C₇₃	2
#477×C₇₃	2
#467×C₇₃	2
#462×C₇₃	2
C₂×C₄⋊C₈×C₇₃	2
#3249	73

subgroup

index

#533×C₇₃

#477×C₇₃

#467×C₇₃

#462×C₇₃

C₂×C₄⋊C₈×C₇₃

#3249

Subgroups — C₇₃×#3249 #22607402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#3249 #22607402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#3249 #22607402