Group #32923113 — subgroups

Subgroups — C₇₃×#13709 #32923113

Order 18688.

154 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 70 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	C₇₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#2178	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#1426	C₂	1	2	yes	maximal, complemented
4	C₇₃×#1422	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1418	C₂	1	2	yes	maximal
6	#508×C₇₃	C₂²	1	1	yes	characteristic
7	#499×C₇₃	—	2	1	no	—
8	#443×C₇₃	—	2	1	no	—
9	M₆₄×C₇₃	C₂²	1	2	yes	—
10	#368×C₇₃	C₂²	1	1	yes	characteristic
11	#367×C₇₃	C₂²	1	2	yes	—
12	#366×C₇₃	—	2	1	no	—
13	#359×C₇₃	—	2	1	no	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#135×C₇₃	—	2	1	no	—
16	#131×C₇₃	—	4	1	no	—
17	#125×C₇₃	—	4	1	no	—
18	Q₃₂×C₇₃	Q₈	1	1	yes	characteristic
19	SD₃₂×C₇₃	—	2	1	no	—
20	D₁₆×C₇₃	Q₈	1	1	yes	characteristic
21	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
23	#108×C₇₃	—	2	1	no	—
24	#107×C₇₃	—	2	2	no	—
25	#106×C₇₃	—	4	1	no	—
26	#104×C₇₃	—	4	1	no	—
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₃₂×C₇₃	—	2	2	no	—
29	#41×C₇₃	—	4	1	no	—
30	Q₁₆×C₇₃	—	2	1	no	—
31	SD₁₆×C₇₃	—	4	1	no	—
32	D₈×C₇₃	—	2	1	no	—
33	M₁₆×C₇₃	—	4	1	no	—
34	M₁₆×C₇₃	—	8	1	no	—
35	C₂×C₈×C₇₃	—	8	1	no	—
36	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄⋊C₄×C₇₃	—	8	2	no	—
40	C₄²×C₇₃	D₈	1	1	yes	characteristic
41	C₁₆×C₇₃	—	2	1	no	—
42	C₁₆×C₇₃	C₂×Q₈	1	1	yes	characteristic
43	C₁₆×C₇₃	#41	1	1	yes	characteristic
44	Q₈×C₇₃	—	4	1	no	—
45	D₄×C₇₃	—	4	1	no	—
46	D₄×C₇₃	—	8	1	no	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	8	1	no	—
49	C₂×C₄×C₇₃	—	8	2	no	—
50	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
51	C₈×C₇₃	—	8	1	no	—
52	C₈×C₇₃	SD₃₂	1	1	yes	characteristic
53	C₈×C₇₃	SD₃₂	1	1	yes	characteristic
54	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₈×C₇₃	#137	1	1	yes	characteristic
56	C₂²×C₇₃	—	8	1	no	—
57	C₂²×C₇₃	#509	1	1	yes	characteristic
58	C₄×C₇₃	—	4	1	no	—
59	C₄×C₇₃	—	8	1	no	—
60	C₄×C₇₃	—	16	2	no	—
61	C₄×C₇₃	#474	1	1	yes	characteristic
62	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
63	#13709	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₇₃	—	2	1	no	—
65	C₂×C₇₃	—	16	1	no	—
66	C₂×C₇₃	#2226	1	1	yes	characteristic
67	#2253	C₂×C₇₃	1	1	yes	characteristic, complemented
68	#2178	C₂×C₇₃	1	1	yes	characteristic
69	#1426	C₂×C₇₃	1	2	yes	complemented
70	#1422	C₂×C₇₃	1	1	yes	characteristic, complemented
71	#1418	C₂×C₇₃	1	2	yes	—
72	C₇₃	#13709	1	1	yes	characteristic, direct product factor, Sylow
73	#508	C₂²×C₇₃	1	1	yes	characteristic
74	#499	—	2	1	no	—
75	#443	—	2	1	no	—
76	M₆₄	C₂²×C₇₃	1	2	yes	—
77	#368	C₂²×C₇₃	1	1	yes	characteristic
78	#367	C₂²×C₇₃	1	2	yes	—
79	#366	—	2	1	no	—
80	#359	—	2	1	no	—
81	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
82	#135	—	2	1	no	—
83	#131	—	4	1	no	—
84	#125	—	4	1	no	—
85	Q₃₂	Q₈×C₇₃	1	1	yes	characteristic
86	SD₃₂	—	2	1	no	—
87	D₁₆	Q₈×C₇₃	1	1	yes	characteristic
88	C₂×C₁₆	D₄×C₇₃	1	1	yes	characteristic
89	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
90	#108	—	2	1	no	—
91	#107	—	2	2	no	—
92	#106	—	4	1	no	—
93	#104	—	4	1	no	—
94	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
95	C₃₂	—	2	2	no	—
96	#41	—	4	1	no	—
97	Q₁₆	—	2	1	no	—
98	SD₁₆	—	4	1	no	—
99	D₈	—	2	1	no	—
100	M₁₆	—	4	1	no	—
101	M₁₆	—	8	1	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	D₈×C₇₃	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄²	D₈×C₇₃	1	1	yes	characteristic
108	C₁₆	—	2	1	no	—
109	C₁₆	C₂×Q₈×C₇₃	1	1	yes	characteristic
110	C₁₆	#41×C₇₃	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	8	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
118	C₈	—	8	1	no	—
119	C₈	SD₃₂×C₇₃	1	1	yes	characteristic
120	C₈	SD₃₂×C₇₃	1	1	yes	characteristic
121	C₈	C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
122	C₈	#137×C₇₃	1	1	yes	characteristic
123	C₂²	—	8	1	no	—
124	C₂²	#509×C₇₃	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	16	2	no	—
128	C₄	#474×C₇₃	1	1	yes	characteristic
129	C₄	C₂×SD₃₂×C₇₃	1	1	yes	characteristic
130	C₂	—	2	1	no	—
131	C₂	—	16	1	no	—
132	C₂	C₇₃×#2226	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃×#2253

C₂

yes

maximal, characteristic, complemented

C₇₃×#2178

C₂

yes

maximal, characteristic

C₇₃×#1426

C₂

yes

maximal, complemented

C₇₃×#1422

C₂

yes

maximal, characteristic, complemented

C₇₃×#1418

C₂

yes

maximal

#508×C₇₃

C₂²

yes

characteristic

#499×C₇₃

—

#443×C₇₃

—

M₆₄×C₇₃

C₂²

yes

—

#368×C₇₃

C₂²

yes

characteristic

#367×C₇₃

C₂²

yes

—

#366×C₇₃

—

#359×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#135×C₇₃

—

#131×C₇₃

—

#125×C₇₃

—

Q₃₂×C₇₃

Q₈

yes

characteristic

SD₃₂×C₇₃

—

D₁₆×C₇₃

Q₈

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#108×C₇₃

—

#107×C₇₃

—

#106×C₇₃

—

#104×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

#41×C₇₃

—

Q₁₆×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×Q₈

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

characteristic

C₈×C₇₃

SD₃₂

yes

characteristic

C₈×C₇₃

C₂²⋊Q₈

yes

characteristic

C₈×C₇₃

#137

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

#509

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#474

yes

characteristic

C₄×C₇₃

C₂×SD₃₂

yes

characteristic

#13709

C₇₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#2226

yes

characteristic

#2253

C₂×C₇₃

yes

characteristic, complemented

#2178

C₂×C₇₃

yes

characteristic

#1426

C₂×C₇₃

yes

complemented

#1422

C₂×C₇₃

yes

characteristic, complemented

#1418

C₂×C₇₃

yes

—

C₇₃

#13709

yes

characteristic, direct product factor, Sylow

#508

C₂²×C₇₃

yes

characteristic

#499

—

#443

—

M₆₄

C₂²×C₇₃

yes

—

#368

C₂²×C₇₃

yes

characteristic

#367

C₂²×C₇₃

yes

—

#366

—

#359

—

C₄×C₁₆

C₂²×C₇₃

yes

—

—

—

Q₃₂

Q₈×C₇₃

yes

characteristic

SD₃₂

—

D₁₆

Q₈×C₇₃

yes

characteristic

C₂×C₁₆

D₄×C₇₃

yes

characteristic

C₂×C₁₆

C₂³×C₇₃

yes

—

—

—

—

C₄×C₈

D₄×C₇₃

yes

characteristic

C₃₂

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

D₈×C₇₃

yes

characteristic

104

C₂×C₈

C₂×D₄×C₇₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

D₈×C₇₃

yes

characteristic

108

C₁₆

—

109

C₁₆

C₂×Q₈×C₇₃

yes

characteristic

110

C₁₆

#41×C₇₃

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₈×C₇₃

yes

characteristic

118

C₈

—

119

C₈

SD₃₂×C₇₃

yes

characteristic

120

C₈

SD₃₂×C₇₃

yes

characteristic

121

C₈

C₂²⋊Q₈×C₇₃

yes

characteristic

122

C₈

#137×C₇₃

yes

characteristic

123

C₂²

—

124

C₂²

#509×C₇₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#474×C₇₃

yes

characteristic

129

C₄

C₂×SD₃₂×C₇₃

yes

characteristic

130

C₂

—

131

C₂

—

132

C₂

C₇₃×#2226

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₇₃	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₇₃
prime radical	#508×C₇₃	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

subgroup

G/H

center, socle

C₂×C₇₃

#2226

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₇₃

prime radical

#508×C₇₃

C₂²

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₇₃ ◁ C₂×C₄×C₇₃ ◁ C₄×C₈×C₇₃ ◁ C₄×C₁₆×C₇₃ ◁ G

N	H ≅ G/N	type
1. C₇₃×#2253	C₂	semidirect
3. C₇₃×#1426	C₂	semidirect
4. C₇₃×#1422	C₂	semidirect
63. #13709	C₇₃	direct (N × H)
67. #2253	C₂×C₇₃	semidirect
69. #1426	C₂×C₇₃	semidirect
70. #1422	C₂×C₇₃	semidirect
72. C₇₃	#13709	direct (N × H)

H ≅ G/N

type

1. C₇₃×#2253

C₂

semidirect

3. C₇₃×#1426

C₂

semidirect

4. C₇₃×#1422

C₂

semidirect

63. #13709

C₇₃

direct (N × H)

67. #2253

C₂×C₇₃

semidirect

69. #1426

C₂×C₇₃

semidirect

70. #1422

C₂×C₇₃

semidirect

72. C₇₃

#13709

direct (N × H)

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

subgroup	index
C₇₃×#2253	2
C₇₃×#2178	2
C₇₃×#1426	2
C₇₃×#1422	2
C₇₃×#1418	2
#13709	73

subgroup

index

C₇₃×#2253

C₇₃×#2178

C₇₃×#1426

C₇₃×#1422

C₇₃×#1418

#13709

Subgroups — C₇₃×#13709 #32923113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C73×#13709 #32923113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#13709 #32923113