Group #33006530 — subgroups

Subgroups — #33006530

Order 18688.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×C₃₂×D₇₃	C₂	1	2	yes	maximal, complemented
2	C₁₆×D₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#22608068	C₂	1	2	yes	maximal, complemented
4	C₄×#14556497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
7	C₈×D₂₉₂	C₄	1	1	yes	characteristic
8	C₂×#14556497	C₂²	1	1	yes	characteristic
9	C₂×#14556497	C₂²	1	1	yes	characteristic
10	C₃₂×D₇₃	—	2	4	no	—
11	C₄⋊#11924003	C₄	1	1	yes	characteristic
12	#17988366	C₄	1	2	yes	—
13	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
16	C₄×D₂₉₂	C₈	1	1	yes	characteristic
17	C₂×#11924003	C₂×C₄	1	2	yes	—
18	C₁₆×D₇₃	—	2	2	no	—
19	C₁₆×D₇₃	—	2	2	no	—
20	C₄⋊#10596831	C₈	1	1	yes	characteristic
21	#14556503	C₈	1	2	yes	—
22	#14556497	—	2	1	no	—
23	#14556497	D₄	1	2	yes	complemented
24	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₃₂×C₇₃	—	2	2	no	—
28	C₂×D₂₉₂	C₁₆	1	1	yes	characteristic
29	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
30	C₂×#10596831	C₂×C₈	1	2	yes	—
31	C₈×D₇₃	—	2	2	no	—
32	C₈×D₇₃	—	2	2	no	—
33	C₄⋊Dic₇₃	C₁₆	1	1	yes	characteristic
34	#11924005	C₁₆	1	2	yes	—
35	#11924003	—	2	2	no	—
36	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₁₆×C₇₃	—	2	1	no	—
40	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₃	#41	1	1	yes	characteristic
42	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
47	D₂₉₂	C₃₂	1	4	yes	complemented
48	C₄×D₇₃	—	2	2	no	—
49	C₄×D₇₃	—	2	2	no	—
50	C₈×C₇₃	—	2	1	no	—
51	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
52	C₈×C₇₃	#131	1	1	yes	characteristic
53	#10596831	—	2	2	no	—
54	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
55	C₂×D₇₃	—	2	2	no	—
56	C₂×D₇₃	—	2	2	no	—
57	C₂×D₇₃	C₂×C₃₂	1	4	yes	complemented
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	C₂×C₃₂	1	2	yes	complemented
60	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₇₃	#504	1	1	yes	characteristic
62	Dic₇₃	—	2	2	no	—
63	D₄×C₃₂	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
65	C₂×C₇₃	C₂²×C₃₂	1	1	yes	characteristic
66	C₂×C₇₃	#2258	1	1	yes	characteristic
67	D₇₃	—	2	4	no	—
68	C₂²×C₃₂	—	73	2	no	—
69	D₄×C₁₆	—	73	1	no	—
70	C₄⋊C₃₂	D₇₃	1	1	yes	characteristic, complemented
71	C₂²⋊C₃₂	—	73	2	no	—
72	C₄×C₃₂	—	73	1	no	—
73	C₇₃	D₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	—	73	2	no	—
75	C₈×D₄	—	73	1	no	—
76	C₂×C₃₂	—	73	1	no	—
77	C₂×C₃₂	—	73	1	no	—
78	C₂×C₃₂	—	146	4	no	—
79	C₂×C₃₂	C₂×D₇₃	1	2	yes	—
80	C₄⋊C₁₆	—	73	1	no	—
81	C₂²⋊C₁₆	—	73	2	no	—
82	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
83	C₂²×C₈	—	73	2	no	—
84	C₄×D₄	—	73	1	no	—
85	C₂×C₁₆	—	73	2	no	—
86	C₂×C₁₆	—	146	2	no	—
87	C₂×C₁₆	—	146	2	no	—
88	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
89	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
90	C₄⋊C₈	—	73	1	no	—
91	C₂²⋊C₈	—	73	2	no	—
92	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
93	C₃₂	—	2	2	no	—
94	C₃₂	—	73	2	no	—
95	C₃₂	—	146	1	no	—
96	C₂×D₄	—	73	1	no	—
97	C₂²×C₄	—	73	2	no	—
98	C₂×C₈	—	73	2	no	—
99	C₂×C₈	—	146	2	no	—
100	C₂×C₈	—	146	2	no	—
101	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
102	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
103	C₄⋊C₄	—	73	1	no	—
104	C₂²⋊C₄	—	73	2	no	—
105	C₄²	C₈×D₇₃	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	—	146	2	no	—
108	C₁₆	D₄×D₇₃	1	1	yes	characteristic
109	C₁₆	#11924029	1	1	yes	characteristic
110	C₂³	—	73	2	no	—
111	D₄	—	73	4	no	—
112	C₂×C₄	—	73	2	no	—
113	C₂×C₄	—	146	2	no	—
114	C₂×C₄	—	146	2	no	—
115	C₂×C₄	C₁₆×D₇₃	1	1	yes	characteristic
116	C₂×C₄	C₁₆×D₇₃	1	1	yes	characteristic
117	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
118	C₈	—	2	1	no	—
119	C₈	—	146	2	no	—
120	C₈	#14556553	1	1	yes	characteristic
121	C₈	#14556599	1	1	yes	characteristic
122	C₂²	—	73	4	no	—
123	C₂²	—	146	2	no	—
124	C₂²	—	146	2	no	—
125	C₂²	C₂×C₁₆×D₇₃	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	146	2	no	—
128	C₄	C₃₂×D₇₃	1	2	yes	complemented
129	C₄	#17988652	1	1	yes	characteristic
130	C₄	#17989005	1	1	yes	characteristic
131	C₂	—	146	4	no	—
132	C₂	#22611750	1	1	yes	characteristic
133	C₂	C₂×C₃₂×D₇₃	1	1	yes	characteristic
134	C₂	#22612226	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₂×D₇₃

C₂

yes

maximal, complemented

C₁₆×D₂₉₂

C₂

yes

maximal, characteristic

#22608068

C₂

yes

maximal, complemented

C₄×#14556497

C₂

yes

maximal, characteristic, complemented

C₇₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₃

C₂²

yes

—

C₈×D₂₉₂

C₄

yes

characteristic

C₂×#14556497

C₂²

yes

characteristic

C₂×#14556497

C₂²

yes

characteristic

C₃₂×D₇₃

—

C₄⋊#11924003

C₄

yes

characteristic

#17988366

C₄

yes

—

C₂×C₃₂×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄×D₂₉₂

C₈

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

C₁₆×D₇₃

—

C₁₆×D₇₃

—

C₄⋊#10596831

C₈

yes

characteristic

#14556503

C₈

yes

—

#14556497

—

#14556497

D₄

yes

complemented

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

—

C₂×D₂₉₂

C₁₆

yes

characteristic

C₂×C₄×D₇₃

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

C₁₆

yes

characteristic

#11924005

C₁₆

yes

—

#11924003

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

C₂²×D₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

D₂₉₂

C₃₂

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄×D₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×C₃₂

yes

complemented

C₄×C₇₃

—

C₄×C₇₃

C₂×C₃₂

yes

complemented

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

D₄×C₃₂

—

maximal, Sylow

C₂×C₇₃

D₄×C₁₆

yes

characteristic

C₂×C₇₃

C₂²×C₃₂

yes

characteristic

C₂×C₇₃

#2258

yes

characteristic

D₇₃

—

C₂²×C₃₂

—

D₄×C₁₆

—

C₄⋊C₃₂

D₇₃

yes

characteristic, complemented

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₇₃

D₄×C₃₂

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₈×D₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

C₂×C₃₂

C₂×D₇₃

yes

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₄×D₇₃

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

146

—

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

146

—

100

C₂×C₈

—

146

—

101

C₂×C₈

C₈×D₇₃

yes

characteristic

102

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

C₈×D₇₃

yes

characteristic

106

C₁₆

—

107

C₁₆

—

146

—

108

C₁₆

D₄×D₇₃

yes

characteristic

109

C₁₆

#11924029

yes

characteristic

110

C₂³

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

146

—

114

C₂×C₄

—

146

—

115

C₂×C₄

C₁₆×D₇₃

yes

characteristic

116

C₂×C₄

C₁₆×D₇₃

yes

characteristic

117

C₂×C₄

C₂×C₈×D₇₃

yes

characteristic

118

C₈

—

119

C₈

—

146

—

120

C₈

#14556553

yes

characteristic

121

C₈

#14556599

yes

characteristic

122

C₂²

—

123

C₂²

—

146

—

124

C₂²

—

146

—

125

C₂²

C₂×C₁₆×D₇₃

yes

characteristic

126

C₄

—

127

C₄

—

146

—

128

C₄

C₃₂×D₇₃

yes

complemented

129

C₄

#17988652

yes

characteristic

130

C₄

#17989005

yes

characteristic

131

C₂

—

146

—

132

C₂

#22611750

yes

characteristic

133

C₂

C₂×C₃₂×D₇₃

yes

characteristic

134

C₂

#22612226

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₃
commutator	C₂×C₇₃	C₂²×C₃₂
Fitting	C₇₃×C₄⋊C₃₂	C₂
socle	C₂²×C₇₃	C₂²×C₁₆
prime radical	C₂×D₂₉₂	C₁₆
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	C₄⋊C₃₂	D₇₃
hypocenter	C₇₃	D₄×C₃₂
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₃

commutator

C₂×C₇₃

C₂²×C₃₂

Fitting

C₇₃×C₄⋊C₃₂

C₂

socle

C₂²×C₇₃

C₂²×C₁₆

prime radical

C₂×D₂₉₂

C₁₆

cosocle

C₂×C₁₆×C₇₃

C₂³

hypercenter

C₄⋊C₃₂

D₇₃

hypocenter

C₇₃

D₄×C₃₂

LCS

G ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
1. C₂×C₃₂×D₇₃	C₂	semidirect
3. #22608068	C₂	semidirect
4. C₄×#14556497	C₂	semidirect
5. C₇₃×C₄⋊C₃₂	C₂	semidirect
23. #14556497	D₄	semidirect
47. D₂₉₂	C₃₂	semidirect
57. C₂×D₇₃	C₂×C₃₂	semidirect
59. C₄×C₇₃	C₂×C₃₂	semidirect
70. C₄⋊C₃₂	D₇₃	semidirect
73. C₇₃	D₄×C₃₂	semidirect
128. C₄	C₃₂×D₇₃	semidirect

H ≅ G/N

type

1. C₂×C₃₂×D₇₃

C₂

semidirect

3. #22608068

C₂

semidirect

4. C₄×#14556497

C₂

semidirect

5. C₇₃×C₄⋊C₃₂

C₂

semidirect

23. #14556497

D₄

semidirect

47. D₂₉₂

C₃₂

semidirect

57. C₂×D₇₃

C₂×C₃₂

semidirect

59. C₄×C₇₃

C₂×C₃₂

semidirect

70. C₄⋊C₃₂

D₇₃

semidirect

73. C₇₃

D₄×C₃₂

semidirect

128. C₄

C₃₂×D₇₃

semidirect

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

subgroup	index
C₂×C₃₂×D₇₃	2
C₁₆×D₂₉₂	2
#22608068	2
C₄×#14556497	2
C₇₃×C₄⋊C₃₂	2
D₄×C₃₂	73

subgroup

index

C₂×C₃₂×D₇₃

C₁₆×D₂₉₂

#22608068

C₄×#14556497

C₇₃×C₄⋊C₃₂

D₄×C₃₂

Subgroups — #33006530

Conjugacy classes of subgroups

Complements

Maximal subgroups