Group #33112529 — subgroups

Subgroups — #33112529

Order 18688.

984 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 255 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	C₄×#14556588	C₂	1	1	yes	maximal, characteristic
2	#22610222	C₂	1	12	yes	maximal, complemented
3	#22610074	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1839	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×D₇₃	C₂²	1	4	yes	—
6	C₂×#14556588	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₉₂	C₂²	1	6	yes	—
8	C₂×#14556503	C₂²	1	12	yes	—
9	#17988489	C₄	1	6	yes	—
10	#17988457	C₄	1	1	yes	characteristic
11	#17988342	C₂²	1	6	yes	—
12	C₄³×C₇₃	C₂²	1	1	yes	characteristic
13	#336×C₇₃	C₂²	1	6	yes	—
14	C₂²×D₂₉₂	C₂×C₄	1	6	yes	—
15	C₂²×C₄×D₇₃	C₂³	1	4	yes	—
16	C₂²×#10596831	C₂³	1	4	yes	—
17	C₂×C₈×D₇₃	—	4	12	no	—
18	#14556588	C₈	1	8	yes	complemented
19	C₄×D₂₉₂	—	4	6	no	—
20	C₄×D₂₉₂	—	4	6	no	—
21	C₂×C₄⋊Dic₇₃	C₂×C₄	1	6	yes	—
22	C₂×#11924005	C₂×C₄	1	12	yes	—
23	#14556503	—	4	12	no	—
24	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₃	C₂³	1	3	yes	—
28	C₂³×D₇₃	C₂²×C₄	1	4	yes	—
29	C₂×D₂₉₂	—	4	6	no	—
30	C₂×D₂₉₂	—	4	6	no	—
31	C₂×D₂₉₂	C₂×C₈	1	24	yes	complemented
32	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
33	C₂×C₄×D₇₃	—	2	12	no	—
34	C₂×C₄×D₇₃	—	2	12	no	—
35	C₂×#10596831	—	2	12	no	—
36	C₈×D₇₃	—	4	16	no	—
37	C₄⋊Dic₇₃	—	4	6	no	—
38	#11924005	—	4	12	no	—
39	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
41	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	12	no	—
43	C₄²×C₇₃	—	2	6	no	—
44	C₄²×C₇₃	—	4	3	no	—
45	C₄²×C₇₃	C₂×C₈	1	4	yes	complemented
46	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
47	C₂²×D₇₃	—	2	12	no	—
48	C₂²×D₇₃	—	2	12	no	—
49	C₂²×D₇₃	C₂²×C₈	1	8	yes	—
50	C₂×C₄×C₇₃	—	2	3	no	—
51	C₂×C₄×C₇₃	—	2	3	no	—
52	C₂×C₄×C₇₃	—	2	3	no	—
53	C₂×C₄×C₇₃	—	2	6	no	—
54	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
55	C₂×C₄×C₇₃	H(5,𝔽₂)	1	6	yes	—
56	C₂×Dic₇₃	—	2	12	no	—
57	D₂₉₂	—	4	24	no	—
58	C₄×D₇₃	—	4	4	no	—
59	C₄×D₇₃	—	4	12	no	—
60	C₈×C₇₃	—	4	4	no	—
61	#10596831	—	4	4	no	—
62	C₂²×C₇₃	#518	1	3	yes	—
63	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
64	C₂²×C₇₃	#568	1	3	yes	—
65	C₂×D₇₃	—	2	24	no	—
66	C₂×D₇₃	—	4	4	no	—
67	C₂×D₇₃	—	4	12	no	—
68	C₄×C₇₃	—	2	6	no	—
69	C₄×C₇₃	—	4	3	no	—
70	C₄×C₇₃	#560	1	4	yes	—
71	Dic₇₃	—	4	4	no	—
72	#23355	—	73	1	no	maximal, Sylow
73	C₂×C₇₃	#2380	1	1	yes	characteristic
74	C₂×C₇₃	#2972	1	3	yes	—
75	C₂×C₇₃	#2980	1	3	yes	—
76	D₇₃	—	4	8	no	—
77	C₄×#127	—	73	1	no	—
78	#1869	—	73	12	no	—
79	#1839	—	73	1	no	—
80	#1839	D₇₃	1	1	yes	characteristic, complemented
81	C₇₃	#23355	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₈	—	73	4	no	—
83	C₂×#127	—	73	1	no	—
84	C₂×C₄×D₄	—	73	6	no	—
85	C₂×C₂²⋊C₈	—	73	12	no	—
86	#390	—	73	6	no	—
87	#384	—	73	1	no	—
88	C₄³	C₂×D₇₃	1	1	yes	characteristic
89	#336	—	73	6	no	—
90	#336	C₂×D₇₃	1	6	yes	—
91	C₂²×D₄	—	73	6	no	—
92	C₂³×C₄	—	73	4	no	—
93	C₂²×C₈	—	292	12	no	—
94	C₂²×C₈	—	73	4	no	—
95	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
96	#127	—	73	8	no	—
97	C₄×D₄	—	292	6	no	—
98	C₄×D₄	—	292	6	no	—
99	C₂×C₄⋊C₄	—	73	6	no	—
100	C₂×C₂²⋊C₄	—	73	12	no	—
101	C₂×C₄²	C₄×D₇₃	1	3	yes	—
102	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
103	C₂×C₄²	C₂²×D₇₃	1	3	yes	—
104	C₂²⋊C₈	—	292	12	no	—
105	C₂⁴	—	73	4	no	—
106	C₂×D₄	—	292	6	no	—
107	C₂×D₄	—	292	6	no	—
108	C₂×D₄	—	73	24	no	—
109	C₂²×C₄	—	73	4	no	—
110	C₂²×C₄	—	146	12	no	—
111	C₂²×C₄	—	146	12	no	—
112	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
113	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
114	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
115	C₂×C₈	—	2	12	no	—
116	C₂×C₈	—	292	16	no	—
117	C₂×C₈	—	146	12	no	—
118	C₄⋊C₄	—	292	6	no	—
119	C₂²⋊C₄	—	292	12	no	—
120	C₄²	—	2	6	no	—
121	C₄²	—	4	3	no	—
122	C₄²	C₈×D₇₃	1	4	yes	complemented
123	C₂³	—	73	8	no	—
124	C₂³	—	146	12	no	—
125	C₂³	—	146	12	no	—
126	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
127	D₄	—	292	24	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	2	3	no	—
130	C₂×C₄	—	2	3	no	—
131	C₂×C₄	—	2	6	no	—
132	C₂×C₄	—	292	4	no	—
133	C₂×C₄	—	292	12	no	—
134	C₂×C₄	—	146	12	no	—
135	C₂×C₄	C₂×C₈×D₇₃	1	6	yes	—
136	C₂×C₄	#14556637	1	6	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	292	4	no	—
139	C₂²	—	292	4	no	—
140	C₂²	—	292	12	no	—
141	C₂²	—	146	24	no	—
142	C₂²	#17989061	1	3	yes	—
143	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
144	C₂²	#17989353	1	3	yes	—
145	C₄	—	2	6	no	—
146	C₄	—	4	3	no	—
147	C₄	—	292	4	no	—
148	C₄	#17989328	1	4	yes	—
149	C₂	—	292	8	no	—
150	C₂	#22613107	1	1	yes	characteristic
151	C₂	#22618954	1	3	yes	—
152	C₂	#22619001	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556588

C₂

yes

maximal, characteristic

#22610222

C₂

yes

maximal, complemented

#22610074

C₂

yes

maximal, characteristic, complemented

C₇₃×#1839

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₇₃

C₂²

yes

—

C₂×#14556588

C₄

yes

characteristic

C₂×C₄×D₂₉₂

C₂²

yes

—

C₂×#14556503

C₂²

yes

—

#17988489

C₄

yes

—

#17988457

C₄

yes

characteristic

#17988342

C₂²

yes

—

C₄³×C₇₃

C₂²

yes

characteristic

#336×C₇₃

C₂²

yes

—

C₂²×D₂₉₂

C₂×C₄

yes

—

C₂²×C₄×D₇₃

C₂³

yes

—

C₂²×#10596831

C₂³

yes

—

C₂×C₈×D₇₃

—

#14556588

C₈

yes

complemented

C₄×D₂₉₂

—

C₄×D₂₉₂

—

C₂×C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂×#11924005

C₂×C₄

yes

—

#14556503

—

C₂²×C₈×C₇₃

C₂³

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₃

C₂³

yes

—

C₂³×D₇₃

C₂²×C₄

yes

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₈

yes

complemented

C₂²×Dic₇₃

C₂²×C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

—

C₂×#10596831

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

—

C₂²×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

complemented

C₂³×C₇₃

C₂³×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₂²×C₈

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₈

yes

—

C₂×C₄×C₇₃

H(5,𝔽₂)

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#518

yes

—

C₂²×C₇₃

C₂³×C₈

yes

characteristic

C₂²×C₇₃

#568

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#560

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2980

yes

—

D₇₃

—

—

—

—

D₇₃

yes

characteristic, complemented

C₇₃

#23355

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×#127

—

C₂×C₄×D₄

—

C₂×C₂²⋊C₈

—

#390

—

#384

—

C₄³

C₂×D₇₃

yes

characteristic

#336

—

#336

C₂×D₇₃

yes

—

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

292

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₇₃

yes

—

#127

—

C₄×D₄

—

292

—

C₄×D₄

—

292

—

C₂×C₄⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₄²

C₄×D₇₃

yes

—

102

C₂×C₄²

C₄×D₇₃

yes

characteristic

103

C₂×C₄²

C₂²×D₇₃

yes

—

104

C₂²⋊C₈

—

292

—

105

C₂⁴

—

106

C₂×D₄

—

292

—

107

C₂×D₄

—

292

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

146

—

111

C₂²×C₄

—

146

—

112

C₂²×C₄

C₂×C₄×D₇₃

yes

—

113

C₂²×C₄

C₂×C₄×D₇₃

yes

—

114

C₂²×C₄

C₂³×D₇₃

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

292

—

117

C₂×C₈

—

146

—

118

C₄⋊C₄

—

292

—

119

C₂²⋊C₄

—

292

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₈×D₇₃

yes

complemented

123

C₂³

—

124

C₂³

—

146

—

125

C₂³

—

146

—

126

C₂³

C₂²×C₄×D₇₃

yes

characteristic

127

D₄

—

292

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

146

—

135

C₂×C₄

C₂×C₈×D₇₃

yes

—

136

C₂×C₄

#14556637

yes

—

137

C₈

—

138

C₈

—

292

—

139

C₂²

—

292

—

140

C₂²

—

292

—

141

C₂²

—

146

—

142

C₂²

#17989061

yes

—

143

C₂²

C₂²×C₈×D₇₃

yes

characteristic

144

C₂²

#17989353

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

292

—

148

C₄

#17989328

yes

—

149

C₂

—

292

—

150

C₂

#22613107

yes

characteristic

151

C₂

#22618954

yes

—

152

C₂

#22619001

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂³×C₈
Fitting	C₇₃×#1839	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂×#14556588	C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1839	D₇₃
hypocenter	C₇₃	#23355
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1839

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₇₃

commutator

C₂²×C₇₃

C₂³×C₈

Fitting

C₇₃×#1839

C₂

socle

C₂³×C₇₃

C₂³×C₄

prime radical

C₂×#14556588

C₄

cosocle

C₂²×C₄×C₇₃

C₂⁴

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ #1839

N	H ≅ G/N	type
2. #22610222	C₂	semidirect
3. #22610074	C₂	semidirect
4. C₇₃×#1839	C₂	semidirect
18. #14556588	C₈	semidirect
31. C₂×D₂₉₂	C₂×C₈	semidirect
45. C₄²×C₇₃	C₂×C₈	semidirect
80. #1839	D₇₃	semidirect
81. C₇₃	#23355	semidirect
122. C₄²	C₈×D₇₃	semidirect

H ≅ G/N

type

2. #22610222

C₂

semidirect

3. #22610074

C₂

semidirect

4. C₇₃×#1839

C₂

semidirect

18. #14556588

C₈

semidirect

31. C₂×D₂₉₂

C₂×C₈

semidirect

45. C₄²×C₇₃

C₂×C₈

semidirect

80. #1839

D₇₃

semidirect

81. C₇₃

#23355

semidirect

122. C₄²

C₈×D₇₃

semidirect

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

subgroup	index
C₄×#14556588	2
#22610222	2
#22610074	2
C₇₃×#1839	2
#23355	73

subgroup

index

C₇₃×#1839

Subgroups — #33112529

Conjugacy classes of subgroups

Complements

Maximal subgroups