Group #33000759 — subgroups

Subgroups — #33000759

Order 18688.

318 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22612186	C₂	1	2	yes	maximal, complemented
2	#22612154	C₂	1	2	yes	maximal, complemented
3	#22609528	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607849	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1712	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556535	—	2	4	no	—
7	#17988992	C₂²	1	1	yes	characteristic
8	#17988985	—	2	1	no	—
9	#17988967	C₂²	1	1	yes	characteristic
10	#17988938	—	2	1	no	—
11	#17988388	—	2	4	no	—
12	C₄×#11924003	C₂²	1	2	yes	—
13	#493×C₇₃	—	2	1	no	—
14	#493×C₇₃	—	2	1	no	—
15	#493×C₇₃	C₂²	1	2	yes	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	2	2	no	—
18	#14556591	—	2	1	no	—
19	#14556591	—	2	1	no	—
20	#14556535	—	4	8	no	—
21	C₂×#11924003	D₄	1	4	yes	—
22	#14556525	—	2	2	no	—
23	#14556523	—	4	2	no	—
24	#14556516	—	4	2	no	—
25	C₂×D₈×C₇₃	—	2	2	no	—
26	C₂×D₈×C₇₃	—	2	2	no	—
27	C₂×D₈×C₇₃	—	4	2	no	—
28	C₂×D₈×C₇₃	—	4	2	no	—
29	#127×C₇₃	—	2	2	no	—
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₄₆	—	4	1	no	—
34	C₂×Dic₁₄₆	—	4	1	no	—
35	Dic₂₉₂	—	4	4	no	—
36	C₄⋊Dic₇₃	—	4	1	no	—
37	C₄⋊Dic₇₃	—	4	1	no	—
38	C₄⋊Dic₇₃	—	4	1	no	—
39	C₄⋊Dic₇₃	—	4	1	no	—
40	#11924003	—	2	4	no	—
41	C₂×D₄×C₇₃	—	4	2	no	—
42	C₂×D₄×C₇₃	—	4	2	no	—
43	C₂×D₄×C₇₃	—	4	2	no	—
44	D₈×C₇₃	—	4	4	no	—
45	D₈×C₇₃	—	4	4	no	—
46	D₈×C₇₃	—	8	4	no	—
47	D₈×C₇₃	—	8	4	no	—
48	C₂×C₈×C₇₃	D₈	1	2	yes	—
49	C₂×C₈×C₇₃	D₈	1	2	yes	—
50	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
51	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₇₃	—	8	2	no	—
53	D₄×C₇₃	—	8	2	no	—
54	D₄×C₇₃	—	8	2	no	—
55	D₄×C₇₃	—	8	4	no	—
56	D₄×C₇₃	—	8	4	no	—
57	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₇₃	—	8	1	no	—
61	C₂×Dic₇₃	—	8	1	no	—
62	Dic₁₄₆	—	8	2	no	—
63	Dic₁₄₆	—	8	2	no	—
64	C₈×C₇₃	—	2	2	no	—
65	C₈×C₇₃	SD₃₂	1	4	yes	complemented
66	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
67	C₂²×C₇₃	—	16	2	no	—
68	C₂²×C₇₃	—	16	2	no	—
69	C₂²×C₇₃	—	16	2	no	—
70	C₂²×C₇₃	#493	1	1	yes	characteristic
71	C₄×C₇₃	#496	1	2	yes	—
72	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
73	C₄×C₇₃	#509	1	2	yes	—
74	Dic₇₃	—	16	1	no	—
75	Dic₇₃	—	16	1	no	—
76	#12529	—	73	1	no	maximal, Sylow
77	C₂×C₇₃	—	32	2	no	—
78	C₂×C₇₃	#1721	1	1	yes	characteristic
79	C₂×C₇₃	#2248	1	1	yes	characteristic
80	C₂×C₇₃	#2250	1	1	yes	characteristic
81	#2248	—	73	2	no	—
82	#2240	—	73	2	no	—
83	#1712	D₇₃	1	1	yes	characteristic
84	#1707	—	73	1	no	—
85	#1312	—	73	1	no	—
86	C₇₃	#12529	1	1	yes	characteristic, complemented, Sylow
87	C₂×SD₃₂	—	146	4	no	—
88	#500	—	73	1	no	—
89	#498	—	146	1	no	—
90	#494	—	73	1	no	—
91	#493	—	2	1	no	—
92	#493	—	2	1	no	—
93	#493	C₂×D₇₃	1	2	yes	—
94	#487	—	146	1	no	—
95	#357	—	146	4	no	—
96	C₄×C₁₆	—	73	2	no	—
97	C₈²	C₂×D₇₃	1	1	yes	characteristic
98	C₂×Q₁₆	—	146	2	no	—
99	C₂×D₈	—	2	2	no	—
100	C₂×D₈	—	2	2	no	—
101	C₂×D₈	—	4	2	no	—
102	C₂×D₈	—	4	2	no	—
103	C₄⋊Q₈	—	146	1	no	—
104	C₄⋊Q₈	—	146	1	no	—
105	#127	—	2	2	no	—
106	SD₃₂	—	292	8	no	—
107	C₂×C₁₆	—	73	4	no	—
108	#107	—	146	2	no	—
109	#106	—	292	2	no	—
110	#103	—	292	2	no	—
111	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
112	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
114	C₂×Q₈	—	292	1	no	—
115	C₂×Q₈	—	292	1	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	2	no	—
118	C₂×D₄	—	4	2	no	—
119	Q₁₆	—	292	4	no	—
120	D₈	—	4	4	no	—
121	D₈	—	4	4	no	—
122	D₈	—	8	4	no	—
123	D₈	—	8	4	no	—
124	C₂×C₈	#11924013	1	2	yes	—
125	C₂×C₈	#11924013	1	2	yes	—
126	C₂×C₈	D₄×D₇₃	1	2	yes	—
127	C₄⋊C₄	—	292	1	no	—
128	C₄⋊C₄	—	292	1	no	—
129	C₄⋊C₄	—	292	1	no	—
130	C₄⋊C₄	—	292	1	no	—
131	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
132	C₁₆	—	146	4	no	—
133	C₂³	—	8	2	no	—
134	Q₈	—	584	2	no	—
135	Q₈	—	584	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	D₄	—	8	4	no	—
139	D₄	—	8	4	no	—
140	C₂×C₄	—	584	1	no	—
141	C₂×C₄	—	584	1	no	—
142	C₂×C₄	#14556587	1	1	yes	characteristic
143	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
144	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	#14556535	1	4	yes	complemented
147	C₈	#14556615	1	4	yes	—
148	C₂²	—	16	2	no	—
149	C₂²	—	16	2	no	—
150	C₂²	—	16	2	no	—
151	C₂²	#17988963	1	1	yes	characteristic
152	C₄	—	1168	1	no	—
153	C₄	—	1168	1	no	—
154	C₄	#17988975	1	2	yes	—
155	C₄	C₂×#14556535	1	2	yes	—
156	C₄	#17989023	1	2	yes	—
157	C₂	—	32	2	no	—
158	C₂	#22609598	1	1	yes	characteristic
159	C₂	#22612186	1	1	yes	characteristic
160	C₂	#22612195	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612186

C₂

yes

maximal, complemented

#22612154

C₂

yes

maximal, complemented

#22609528

C₂

yes

maximal, characteristic, complemented

#22607849

C₂

yes

maximal, characteristic, complemented

C₇₃×#1712

C₂

yes

maximal, characteristic

C₂×#14556535

—

#17988992

C₂²

yes

characteristic

#17988985

—

#17988967

C₂²

yes

—

—

C₂²

yes

—

#493×C₇₃

—

#493×C₇₃

—

#493×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

—

—

—

D₄

yes

—

#14556525

—

#14556523

—

#14556516

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

SD₃₂

yes

complemented

C₈×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

#496

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

C₄×C₇₃

#509

yes

—

Dic₇₃

—

Dic₇₃

—

#12529

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

#1707

—

#1312

—

C₇₃

#12529

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

146

—

#500

—

#498

—

146

—

—

—

—

C₂×D₇₃

yes

—

#487

—

146

—

#357

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×D₈

—

100

C₂×D₈

—

101

C₂×D₈

—

102

C₂×D₈

—

103

C₄⋊Q₈

—

146

—

104

C₄⋊Q₈

—

146

—

105

#127

—

106

SD₃₂

—

292

—

107

C₂×C₁₆

—

108

#107

—

146

—

109

#106

—

292

—

110

#103

—

292

—

111

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

112

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

113

C₄×C₈

C₂²×D₇₃

yes

characteristic

114

C₂×Q₈

—

292

—

115

C₂×Q₈

—

292

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

Q₁₆

—

292

—

120

D₈

—

121

D₈

—

122

D₈

—

123

D₈

—

124

C₂×C₈

#11924013

yes

—

125

C₂×C₈

#11924013

yes

—

126

C₂×C₈

D₄×D₇₃

yes

—

127

C₄⋊C₄

—

292

—

128

C₄⋊C₄

—

292

—

129

C₄⋊C₄

—

292

—

130

C₄⋊C₄

—

292

—

131

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

132

C₁₆

—

146

—

133

C₂³

—

134

Q₈

—

584

—

135

Q₈

—

584

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

584

—

141

C₂×C₄

—

584

—

142

C₂×C₄

#14556587

yes

characteristic

143

C₂×C₄

C₂×#11924013

yes

characteristic

144

C₂×C₄

C₂×#11924013

yes

characteristic

145

C₈

—

146

C₈

#14556535

yes

complemented

147

C₈

#14556615

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#17988963

yes

characteristic

152

C₄

—

1168

—

153

C₄

—

1168

—

154

C₄

#17988975

yes

—

155

C₄

C₂×#14556535

yes

—

156

C₄

#17989023

yes

—

157

C₂

—

158

C₂

#22609598

yes

characteristic

159

C₂

#22612186

yes

characteristic

160

C₂

#22612195

yes

characteristic

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting, prime radical	C₇₃×#1712	C₂
socle	C₂²×C₇₃	#493
hypercenter	#1712	D₇₃
hypocenter	C₇₃	#12529
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting, prime radical

C₇₃×#1712

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #22612186	C₂	semidirect
2. #22612154	C₂	semidirect
3. #22609528	C₂	semidirect
4. #22607849	C₂	semidirect
65. C₈×C₇₃	SD₃₂	semidirect
86. C₇₃	#12529	semidirect
146. C₈	#14556535	semidirect

H ≅ G/N

type

1. #22612186

C₂

semidirect

2. #22612154

C₂

semidirect

3. #22609528

C₂

semidirect

4. #22607849

C₂

semidirect

65. C₈×C₇₃

SD₃₂

semidirect

86. C₇₃

#12529

semidirect

146. C₈

#14556535

semidirect

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

subgroup	index
#22612186	2
#22612154	2
#22609528	2
#22607849	2
C₇₃×#1712	2
#12529	73

subgroup

index

C₇₃×#1712

Subgroups — #33000759

Conjugacy classes of subgroups

Complements

Maximal subgroups