Group #33000739 — subgroups

Subgroups — #33000739

Order 18688.

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

normal subgroups only

This group has 296 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
2	#22612160	C₂	1	2	yes	maximal, complemented
3	#22609549	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607849	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1706	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556535	—	2	4	no	—
7	#17988992	—	2	1	no	—
8	#17988992	—	2	1	no	—
9	#17988967	C₂²	1	2	yes	—
10	#17988390	—	2	4	no	—
11	C₄×#11924003	C₂²	1	2	yes	—
12	#500×C₇₃	C₂²	1	1	yes	characteristic
13	#493×C₇₃	C₂²	1	1	yes	characteristic
14	#492×C₇₃	—	2	1	no	—
15	#486×C₇₃	—	2	1	no	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	2	2	no	—
18	C₂×Dic₂₉₂	—	2	2	no	—
19	#14556591	—	2	2	no	—
20	#14556535	—	4	8	no	—
21	C₂×#11924003	D₄	1	4	yes	—
22	#14556525	—	4	2	no	—
23	#14556525	—	4	2	no	—
24	C₂×SD₁₆×C₇₃	—	4	2	no	—
25	C₂×D₈×C₇₃	—	2	2	no	—
26	C₄⋊Q₈×C₇₃	—	2	1	no	—
27	#127×C₇₃	—	2	1	no	—
28	#107×C₇₃	—	2	2	no	—
29	#102×C₇₃	—	4	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	2	no	—
34	Dic₂₉₂	—	4	4	no	—
35	Dic₂₉₂	—	4	4	no	—
36	C₄⋊Dic₇₃	—	4	2	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	#11924003	—	2	4	no	—
39	C₂×Q₈×C₇₃	—	4	1	no	—
40	C₂×D₄×C₇₃	—	4	1	no	—
41	C₂×D₄×C₇₃	—	4	1	no	—
42	C₂×D₄×C₇₃	—	4	1	no	—
43	SD₁₆×C₇₃	—	8	4	no	—
44	D₈×C₇₃	—	4	4	no	—
45	C₂×C₈×C₇₃	D₈	1	2	yes	—
46	C₂×C₈×C₇₃	D₈	1	2	yes	—
47	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
48	C₄⋊C₄×C₇₃	—	4	1	no	—
49	C₄⋊C₄×C₇₃	—	4	1	no	—
50	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₇₃	—	8	1	no	—
52	Q₈×C₇₃	—	8	2	no	—
53	D₄×C₇₃	—	8	2	no	—
54	D₄×C₇₃	—	8	2	no	—
55	D₄×C₇₃	—	8	2	no	—
56	C₂×C₄×C₇₃	—	8	1	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	2	no	—
61	Dic₁₄₆	—	8	2	no	—
62	Dic₁₄₆	—	8	2	no	—
63	C₈×C₇₃	—	2	2	no	—
64	C₈×C₇₃	SD₃₂	1	4	yes	complemented
65	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
66	C₂²×C₇₃	—	16	1	no	—
67	C₂²×C₇₃	—	16	1	no	—
68	C₂²×C₇₃	—	16	1	no	—
69	C₂²×C₇₃	#493	1	1	yes	characteristic
70	C₄×C₇₃	—	16	1	no	—
71	C₄×C₇₃	#496	1	2	yes	—
72	C₄×C₇₃	C₂×SD₃₂	1	2	yes	—
73	C₄×C₇₃	#510	1	2	yes	—
74	Dic₇₃	—	16	2	no	—
75	#12526	—	73	1	no	maximal, Sylow
76	C₂×C₇₃	—	32	1	no	—
77	C₂×C₇₃	#1721	1	1	yes	characteristic
78	C₂×C₇₃	#2248	1	1	yes	characteristic
79	C₂×C₇₃	#2251	1	1	yes	characteristic
80	#2248	—	73	2	no	—
81	#2241	—	73	2	no	—
82	#1713	—	73	1	no	—
83	#1706	D₇₃	1	1	yes	characteristic
84	#1312	—	73	1	no	—
85	C₇₃	#12526	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₃₂	—	146	4	no	—
87	#500	—	146	1	no	—
88	#500	—	146	1	no	—
89	#500	C₂×D₇₃	1	1	yes	characteristic
90	#494	—	73	2	no	—
91	#493	C₂×D₇₃	1	1	yes	characteristic
92	#492	—	2	1	no	—
93	#486	—	2	1	no	—
94	#358	—	146	4	no	—
95	C₄×C₁₆	—	73	2	no	—
96	C₈²	C₂×D₇₃	1	1	yes	characteristic
97	C₂×Q₁₆	—	146	2	no	—
98	C₂×Q₁₆	—	146	2	no	—
99	C₂×SD₁₆	—	4	2	no	—
100	C₂×D₈	—	2	2	no	—
101	C₄⋊Q₈	—	2	1	no	—
102	C₄⋊Q₈	—	146	2	no	—
103	#127	—	2	1	no	—
104	SD₃₂	—	292	8	no	—
105	C₂×C₁₆	—	73	4	no	—
106	#107	—	2	2	no	—
107	#107	—	292	2	no	—
108	#107	—	292	2	no	—
109	#102	—	4	2	no	—
110	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
113	C₂×Q₈	—	4	1	no	—
114	C₂×Q₈	—	292	2	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	4	1	no	—
117	C₂×D₄	—	4	1	no	—
118	Q₁₆	—	292	4	no	—
119	Q₁₆	—	292	4	no	—
120	SD₁₆	—	8	4	no	—
121	D₈	—	4	4	no	—
122	C₂×C₈	#11924013	1	2	yes	—
123	C₂×C₈	#11924013	1	2	yes	—
124	C₂×C₈	D₄×D₇₃	1	2	yes	—
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	292	2	no	—
128	C₄⋊C₄	—	292	2	no	—
129	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
130	C₁₆	—	146	4	no	—
131	C₂³	—	8	1	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	584	2	no	—
134	Q₈	—	584	2	no	—
135	D₄	—	8	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	C₂×C₄	—	8	1	no	—
139	C₂×C₄	—	584	2	no	—
140	C₂×C₄	#14556587	1	1	yes	characteristic
141	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
142	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	#14556535	1	4	yes	complemented
145	C₈	#14556618	1	4	yes	—
146	C₂²	—	16	1	no	—
147	C₂²	—	16	1	no	—
148	C₂²	—	16	1	no	—
149	C₂²	#17988963	1	1	yes	characteristic
150	C₄	—	16	1	no	—
151	C₄	—	1168	2	no	—
152	C₄	#17988973	1	2	yes	—
153	C₄	C₂×#14556535	1	2	yes	—
154	C₄	#17989027	1	2	yes	—
155	C₂	—	32	1	no	—
156	C₂	#22609595	1	1	yes	characteristic
157	C₂	#22612186	1	1	yes	characteristic
158	C₂	#22612203	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612186

C₂

yes

maximal

#22612160

C₂

yes

maximal, complemented

#22609549

C₂

yes

maximal, characteristic, complemented

#22607849

C₂

yes

maximal, characteristic, complemented

C₇₃×#1706

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#17988390

—

C₄×#11924003

C₂²

yes

—

#500×C₇₃

C₂²

yes

characteristic

#493×C₇₃

C₂²

yes

characteristic

#492×C₇₃

—

#486×C₇₃

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

#14556591

—

#14556535

—

C₂×#11924003

D₄

yes

—

#14556525

—

#14556525

—

C₂×SD₁₆×C₇₃

—

C₂×D₈×C₇₃

—

C₄⋊Q₈×C₇₃

—

#127×C₇₃

—

#107×C₇₃

—

#102×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

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

—

C₄×C₇₃

#496

yes

—

C₄×C₇₃

C₂×SD₃₂

yes

—

C₄×C₇₃

#510

yes

—

Dic₇₃

—

#12526

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

D₇₃

yes

characteristic

#1312

—

C₇₃

#12526

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

146

—

#500

—

146

—

#500

—

146

—

#500

C₂×D₇₃

yes

characteristic

#494

—

#493

C₂×D₇₃

yes

—

—

—

146

—

C₄×C₁₆

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×SD₁₆

—

100

C₂×D₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

146

—

103

#127

—

104

SD₃₂

—

292

—

105

C₂×C₁₆

—

106

#107

—

107

#107

—

292

—

108

#107

—

292

—

109

#102

—

110

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

111

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

112

C₄×C₈

C₂²×D₇₃

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

292

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

Q₁₆

—

292

—

119

Q₁₆

—

292

—

120

SD₁₆

—

121

D₈

—

122

C₂×C₈

#11924013

yes

—

123

C₂×C₈

#11924013

yes

—

124

C₂×C₈

D₄×D₇₃

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

292

—

128

C₄⋊C₄

—

292

—

129

C₄²

C₂×C₂²⋊D₇₃

yes

characteristic

130

C₁₆

—

146

—

131

C₂³

—

132

Q₈

—

133

Q₈

—

584

—

134

Q₈

—

584

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

584

—

140

C₂×C₄

#14556587

yes

characteristic

141

C₂×C₄

C₂×#11924013

yes

characteristic

142

C₂×C₄

C₂×#11924013

yes

characteristic

143

C₈

—

144

C₈

#14556535

yes

complemented

145

C₈

#14556618

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#17988963

yes

characteristic

150

C₄

—

151

C₄

—

1168

—

152

C₄

#17988973

yes

—

153

C₄

C₂×#14556535

yes

—

154

C₄

#17989027

yes

—

155

C₂

—

156

C₂

#22609595

yes

characteristic

157

C₂

#22612186

yes

characteristic

158

C₂

#22612203

yes

characteristic

	subgroup	G/H
center	C₂²	#17988963
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1706	C₂
socle	C₂²×C₇₃	#493
prime radical	#493×C₇₃	C₂²
hypercenter	#1706	D₇₃
hypocenter	C₇₃	#12526
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

subgroup

G/H

center

C₂²

#17988963

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1706

C₂

socle

C₂²×C₇₃

#493

prime radical

#493×C₇₃

C₂²

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

N	H ≅ G/N	type
2. #22612160	C₂	semidirect
3. #22609549	C₂	semidirect
4. #22607849	C₂	semidirect
64. C₈×C₇₃	SD₃₂	semidirect
85. C₇₃	#12526	semidirect
144. C₈	#14556535	semidirect

H ≅ G/N

type

2. #22612160

C₂

semidirect

3. #22609549

C₂

semidirect

4. #22607849

C₂

semidirect

64. C₈×C₇₃

SD₃₂

semidirect

85. C₇₃

#12526

semidirect

144. C₈

#14556535

semidirect

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

subgroup	index
#22612186	2
#22612160	2
#22609549	2
#22607849	2
C₇₃×#1706	2
#12526	73

subgroup

index

C₇₃×#1706

Subgroups — #33000739

Conjugacy classes of subgroups

Complements

Maximal subgroups