Group #33000238 — subgroups

Subgroups — #33000238

Order 18688.

248 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22609288	C₂	1	2	yes	maximal, complemented
3	#22608003	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607977	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2246	C₂	1	1	yes	maximal, characteristic
6	#17988784	—	2	2	no	—
7	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
8	C₄⋊#11924003	C₂²	1	1	yes	characteristic
9	#17988405	—	2	1	no	—
10	C₁₆×Dic₇₃	—	2	1	no	—
11	#17988340	C₂²	1	2	yes	—
12	#17988331	—	2	2	no	—
13	C₂×D₁₆×C₇₃	—	2	1	no	—
14	C₂×D₁₆×C₇₃	—	2	1	no	—
15	#493×C₇₃	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#11924016	—	4	2	no	—
18	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
19	C₂×#11924003	—	2	1	no	—
20	C₄⋊#10596831	—	2	2	no	—
21	C₄⋊#10596831	D₄	1	1	yes	characteristic
22	#14556520	—	2	1	no	—
23	#14556512	—	4	2	no	—
24	C₈×Dic₇₃	—	2	1	no	—
25	C₂×D₈×C₇₃	—	2	2	no	—
26	C₂×D₈×C₇₃	—	2	2	no	—
27	#127×C₇₃	—	2	2	no	—
28	D₁₆×C₇₃	—	4	2	no	—
29	D₁₆×C₇₃	—	4	2	no	—
30	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
34	#11924016	—	8	4	no	—
35	C₂×#10596831	—	2	1	no	—
36	C₂×#10596831	—	4	2	no	—
37	C₄⋊Dic₇₃	D₈	1	1	yes	characteristic
38	#11924007	—	2	1	no	—
39	C₄×Dic₇₃	—	2	1	no	—
40	#11924003	—	4	1	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	2	no	—
45	D₈×C₇₃	—	4	2	no	—
46	D₈×C₇₃	—	8	2	no	—
47	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
49	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₇₃	—	2	1	no	—
51	C₁₆×C₇₃	SD₁₆	1	2	yes	complemented
52	C₂³×C₇₃	—	8	2	no	—
53	D₄×C₇₃	—	8	2	no	—
54	D₄×C₇₃	—	8	2	no	—
55	D₄×C₇₃	—	8	2	no	—
56	D₄×C₇₃	—	8	2	no	—
57	D₄×C₇₃	—	16	2	no	—
58	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
61	C₂×Dic₇₃	—	2	1	no	—
62	C₂×Dic₇₃	—	2	1	no	—
63	Dic₁₄₆	—	2	1	no	—
64	Dic₁₄₆	D₁₆	1	2	yes	complemented
65	C₈×C₇₃	—	2	1	no	—
66	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₈×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	#10596831	—	4	1	no	—
69	#10596831	—	8	2	no	—
70	C₂²×C₇₃	—	16	2	no	—
71	C₂²×C₇₃	—	16	2	no	—
72	C₂²×C₇₃	—	16	2	no	—
73	C₂²×C₇₃	#466	1	1	yes	characteristic
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#460	1	1	yes	characteristic
76	C₄×C₇₃	#472	1	1	yes	characteristic
77	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
78	C₄×C₇₃	#508	1	1	yes	characteristic
79	Dic₇₃	—	2	2	no	—
80	Dic₇₃	—	4	1	no	—
81	#12439	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	—	32	2	no	—
83	C₂×C₇₃	#1668	1	1	yes	characteristic
84	C₂×C₇₃	#2215	1	1	yes	characteristic
85	C₂×C₇₃	#2221	1	1	yes	characteristic
86	#2246	D₇₃	1	1	yes	characteristic
87	Q₈×C₁₆	—	73	1	no	—
88	#1668	—	73	2	no	—
89	#1371	—	73	1	no	—
90	#1361	—	73	2	no	—
91	C₇₃	#12439	1	1	yes	characteristic, complemented, Sylow
92	C₂×D₁₆	—	2	1	no	—
93	C₂×D₁₆	—	2	1	no	—
94	#493	C₂×D₇₃	1	2	yes	—
95	#460	—	146	2	no	—
96	C₈×Q₈	—	73	1	no	—
97	C₄⋊C₁₆	—	73	1	no	—
98	C₄⋊C₁₆	—	146	1	no	—
99	C₄×C₁₆	—	146	1	no	—
100	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
101	#335	—	73	2	no	—
102	#331	—	146	2	no	—
103	C₂×SD₁₆	—	292	2	no	—
104	C₂×D₈	—	2	2	no	—
105	C₂×D₈	—	2	2	no	—
106	#127	—	2	2	no	—
107	C₄×Q₈	—	73	1	no	—
108	D₁₆	—	4	2	no	—
109	D₁₆	—	4	2	no	—
110	C₂×C₁₆	—	146	1	no	—
111	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
112	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
113	C₄⋊C₈	—	73	1	no	—
114	C₄⋊C₈	—	146	1	no	—
115	C₄⋊C₈	—	146	2	no	—
116	#102	—	292	2	no	—
117	C₄×C₈	—	146	1	no	—
118	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
119	C₂×Q₈	—	73	1	no	—
120	C₂×D₄	—	4	2	no	—
121	C₂×D₄	—	4	2	no	—
122	C₂×D₄	—	4	2	no	—
123	SD₁₆	—	584	4	no	—
124	D₈	—	4	2	no	—
125	D₈	—	4	2	no	—
126	D₈	—	8	2	no	—
127	C₂×C₈	—	292	2	no	—
128	C₂×C₈	—	146	1	no	—
129	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂×C₈	#11924027	1	1	yes	characteristic
131	C₄⋊C₄	—	73	1	no	—
132	C₄⋊C₄	—	146	1	no	—
133	C₄²	—	146	1	no	—
134	C₄²	D₄×D₇₃	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	—	292	1	no	—
137	C₁₆	#11924016	1	2	yes	complemented
138	C₂³	—	8	2	no	—
139	Q₈	—	73	2	no	—
140	Q₈	—	146	1	no	—
141	D₄	—	8	2	no	—
142	D₄	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	2	no	—
145	D₄	—	16	2	no	—
146	C₂×C₄	—	146	1	no	—
147	C₂×C₄	—	146	1	no	—
148	C₂×C₄	#14556562	1	1	yes	characteristic
149	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
150	C₂×C₄	#14556610	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	292	1	no	—
153	C₈	—	584	2	no	—
154	C₈	C₂×#11924016	1	1	yes	characteristic
155	C₈	#14556617	1	1	yes	characteristic
156	C₂²	—	16	2	no	—
157	C₂²	—	16	2	no	—
158	C₂²	—	16	2	no	—
159	C₂²	#17988820	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	292	1	no	—
162	C₄	—	146	2	no	—
163	C₄	#17988784	1	1	yes	characteristic
164	C₄	#17988854	1	1	yes	characteristic
165	C₄	D₁₆×D₇₃	1	1	yes	characteristic
166	C₄	#17989016	1	1	yes	characteristic
167	C₂	—	32	2	no	—
168	C₂	#22609288	1	1	yes	characteristic
169	C₂	#22612014	1	1	yes	characteristic
170	C₂	#22612048	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄₆

C₂

yes

maximal, characteristic, complemented

#22609288

C₂

yes

maximal, complemented

#22608003

C₂

yes

maximal, characteristic, complemented

#22607977

C₂

yes

maximal, complemented

C₇₃×#2246

C₂

yes

maximal, characteristic

#17988784

—

C₈×Dic₁₄₆

C₂²

yes

characteristic

C₄⋊#11924003

C₂²

yes

characteristic

#17988405

—

C₁₆×Dic₇₃

—

#17988340

C₂²

yes

—

#17988331

—

C₂×D₁₆×C₇₃

—

C₂×D₁₆×C₇₃

—

#493×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×#11924016

—

C₄×Dic₁₄₆

D₄

yes

—

—

D₄

yes

characteristic

#14556520

—

#14556512

—

C₈×Dic₇₃

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

D₁₆×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

D₈

yes

—

—

—

C₄⋊Dic₇₃

D₈

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

#11924003

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

SD₁₆

yes

complemented

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#121

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

#135

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

D₁₆

yes

complemented

C₈×C₇₃

—

C₈×C₇₃

C₂×SD₁₆

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#466

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

#508

yes

characteristic

Dic₇₃

—

Dic₇₃

—

#12439

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

characteristic

Q₈×C₁₆

—

#1668

—

#1371

—

#1361

—

C₇₃

#12439

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

—

#493

C₂×D₇₃

yes

—

#460

—

146

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

146

—

C₄×C₁₆

—

146

—

100

C₄×C₁₆

C₂×D₇₃

yes

characteristic

101

#335

—

102

#331

—

146

—

103

C₂×SD₁₆

—

292

—

104

C₂×D₈

—

105

C₂×D₈

—

106

#127

—

107

C₄×Q₈

—

108

D₁₆

—

109

D₁₆

—

110

C₂×C₁₆

—

146

—

111

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

112

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

146

—

115

C₄⋊C₈

—

146

—

116

#102

—

292

—

117

C₄×C₈

—

146

—

118

C₄×C₈

C₂²×D₇₃

yes

characteristic

119

C₂×Q₈

—

120

C₂×D₄

—

121

C₂×D₄

—

122

C₂×D₄

—

123

SD₁₆

—

584

—

124

D₈

—

125

D₈

—

126

D₈

—

127

C₂×C₈

—

292

—

128

C₂×C₈

—

146

—

129

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

130

C₂×C₈

#11924027

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

146

—

133

C₄²

—

146

—

134

C₄²

D₄×D₇₃

yes

characteristic

135

C₁₆

—

136

C₁₆

—

292

—

137

C₁₆

#11924016

yes

complemented

138

C₂³

—

139

Q₈

—

140

Q₈

—

146

—

141

D₄

—

142

D₄

—

143

D₄

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

146

—

147

C₂×C₄

—

146

—

148

C₂×C₄

#14556562

yes

characteristic

149

C₂×C₄

D₈×D₇₃

yes

characteristic

150

C₂×C₄

#14556610

yes

characteristic

151

C₈

—

152

C₈

—

292

—

153

C₈

—

584

—

154

C₈

C₂×#11924016

yes

characteristic

155

C₈

#14556617

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#17988820

yes

characteristic

160

C₄

—

161

C₄

—

292

—

162

C₄

—

146

—

163

C₄

#17988784

yes

characteristic

164

C₄

#17988854

yes

characteristic

165

C₄

D₁₆×D₇₃

yes

characteristic

166

C₄

#17989016

yes

characteristic

167

C₂

—

168

C₂

#22609288

yes

characteristic

169

C₂

#22612014

yes

characteristic

170

C₂

#22612048

yes

characteristic

	subgroup	G/H
center	C₂²	#17988820
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting, prime radical	C₇₃×#2246	C₂
socle	C₂²×C₇₃	#466
hypercenter	#2246	D₇₃
hypocenter	C₇₃	#12439
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#17988820

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting, prime radical

C₇₃×#2246

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. C₁₆×Dic₁₄₆	C₂	semidirect
2. #22609288	C₂	semidirect
3. #22608003	C₂	semidirect
4. #22607977	C₂	semidirect
51. C₁₆×C₇₃	SD₁₆	semidirect
64. Dic₁₄₆	D₁₆	semidirect
91. C₇₃	#12439	semidirect
137. C₁₆	#11924016	semidirect

H ≅ G/N

type

1. C₁₆×Dic₁₄₆

C₂

semidirect

2. #22609288

C₂

semidirect

3. #22608003

C₂

semidirect

4. #22607977

C₂

semidirect

51. C₁₆×C₇₃

SD₁₆

semidirect

64. Dic₁₄₆

D₁₆

semidirect

91. C₇₃

#12439

semidirect

137. C₁₆

#11924016

semidirect

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

subgroup	index
C₁₆×Dic₁₄₆	2
#22609288	2
#22608003	2
#22607977	2
C₇₃×#2246	2
#12439	73

subgroup

index

C₁₆×Dic₁₄₆

C₇₃×#2246

Subgroups — #33000238

Conjugacy classes of subgroups

Complements

Maximal subgroups