Group #33007001 — subgroups

Subgroups — #33007001

Order 18688.

232 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 71 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×#17988427	C₂	1	2	yes	maximal, complemented
2	C₂×#17988425	C₂	1	2	yes	maximal
3	#22612184	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608065	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2249	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₈₄	—	2	1	no	—
7	C₂×Dic₅₈₄	C₂²	1	2	yes	—
8	#17988967	—	2	1	no	—
9	#17988427	—	2	4	no	—
10	#17988425	—	2	4	no	—
11	C₂×#14556497	C₂²	1	2	yes	—
12	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×SD₃₂×C₇₃	—	2	1	no	—
14	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#495×C₇₃	—	2	1	no	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	2	2	no	—
18	C₂×Dic₂₉₂	—	4	1	no	—
19	#14556591	—	4	1	no	—
20	Dic₅₈₄	—	2	2	no	—
21	Dic₅₈₄	—	2	2	no	—
22	Dic₅₈₄	—	4	2	no	—
23	#14556497	D₄	1	4	yes	—
24	C₂×Q₁₆×C₇₃	—	2	1	no	—
25	C₂×SD₁₆×C₇₃	—	4	1	no	—
26	C₂×D₈×C₇₃	—	2	1	no	—
27	#124×C₇₃	—	4	1	no	—
28	Q₃₂×C₇₃	—	2	2	no	—
29	SD₃₂×C₇₃	—	4	2	no	—
30	D₁₆×C₇₃	—	2	2	no	—
31	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂×Dic₁₄₆	—	4	2	no	—
35	Dic₂₉₂	—	4	2	no	—
36	Dic₂₉₂	—	4	2	no	—
37	Dic₂₉₂	—	8	2	no	—
38	C₄⋊Dic₇₃	—	8	2	no	—
39	C₂×Q₈×C₇₃	—	4	1	no	—
40	C₂×D₄×C₇₃	—	4	1	no	—
41	Q₁₆×C₇₃	—	4	1	no	—
42	Q₁₆×C₇₃	—	4	1	no	—
43	SD₁₆×C₇₃	—	8	2	no	—
44	D₈×C₇₃	—	4	1	no	—
45	D₈×C₇₃	—	4	1	no	—
46	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊C₄×C₇₃	—	8	2	no	—
49	C₄²×C₇₃	D₈	1	1	yes	characteristic
50	C₁₆×C₇₃	D₈	1	2	yes	—
51	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
52	C₂³×C₇₃	—	8	1	no	—
53	Q₈×C₇₃	—	8	1	no	—
54	Q₈×C₇₃	—	8	1	no	—
55	D₄×C₇₃	—	8	1	no	—
56	D₄×C₇₃	—	8	1	no	—
57	C₂×C₄×C₇₃	—	8	1	no	—
58	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
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	D₁₆	1	2	yes	—
64	C₈×C₇₃	#127	1	1	yes	characteristic
65	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
66	C₂²×C₇₃	—	16	1	no	—
67	C₂²×C₇₃	—	16	2	no	—
68	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	—	16	1	no	—
71	C₄×C₇₃	#493	1	1	yes	characteristic
72	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
73	Dic₇₃	—	16	2	no	—
74	#13730	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	32	1	no	—
76	C₂×C₇₃	#2246	1	1	yes	characteristic
77	C₂×C₇₃	#2264	1	2	yes	—
78	C₂×Q₆₄	—	73	2	no	—
79	C₂×SD₆₄	—	73	2	no	—
80	#2249	D₇₃	1	1	yes	characteristic
81	#2247	—	73	1	no	—
82	#1397	—	73	1	no	—
83	C₇₃	#13730	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	—	73	2	no	—
85	C₂×Q₃₂	—	146	1	no	—
86	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
87	C₂×SD₃₂	—	2	1	no	—
88	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
89	#495	—	2	1	no	—
90	#494	—	146	1	no	—
91	Q₆₄	—	146	4	no	—
92	SD₆₄	—	146	4	no	—
93	C₂×C₃₂	—	73	2	no	—
94	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
95	C₂×Q₁₆	—	2	1	no	—
96	C₂×Q₁₆	—	292	1	no	—
97	C₂×Q₁₆	—	146	2	no	—
98	C₂×SD₁₆	—	4	1	no	—
99	C₂×D₈	—	2	1	no	—
100	C₄⋊Q₈	—	292	1	no	—
101	#124	—	4	1	no	—
102	Q₃₂	—	2	2	no	—
103	Q₃₂	—	292	2	no	—
104	Q₃₂	—	146	2	no	—
105	Q₃₂	—	146	2	no	—
106	SD₃₂	—	4	2	no	—
107	D₁₆	—	2	2	no	—
108	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
111	C₃₂	—	73	4	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	292	2	no	—
114	C₂×D₄	—	4	1	no	—
115	Q₁₆	—	4	1	no	—
116	Q₁₆	—	4	1	no	—
117	Q₁₆	—	292	2	no	—
118	Q₁₆	—	292	2	no	—
119	Q₁₆	—	584	2	no	—
120	SD₁₆	—	8	2	no	—
121	D₈	—	4	1	no	—
122	D₈	—	4	1	no	—
123	C₂×C₈	#11924013	1	1	yes	characteristic
124	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
125	C₄⋊C₄	—	584	2	no	—
126	C₂²⋊C₄	—	8	2	no	—
127	C₄²	#11924013	1	1	yes	characteristic
128	C₁₆	#11924013	1	2	yes	—
129	C₁₆	D₄×D₇₃	1	2	yes	—
130	C₂³	—	8	1	no	—
131	Q₈	—	8	1	no	—
132	Q₈	—	8	1	no	—
133	Q₈	—	584	2	no	—
134	Q₈	—	584	2	no	—
135	D₄	—	8	1	no	—
136	D₄	—	8	1	no	—
137	C₂×C₄	—	8	1	no	—
138	C₂×C₄	—	584	2	no	—
139	C₂×C₄	#14556532	1	2	yes	—
140	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
141	C₈	#14556532	1	2	yes	—
142	C₈	#14556587	1	1	yes	characteristic
143	C₈	C₂×#11924013	1	1	yes	characteristic
144	C₂²	—	16	2	no	—
145	C₂²	—	16	1	no	—
146	C₂²	C₂×#14556532	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	16	1	no	—
149	C₄	—	1168	2	no	—
150	C₄	#17988963	1	1	yes	characteristic
151	C₄	C₂×#14556532	1	1	yes	characteristic
152	C₂	—	32	1	no	—
153	C₂	#22612180	1	1	yes	characteristic
154	C₂	#22612248	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17988427

C₂

yes

maximal, complemented

C₂×#17988425

C₂

yes

maximal

#22612184

C₂

yes

maximal, characteristic, complemented

#22608065

C₂

yes

maximal, characteristic, complemented

C₇₃×#2249

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

—

C₂×Dic₅₈₄

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×Q₃₂×C₇₃

C₂²

yes

characteristic

C₂×SD₃₂×C₇₃

—

C₂×D₁₆×C₇₃

C₂²

yes

characteristic

#495×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

#14556591

—

Dic₅₈₄

—

Dic₅₈₄

—

Dic₅₈₄

—

#14556497

D₄

yes

—

C₂×Q₁₆×C₇₃

—

C₂×SD₁₆×C₇₃

—

C₂×D₈×C₇₃

—

#124×C₇₃

—

Q₃₂×C₇₃

—

SD₃₂×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

SD₁₆×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

D₈

yes

—

C₁₆×C₇₃

C₂×D₄

yes

—

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

D₁₆

yes

—

C₈×C₇₃

#127

yes

characteristic

C₈×C₇₃

C₂×D₈

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#493

yes

characteristic

C₄×C₇₃

C₂×D₁₆

yes

characteristic

Dic₇₃

—

#13730

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#2246

yes

characteristic

C₂×C₇₃

#2264

yes

—

C₂×Q₆₄

—

C₂×SD₆₄

—

#2249

D₇₃

yes

characteristic

#2247

—

#1397

—

C₇₃

#13730

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

146

—

C₂×Q₃₂

C₂×D₇₃

yes

characteristic

C₂×SD₃₂

—

C₂×D₁₆

C₂×D₇₃

yes

characteristic

#495

—

#494

—

146

—

Q₆₄

—

146

—

SD₆₄

—

146

—

C₂×C₃₂

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

146

—

C₂×SD₁₆

—

C₂×D₈

—

100

C₄⋊Q₈

—

292

—

101

#124

—

102

Q₃₂

—

103

Q₃₂

—

292

—

104

Q₃₂

—

146

—

105

Q₃₂

—

146

—

106

SD₃₂

—

107

D₁₆

—

108

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

109

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

110

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

111

C₃₂

—

112

C₂×Q₈

—

113

C₂×Q₈

—

292

—

114

C₂×D₄

—

115

Q₁₆

—

116

Q₁₆

—

117

Q₁₆

—

292

—

118

Q₁₆

—

292

—

119

Q₁₆

—

584

—

120

SD₁₆

—

121

D₈

—

122

D₈

—

123

C₂×C₈

#11924013

yes

characteristic

124

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

125

C₄⋊C₄

—

584

—

126

C₂²⋊C₄

—

127

C₄²

#11924013

yes

characteristic

128

C₁₆

#11924013

yes

—

129

C₁₆

D₄×D₇₃

yes

—

130

C₂³

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

584

—

134

Q₈

—

584

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

584

—

139

C₂×C₄

#14556532

yes

—

140

C₂×C₄

C₂×#11924013

yes

characteristic

141

C₈

#14556532

yes

—

142

C₈

#14556587

yes

characteristic

143

C₈

C₂×#11924013

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×#14556532

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

1168

—

150

C₄

#17988963

yes

characteristic

151

C₄

C₂×#14556532

yes

characteristic

152

C₂

—

153

C₂

#22612180

yes

characteristic

154

C₂

#22612248

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556532
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#2249	C₂
socle	C₂²×C₇₃	C₂×D₁₆
prime radical	C₂×D₁₆×C₇₃	C₂²
hypercenter	#2249	D₇₃
hypocenter	C₇₃	#13730
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2249

subgroup

G/H

center

C₂²

C₂×#14556532

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#2249

C₂

socle

C₂²×C₇₃

C₂×D₁₆

prime radical

C₂×D₁₆×C₇₃

C₂²

D₇₃

C₇₃

G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2249

N	H ≅ G/N	type
1. C₂×#17988427	C₂	semidirect
3. #22612184	C₂	semidirect
4. #22608065	C₂	semidirect
83. C₇₃	#13730	semidirect

H ≅ G/N

type

1. C₂×#17988427

C₂

semidirect

3. #22612184

C₂

semidirect

4. #22608065

C₂

semidirect

83. C₇₃

#13730

semidirect

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

subgroup	index
C₂×#17988427	2
C₂×#17988425	2
#22612184	2
#22608065	2
C₇₃×#2249	2
#13730	73

subgroup

index

C₇₃×#2249

Subgroups — #33007001

Conjugacy classes of subgroups

Complements

Maximal subgroups