Group #33006430 — subgroups

Subgroups — #33006430

Order 18688.

290 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂²×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988421	C₂	1	1	yes	maximal, characteristic
3	#22608111	C₂	1	4	yes	maximal, complemented
4	C₇₃×#2158	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₉₂	—	2	1	no	—
6	C₂×Dic₅₈₄	—	2	1	no	—
7	C₂×Dic₅₈₄	—	2	2	no	—
8	C₂×Dic₅₈₄	C₄	1	4	yes	complemented
9	C₂×Dic₅₈₄	C₂²	1	2	yes	—
10	#17988421	—	2	2	no	—
11	C₂×#14556497	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	#427×C₇₃	—	2	1	no	—
14	#367×C₇₃	—	2	1	no	—
15	#366×C₇₃	C₂²	1	2	yes	—
16	C₂²×Dic₁₄₆	—	4	1	no	—
17	C₂×Dic₂₉₂	—	2	4	no	—
18	C₂×Dic₂₉₂	—	2	2	no	—
19	C₂×Dic₂₉₂	—	4	1	no	—
20	C₂×Dic₂₉₂	—	4	2	no	—
21	Dic₅₈₄	—	2	4	no	—
22	Dic₅₈₄	—	2	2	no	—
23	Dic₅₈₄	C₂×C₄	1	4	yes	—
24	#14556497	—	2	2	no	—
25	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
26	#117×C₇₃	—	4	1	no	—
27	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
30	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
31	#107×C₇₃	—	2	2	no	—
32	#106×C₇₃	—	4	1	no	—
33	C₂×Dic₁₄₆	—	4	4	no	—
34	C₂×Dic₁₄₆	—	4	2	no	—
35	C₂×Dic₁₄₆	—	8	1	no	—
36	C₂×Dic₁₄₆	—	8	2	no	—
37	C₂²×Dic₇₃	—	8	1	no	—
38	Dic₂₉₂	—	2	4	no	—
39	Dic₂₉₂	—	4	4	no	—
40	Dic₂₉₂	—	4	2	no	—
41	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₇₃	—	4	2	no	—
47	C₂²⋊C₄×C₇₃	—	8	1	no	—
48	C₄²×C₇₃	—	8	1	no	—
49	C₁₆×C₇₃	C₂²⋊C₄	1	2	yes	—
50	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₇₃	D₁₆	1	1	yes	characteristic
53	C₂×C₄×C₇₃	—	8	2	no	—
54	C₂×C₄×C₇₃	#102	1	2	yes	—
55	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
56	C₂×C₄×C₇₃	SD₃₂	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₇₃	—	8	4	no	—
59	C₂×Dic₇₃	—	8	2	no	—
60	Dic₁₄₆	—	4	4	no	—
61	Dic₁₄₆	—	8	4	no	—
62	Dic₁₄₆	—	8	2	no	—
63	C₈×C₇₃	#102	1	2	yes	—
64	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₇₃	—	2	2	no	—
67	C₂²×C₇₃	#357	1	2	yes	—
68	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
69	C₄×C₇₃	—	16	2	no	—
70	C₄×C₇₃	#357	1	2	yes	—
71	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
72	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
73	Dic₇₃	—	8	4	no	—
74	#13618	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	—	2	2	no	—
76	C₂×C₇₃	C₂×#357	1	1	yes	characteristic
77	C₂×C₇₃	#2264	1	2	yes	—
78	C₂²×Q₃₂	—	73	1	no	—
79	C₂×M₆₄	—	73	1	no	—
80	#2158	D₇₃	1	1	yes	characteristic
81	#1416	—	73	4	no	—
82	C₇₃	#13618	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₁₆	—	146	1	no	—
84	C₂×Q₃₂	—	73	4	no	—
85	C₂×Q₃₂	—	73	2	no	—
86	C₂×Q₃₂	—	146	2	no	—
87	C₂×Q₃₂	—	146	1	no	—
88	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
89	#427	—	2	1	no	—
90	M₆₄	—	146	2	no	—
91	C₂×C₃₂	—	73	2	no	—
92	#367	—	2	1	no	—
93	#366	C₂×D₇₃	1	2	yes	—
94	C₂²×Q₈	—	292	1	no	—
95	C₂×Q₁₆	—	292	1	no	—
96	C₂×Q₁₆	—	292	2	no	—
97	C₂×Q₁₆	—	146	4	no	—
98	C₂×Q₁₆	—	146	2	no	—
99	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
100	#117	—	4	1	no	—
101	Q₃₂	—	73	4	no	—
102	Q₃₂	—	146	4	no	—
103	Q₃₂	—	146	2	no	—
104	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
105	C₂×C₁₆	D₂₉₂	1	2	yes	—
106	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
107	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
108	#107	—	2	2	no	—
109	#106	—	4	1	no	—
110	C₃₂	—	146	2	no	—
111	C₂×Q₈	—	292	4	no	—
112	C₂×Q₈	—	292	2	no	—
113	C₂×Q₈	—	584	1	no	—
114	C₂×Q₈	—	584	2	no	—
115	C₂²×C₄	—	584	1	no	—
116	C₂²×C₄	#11924013	1	1	yes	characteristic
117	Q₁₆	—	292	4	no	—
118	Q₁₆	—	292	2	no	—
119	Q₁₆	—	146	4	no	—
120	C₂×C₈	#11924005	1	2	yes	—
121	C₂×C₈	#11924013	1	1	yes	characteristic
122	C₂×C₈	#11924015	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
124	C₄⋊C₄	—	4	2	no	—
125	C₂²⋊C₄	—	8	1	no	—
126	C₄²	—	8	1	no	—
127	C₁₆	#11924005	1	2	yes	—
128	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
129	C₁₆	C₂×D₂₉₂	1	1	yes	characteristic
130	C₂³	#14556532	1	1	yes	characteristic
131	Q₈	—	292	4	no	—
132	Q₈	—	584	4	no	—
133	Q₈	—	584	2	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	584	4	no	—
136	C₂×C₄	—	584	2	no	—
137	C₂×C₄	#14556511	1	2	yes	—
138	C₂×C₄	#14556532	1	1	yes	characteristic
139	C₂×C₄	#14556534	1	2	yes	—
140	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
141	C₈	#14556511	1	2	yes	—
142	C₈	C₂×#11924005	1	1	yes	characteristic
143	C₈	C₂×#11924015	1	1	yes	characteristic
144	C₂²	—	2	2	no	—
145	C₂²	#17988387	1	2	yes	—
146	C₂²	C₂×#14556532	1	1	yes	characteristic
147	C₄	—	16	2	no	—
148	C₄	—	584	4	no	—
149	C₄	#17988387	1	2	yes	—
150	C₄	C₂×#14556511	1	1	yes	characteristic
151	C₄	C₂×#14556534	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	C₂×#17988387	1	1	yes	characteristic
154	C₂	#22612248	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₈₄

C₂

yes

maximal, characteristic

C₂×#17988421

C₂

yes

maximal, characteristic

#22608111

C₂

yes

maximal, complemented

C₇₃×#2158

C₂

yes

maximal, characteristic

C₂²×Dic₂₉₂

—

C₂×Dic₅₈₄

—

C₂×Dic₅₈₄

—

C₂×Dic₅₈₄

C₄

yes

complemented

C₂×Dic₅₈₄

C₂²

yes

—

#17988421

—

C₂×#14556497

C₂²

yes

—

C₂²×C₁₆×C₇₃

C₂²

yes

characteristic

#427×C₇₃

—

#367×C₇₃

—

#366×C₇₃

C₂²

yes

—

C₂²×Dic₁₄₆

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

Dic₅₈₄

—

Dic₅₈₄

—

Dic₅₈₄

C₂×C₄

yes

—

#14556497

—

C₂²×C₈×C₇₃

D₄

yes

characteristic

#117×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

—

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#106×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₂²×C₄×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

SD₁₆

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₁₆×C₇₃

C₂²⋊C₄

yes

—

C₁₆×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₂³×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

SD₃₂

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

#102

yes

—

C₈×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

#357

yes

—

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#357

yes

—

C₄×C₇₃

C₂×#102

yes

characteristic

C₄×C₇₃

C₂×SD₃₂

yes

characteristic

Dic₇₃

—

#13618

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×#357

yes

characteristic

C₂×C₇₃

#2264

yes

—

C₂²×Q₃₂

—

C₂×M₆₄

—

#2158

D₇₃

yes

characteristic

#1416

—

C₇₃

#13618

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

146

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

146

—

C₂×Q₃₂

—

146

—

C₂²×C₁₆

C₂×D₇₃

yes

characteristic

#427

—

M₆₄

—

146

—

C₂×C₃₂

—

#367

—

#366

C₂×D₇₃

yes

—

C₂²×Q₈

—

292

—

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂²×C₈

C₂²⋊D₇₃

yes

characteristic

100

#117

—

101

Q₃₂

—

102

Q₃₂

—

146

—

103

Q₃₂

—

146

—

104

C₂×C₁₆

C₄×D₇₃

yes

—

105

C₂×C₁₆

D₂₉₂

yes

—

106

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

107

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

108

#107

—

109

#106

—

110

C₃₂

—

146

—

111

C₂×Q₈

—

292

—

112

C₂×Q₈

—

292

—

113

C₂×Q₈

—

584

—

114

C₂×Q₈

—

584

—

115

C₂²×C₄

—

584

—

116

C₂²×C₄

#11924013

yes

characteristic

117

Q₁₆

—

292

—

118

Q₁₆

—

292

—

119

Q₁₆

—

146

—

120

C₂×C₈

#11924005

yes

—

121

C₂×C₈

#11924013

yes

characteristic

122

C₂×C₈

#11924015

yes

—

123

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

126

C₄²

—

127

C₁₆

#11924005

yes

—

128

C₁₆

C₂×C₄×D₇₃

yes

characteristic

129

C₁₆

C₂×D₂₉₂

yes

characteristic

130

C₂³

#14556532

yes

characteristic

131

Q₈

—

292

—

132

Q₈

—

584

—

133

Q₈

—

584

—

134

C₂×C₄

—

135

C₂×C₄

—

584

—

136

C₂×C₄

—

584

—

137

C₂×C₄

#14556511

yes

—

138

C₂×C₄

#14556532

yes

characteristic

139

C₂×C₄

#14556534

yes

—

140

C₂×C₄

C₂×#11924013

yes

characteristic

141

C₈

#14556511

yes

—

142

C₈

C₂×#11924005

yes

characteristic

143

C₈

C₂×#11924015

yes

characteristic

144

C₂²

—

145

C₂²

#17988387

yes

—

146

C₂²

C₂×#14556532

yes

characteristic

147

C₄

—

148

C₄

—

584

—

149

C₄

#17988387

yes

—

150

C₄

C₂×#14556511

yes

characteristic

151

C₄

C₂×#14556534

yes

characteristic

152

C₂

—

153

C₂

C₂×#17988387

yes

characteristic

154

C₂

#22612248

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556532
commutator	C₁₆×C₇₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#2158	C₂
socle	C₂²×C₇₃	C₂×D₁₆
prime radical	C₂³×C₇₃	D₁₆
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	#2158	D₇₃
hypocenter	C₇₃	#13618
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #2158

subgroup

G/H

center

C₂²

C₂×#14556532

commutator

C₁₆×C₇₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#2158

C₂

socle

C₂²×C₇₃

C₂×D₁₆

prime radical

C₂³×C₇₃

D₁₆

cosocle

C₂×C₁₆×C₇₃

C₂³

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ #2158

N	H ≅ G/N	type
3. #22608111	C₂	semidirect
8. C₂×Dic₅₈₄	C₄	semidirect
82. C₇₃	#13618	semidirect

H ≅ G/N

type

3. #22608111

C₂

semidirect

8. C₂×Dic₅₈₄

C₄

semidirect

82. C₇₃

#13618

semidirect

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

subgroup	index
C₂²×Dic₅₈₄	2
C₂×#17988421	2
#22608111	2
C₇₃×#2158	2
#13618	73

subgroup

index

C₂²×Dic₅₈₄

C₂×#17988421

#22608111

C₇₃×#2158

#13618

Subgroups — #33006430

Conjugacy classes of subgroups

Complements

Maximal subgroups