Group #32997739 — subgroups

Subgroups — #32997739

Order 18688.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22611813	C₂	1	1	yes	maximal, characteristic
2	#22611767	C₂	1	2	yes	maximal, complemented
3	#22607903	C₂	1	2	yes	maximal, complemented
4	C₁₆×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989019	C₄	1	1	yes	characteristic
7	#17989005	—	2	2	no	—
8	#17988705	C₂²	1	2	yes	—
9	#17988417	C₄	1	1	yes	characteristic
10	#17988395	C₄	1	2	yes	—
11	#17988372	—	2	2	no	—
12	C₄×#11924003	C₂²	1	1	yes	characteristic
13	C₈×#10596831	C₂²	1	1	yes	characteristic
14	#364×C₇₃	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	#14556613	C₂×C₄	1	2	yes	—
17	#14556597	—	2	2	no	—
18	Dic₅₈₄	C₈	1	1	yes	characteristic, complemented
19	#14556536	C₈	1	2	yes	complemented
20	D₁₁₆₈	C₈	1	1	yes	characteristic, complemented
21	#14556530	—	4	2	no	—
22	C₂×#11924003	D₄	1	1	yes	characteristic
23	C₂×#11924003	D₄	1	1	yes	characteristic
24	C₁₆×D₇₃	—	4	2	no	—
25	#14556527	C₂×C₄	1	2	yes	—
26	#14556519	—	2	2	no	—
27	C₄×#10596831	—	2	1	no	—
28	M₃₂×C₇₃	—	2	2	no	—
29	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	#11924028	—	2	2	no	—
33	Dic₂₉₂	C₂×C₈	1	2	yes	—
34	#11924017	—	2	2	no	—
35	D₅₈₄	C₂×C₈	1	2	yes	—
36	#11924012	—	2	2	no	—
37	#11924011	—	4	2	no	—
38	C₂×#10596831	—	2	2	no	—
39	C₈×D₇₃	—	4	2	no	—
40	#11924003	—	2	1	no	—
41	#11924003	D₈	1	1	yes	characteristic
42	#11924003	D₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₇₃	#41	1	1	yes	characteristic
46	C₁₆×C₇₃	—	2	1	no	—
47	C₁₆×C₇₃	—	4	2	no	—
48	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
52	C₂²⋊D₇₃	—	4	2	no	—
53	D₂₉₂	—	2	2	no	—
54	C₄×D₇₃	—	4	2	no	—
55	Dic₁₄₆	—	2	2	no	—
56	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₇₃	#131	1	1	yes	characteristic
58	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
59	C₈×C₇₃	#135	1	1	yes	characteristic
60	#10596831	—	2	4	no	—
61	#10596831	—	4	2	no	—
62	C₂²×C₇₃	#443	1	1	yes	characteristic
63	C₂×D₇₃	—	4	2	no	—
64	C₄×C₇₃	—	2	2	no	—
65	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
66	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
67	Dic₇₃	—	4	2	no	—
68	#12046	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	—	2	1	no	—
70	C₂×C₇₃	C₈×D₈	1	1	yes	characteristic
71	D₇₃	—	8	2	no	—
72	#2178	—	73	1	no	—
73	#2170	—	73	2	no	—
74	#1373	D₇₃	1	1	yes	characteristic, complemented
75	#1335	—	73	2	no	—
76	C₈×C₁₆	—	73	1	no	—
77	C₇₃	#12046	1	1	yes	characteristic, complemented, Sylow
78	#508	—	73	1	no	—
79	#504	—	146	2	no	—
80	#443	—	73	2	no	—
81	#368	—	73	1	no	—
82	#364	C₂×D₇₃	1	2	yes	—
83	#359	—	73	2	no	—
84	#350	—	146	2	no	—
85	C₄×C₁₆	—	73	1	no	—
86	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
87	C₈²	—	73	1	no	—
88	#135	—	73	2	no	—
89	#131	—	146	2	no	—
90	Q₃₂	—	73	1	no	—
91	SD₃₂	—	73	2	no	—
92	D₁₆	—	73	1	no	—
93	M₃₂	—	2	2	no	—
94	M₃₂	—	292	2	no	—
95	C₂×C₁₆	—	292	2	no	—
96	C₂×C₁₆	—	73	1	no	—
97	C₂×C₁₆	—	73	1	no	—
98	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
99	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
100	#108	—	73	2	no	—
101	#104	—	146	2	no	—
102	C₄×C₈	—	146	1	no	—
103	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	#41	—	146	2	no	—
105	Q₁₆	—	73	2	no	—
106	SD₁₆	—	146	2	no	—
107	D₈	—	73	2	no	—
108	M₁₆	—	292	2	no	—
109	M₁₆	—	146	2	no	—
110	C₂×C₈	—	292	2	no	—
111	C₂×C₈	—	146	2	no	—
112	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
113	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
114	C₄²	#11924029	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	—	73	1	no	—
118	C₁₆	—	73	1	no	—
119	C₁₆	—	146	1	no	—
120	C₁₆	C₈×D₇₃	1	1	yes	characteristic
121	C₁₆	C₈×D₇₃	1	1	yes	characteristic
122	Q₈	—	146	2	no	—
123	D₄	—	292	2	no	—
124	D₄	—	146	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	292	2	no	—
127	C₂×C₄	#14556553	1	1	yes	characteristic
128	C₈	—	292	2	no	—
129	C₈	—	146	4	no	—
130	C₈	C₂×C₈×D₇₃	1	1	yes	characteristic
131	C₈	#14556599	1	1	yes	characteristic
132	C₈	D₈×D₇₃	1	1	yes	characteristic
133	C₈	#14556614	1	1	yes	characteristic
134	C₂²	—	292	2	no	—
135	C₂²	#17988703	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	292	2	no	—
138	C₄	#17988652	1	1	yes	characteristic
139	C₄	#17988669	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	584	2	no	—
142	C₂	#22608725	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611813

C₂

yes

maximal, characteristic

#22611767

C₂

yes

maximal, complemented

#22607903

C₂

yes

maximal, complemented

C₁₆×#10596831

C₂

yes

maximal, characteristic, complemented

C₇₃×#1373

C₂

yes

maximal, characteristic, complemented

#17989019

C₄

yes

characteristic

#17989005

—

#17988705

C₂²

yes

—

#17988417

C₄

yes

characteristic

#17988395

C₄

yes

—

#17988372

—

C₄×#11924003

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

#364×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

#14556613

C₂×C₄

yes

—

#14556597

—

Dic₅₈₄

C₈

yes

characteristic, complemented

#14556536

C₈

yes

complemented

D₁₁₆₈

C₈

yes

characteristic, complemented

#14556530

—

C₂×#11924003

D₄

yes

characteristic

C₂×#11924003

D₄

yes

characteristic

C₁₆×D₇₃

—

#14556527

C₂×C₄

yes

—

#14556519

—

C₄×#10596831

—

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

#11924028

—

Dic₂₉₂

C₂×C₈

yes

—

#11924017

—

D₅₈₄

C₂×C₈

yes

—

#11924012

—

#11924011

—

C₂×#10596831

—

C₈×D₇₃

—

#11924003

—

#11924003

D₈

yes

characteristic

#11924003

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

#41

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂²⋊D₇₃

—

D₂₉₂

—

C₄×D₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₂²×C₈

yes

characteristic

C₈×C₇₃

#131

yes

characteristic

C₈×C₇₃

C₂×D₈

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

#443

yes

characteristic

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

C₄×D₈

yes

characteristic

Dic₇₃

—

#12046

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₈×D₈

yes

characteristic

D₇₃

—

#2178

—

#2170

—

#1373

D₇₃

yes

characteristic, complemented

#1335

—

C₈×C₁₆

—

C₇₃

#12046

yes

characteristic, complemented, Sylow

#508

—

#504

—

146

—

#443

—

#368

—

#364

C₂×D₇₃

yes

—

#359

—

#350

—

146

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

—

#135

—

#131

—

146

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

292

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₄×D₇₃

yes

characteristic

100

#108

—

101

#104

—

146

—

102

C₄×C₈

—

146

—

103

C₄×C₈

C₂²×D₇₃

yes

characteristic

104

#41

—

146

—

105

Q₁₆

—

106

SD₁₆

—

146

—

107

D₈

—

108

M₁₆

—

292

—

109

M₁₆

—

146

—

110

C₂×C₈

—

292

—

111

C₂×C₈

—

146

—

112

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

113

C₂×C₈

D₄×D₇₃

yes

characteristic

114

C₄²

#11924029

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

146

—

120

C₁₆

C₈×D₇₃

yes

characteristic

121

C₁₆

C₈×D₇₃

yes

characteristic

122

Q₈

—

146

—

123

D₄

—

292

—

124

D₄

—

146

—

125

C₂×C₄

—

126

C₂×C₄

—

292

—

127

C₂×C₄

#14556553

yes

characteristic

128

C₈

—

292

—

129

C₈

—

146

—

130

C₈

C₂×C₈×D₇₃

yes

characteristic

131

C₈

#14556599

yes

characteristic

132

C₈

D₈×D₇₃

yes

characteristic

133

C₈

#14556614

yes

characteristic

134

C₂²

—

292

—

135

C₂²

#17988703

yes

characteristic

136

C₄

—

137

C₄

—

292

—

138

C₄

#17988652

yes

characteristic

139

C₄

#17988669

yes

characteristic

140

C₂

—

141

C₂

—

584

—

142

C₂

#22608725

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₇₃
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1373	C₂
socle	C₂×C₇₃	C₈×D₈
prime radical	#17989019	C₄
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#1373	D₇₃
hypocenter	C₇₃	#12046
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₇₃

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1373

C₂

socle

C₂×C₇₃

C₈×D₈

prime radical

#17989019

C₄

cosocle

C₄×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #22611767	C₂	semidirect
3. #22607903	C₂	semidirect
4. C₁₆×#10596831	C₂	semidirect
5. C₇₃×#1373	C₂	semidirect
18. Dic₅₈₄	C₈	semidirect
19. #14556536	C₈	semidirect
20. D₁₁₆₈	C₈	semidirect
74. #1373	D₇₃	semidirect
77. C₇₃	#12046	semidirect

H ≅ G/N

type

2. #22611767

C₂

semidirect

3. #22607903

C₂

semidirect

4. C₁₆×#10596831

C₂

semidirect

5. C₇₃×#1373

C₂

semidirect

18. Dic₅₈₄

C₈

semidirect

19. #14556536

C₈

semidirect

20. D₁₁₆₈

C₈

semidirect

74. #1373

D₇₃

semidirect

77. C₇₃

#12046

semidirect

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

subgroup	index
#22611813	2
#22611767	2
#22607903	2
C₁₆×#10596831	2
C₇₃×#1373	2
#12046	73

subgroup

index

#22611813

#22611767

#22607903

C₁₆×#10596831

C₇₃×#1373

#12046

Subgroups — #32997739

Conjugacy classes of subgroups

Complements

Maximal subgroups