Group #32997875 — subgroups

Subgroups — #32997875

Order 18688.

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

normal subgroups only

This group has 204 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	#22611818	C₂	1	1	yes	maximal, characteristic
2	#22611774	C₂	1	2	yes	maximal, complemented
3	#22607999	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607908	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989019	C₄	1	1	yes	characteristic
7	#17989003	—	2	2	no	—
8	#17988705	C₂²	1	2	yes	—
9	#17988417	C₄	1	1	yes	characteristic
10	#17988408	C₂²	1	2	yes	—
11	#17988395	C₄	1	2	yes	—
12	#17988370	—	2	2	no	—
13	#346×C₇₃	C₂²	1	1	yes	characteristic
14	#346×C₇₃	C₂²	1	1	yes	characteristic
15	#322×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	#14556531	—	2	2	no	—
22	#14556530	—	4	2	no	—
23	C₁₆×D₇₃	—	4	2	no	—
24	#14556527	C₂×C₄	1	2	yes	—
25	#14556519	—	2	2	no	—
26	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
29	C₄×C₈×C₇₃	—	2	1	no	—
30	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	#11924028	—	2	2	no	—
32	Dic₂₉₂	C₂×C₈	1	2	yes	—
33	#11924017	—	2	2	no	—
34	D₅₈₄	C₂×C₈	1	2	yes	—
35	#11924012	—	2	2	no	—
36	#11924011	—	4	2	no	—
37	C₈×D₇₃	—	4	2	no	—
38	#11924003	—	4	2	no	—
39	C₂×C₈×C₇₃	—	4	1	no	—
40	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₇₃	#41	1	1	yes	characteristic
43	C₁₆×C₇₃	—	2	1	no	—
44	C₁₆×C₇₃	—	2	2	no	—
45	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
49	C₂²⋊D₇₃	—	4	2	no	—
50	D₂₉₂	—	2	2	no	—
51	C₄×D₇₃	—	4	2	no	—
52	Dic₁₄₆	—	2	2	no	—
53	C₈×C₇₃	—	8	1	no	—
54	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₇₃	#131	1	1	yes	characteristic
56	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
57	#10596831	—	4	2	no	—
58	C₂²×C₇₃	#444	1	1	yes	characteristic
59	C₂×D₇₃	—	4	2	no	—
60	C₄×C₇₃	—	4	1	no	—
61	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₇₃	#442	1	1	yes	characteristic
63	Dic₇₃	—	4	2	no	—
64	#12068	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	—	2	1	no	—
66	C₂×C₇₃	#1580	1	1	yes	characteristic
67	D₇₃	—	8	2	no	—
68	#2179	—	73	1	no	—
69	#2171	—	73	2	no	—
70	#1369	—	73	1	no	—
71	#1336	—	73	2	no	—
72	#1313	D₇₃	1	1	yes	characteristic, complemented
73	C₇₃	#12068	1	1	yes	characteristic, complemented, Sylow
74	#508	—	73	1	no	—
75	#504	—	146	2	no	—
76	#443	—	73	2	no	—
77	#368	—	73	1	no	—
78	#364	—	73	2	no	—
79	#359	—	73	2	no	—
80	#350	—	146	2	no	—
81	#346	C₂×D₇₃	1	1	yes	characteristic
82	#346	C₂×D₇₃	1	1	yes	characteristic
83	#322	C₂×D₇₃	1	1	yes	characteristic
84	#135	—	73	2	no	—
85	#131	—	146	2	no	—
86	Q₃₂	—	73	1	no	—
87	SD₃₂	—	73	2	no	—
88	D₁₆	—	73	1	no	—
89	M₃₂	—	292	2	no	—
90	M₃₂	—	146	2	no	—
91	C₂×C₁₆	—	292	2	no	—
92	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
94	C₂×C₁₆	D₂₉₂	1	2	yes	—
95	#108	—	73	2	no	—
96	#104	—	146	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
99	#41	—	146	2	no	—
100	Q₁₆	—	73	2	no	—
101	SD₁₆	—	146	2	no	—
102	D₈	—	73	2	no	—
103	M₁₆	—	292	2	no	—
104	M₁₆	—	146	2	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	292	2	no	—
107	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
108	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
109	C₄²	#11924028	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	292	2	no	—
113	C₁₆	C₈×D₇₃	1	1	yes	characteristic
114	C₁₆	C₈×D₇₃	1	1	yes	characteristic
115	Q₈	—	146	2	no	—
116	D₄	—	292	2	no	—
117	D₄	—	146	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	292	2	no	—
120	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
121	C₈	—	8	1	no	—
122	C₈	—	292	2	no	—
123	C₈	C₂×C₈×D₇₃	1	1	yes	characteristic
124	C₈	#14556597	1	1	yes	characteristic
125	C₈	#14556619	1	2	yes	—
126	C₂²	—	292	2	no	—
127	C₂²	#17988710	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	292	2	no	—
130	C₄	C₈×D₂₉₂	1	1	yes	characteristic
131	C₄	#17988700	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	584	2	no	—
134	C₂	#22608756	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22611818

C₂

yes

maximal, characteristic

#22611774

C₂

yes

maximal, complemented

#22607999

C₂

yes

maximal, characteristic, complemented

#22607908

C₂

yes

maximal, complemented

C₇₃×#1313

C₂

yes

maximal, characteristic, complemented

#17989019

C₄

yes

characteristic

#17989003

—

#17988705

C₂²

yes

—

#17988417

C₄

yes

characteristic

#17988408

C₂²

yes

—

#17988395

C₄

yes

—

#17988370

—

#346×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₂²

yes

characteristic

#322×C₇₃

C₂²

yes

characteristic

#14556613

C₂×C₄

yes

—

#14556597

—

Dic₅₈₄

C₈

yes

characteristic, complemented

#14556536

C₈

yes

complemented

D₁₁₆₈

C₈

yes

characteristic, complemented

#14556531

—

#14556530

—

C₁₆×D₇₃

—

#14556527

C₂×C₄

yes

—

#14556519

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

#11924028

—

Dic₂₉₂

C₂×C₈

yes

—

#11924017

—

D₅₈₄

C₂×C₈

yes

—

#11924012

—

#11924011

—

C₈×D₇₃

—

#11924003

—

C₂×C₈×C₇₃

—

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

—

C₈×C₇₃

C₂²×C₈

yes

C₈×C₇₃

yes

C₈×C₇₃

yes

—

#10596831

—

C₂²×C₇₃

#444

yes

characteristic

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

#442

yes

characteristic

Dic₇₃

—

#12068

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1580

yes

characteristic

D₇₃

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₇₃

#12068

yes

characteristic, complemented, Sylow

#508

—

#504

—

146

—

—

—

—

—

—

146

—

#346

C₂×D₇₃

yes

characteristic

#346

C₂×D₇₃

yes

characteristic

#322

C₂×D₇₃

yes

characteristic

#135

—

#131

—

146

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

292

—

M₃₂

—

146

—

C₂×C₁₆

—

292

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

D₂₉₂

yes

—

#108

—

#104

—

146

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

#41

—

146

—

100

Q₁₆

—

101

SD₁₆

—

146

—

102

D₈

—

103

M₁₆

—

292

—

104

M₁₆

—

146

—

105

C₂×C₈

—

106

C₂×C₈

—

292

—

107

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

108

C₂×C₈

C₂×D₂₉₂

yes

characteristic

109

C₄²

#11924028

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

292

—

113

C₁₆

C₈×D₇₃

yes

characteristic

114

C₁₆

C₈×D₇₃

yes

characteristic

115

Q₈

—

146

—

116

D₄

—

292

—

117

D₄

—

146

—

118

C₂×C₄

—

119

C₂×C₄

—

292

—

120

C₂×C₄

C₄×D₂₉₂

yes

characteristic

121

C₈

—

122

C₈

—

292

—

123

C₈

C₂×C₈×D₇₃

yes

characteristic

124

C₈

#14556597

yes

characteristic

125

C₈

#14556619

yes

—

126

C₂²

—

292

—

127

C₂²

#17988710

yes

characteristic

128

C₄

—

129

C₄

—

292

—

130

C₄

C₈×D₂₉₂

yes

characteristic

131

C₄

#17988700

yes

characteristic

132

C₂

—

133

C₂

—

584

—

134

C₂

#22608756

yes

characteristic

	subgroup	G/H
center	C₄	#17988700
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1313	C₂
socle	C₂×C₇₃	#1580
prime radical	#17989019	C₄
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#1313	D₇₃
hypocenter	C₇₃	#12068
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

subgroup

G/H

center

C₄

#17988700

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1313

C₂

C₂×C₇₃

C₄

C₄×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

N	H ≅ G/N	type
2. #22611774	C₂	semidirect
3. #22607999	C₂	semidirect
4. #22607908	C₂	semidirect
5. C₇₃×#1313	C₂	semidirect
18. Dic₅₈₄	C₈	semidirect
19. #14556536	C₈	semidirect
20. D₁₁₆₈	C₈	semidirect
72. #1313	D₇₃	semidirect
73. C₇₃	#12068	semidirect

H ≅ G/N

type

2. #22611774

C₂

semidirect

3. #22607999

C₂

semidirect

4. #22607908

C₂

semidirect

5. C₇₃×#1313

C₂

semidirect

18. Dic₅₈₄

C₈

semidirect

19. #14556536

C₈

semidirect

20. D₁₁₆₈

C₈

semidirect

72. #1313

D₇₃

semidirect

73. C₇₃

#12068

semidirect

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

subgroup	index
#22611818	2
#22611774	2
#22607999	2
#22607908	2
C₇₃×#1313	2
#12068	73

subgroup

index

C₇₃×#1313

Subgroups — #32997875

Conjugacy classes of subgroups

Complements

Maximal subgroups