Group #32998010 — subgroups

Subgroups — #32998010

Order 18688.

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

normal subgroups only

This group has 226 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	#22611808	C₂	1	1	yes	maximal, characteristic
2	#22608846	C₂	1	2	yes	maximal, complemented
3	#22607997	C₂	1	1	yes	maximal, characteristic, complemented
4	#22607899	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₆₈	C₄	1	1	yes	characteristic
7	C₄×D₅₈₄	C₂²	1	2	yes	—
8	#17988666	—	2	2	no	—
9	#17988415	C₄	1	1	yes	characteristic
10	#17988389	C₄	1	2	yes	—
11	#17988338	C₂²	1	2	yes	—
12	#17988316	—	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	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
17	C₂×#11924011	—	4	2	no	—
18	C₄×D₂₉₂	—	2	2	no	—
19	D₁₁₆₈	—	2	2	no	—
20	#14556525	C₂×C₄	1	2	yes	—
21	C₄⋊#10596831	—	2	2	no	—
22	#14556513	—	2	2	no	—
23	#14556503	—	4	2	no	—
24	C₂×C₁₆×C₇₃	—	2	1	no	—
25	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	D₄	1	2	yes	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×D₂₉₂	—	2	2	no	—
30	C₂×C₄×D₇₃	—	4	2	no	—
31	D₅₈₄	—	2	2	no	—
32	D₅₈₄	—	2	2	no	—
33	#11924011	—	8	4	no	—
34	C₂×#10596831	—	4	2	no	—
35	C₄⋊Dic₇₃	—	2	2	no	—
36	#11924005	—	4	2	no	—
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
40	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₃	—	2	1	no	—
42	C₁₆×C₇₃	—	4	1	no	—
43	C₁₆×C₇₃	M₁₆	1	2	yes	complemented
44	C₂²×D₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₇₃	—	4	2	no	—
48	D₂₉₂	—	4	2	no	—
49	D₂₉₂	—	4	2	no	—
50	C₄×D₇₃	—	8	2	no	—
51	C₄×D₇₃	—	8	2	no	—
52	C₈×C₇₃	—	2	1	no	—
53	C₈×C₇₃	—	4	2	no	—
54	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₇₃	#131	1	1	yes	characteristic
56	#10596831	—	8	2	no	—
57	C₂²×C₇₃	#442	1	1	yes	characteristic
58	C₂×D₇₃	—	8	2	no	—
59	C₂×D₇₃	—	8	2	no	—
60	C₂×D₇₃	—	8	2	no	—
61	C₄×C₇₃	—	2	2	no	—
62	C₄×C₇₃	#436	1	1	yes	characteristic
63	C₄×C₇₃	#444	1	1	yes	characteristic
64	Dic₇₃	—	8	2	no	—
65	#12089	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	#1594	1	1	yes	characteristic
67	C₂×C₇₃	AGL(1,ℤ/16)	1	2	yes	—
68	D₇₃	—	16	2	no	—
69	#2177	—	73	1	no	—
70	#1594	—	73	2	no	—
71	#1368	—	73	1	no	—
72	#1333	—	73	2	no	—
73	#1313	D₇₃	1	1	yes	characteristic, complemented
74	C₇₃	#12089	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	—	73	1	no	—
76	C₄×D₈	—	73	2	no	—
77	#436	—	146	2	no	—
78	#367	—	73	1	no	—
79	#357	—	73	2	no	—
80	#346	C₂×D₇₃	1	1	yes	characteristic
81	#346	C₂×D₇₃	1	1	yes	characteristic
82	#334	—	73	2	no	—
83	D₄⋊C₈	—	146	2	no	—
84	#322	C₂×D₇₃	1	1	yes	characteristic
85	C₂×D₈	—	73	2	no	—
86	C₂×M₁₆	—	292	2	no	—
87	C₄×D₄	—	146	2	no	—
88	D₁₆	—	146	2	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
92	#107	—	73	2	no	—
93	C₄⋊C₈	—	146	2	no	—
94	#102	—	146	2	no	—
95	C₂²⋊C₈	—	292	2	no	—
96	C₄×C₈	D₂₉₂	1	2	yes	—
97	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
98	C₂×D₄	—	146	2	no	—
99	C₂²×C₄	—	292	2	no	—
100	D₈	—	146	2	no	—
101	D₈	—	146	2	no	—
102	M₁₆	—	584	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	292	2	no	—
105	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
106	C₂×C₈	#11924028	1	1	yes	characteristic
107	C₄⋊C₄	—	146	2	no	—
108	C₂²⋊C₄	—	292	2	no	—
109	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	#11924011	1	2	yes	complemented
113	C₂³	—	292	2	no	—
114	D₄	—	292	2	no	—
115	D₄	—	292	2	no	—
116	C₂×C₄	—	292	2	no	—
117	C₂×C₄	—	584	2	no	—
118	C₂×C₄	—	584	2	no	—
119	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
120	C₂×C₄	#14556619	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	4	2	no	—
123	C₈	—	584	2	no	—
124	C₈	C₂×#11924011	1	1	yes	characteristic
125	C₈	#14556597	1	1	yes	characteristic
126	C₂²	—	584	2	no	—
127	C₂²	—	584	2	no	—
128	C₂²	—	584	2	no	—
129	C₂²	#17988700	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	584	2	no	—
132	C₄	#17988666	1	1	yes	characteristic
133	C₄	#17988710	1	1	yes	characteristic
134	C₂	—	1168	2	no	—
135	C₂	#22608846	1	1	yes	characteristic
136	C₂	#22611831	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22611808

C₂

yes

maximal, characteristic

#22608846

C₂

yes

maximal, complemented

#22607997

C₂

yes

maximal, characteristic, complemented

#22607899

C₂

yes

maximal, complemented

C₇₃×#1313

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₆₈

C₄

yes

characteristic

C₄×D₅₈₄

C₂²

yes

—

#17988666

—

#17988415

C₄

yes

characteristic

#17988389

C₄

yes

—

#17988338

C₂²

yes

—

#17988316

—

#346×C₇₃

C₂²

yes

characteristic

#346×C₇₃

C₂²

yes

characteristic

#322×C₇₃

C₂²

yes

characteristic

C₂×D₅₈₄

C₂×C₄

yes

—

C₂×#11924011

—

C₄×D₂₉₂

—

D₁₁₆₈

—

#14556525

C₂×C₄

yes

—

C₄⋊#10596831

—

#14556513

—

#14556503

—

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×D₂₉₂

—

C₂×C₄×D₇₃

—

D₅₈₄

—

D₅₈₄

—

#11924011

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924005

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

complemented

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

#442

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₇₃

#12089

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

#436

—

146

—

#367

—

#357

—

#346

C₂×D₇₃

yes

characteristic

#346

C₂×D₇₃

yes

characteristic

#334

—

D₄⋊C₈

—

146

—

#322

C₂×D₇₃

yes

characteristic

C₂×D₈

—

C₂×M₁₆

—

292

—

C₄×D₄

—

146

—

D₁₆

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₄×D₇₃

yes

characteristic

#107

—

C₄⋊C₈

—

146

—

#102

—

146

—

C₂²⋊C₈

—

292

—

C₄×C₈

D₂₉₂

yes

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×D₄

—

146

—

C₂²×C₄

—

292

—

100

D₈

—

146

—

101

D₈

—

146

—

102

M₁₆

—

584

—

103

C₂×C₈

—

104

C₂×C₈

—

292

—

105

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

106

C₂×C₈

#11924028

yes

characteristic

107

C₄⋊C₄

—

146

—

108

C₂²⋊C₄

—

292

—

109

C₄²

C₂×D₂₉₂

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

#11924011

yes

complemented

113

C₂³

—

292

—

114

D₄

—

292

—

115

D₄

—

292

—

116

C₂×C₄

—

292

—

117

C₂×C₄

—

584

—

118

C₂×C₄

—

584

—

119

C₂×C₄

C₄×D₂₉₂

yes

characteristic

120

C₂×C₄

#14556619

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

584

—

124

C₈

C₂×#11924011

yes

characteristic

125

C₈

#14556597

yes

characteristic

126

C₂²

—

584

—

127

C₂²

—

584

—

128

C₂²

—

584

—

129

C₂²

#17988700

yes

characteristic

130

C₄

—

131

C₄

—

584

—

132

C₄

#17988666

yes

characteristic

133

C₄

#17988710

yes

characteristic

134

C₂

—

1168

—

135

C₂

#22608846

yes

characteristic

136

C₂

#22611831

yes

—

	subgroup	G/H
center	C₂²	#17988700
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1313	C₂
socle	C₂²×C₇₃	#442
prime radical	C₂×D₁₁₆₈	C₄
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#1313	D₇₃
hypocenter	C₇₃	#12089
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#17988700

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1313

C₂

socle

C₂²×C₇₃

#442

prime radical

C₂×D₁₁₆₈

C₄

cosocle

C₄×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
2. #22608846	C₂	semidirect
3. #22607997	C₂	semidirect
4. #22607899	C₂	semidirect
5. C₇₃×#1313	C₂	semidirect
43. C₁₆×C₇₃	M₁₆	semidirect
73. #1313	D₇₃	semidirect
74. C₇₃	#12089	semidirect
112. C₁₆	#11924011	semidirect

H ≅ G/N

type

2. #22608846

C₂

semidirect

3. #22607997

C₂

semidirect

4. #22607899

C₂

semidirect

5. C₇₃×#1313

C₂

semidirect

43. C₁₆×C₇₃

M₁₆

semidirect

73. #1313

D₇₃

semidirect

74. C₇₃

#12089

semidirect

112. C₁₆

#11924011

semidirect

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

subgroup	index
#22611808	2
#22608846	2
#22607997	2
#22607899	2
C₇₃×#1313	2
#12089	73

subgroup

index

C₇₃×#1313

Subgroups — #32998010

Conjugacy classes of subgroups

Complements

Maximal subgroups