Group #32998009 — subgroups

Subgroups — #32998009

Order 18688.

226 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 77 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	#22608845	C₂	1	2	yes	maximal, complemented
2	#22607997	C₂	1	1	yes	maximal, characteristic, complemented
3	#22607898	C₂	1	2	yes	maximal, complemented
4	#22607851	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2177	C₂	1	1	yes	maximal, characteristic
6	#17988665	—	2	2	no	—
7	#17988363	C₂²	1	1	yes	characteristic
8	#17988338	C₂²	1	2	yes	—
9	D₄⋊#10596831	—	2	2	no	—
10	#17988308	C₂²	1	1	yes	characteristic
11	C₂×D₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₄×D₈×C₇₃	C₂²	1	2	yes	—
13	#367×C₇₃	C₄	1	1	yes	characteristic
14	#357×C₇₃	C₄	1	2	yes	—
15	#346×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924012	—	4	2	no	—
17	C₂×#11924003	—	2	1	no	—
18	C₄⋊#10596831	—	2	2	no	—
19	C₂²⋊#10596831	—	4	2	no	—
20	C₄×#10596831	D₄	1	2	yes	—
21	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	—
22	C₄×D₄×C₇₃	—	2	2	no	—
23	D₁₆×C₇₃	—	2	2	no	—
24	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
26	#107×C₇₃	C₂×C₄	1	2	yes	—
27	#102×C₇₃	—	2	2	no	—
28	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	#11924012	—	8	4	no	—
30	C₂×#10596831	—	2	2	no	—
31	C₂×#10596831	—	4	2	no	—
32	#11924003	—	4	1	no	—
33	C₂×D₄×C₇₃	—	2	2	no	—
34	C₂²×C₄×C₇₃	—	4	2	no	—
35	D₈×C₇₃	—	2	2	no	—
36	D₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	2	2	no	—
40	C₂²⋊C₄×C₇₃	—	4	2	no	—
41	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₃	—	2	1	no	—
43	C₁₆×C₇₃	M₁₆	1	2	yes	complemented
44	C₂³×C₇₃	—	4	2	no	—
45	D₄×C₇₃	—	4	2	no	—
46	D₄×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	8	2	no	—
49	C₂×C₄×C₇₃	—	8	2	no	—
50	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
52	C₈×C₇₃	—	2	1	no	—
53	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
54	C₈×C₇₃	#131	1	1	yes	characteristic
55	#10596831	—	4	2	no	—
56	#10596831	—	8	2	no	—
57	C₂²×C₇₃	—	8	2	no	—
58	C₂²×C₇₃	—	8	2	no	—
59	C₂²×C₇₃	—	8	2	no	—
60	C₂²×C₇₃	#442	1	1	yes	characteristic
61	C₄×C₇₃	—	2	2	no	—
62	C₄×C₇₃	—	8	2	no	—
63	C₄×C₇₃	#436	1	1	yes	characteristic
64	C₄×C₇₃	#444	1	1	yes	characteristic
65	#12089	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	16	2	no	—
67	C₂×C₇₃	#1594	1	1	yes	characteristic
68	C₂×C₇₃	AGL(1,ℤ/16)	1	2	yes	—
69	#2177	D₇₃	1	1	yes	characteristic
70	#1594	—	73	2	no	—
71	#1368	—	73	1	no	—
72	#1333	—	73	2	no	—
73	#1313	—	73	1	no	—
74	C₇₃	#12089	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	Dic₇₃	1	1	yes	characteristic
76	C₄×D₈	C₂×D₇₃	1	2	yes	—
77	#436	—	146	2	no	—
78	#367	Dic₇₃	1	1	yes	characteristic
79	#357	Dic₇₃	1	2	yes	—
80	#346	—	73	1	no	—
81	#346	C₂×D₇₃	1	1	yes	characteristic
82	#334	—	73	2	no	—
83	D₄⋊C₈	—	146	2	no	—
84	#322	—	73	1	no	—
85	C₂×D₈	C₂×Dic₇₃	1	2	yes	—
86	C₂×M₁₆	—	292	2	no	—
87	C₄×D₄	—	2	2	no	—
88	D₁₆	—	2	2	no	—
89	C₂×C₁₆	—	146	1	no	—
90	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
91	C₂×C₁₆	C₂×Dic₇₃	1	1	yes	characteristic
92	#107	C₂×Dic₇₃	1	2	yes	—
93	C₄⋊C₈	—	146	2	no	—
94	#102	—	2	2	no	—
95	C₂²⋊C₈	—	292	2	no	—
96	C₄×C₈	—	73	2	no	—
97	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
98	C₂×D₄	—	2	2	no	—
99	C₂²×C₄	—	4	2	no	—
100	D₈	—	2	2	no	—
101	D₈	—	2	2	no	—
102	M₁₆	—	584	4	no	—
103	C₂×C₈	—	292	2	no	—
104	C₂×C₈	—	146	2	no	—
105	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
106	C₂×C₈	#11924027	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	4	2	no	—
109	C₄²	D₄×D₇₃	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	292	1	no	—
112	C₁₆	#11924012	1	2	yes	complemented
113	C₂³	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	8	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
120	C₂×C₄	#14556615	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	292	2	no	—
123	C₈	—	584	2	no	—
124	C₈	C₂×#11924012	1	1	yes	characteristic
125	C₈	#14556598	1	1	yes	characteristic
126	C₂²	—	8	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	2	no	—
129	C₂²	#17988699	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	8	2	no	—
132	C₄	#17988665	1	1	yes	characteristic
133	C₄	#17988709	1	1	yes	characteristic
134	C₂	—	16	2	no	—
135	C₂	#22608845	1	1	yes	characteristic
136	C₂	#22611830	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22608845

C₂

yes

maximal, complemented

#22607997

C₂

yes

maximal, characteristic, complemented

#22607898

C₂

yes

maximal, complemented

#22607851

C₂

yes

maximal, characteristic, complemented

C₇₃×#2177

C₂

yes

maximal, characteristic

#17988665

—

#17988363

C₂²

yes

characteristic

#17988338

C₂²

yes

—

D₄⋊#10596831

—

#17988308

C₂²

yes

characteristic

C₂×D₁₆×C₇₃

C₄

yes

characteristic

C₄×D₈×C₇₃

C₂²

yes

—

#367×C₇₃

C₄

yes

characteristic

#357×C₇₃

C₄

yes

—

#346×C₇₃

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×D₈×C₇₃

C₂×C₄

yes

—

C₄×D₄×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

#107×C₇₃

C₂×C₄

yes

—

#102×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

—

—

—

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

#41

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

M₁₆

yes

complemented

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×D₄

yes

characteristic

C₂×C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₈×C₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

C₈×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#442

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1594

yes

characteristic

C₂×C₇₃

AGL(1,ℤ/16)

yes

—

#2177

D₇₃

yes

—

—

—

—

C₇₃

#12089

yes

characteristic, complemented, Sylow

C₂×D₁₆

Dic₇₃

yes

characteristic

C₄×D₈

C₂×D₇₃

yes

—

#436

—

146

—

#367

Dic₇₃

yes

characteristic

#357

Dic₇₃

yes

—

#346

—

#346

C₂×D₇₃

yes

characteristic

#334

—

D₄⋊C₈

—

146

—

#322

—

C₂×D₈

C₂×Dic₇₃

yes

—

C₂×M₁₆

—

292

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

C₂×C₁₆

C₂×Dic₇₃

yes

characteristic

#107

C₂×Dic₇₃

yes

—

C₄⋊C₈

—

146

—

#102

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

100

D₈

—

101

D₈

—

102

M₁₆

—

584

—

103

C₂×C₈

—

292

—

104

C₂×C₈

—

146

—

105

C₂×C₈

C₂²×Dic₇₃

yes

characteristic

106

C₂×C₈

#11924027

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

D₄×D₇₃

yes

characteristic

110

C₁₆

—

111

C₁₆

—

292

—

112

C₁₆

#11924012

yes

complemented

113

C₂³

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

D₄×Dic₇₃

yes

characteristic

120

C₂×C₄

#14556615

yes

—

121

C₈

—

122

C₈

—

292

—

123

C₈

—

584

—

124

C₈

C₂×#11924012

yes

characteristic

125

C₈

#14556598

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#17988699

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

#17988665

yes

characteristic

133

C₄

#17988709

yes

characteristic

134

C₂

—

135

C₂

#22608845

yes

characteristic

136

C₂

#22611830

yes

—

	subgroup	G/H
center	C₂²	#17988699
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2177	C₂
socle	C₂²×C₇₃	#442
prime radical	C₂×D₁₆×C₇₃	C₄
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	#2177	D₇₃
hypocenter	C₇₃	#12089
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

subgroup

G/H

center

C₂²

#17988699

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2177

C₂

socle

C₂²×C₇₃

#442

prime radical

C₂×D₁₆×C₇₃

C₄

cosocle

C₄×C₈×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2177

N	H ≅ G/N	type
1. #22608845	C₂	semidirect
2. #22607997	C₂	semidirect
3. #22607898	C₂	semidirect
4. #22607851	C₂	semidirect
43. C₁₆×C₇₃	M₁₆	semidirect
74. C₇₃	#12089	semidirect
112. C₁₆	#11924012	semidirect

H ≅ G/N

type

1. #22608845

C₂

semidirect

2. #22607997

C₂

semidirect

3. #22607898

C₂

semidirect

4. #22607851

C₂

semidirect

43. C₁₆×C₇₃

M₁₆

semidirect

74. C₇₃

#12089

semidirect

112. C₁₆

#11924012

semidirect

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

subgroup	index
#22608845	2
#22607997	2
#22607898	2
#22607851	2
C₇₃×#2177	2
#12089	73

subgroup

index

C₇₃×#2177

Subgroups — #32998009

Conjugacy classes of subgroups

Complements

Maximal subgroups