Group #34000959 — subgroups

Subgroups — #34000959

Order 18688.

150 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 38 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	#22625045	C₂	1	1	yes	maximal, characteristic, complemented
2	#22624991	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612014	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556662	—	2	1	no	—
5	C₂×#14556641	—	2	1	no	—
6	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
7	#17988820	—	2	1	no	—
8	#17988413	C₄	1	1	yes	characteristic
9	#17988388	—	2	1	no	—
10	C₂×D₁₆×C₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924038	—	4	1	no	—
12	#14556662	—	2	2	no	—
13	#14556662	—	4	1	no	—
14	#14556641	—	4	1	no	—
15	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
16	#14556562	—	4	1	no	—
17	C₂×#11924003	D₄	1	1	yes	characteristic
18	C₁₆×D₇₃	—	4	1	no	—
19	#14556525	—	2	1	no	—
20	#14556512	—	4	1	no	—
21	C₂×D₈×C₇₃	—	2	1	no	—
22	D₁₆×C₇₃	—	4	1	no	—
23	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
24	#11924038	—	4	2	no	—
25	#11924038	—	8	1	no	—
26	C₂×#10596833	—	8	1	no	—
27	C₂×C₂²⋊D₇₃	—	8	1	no	—
28	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
29	C₂×#10596831	SD₁₆	1	1	yes	characteristic
30	C₈×D₇₃	—	2	1	no	—
31	C₈×D₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	4	1	no	—
33	C₂²⋊Dic₇₃	—	8	1	no	—
34	#11924003	—	2	1	no	—
35	C₂×D₄×C₇₃	—	4	1	no	—
36	D₈×C₇₃	—	4	1	no	—
37	D₈×C₇₃	—	4	1	no	—
38	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₁₆×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	—	8	1	no	—
41	C₂²×D₇₃	SD₃₂	1	1	yes	characteristic
42	D₄×C₇₃	—	8	1	no	—
43	D₄×C₇₃	—	8	1	no	—
44	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
45	C₂²⋊D₇₃	—	16	2	no	—
46	C₂×Dic₇₃	—	8	1	no	—
47	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
48	C₄×D₇₃	—	2	1	no	—
49	C₄×D₇₃	—	2	1	no	—
50	#10596833	—	8	2	no	—
51	C₈×C₇₃	#99	1	1	yes	characteristic
52	C₈×C₇₃	#104	1	1	yes	characteristic
53	#10596831	—	2	1	no	—
54	C₂²×C₇₃	—	16	1	no	—
55	C₂²×C₇₃	—	16	1	no	—
56	C₂²×C₇₃	—	16	1	no	—
57	C₂²×C₇₃	#357	1	1	yes	characteristic
58	C₂×D₇₃	—	2	1	no	—
59	C₂×D₇₃	—	2	1	no	—
60	C₂×D₇₃	—	2	1	no	—
61	C₄×C₇₃	#327	1	1	yes	characteristic
62	C₄×C₇₃	#360	1	1	yes	characteristic
63	Dic₇₃	—	2	1	no	—
64	Dic₇₃	—	16	1	no	—
65	#7420	—	73	1	no	maximal, Sylow
66	C₂×C₇₃	—	32	1	no	—
67	C₂×C₇₃	#1347	1	1	yes	characteristic
68	C₂×C₇₃	#1417	1	1	yes	characteristic
69	C₂×C₇₃	#1419	1	1	yes	characteristic
70	D₇₃	—	4	1	no	—
71	#2215	—	73	1	no	—
72	C₂²⋊C₃₂	—	73	1	no	—
73	#1381	—	73	1	no	—
74	C₇₃	#7420	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	#64408	1	1	yes	characteristic
76	C₂²×C₁₆	—	73	1	no	—
77	#466	—	146	1	no	—
78	C₂×#108	—	146	1	no	—
79	C₂×C₃₂	—	146	1	no	—
80	#366	—	73	1	no	—
81	#357	—	146	1	no	—
82	C₂×D₈	—	2	1	no	—
83	C₂×M₁₆	—	292	1	no	—
84	C₂²×C₈	—	73	1	no	—
85	#121	—	292	1	no	—
86	D₁₆	—	4	1	no	—
87	C₂×C₁₆	—	292	1	no	—
88	C₂×C₁₆	—	73	1	no	—
89	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
90	#108	—	292	1	no	—
91	#108	—	146	2	no	—
92	#107	—	146	1	no	—
93	#102	—	292	1	no	—
94	C₃₂	—	292	1	no	—
95	C₂×D₄	—	4	1	no	—
96	C₂×D₄	—	584	1	no	—
97	C₂²×C₄	—	73	1	no	—
98	D₈	—	4	1	no	—
99	D₈	—	4	1	no	—
100	M₁₆	—	292	2	no	—
101	M₁₆	—	584	1	no	—
102	C₂×C₈	—	584	1	no	—
103	C₂×C₈	—	73	1	no	—
104	C₂×C₈	—	146	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
107	C₄⋊C₄	—	292	1	no	—
108	C₂²⋊C₄	—	584	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	146	1	no	—
111	C₂³	—	8	1	no	—
112	C₂³	—	73	1	no	—
113	D₄	—	8	1	no	—
114	D₄	—	8	1	no	—
115	D₄	—	1168	2	no	—
116	C₂×C₄	—	584	1	no	—
117	C₂×C₄	—	73	1	no	—
118	C₂×C₄	—	146	1	no	—
119	C₂×C₄	—	146	1	no	—
120	C₂×C₄	#14556653	1	1	yes	characteristic
121	C₈	—	584	2	no	—
122	C₈	—	146	1	no	—
123	C₈	#14556649	1	1	yes	characteristic
124	C₈	#14556655	1	1	yes	characteristic
125	C₂²	—	16	1	no	—
126	C₂²	—	16	1	no	—
127	C₂²	—	16	1	no	—
128	C₂²	—	146	1	no	—
129	C₂²	—	146	1	no	—
130	C₂²	—	146	1	no	—
131	C₂²	#17989499	1	1	yes	characteristic
132	C₄	—	1168	1	no	—
133	C₄	—	146	1	no	—
134	C₄	#17989438	1	1	yes	characteristic
135	C₄	#17989503	1	1	yes	characteristic
136	C₂	—	32	1	no	—
137	C₂	—	292	1	no	—
138	C₂	#22624917	1	1	yes	characteristic
139	C₂	#22625082	1	1	yes	characteristic
140	C₂	#22625086	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625045

C₂

yes

maximal, characteristic, complemented

#22624991

C₂

yes

maximal, characteristic, complemented

#22612014

C₂

yes

maximal, characteristic

C₂×#14556662

—

C₂×#14556641

—

C₂×C₁₆×D₇₃

C₂²

yes

characteristic

#17988820

—

#17988413

C₄

yes

characteristic

#17988388

—

C₂×D₁₆×C₇₃

C₄

yes

—

—

—

—

C₂×C₈×D₇₃

D₄

yes

characteristic

#14556562

—

C₂×#11924003

D₄

yes

characteristic

C₁₆×D₇₃

—

#14556525

—

#14556512

—

C₂×D₈×C₇₃

—

D₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

—

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

D₈

yes

characteristic

C₂×#10596831

SD₁₆

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

#11924003

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂²×D₇₃

SD₃₂

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₁₆

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#357

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7420

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2215

—

C₂²⋊C₃₂

—

#1381

—

C₇₃

#7420

yes

characteristic, complemented, Sylow

C₂×D₁₆

#64408

yes

characteristic

C₂²×C₁₆

—

#466

—

146

—

C₂×#108

—

146

—

C₂×C₃₂

—

146

—

#366

—

#357

—

146

—

C₂×D₈

—

C₂×M₁₆

—

292

—

C₂²×C₈

—

#121

—

292

—

D₁₆

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#64408

yes

characteristic

#108

—

292

—

#108

—

146

—

#107

—

146

—

#102

—

292

—

C₃₂

—

292

—

C₂×D₄

—

C₂×D₄

—

584

—

C₂²×C₄

—

D₈

—

D₈

—

100

M₁₆

—

292

—

101

M₁₆

—

584

—

102

C₂×C₈

—

584

—

103

C₂×C₈

—

104

C₂×C₈

—

146

—

105

C₂×C₈

—

146

—

106

C₂×C₈

C₂²⋊#64408

yes

characteristic

107

C₄⋊C₄

—

292

—

108

C₂²⋊C₄

—

584

—

109

C₁₆

—

110

C₁₆

—

146

—

111

C₂³

—

112

C₂³

—

113

D₄

—

114

D₄

—

115

D₄

—

1168

—

116

C₂×C₄

—

584

—

117

C₂×C₄

—

118

C₂×C₄

—

146

—

119

C₂×C₄

—

146

—

120

C₂×C₄

#14556653

yes

characteristic

121

C₈

—

584

—

122

C₈

—

146

—

123

C₈

#14556649

yes

characteristic

124

C₈

#14556655

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

146

—

129

C₂²

—

146

—

130

C₂²

—

146

—

131

C₂²

#17989499

yes

characteristic

132

C₄

—

1168

—

133

C₄

—

146

—

134

C₄

#17989438

yes

characteristic

135

C₄

#17989503

yes

characteristic

136

C₂

—

137

C₂

—

292

—

138

C₂

#22624917

yes

characteristic

139

C₂

#22625082

yes

characteristic

140

C₂

#22625086

yes

characteristic

	subgroup	G/H
center	C₂²	#17989499
commutator	C₂×C₁₆×C₇₃	C₂×C₄
Frattini	C₂×C₁₆	C₂×#64408
Fitting	C₂×D₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#357
prime radical	#22612014	C₂
cosocle	C₂×C₁₆×D₇₃	C₂²
hypercenter	C₂×D₁₆	#64408
hypocenter	C₇₃	#7420
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

#17989499

commutator

C₂×C₁₆×C₇₃

C₂×C₄

Frattini

C₂×C₁₆

C₂×#64408

Fitting

C₂×D₁₆×C₇₃

C₄

C₂²×C₇₃

C₂

C₂×C₁₆×D₇₃

C₂²

C₂×D₁₆

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. #22625045	C₂	semidirect
2. #22624991	C₂	semidirect
74. C₇₃	#7420	semidirect

H ≅ G/N

type

1. #22625045

C₂

semidirect

2. #22624991

C₂

semidirect

74. C₇₃

#7420

semidirect

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

subgroup	index
#22625045	2
#22624991	2
#22612014	2
#7420	73

subgroup

index

Subgroups — #34000959

Conjugacy classes of subgroups

Complements

Maximal subgroups