Group #34000494 — subgroups

Subgroups — #34000494

Order 18688.

190 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 66 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	#22625493	C₂	1	2	yes	maximal
2	C₂×#17988405	C₂	1	1	yes	maximal, characteristic
3	#17989608	—	2	2	no	—
4	#17989450	—	2	2	no	—
5	#17989449	C₄	1	4	yes	—
6	C₂²×#11924003	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
8	#17988405	—	2	2	no	—
9	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
10	C₄⋊#10596833	—	2	4	no	—
11	C₂²⋊#10596833	—	4	2	no	—
12	#14556646	—	4	2	no	—
13	C₂²×#10596831	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
15	C₂×#11924003	—	2	1	no	—
16	C₂×#11924003	—	4	1	no	—
17	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
18	C₈×Dic₇₃	D₄	1	2	yes	—
19	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₇₃	—	2	1	no	—
21	C₂×C₁₆×C₇₃	—	4	1	no	—
22	C₂×#10596833	—	4	4	no	—
23	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
24	C₂²×Dic₇₃	Q₁₆	1	1	yes	characteristic
25	C₂×#10596831	—	4	1	no	—
26	C₂×#10596831	C₄⋊C₄	1	2	yes	—
27	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
28	C₄×Dic₇₃	SD₁₆	1	2	yes	—
29	#11924003	—	4	1	no	—
30	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₃	—	2	1	no	—
32	C₂×C₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₁₆×C₇₃	—	4	1	no	—
36	C₂³×C₇₃	#107	1	1	yes	characteristic
37	C₂×C₄×C₇₃	—	2	1	no	—
38	C₂×C₄×C₇₃	—	2	1	no	—
39	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
40	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
41	C₂×Dic₇₃	—	2	1	no	—
42	C₂×Dic₇₃	—	2	1	no	—
43	C₂×Dic₇₃	—	4	1	no	—
44	C₂×Dic₇₃	#102	1	2	yes	—
45	C₂×Dic₇₃	#103	1	2	yes	—
46	#10596833	—	8	4	no	—
47	C₈×C₇₃	—	2	1	no	—
48	C₈×C₇₃	#100	1	1	yes	characteristic
49	C₈×C₇₃	#101	1	1	yes	characteristic
50	#10596831	—	2	2	no	—
51	C₂²×C₇₃	—	2	1	no	—
52	C₂²×C₇₃	—	2	1	no	—
53	C₂²×C₇₃	#340	1	1	yes	characteristic
54	C₂²×C₇₃	#368	1	1	yes	characteristic
55	C₂²×C₇₃	#368	1	1	yes	characteristic
56	C₄×C₇₃	—	2	1	no	—
57	C₄×C₇₃	#338	1	1	yes	characteristic
58	C₄×C₇₃	#343	1	1	yes	characteristic
59	Dic₇₃	—	2	2	no	—
60	Dic₇₃	—	2	2	no	—
61	#7242	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	2	1	no	—
63	C₂×C₇₃	—	2	1	no	—
64	C₂×C₇₃	#1292	1	1	yes	characteristic
65	C₂×C₇₃	#1381	1	1	yes	characteristic
66	C₂×C₇₃	#1384	1	1	yes	characteristic
67	C₂×C₄⋊C₁₆	—	73	1	no	—
68	#1568	—	73	2	no	—
69	C₇₃	#7242	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₁₆	—	73	1	no	—
71	C₂²×C₁₆	#64408	1	1	yes	characteristic
72	#432	—	146	2	no	—
73	C₂×C₄×C₈	—	73	1	no	—
74	C₄⋊C₁₆	—	146	2	no	—
75	#335	—	146	2	no	—
76	#334	—	73	4	no	—
77	C₂²×C₈	—	73	1	no	—
78	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
79	C₂×C₄²	—	73	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	4	1	no	—
82	C₂×C₁₆	—	292	1	no	—
83	C₂×C₁₆	—	146	1	no	—
84	C₄⋊C₈	—	146	4	no	—
85	C₂²⋊C₈	—	292	2	no	—
86	#97	—	292	2	no	—
87	C₄×C₈	—	73	2	no	—
88	C₄×C₈	—	73	2	no	—
89	C₂²×C₄	—	73	1	no	—
90	C₂²×C₄	—	73	1	no	—
91	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	292	4	no	—
95	C₂×C₈	—	292	1	no	—
96	C₂×C₈	—	73	2	no	—
97	C₂×C₈	C₄×#64408	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
99	C₄²	—	73	2	no	—
100	C₄²	—	73	2	no	—
101	C₁₆	—	4	1	no	—
102	C₁₆	—	292	1	no	—
103	C₂³	#14556660	1	1	yes	characteristic
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	292	1	no	—
107	C₂×C₄	—	73	2	no	—
108	C₂×C₄	—	73	2	no	—
109	C₂×C₄	—	146	1	no	—
110	C₂×C₄	—	146	1	no	—
111	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
112	C₂×C₄	#14556659	1	1	yes	characteristic
113	C₈	—	2	1	no	—
114	C₈	—	584	4	no	—
115	C₈	—	146	2	no	—
116	C₈	#14556651	1	1	yes	characteristic
117	C₈	#14556652	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#17989461	1	1	yes	characteristic
121	C₂²	#17989515	1	1	yes	characteristic
122	C₂²	#17989516	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	146	2	no	—
125	C₄	—	146	2	no	—
126	C₄	#17989456	1	1	yes	characteristic
127	C₄	#17989467	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#22624801	1	1	yes	characteristic
131	C₂	#22624991	1	1	yes	characteristic
132	C₂	#22624999	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625493

C₂

yes

maximal

C₂×#17988405

C₂

yes

maximal, characteristic

#17989608

—

#17989450

—

#17989449

C₄

yes

—

C₂²×#11924003

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988405

—

C₂²×C₁₆×C₇₃

C₄

yes

—

—

—

Q₈

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×#11924003

—

C₂×#11924003

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

C₂×#10596833

—

C₂²×Dic₇₃

D₈

yes

characteristic

C₂²×Dic₇₃

Q₁₆

yes

characteristic

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂²⋊C₄

yes

—

C₄×Dic₇₃

SD₁₆

yes

—

#11924003

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#107

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#102

yes

—

C₂×Dic₇₃

#103

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#7242

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₄⋊C₁₆

—

#1568

—

C₇₃

#7242

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

#64408

yes

characteristic

#432

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

#335

—

146

—

#334

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

292

—

#97

—

292

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

100

C₄²

—

101

C₁₆

—

102

C₁₆

—

292

—

103

C₂³

#14556660

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

292

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

146

—

110

C₂×C₄

—

146

—

111

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

112

C₂×C₄

#14556659

yes

characteristic

113

C₈

—

114

C₈

—

584

—

115

C₈

—

146

—

116

C₈

#14556651

yes

characteristic

117

C₈

#14556652

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

#17989461

yes

characteristic

121

C₂²

#17989515

yes

characteristic

122

C₂²

#17989516

yes

characteristic

123

C₄

—

124

C₄

—

146

—

125

C₄

—

146

—

126

C₄

#17989456

yes

characteristic

127

C₄

#17989467

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#22624801

yes

characteristic

131

C₂

#22624991

yes

characteristic

132

C₂

#22624999

yes

characteristic

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#107
cosocle	C₂×C₈×Dic₇₃	C₂²
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#7242
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂²×C₁₆×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₂×C₈×Dic₇₃

C₂²

C₂²×C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
69. C₇₃	#7242	semidirect

H ≅ G/N

type

69. C₇₃

#7242

semidirect

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

subgroup	index
#22625493	2
C₂×#17988405	2
#7242	73

subgroup

index

#22625493

C₂×#17988405

#7242

Subgroups — #34000494

Conjugacy classes of subgroups

Complements

Maximal subgroups