Group #34000502 — subgroups

Subgroups — #34000502

Order 18688.

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

normal subgroups only

This group has 200 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	#22625488	C₂	1	2	yes	maximal
2	#22611686	C₂	1	1	yes	maximal, characteristic
3	#17989590	—	2	2	no	—
4	#17989451	C₄	1	4	yes	—
5	#17989450	—	2	2	no	—
6	C₂×#14556531	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
8	#17988405	—	2	2	no	—
9	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
10	C₂×#11924037	—	4	2	no	—
11	#14556658	—	4	2	no	—
12	C₄⋊#10596833	—	2	4	no	—
13	C₂²×#10596831	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
15	#14556531	—	4	1	no	—
16	C₂×#11924003	—	2	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	M₃₂×C₇₃	—	4	1	no	—
21	C₂×C₁₆×C₇₃	—	2	1	no	—
22	#11924037	—	8	4	no	—
23	C₂×#10596833	—	4	4	no	—
24	C₂²×Dic₇₃	SD₁₆	1	2	yes	—
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	D₈	1	1	yes	characteristic
29	C₄×Dic₇₃	Q₁₆	1	1	yes	characteristic
30	#11924003	—	4	1	no	—
31	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₁₆×C₇₃	—	4	1	no	—
37	C₂³×C₇₃	#106	1	1	yes	characteristic
38	C₂×C₄×C₇₃	—	2	1	no	—
39	C₂×C₄×C₇₃	—	2	1	no	—
40	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
41	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
42	C₂×Dic₇₃	—	2	2	no	—
43	C₂×Dic₇₃	—	2	2	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	2	no	—
53	C₂²×C₇₃	#340	1	1	yes	characteristic
54	C₄×C₇₃	—	2	1	no	—
55	C₄×C₇₃	#338	1	1	yes	characteristic
56	C₄×C₇₃	#343	1	1	yes	characteristic
57	Dic₇₃	—	2	2	no	—
58	Dic₇₃	#361	1	2	yes	—
59	Dic₇₃	#362	1	2	yes	—
60	#7245	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	—	4	1	no	—
62	C₂×C₇₃	#1292	1	1	yes	characteristic
63	C₂×C₇₃	#1387	1	2	yes	—
64	#2150	—	73	1	no	—
65	#1567	—	73	2	no	—
66	C₇₃	#7245	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₃₂	—	73	1	no	—
68	C₂×M₃₂	#64408	1	1	yes	characteristic
69	#423	—	146	2	no	—
70	C₂×C₄×C₈	—	73	1	no	—
71	C₄⋊C₁₆	—	146	2	no	—
72	#335	—	73	4	no	—
73	#335	—	146	2	no	—
74	C₂×M₁₆	—	292	2	no	—
75	C₂²×C₈	—	73	1	no	—
76	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
77	C₂×C₄²	—	73	1	no	—
78	M₃₂	—	4	1	no	—
79	M₃₂	—	292	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	146	1	no	—
82	C₄⋊C₈	—	292	2	no	—
83	C₄⋊C₈	—	146	4	no	—
84	C₄×C₈	—	73	2	no	—
85	C₄×C₈	—	73	2	no	—
86	C₂²×C₄	—	73	2	no	—
87	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
88	M₁₆	—	584	4	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	292	4	no	—
92	C₂×C₈	—	292	1	no	—
93	C₂×C₈	—	73	2	no	—
94	C₂×C₈	C₄×#64408	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
96	C₄²	—	73	2	no	—
97	C₄²	—	73	1	no	—
98	C₄²	—	73	1	no	—
99	C₁₆	—	4	1	no	—
100	C₁₆	—	292	1	no	—
101	C₂³	#14556659	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	73	2	no	—
105	C₂×C₄	—	73	2	no	—
106	C₂×C₄	—	146	2	no	—
107	C₂×C₄	—	146	2	no	—
108	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
109	C₂×C₄	#14556660	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	584	4	no	—
112	C₈	—	146	2	no	—
113	C₈	#14556651	1	1	yes	characteristic
114	C₈	#14556652	1	1	yes	characteristic
115	C₂²	—	2	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	#17989461	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	73	2	no	—
120	C₄	—	73	2	no	—
121	C₄	—	146	2	no	—
122	C₄	#17989456	1	1	yes	characteristic
123	C₄	#17989467	1	1	yes	characteristic
124	C₂	—	4	1	no	—
125	C₂	#22624801	1	1	yes	characteristic
126	C₂	#22625006	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625488

C₂

yes

maximal

#22611686

C₂

yes

maximal, characteristic

#17989590

—

#17989451

C₄

yes

—

#17989450

—

C₂×#14556531

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988405

—

C₂×M₃₂×C₇₃

C₄

yes

—

—

—

Q₈

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

#14556531

—

C₂×#11924003

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

—

#11924037

—

C₂×#10596833

—

C₂²×Dic₇₃

SD₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

—

C₄×Dic₇₃

C₂²⋊C₄

yes

—

C₄×Dic₇₃

D₈

yes

characteristic

C₄×Dic₇₃

Q₁₆

yes

characteristic

#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

#106

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

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

#340

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

#361

yes

—

Dic₇₃

#362

yes

—

#7245

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1292

yes

characteristic

C₂×C₇₃

#1387

yes

—

#2150

—

#1567

—

C₇₃

#7245

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

#64408

yes

characteristic

#423

—

146

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

146

—

#335

—

#335

—

146

—

C₂×M₁₆

—

292

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#64408

yes

characteristic

M₁₆

—

584

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

100

C₁₆

—

292

—

101

C₂³

#14556659

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

146

—

107

C₂×C₄

—

146

—

108

C₂×C₄

(C₂×#64408)⋊C₄

yes

characteristic

109

C₂×C₄

#14556660

yes

characteristic

110

C₈

—

111

C₈

—

584

—

112

C₈

—

146

—

113

C₈

#14556651

yes

characteristic

114

C₈

#14556652

yes

characteristic

115

C₂²

—

116

C₂²

—

117

C₂²

#17989461

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

146

—

122

C₄

#17989456

yes

characteristic

123

C₄

#17989467

yes

characteristic

124

C₂

—

125

C₂

#22624801

yes

characteristic

126

C₂

#22625006

yes

—

	subgroup	G/H
center	C₂²	#17989461
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	#340
prime radical	C₂³×C₇₃	#106
cosocle	C₂×C₈×Dic₇₃	C₂²
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#7245
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#17989461

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂×M₃₂×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₂×C₈×Dic₇₃

C₂²

C₂×M₃₂

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
66. C₇₃	#7245	semidirect

H ≅ G/N

type

66. C₇₃

#7245

semidirect

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

subgroup	index
#22625488	2
#22611686	2
#7245	73

subgroup

index

#22625488

#22611686

#7245

Subgroups — #34000502

Conjugacy classes of subgroups

Complements

Maximal subgroups