Group #34005501 — subgroups

Subgroups — #34005501

Order 18688.

354 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 144 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	#22625872	C₂	1	2	yes	maximal
2	#22625489	C₂	1	4	yes	maximal
3	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556662	—	2	4	no	—
5	#17989591	C₂²	1	4	yes	—
6	#17989450	—	2	8	no	—
7	C₄×C₈×D₇₃	C₂²	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₈×#10596831	C₄	1	1	yes	characteristic
10	C₈×#10596831	C₄	1	2	yes	—
11	C₈²×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924038	—	2	4	no	—
13	#14556662	—	4	8	no	—
14	C₄⋊#10596833	—	2	8	no	—
15	C₂×C₈×D₇₃	D₄	1	2	yes	—
16	C₂×C₈×D₇₃	Q₈	1	4	yes	—
17	C₄²×D₇₃	C₂³	1	1	yes	characteristic
18	C₄×#10596831	C₂×C₄	1	2	yes	—
19	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₇₃	D₄	1	4	yes	—
21	C₈×Dic₇₃	Q₈	1	2	yes	—
22	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
24	#11924038	—	4	8	no	—
25	C₂×#10596833	—	4	4	no	—
26	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
28	C₂×#10596831	C₄⋊C₄	1	4	yes	—
29	C₂×#10596831	C₄⋊C₄	1	2	yes	—
30	C₈×D₇₃	—	2	8	no	—
31	C₈×D₇₃	D₈	1	4	yes	—
32	C₈×D₇₃	Q₁₆	1	4	yes	—
33	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
34	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
37	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
41	C₂×Dic₇₃	#127	1	1	yes	characteristic
42	C₂×Dic₇₃	C₄⋊Q₈	1	2	yes	—
43	C₄×D₇₃	—	2	4	no	—
44	C₄×D₇₃	C₂×D₈	1	2	yes	—
45	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
46	#10596833	—	8	4	no	—
47	C₈×C₇₃	#107	1	4	yes	—
48	C₈×C₇₃	#108	1	8	yes	complemented
49	#10596831	—	2	4	no	—
50	#10596831	#107	1	4	yes	—
51	C₂²×C₇₃	#384	1	1	yes	characteristic
52	C₂×D₇₃	—	2	2	no	—
53	C₂×D₇₃	#500	1	2	yes	—
54	C₄×C₇₃	C₂×#107	1	2	yes	—
55	C₄×C₇₃	C₂×#108	1	4	yes	—
56	Dic₇₃	—	2	2	no	—
57	Dic₇₃	#493	1	1	yes	characteristic
58	Dic₇₃	#494	1	1	yes	characteristic
59	#8823	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	#1845	1	1	yes	characteristic
61	C₂×C₇₃	#1848	1	2	yes	—
62	D₇₃	—	2	2	no	—
63	#1848	—	73	2	no	—
64	#1567	—	73	4	no	—
65	C₂×C₈²	—	73	1	no	—
66	C₇₃	#8823	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	146	4	no	—
68	#423	—	73	4	no	—
69	C₂×C₄×C₈	—	73	2	no	—
70	C₂×C₄×C₈	—	73	1	no	—
71	#335	—	146	8	no	—
72	C₈²	—	73	2	no	—
73	C₈²	—	73	1	no	—
74	C₈²	#64408	1	1	yes	characteristic
75	C₂×M₁₆	—	146	4	no	—
76	C₂²×C₈	—	73	2	no	—
77	C₂²×C₈	—	73	4	no	—
78	C₂×C₄²	—	73	1	no	—
79	#108	—	292	8	no	—
80	C₄⋊C₈	—	146	8	no	—
81	C₄×C₈	—	73	4	no	—
82	C₄×C₈	—	73	2	no	—
83	C₄×C₈	—	73	1	no	—
84	C₄×C₈	—	73	2	no	—
85	C₄×C₈	C₂×#64408	1	1	yes	characteristic
86	C₄×C₈	C₂×#64408	1	2	yes	—
87	C₂²×C₄	—	73	1	no	—
88	C₂²×C₄	—	73	2	no	—
89	M₁₆	—	292	8	no	—
90	C₂×C₈	—	292	4	no	—
91	C₂×C₈	—	73	2	no	—
92	C₂×C₈	—	73	4	no	—
93	C₂×C₈	—	73	4	no	—
94	C₂×C₈	—	73	4	no	—
95	C₂×C₈	—	146	8	no	—
96	C₂×C₈	C₄⋊#64408	1	2	yes	—
97	C₂×C₈	C₄⋊#64408	1	4	yes	—
98	C₄²	—	73	2	no	—
99	C₄²	—	73	1	no	—
100	C₄²	C₂²×#64408	1	1	yes	characteristic
101	C₂³	—	73	1	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	—	73	2	no	—
104	C₂×C₄	—	73	2	no	—
105	C₂×C₄	—	73	2	no	—
106	C₂×C₄	—	146	4	no	—
107	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
109	C₈	—	584	4	no	—
110	C₈	—	73	4	no	—
111	C₈	—	146	4	no	—
112	C₈	#14556660	1	4	yes	—
113	C₈	#14556662	1	8	yes	complemented
114	C₂²	—	73	2	no	—
115	C₂²	—	146	2	no	—
116	C₂²	#17989533	1	1	yes	characteristic
117	C₄	—	73	1	no	—
118	C₄	—	73	1	no	—
119	C₄	—	146	2	no	—
120	C₄	C₂×#14556660	1	2	yes	—
121	C₄	C₂×#14556662	1	4	yes	—
122	C₂	—	146	2	no	—
123	C₂	#22625867	1	1	yes	characteristic
124	C₂	#22625872	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625872

C₂

yes

maximal

#22625489

C₂

yes

maximal

C₈²×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556662

—

#17989591

C₂²

yes

—

#17989450

—

C₄×C₈×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₈×#10596831

C₄

yes

characteristic

C₈×#10596831

C₄

yes

—

C₈²×C₇₃

C₄

yes

—

—

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×C₈×D₇₃

Q₈

yes

—

C₄²×D₇₃

C₂³

yes

characteristic

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

—

#11924038

—

C₂×#10596833

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

D₈

yes

—

C₈×D₇₃

Q₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₃

#127

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

—

C₄×D₇₃

—

C₄×D₇₃

C₂×D₈

yes

—

C₄×D₇₃

C₂×Q₁₆

yes

—

#10596833

—

C₈×C₇₃

#107

yes

—

C₈×C₇₃

yes

—

yes

—

C₂²×C₇₃

#384

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#500

yes

—

C₄×C₇₃

C₂×#107

yes

—

C₄×C₇₃

C₂×#108

yes

—

Dic₇₃

—

Dic₇₃

yes

Dic₇₃

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#1848

—

#1567

—

C₂×C₈²

—

C₇₃

#8823

yes

characteristic, complemented, Sylow

C₂×#108

—

146

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#335

—

146

—

C₈²

—

C₈²

—

C₈²

#64408

yes

characteristic

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

292

—

C₄⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₄×C₈

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₄⋊#64408

yes

—

C₂×C₈

C₄⋊#64408

yes

—

C₄²

—

C₄²

—

100

C₄²

C₂²×#64408

yes

characteristic

101

C₂³

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

146

—

107

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

108

C₂×C₄

C₂×C₄⋊#64408

yes

—

109

C₈

—

584

—

110

C₈

—

111

C₈

—

146

—

112

C₈

#14556660

yes

—

113

C₈

#14556662

yes

complemented

114

C₂²

—

115

C₂²

—

146

—

116

C₂²

#17989533

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

146

—

120

C₄

C₂×#14556660

yes

—

121

C₄

C₂×#14556662

yes

—

122

C₂

—

146

—

123

C₂

#22625867

yes

characteristic

124

C₂

#22625872

yes

—

	subgroup	G/H
center	C₂²	#17989533
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	#384
prime radical	C₂²×D₇₃	C₄⋊Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8823
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#17989533

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

socle

C₂²×C₇₃

#384

prime radical

C₂²×D₇₃

C₄⋊Q₈

cosocle

C₄²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
48. C₈×C₇₃	#108	semidirect
66. C₇₃	#8823	semidirect
113. C₈	#14556662	semidirect

H ≅ G/N

type

48. C₈×C₇₃

#108

semidirect

66. C₇₃

#8823

semidirect

113. C₈

#14556662

semidirect

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

subgroup	index
#22625872	2
#22625489	2
C₈²×D₇₃	2
#8823	73

subgroup

index

#22625872

#22625489

C₈²×D₇₃

#8823

Subgroups — #34005501

Conjugacy classes of subgroups

Complements

Maximal subgroups