Group #34005491 — subgroups

Subgroups — #34005491

Order 18688.

422 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 148 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	#22625867	C₂	1	2	yes	maximal
2	#22625852	C₂	1	4	yes	maximal
3	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556660	—	2	4	no	—
5	#17989533	C₂²	1	4	yes	—
6	#17989461	—	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	2	yes	complemented
10	C₈×#10596831	C₄	1	1	yes	characteristic, complemented
11	C₈²×C₇₃	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄⋊#64408	—	2	8	no	—
13	C₂×C₄⋊#64408	—	2	4	no	—
14	#14556660	—	4	8	no	—
15	C₂×C₈×D₇₃	D₄	1	4	yes	—
16	C₂×C₈×D₇₃	D₄	1	2	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	Q₈	1	4	yes	—
21	C₈×Dic₇₃	Q₈	1	2	yes	—
22	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#64408	—	4	4	no	—
25	C₄⋊#64408	—	4	8	no	—
26	C₄⋊#64408	—	4	16	no	—
27	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
29	C₂×#10596831	C₄⋊C₄	1	4	yes	—
30	C₂×#10596831	C₄⋊C₄	1	2	yes	—
31	C₈×D₇₃	—	2	8	no	—
32	C₈×D₇₃	D₈	1	4	yes	—
33	C₈×D₇₃	Q₁₆	1	4	yes	—
34	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
35	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
36	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
37	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
38	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×D₇₃	#127	1	1	yes	characteristic
40	C₂×#64408	—	8	4	no	—
41	C₂×#64408	—	8	8	no	—
42	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
44	C₂×Dic₇₃	C₄⋊Q₈	1	2	yes	—
45	C₂×Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
46	C₄×D₇₃	C₂×D₈	1	2	yes	—
47	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
48	C₄×D₇₃	#135	1	8	yes	—
49	C₈×C₇₃	—	2	4	no	—
50	C₈×C₇₃	#107	1	4	yes	—
51	#10596831	—	2	4	no	—
52	#10596831	#107	1	4	yes	—
53	C₂²×C₇₃	#384	1	1	yes	characteristic
54	C₂×D₇₃	#493	1	1	yes	characteristic
55	C₂×D₇₃	#494	1	1	yes	characteristic
56	C₂×D₇₃	#495	1	4	yes	—
57	#64408	—	16	4	no	—
58	C₄×C₇₃	C₂×#107	1	2	yes	—
59	C₄×C₇₃	#427	1	4	yes	—
60	Dic₇₃	#499	1	4	yes	—
61	Dic₇₃	#500	1	2	yes	—
62	#8816	—	73	1	no	maximal, Sylow
63	C₂×C₇₃	#1845	1	1	yes	characteristic
64	C₂×C₇₃	#1846	1	2	yes	—
65	D₇₃	#1709	1	2	yes	—
66	D₇₃	#1710	1	2	yes	—
67	#1845	—	73	2	no	—
68	#1834	—	73	4	no	—
69	C₂×C₈²	—	73	1	no	—
70	C₇₃	#8816	1	1	yes	characteristic, complemented, Sylow
71	C₂×#107	—	146	4	no	—
72	C₂×C₄×C₈	—	73	2	no	—
73	C₂×C₄×C₈	—	73	1	no	—
74	#384	—	73	4	no	—
75	#340	—	146	8	no	—
76	C₈²	—	73	2	no	—
77	C₈²	—	73	1	no	—
78	C₈²	#64408	1	1	yes	characteristic, complemented
79	C₂²×C₈	—	73	2	no	—
80	C₂²×C₈	—	73	4	no	—
81	C₂×C₄⋊C₄	—	146	8	no	—
82	C₂×C₄⋊C₄	—	146	4	no	—
83	C₂×C₄²	—	73	1	no	—
84	#107	—	292	8	no	—
85	C₄×C₈	—	73	2	no	—
86	C₄×C₈	—	73	4	no	—
87	C₄×C₈	—	73	1	no	—
88	C₄×C₈	—	73	2	no	—
89	C₄×C₈	C₂×#64408	1	2	yes	—
90	C₄×C₈	C₂×#64408	1	1	yes	characteristic
91	C₂²×C₄	—	292	4	no	—
92	C₂²×C₄	—	73	1	no	—
93	C₂²×C₄	—	73	2	no	—
94	C₂×C₈	—	73	2	no	—
95	C₂×C₈	—	73	4	no	—
96	C₂×C₈	—	73	4	no	—
97	C₂×C₈	—	73	4	no	—
98	C₂×C₈	—	146	8	no	—
99	C₂×C₈	C₄⋊#64408	1	4	yes	—
100	C₂×C₈	C₄⋊#64408	1	2	yes	—
101	C₄⋊C₄	—	292	8	no	—
102	C₄⋊C₄	—	292	16	no	—
103	C₄²	—	73	1	no	—
104	C₄²	—	73	2	no	—
105	C₄²	C₂²×#64408	1	1	yes	characteristic
106	C₂³	—	73	1	no	—
107	C₂×C₄	—	584	4	no	—
108	C₂×C₄	—	584	8	no	—
109	C₂×C₄	—	73	1	no	—
110	C₂×C₄	—	73	2	no	—
111	C₂×C₄	—	73	2	no	—
112	C₂×C₄	—	73	2	no	—
113	C₂×C₄	—	73	8	no	—
114	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
116	C₈	—	2	4	no	—
117	C₈	—	73	4	no	—
118	C₈	—	146	4	no	—
119	C₈	#14556660	1	4	yes	—
120	C₂²	—	73	1	no	—
121	C₂²	—	73	1	no	—
122	C₂²	—	73	4	no	—
123	C₂²	#17989533	1	1	yes	characteristic
124	C₄	—	73	2	no	—
125	C₄	—	73	4	no	—
126	C₄	—	1168	4	no	—
127	C₄	C₂×#14556660	1	2	yes	—
128	C₄	#17989597	1	4	yes	—
129	C₂	—	73	2	no	—
130	C₂	—	73	2	no	—
131	C₂	#22625867	1	1	yes	characteristic
132	C₂	#22625868	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625867

C₂

yes

maximal

#22625852

C₂

yes

maximal

C₈²×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556660

—

#17989533

C₂²

yes

—

#17989461

—

C₄×C₈×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₈×#10596831

C₄

yes

complemented

C₈×#10596831

C₄

yes

characteristic, complemented

C₈²×C₇₃

C₄

yes

characteristic, complemented

C₂×C₄⋊#64408

—

C₂×C₄⋊#64408

—

#14556660

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×C₈×D₇₃

D₄

yes

—

C₄²×D₇₃

C₂³

yes

characteristic

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²×#64408

—

C₄⋊#64408

—

C₄⋊#64408

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

C₂×D₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

D₈

yes

—

C₈×D₇₃

Q₁₆

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

C₂×Q₈

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

C₂²×D₇₃

yes

—

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₃

C₄⋊Q₈

yes

—

C₂×Dic₇₃

C₄⋊Q₈

yes

characteristic

C₄×D₇₃

C₂×D₈

yes

—

C₄×D₇₃

C₂×Q₁₆

yes

—

C₄×D₇₃

#135

yes

—

C₈×C₇₃

—

C₈×C₇₃

#107

yes

—

#10596831

—

#10596831

#107

yes

—

C₂²×C₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

yes

C₂×D₇₃

yes

—

#64408

—

C₄×C₇₃

C₂×#107

yes

—

C₄×C₇₃

#427

yes

—

Dic₇₃

#499

yes

—

Dic₇₃

#500

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

#1709

yes

—

D₇₃

#1710

yes

—

#1845

—

#1834

—

C₂×C₈²

—

C₇₃

#8816

yes

characteristic, complemented, Sylow

C₂×#107

—

146

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#384

—

#340

—

146

—

C₈²

—

C₈²

—

C₈²

#64408

yes

characteristic, complemented

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

#107

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

292

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₄⋊#64408

yes

—

100

C₂×C₈

C₄⋊#64408

yes

—

101

C₄⋊C₄

—

292

—

102

C₄⋊C₄

—

292

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²×#64408

yes

characteristic

106

C₂³

—

107

C₂×C₄

—

584

—

108

C₂×C₄

—

584

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

115

C₂×C₄

C₂×C₄⋊#64408

yes

—

116

C₈

—

117

C₈

—

118

C₈

—

146

—

119

C₈

#14556660

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#17989533

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

1168

—

127

C₄

C₂×#14556660

yes

—

128

C₄

#17989597

yes

—

129

C₂

—

130

C₂

—

131

C₂

#22625867

yes

characteristic

132

C₂

#22625868

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	#127
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8816
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

C₂²×C₇₃

C₂²×D₇₃

C₄²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
9. C₈×#10596831	C₄	semidirect
10. C₈×#10596831	C₄	semidirect
11. C₈²×C₇₃	C₄	semidirect
70. C₇₃	#8816	semidirect
78. C₈²	#64408	semidirect

H ≅ G/N

type

9. C₈×#10596831

C₄

semidirect

10. C₈×#10596831

C₄

semidirect

11. C₈²×C₇₃

C₄

semidirect

70. C₇₃

#8816

semidirect

78. C₈²

#64408

semidirect

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

subgroup	index
#22625867	2
#22625852	2
C₈²×D₇₃	2
#8816	73

subgroup

index

#22625867

#22625852

C₈²×D₇₃

#8816

Subgroups — #34005491

Conjugacy classes of subgroups

Complements

Maximal subgroups