Group #34033090 — subgroups

Subgroups — #34033090

Order 18688.

124 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 35 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	#22627018	C₂	1	1	yes	maximal, characteristic
2	#22626998	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×#64408	C₂	1	1	yes	maximal, characteristic
4	#17989694	—	2	1	no	—
5	#17989677	—	2	1	no	—
6	Q₁₆×#64408	—	2	1	no	—
7	#17989513	C₄	1	1	yes	characteristic
8	#17989500	—	2	1	no	—
9	C₁₆×#64408	C₂²	1	1	yes	characteristic
10	Q₃₂×D₇₃	C₄	1	1	yes	characteristic
11	C₄⋊#10596842	—	4	1	no	—
12	#14556681	—	2	2	no	—
13	Q₈×#64408	—	4	1	no	—
14	#14556664	D₄	1	1	yes	characteristic
15	#14556660	—	2	1	no	—
16	#14556654	—	4	1	no	—
17	C₈×#64408	D₄	1	1	yes	characteristic
18	Q₁₆×D₇₃	—	2	1	no	—
19	Dic₅₈₄	C₈	1	1	yes	characteristic, complemented
20	#14556537	—	2	1	no	—
21	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
22	Q₃₂×C₇₃	C₈	1	1	yes	characteristic, complemented
23	C₂×#10596842	—	8	1	no	—
24	#11924036	SD₁₆	1	1	yes	characteristic
25	C₄⋊#64408	—	4	1	no	—
26	C₄⋊#64408	—	8	1	no	—
27	C₄×#64408	—	8	1	no	—
28	C₄×#64408	D₈	1	1	yes	characteristic
29	#11924031	—	2	1	no	—
30	Q₈×D₇₃	—	4	1	no	—
31	Dic₂₉₂	—	2	1	no	—
32	#11924018	—	4	1	no	—
33	C₈×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
34	#11924003	M₁₆	1	1	yes	characteristic
35	Q₁₆×C₇₃	—	2	1	no	—
36	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
37	C₂×#64408	—	8	1	no	—
38	C₂×#64408	D₁₆	1	1	yes	characteristic
39	C₂×#64408	SD₃₂	1	1	yes	characteristic
40	#10596842	—	16	1	no	—
41	Q₈×C₇₃	—	4	1	no	—
42	C₄×D₇₃	—	8	1	no	—
43	C₄×D₇₃	#102	1	1	yes	characteristic
44	Dic₁₄₆	—	4	1	no	—
45	Dic₁₄₆	—	8	1	no	—
46	#10596833	—	2	1	no	—
47	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
48	#10596831	#104	1	1	yes	characteristic
49	C₂×D₇₃	#357	1	1	yes	characteristic
50	#64408	—	2	1	no	—
51	#64408	—	2	1	no	—
52	#64408	—	16	1	no	—
53	C₄×C₇₃	—	8	1	no	—
54	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
55	Dic₇₃	—	8	1	no	—
56	Dic₇₃	#359	1	1	yes	characteristic
57	#7386	—	73	1	no	maximal, Sylow
58	C₂×C₇₃	#1331	1	1	yes	characteristic
59	D₇₃	#1418	1	1	yes	characteristic
60	D₇₃	#1420	1	1	yes	characteristic
61	C₄×Q₃₂	—	73	1	no	—
62	#1397	—	73	1	no	—
63	#1372	—	73	1	no	—
64	C₇₃	#7386	1	1	yes	characteristic, complemented, Sylow
65	C₂×Q₃₂	—	73	1	no	—
66	C₄×Q₁₆	—	146	1	no	—
67	C₂×C₃₂	—	146	1	no	—
68	#366	—	73	1	no	—
69	#358	—	146	1	no	—
70	C₄×C₁₆	—	73	1	no	—
71	#335	—	146	1	no	—
72	C₂×Q₁₆	—	146	1	no	—
73	C₄×Q₈	—	292	1	no	—
74	Q₃₂	—	73	1	no	—
75	Q₃₂	—	146	1	no	—
76	Q₃₂	#10596842	1	1	yes	characteristic, complemented
77	C₂×C₁₆	—	73	1	no	—
78	C₂×C₁₆	—	73	1	no	—
79	#107	—	146	1	no	—
80	C₄⋊C₈	—	292	1	no	—
81	#103	—	292	1	no	—
82	C₄×C₈	—	73	1	no	—
83	C₃₂	—	146	2	no	—
84	C₂×Q₈	—	292	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	292	1	no	—
87	Q₁₆	—	146	1	no	—
88	C₂×C₈	—	584	1	no	—
89	C₂×C₈	—	73	1	no	—
90	C₂×C₈	—	73	1	no	—
91	C₄⋊C₄	—	292	1	no	—
92	C₄⋊C₄	—	584	1	no	—
93	C₄²	—	584	1	no	—
94	C₄²	—	73	1	no	—
95	C₁₆	—	73	1	no	—
96	C₁₆	—	146	1	no	—
97	C₁₆	C₂×#10596842	1	1	yes	characteristic
98	Q₈	—	4	1	no	—
99	Q₈	—	292	1	no	—
100	Q₈	—	584	1	no	—
101	C₂×C₄	—	584	1	no	—
102	C₂×C₄	—	584	1	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	—	73	1	no	—
105	C₂×C₄	—	73	1	no	—
106	C₈	—	73	1	no	—
107	C₈	—	1168	1	no	—
108	C₈	—	146	1	no	—
109	C₈	C₂²⋊#10596842	1	1	yes	characteristic
110	C₂²	—	73	1	no	—
111	C₄	—	8	1	no	—
112	C₄	—	584	1	no	—
113	C₄	—	73	1	no	—
114	C₄	—	1168	1	no	—
115	C₄	—	146	1	no	—
116	C₄	—	146	1	no	—
117	C₄	D₄⋊#10596842	1	1	yes	characteristic
118	C₂	—	73	1	no	—
119	C₂	—	73	1	no	—
120	C₂	#22626977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627018

C₂

yes

maximal, characteristic

#22626998

C₂

yes

maximal, characteristic

Q₃₂×#64408

C₂

yes

maximal, characteristic

#17989694

—

#17989677

—

Q₁₆×#64408

—

#17989513

C₄

yes

characteristic

#17989500

—

C₁₆×#64408

C₂²

yes

characteristic

Q₃₂×D₇₃

C₄

yes

—

—

—

D₄

yes

—

—

D₄

yes

characteristic

Q₁₆×D₇₃

—

Dic₅₈₄

C₈

yes

characteristic, complemented

#14556537

—

C₁₆×D₇₃

C₂×C₄

yes

characteristic

Q₃₂×C₇₃

C₈

yes

characteristic, complemented

C₂×#10596842

—

#11924036

SD₁₆

yes

—

—

—

D₈

yes

characteristic

#11924031

—

Q₈×D₇₃

—

Dic₂₉₂

—

#11924018

—

C₈×D₇₃

C₂²⋊C₄

yes

characteristic

#11924003

M₁₆

yes

characteristic

Q₁₆×C₇₃

—

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₂×#64408

—

C₂×#64408

D₁₆

yes

characteristic

C₂×#64408

SD₃₂

yes

characteristic

#10596842

—

Q₈×C₇₃

—

C₄×D₇₃

—

C₄×D₇₃

#102

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

C₂²⋊C₈

yes

yes

C₂×D₇₃

yes

—

—

—

C₄×C₇₃

—

C₄×C₇₃

D₄⋊C₈

yes

characteristic

Dic₇₃

—

Dic₇₃

yes

—

C₂×C₇₃

yes

D₇₃

yes

D₇₃

yes

C₄×Q₃₂

—

#1397

—

#1372

—

C₇₃

#7386

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₄×Q₁₆

—

146

—

C₂×C₃₂

—

146

—

#366

—

#358

—

146

—

C₄×C₁₆

—

#335

—

146

—

C₂×Q₁₆

—

146

—

C₄×Q₈

—

292

—

Q₃₂

—

Q₃₂

—

146

—

Q₃₂

#10596842

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

146

—

C₄⋊C₈

—

292

—

#103

—

292

—

C₄×C₈

—

C₃₂

—

146

—

C₂×Q₈

—

292

—

Q₁₆

—

Q₁₆

—

292

—

Q₁₆

—

146

—

C₂×C₈

—

584

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

584

—

C₄²

—

584

—

C₄²

—

C₁₆

—

C₁₆

—

146

—

C₁₆

C₂×#10596842

yes

characteristic

Q₈

—

Q₈

—

292

—

100

Q₈

—

584

—

101

C₂×C₄

—

584

—

102

C₂×C₄

—

584

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₈

—

107

C₈

—

1168

—

108

C₈

—

146

—

109

C₈

C₂²⋊#10596842

yes

characteristic

110

C₂²

—

111

C₄

—

112

C₄

—

584

—

113

C₄

—

114

C₄

—

1168

—

115

C₄

—

146

—

116

C₄

—

146

—

117

C₄

D₄⋊#10596842

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#22626977

yes

characteristic

	subgroup	G/H
center	C₂	#22626977
commutator	C₁₆×C₇₃	C₂×C₈
Frattini	C₁₆	C₂×#10596842
Fitting	Q₃₂×C₇₃	C₈
socle	C₂×C₇₃	#1331
prime radical	C₂×D₇₃	#357
cosocle	C₁₆×#64408	C₂²
hypercenter	Q₃₂	#10596842
hypocenter	C₇₃	#7386
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

subgroup

G/H

center

C₂

#22626977

commutator

C₁₆×C₇₃

C₂×C₈

Frattini

C₁₆

C₂×#10596842

Fitting

Q₃₂×C₇₃

C₈

C₂×C₇₃

C₂×D₇₃

C₁₆×#64408

C₂²

Q₃₂

C₇₃

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

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

N	H ≅ G/N	type
19. Dic₅₈₄	C₈	semidirect
22. Q₃₂×C₇₃	C₈	semidirect
64. C₇₃	#7386	semidirect
76. Q₃₂	#10596842	semidirect

H ≅ G/N

type

19. Dic₅₈₄

C₈

semidirect

22. Q₃₂×C₇₃

C₈

semidirect

64. C₇₃

#7386

semidirect

76. Q₃₂

#10596842

semidirect

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

subgroup	index
#22627018	2
#22626998	2
Q₃₂×#64408	2
#7386	73

subgroup

index

#22627018

#22626998

Q₃₂×#64408

#7386

Subgroups — #34033090

Conjugacy classes of subgroups

Complements

Maximal subgroups