Group #34000067 — subgroups

Subgroups — #34000067

Order 18688.

186 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22624961	C₂	1	2	yes	maximal, complemented
2	#22609545	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11924031	—	2	2	no	—
4	#17988964	C₄	1	1	yes	characteristic
5	#17988963	—	2	2	no	—
6	C₈×#10596831	C₂²	1	1	yes	characteristic
7	#493×C₇₃	C₄	1	1	yes	characteristic
8	C₂×#11924031	—	4	2	no	—
9	C₂×D₅₈₄	—	4	1	no	—
10	C₂×#11924013	—	4	2	no	—
11	C₂×#11924013	—	4	2	no	—
12	#14556588	—	2	1	no	—
13	C₄×#10596831	D₄	1	2	yes	—
14	C₂×D₈×C₇₃	—	4	1	no	—
15	#127×C₇₃	—	2	1	no	—
16	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
17	#11924031	—	8	2	no	—
18	C₂×D₂₉₂	—	4	1	no	—
19	C₂×D₂₉₂	—	4	2	no	—
20	D₅₈₄	—	8	2	no	—
21	#11924013	—	8	4	no	—
22	#11924013	—	8	2	no	—
23	#11924013	—	8	2	no	—
24	C₂×#10596831	D₈	1	2	yes	—
25	C₂×#10596831	SD₁₆	1	2	yes	—
26	C₂×D₄×C₇₃	—	4	1	no	—
27	C₂×D₄×C₇₃	—	4	2	no	—
28	D₈×C₇₃	—	8	2	no	—
29	C₂×C₈×C₇₃	—	2	1	no	—
30	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂³×C₇₃	—	8	1	no	—
32	C₂²×D₇₃	—	8	1	no	—
33	D₄×C₇₃	—	8	2	no	—
34	D₄×C₇₃	—	8	2	no	—
35	D₄×C₇₃	—	8	2	no	—
36	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
37	C₂×C₄×C₇₃	#102	1	2	yes	—
38	D₂₉₂	—	8	2	no	—
39	D₂₉₂	—	8	2	no	—
40	D₂₉₂	—	8	2	no	—
41	C₈×C₇₃	—	2	2	no	—
42	#10596831	—	2	2	no	—
43	#10596831	D₁₆	1	2	yes	—
44	#10596831	SD₃₂	1	2	yes	—
45	C₂²×C₇₃	—	16	1	no	—
46	C₂²×C₇₃	—	16	2	no	—
47	C₂²×C₇₃	#331	1	1	yes	characteristic
48	C₂×D₇₃	—	16	1	no	—
49	C₂×D₇₃	—	16	2	no	—
50	C₄×C₇₃	—	2	1	no	—
51	C₄×C₇₃	#357	1	2	yes	—
52	C₄×C₇₃	#361	1	2	yes	—
53	#7096	—	73	1	no	maximal, Sylow
54	C₂×C₇₃	—	32	1	no	—
55	C₂×C₇₃	#1357	1	1	yes	characteristic
56	C₂×C₇₃	#1361	1	2	yes	—
57	D₇₃	—	32	1	no	—
58	#1712	—	73	1	no	—
59	#1370	—	73	2	no	—
60	C₇₃	#7096	1	1	yes	characteristic, complemented, Sylow
61	#493	—	73	1	no	—
62	#493	—	146	2	no	—
63	#493	#64408	1	1	yes	characteristic
64	C₄⋊C₁₆	—	146	2	no	—
65	C₈²	—	73	1	no	—
66	C₂×D₈	—	4	1	no	—
67	C₂×D₈	—	292	1	no	—
68	C₂×D₈	—	292	2	no	—
69	C₂×D₈	—	292	2	no	—
70	#127	—	2	1	no	—
71	#127	—	146	1	no	—
72	C₂×C₁₆	—	292	2	no	—
73	C₄×C₈	—	73	2	no	—
74	C₄×C₈	C₂×#64408	1	1	yes	characteristic
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	—	4	2	no	—
77	C₂×D₄	—	292	1	no	—
78	C₂×D₄	—	292	2	no	—
79	D₈	—	8	2	no	—
80	D₈	—	584	2	no	—
81	D₈	—	584	2	no	—
82	D₈	—	584	2	no	—
83	D₈	—	584	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	73	2	no	—
86	C₂×C₈	—	73	2	no	—
87	C₄²	C₂²⋊#64408	1	1	yes	characteristic
88	C₁₆	—	584	2	no	—
89	C₂³	—	8	1	no	—
90	C₂³	—	584	1	no	—
91	D₄	—	8	2	no	—
92	D₄	—	8	2	no	—
93	D₄	—	8	2	no	—
94	D₄	—	584	2	no	—
95	D₄	—	584	2	no	—
96	D₄	—	584	2	no	—
97	C₂×C₄	#14556651	1	1	yes	characteristic
98	C₂×C₄	#14556653	1	2	yes	—
99	C₈	—	2	2	no	—
100	C₈	—	73	2	no	—
101	C₈	—	73	2	no	—
102	C₈	—	146	2	no	—
103	C₂²	—	16	1	no	—
104	C₂²	—	16	2	no	—
105	C₂²	—	1168	1	no	—
106	C₂²	—	1168	2	no	—
107	C₂²	#17989445	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	#17989499	1	2	yes	—
110	C₄	#17989505	1	2	yes	—
111	C₂	—	32	1	no	—
112	C₂	—	2336	1	no	—
113	C₂	#22624939	1	1	yes	characteristic
114	C₂	#22624945	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22624961

C₂

yes

maximal, complemented

#22609545

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

#17988964

C₄

yes

characteristic

#17988963

—

C₈×#10596831

C₂²

yes

characteristic

#493×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂×D₅₈₄

—

—

—

—

D₄

yes

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924031

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

—

—

—

D₈

yes

—

C₂×#10596831

SD₁₆

yes

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#100

yes

characteristic

C₂×C₄×C₇₃

#102

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

#10596831

—

#10596831

D₁₆

yes

—

#10596831

SD₃₂

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#331

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#357

yes

—

C₄×C₇₃

#361

yes

—

#7096

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1357

yes

characteristic

C₂×C₇₃

#1361

yes

—

D₇₃

—

#1712

—

#1370

—

C₇₃

#7096

yes

characteristic, complemented, Sylow

#493

—

#493

—

146

—

#493

#64408

yes

characteristic

C₄⋊C₁₆

—

146

—

C₈²

—

C₂×D₈

—

C₂×D₈

—

292

—

C₂×D₈

—

292

—

C₂×D₈

—

292

—

#127

—

#127

—

146

—

C₂×C₁₆

—

292

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

D₈

—

D₈

—

584

—

D₈

—

584

—

D₈

—

584

—

D₈

—

584

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂²⋊#64408

yes

characteristic

C₁₆

—

584

—

C₂³

—

C₂³

—

584

—

D₄

—

D₄

—

D₄

—

D₄

—

584

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

#14556651

yes

characteristic

C₂×C₄

#14556653

yes

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

146

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

1168

—

106

C₂²

—

1168

—

107

C₂²

#17989445

yes

characteristic

108

C₄

—

109

C₄

#17989499

yes

—

110

C₄

#17989505

yes

—

111

C₂

—

112

C₂

—

2336

—

113

C₂

#22624939

yes

characteristic

114

C₂

#22624945

yes

—

	subgroup	G/H
center	C₂²	#17989445
commutator	C₄×C₈×C₇₃	C₂×C₄
Frattini	C₄×C₈	C₂×#64408
Fitting	#493×C₇₃	C₄
socle	C₂²×C₇₃	#331
prime radical	#22609545	C₂
cosocle	C₈×#10596831	C₂²
hypercenter	#493	#64408
hypocenter	C₇₃	#7096
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂²

#17989445

commutator

C₄×C₈×C₇₃

C₂×C₄

Frattini

C₄×C₈

C₂×#64408

Fitting

#493×C₇₃

C₄

C₂²×C₇₃

C₂

C₂²

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #22624961	C₂	semidirect
60. C₇₃	#7096	semidirect

H ≅ G/N

type

1. #22624961

C₂

semidirect

60. C₇₃

#7096

semidirect

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

subgroup	index
#22624961	2
#22609545	2
#7096	73

subgroup

index

#22624961

#22609545

#7096

Subgroups — #34000067

Conjugacy classes of subgroups

Complements

Maximal subgroups