Group #34000060 — subgroups

Subgroups — #34000060

Order 18688.

152 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 42 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	#22624954	C₂	1	2	yes	maximal, complemented
2	#22609536	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11924031	—	2	2	no	—
4	#17988967	C₄	1	1	yes	characteristic
5	#17988933	—	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×Dic₂₉₂	—	4	1	no	—
10	#14556591	—	2	1	no	—
11	#14556512	—	4	2	no	—
12	#14556512	—	4	2	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×Dic₁₄₆	—	4	1	no	—
19	Dic₂₉₂	—	8	2	no	—
20	C₂×#10596831	D₈	1	2	yes	—
21	C₂×#10596831	SD₁₆	1	2	yes	—
22	C₄⋊Dic₇₃	—	4	2	no	—
23	C₂×D₄×C₇₃	—	4	1	no	—
24	C₂×D₄×C₇₃	—	4	2	no	—
25	D₈×C₇₃	—	8	2	no	—
26	C₂×C₈×C₇₃	—	2	1	no	—
27	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂³×C₇₃	—	8	1	no	—
29	D₄×C₇₃	—	8	2	no	—
30	D₄×C₇₃	—	8	2	no	—
31	D₄×C₇₃	—	8	2	no	—
32	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
33	C₂×C₄×C₇₃	#102	1	2	yes	—
34	C₂×Dic₇₃	—	8	1	no	—
35	Dic₁₄₆	—	8	2	no	—
36	C₈×C₇₃	—	2	2	no	—
37	#10596831	—	2	2	no	—
38	#10596831	—	2	2	no	—
39	C₂²×C₇₃	—	16	1	no	—
40	C₂²×C₇₃	—	16	2	no	—
41	C₂²×C₇₃	#331	1	1	yes	characteristic
42	C₄×C₇₃	—	2	1	no	—
43	C₄×C₇₃	#359	1	2	yes	—
44	C₄×C₇₃	#360	1	2	yes	—
45	Dic₇₃	—	16	1	no	—
46	#7092	—	73	1	no	maximal, Sylow
47	C₂×C₇₃	—	32	1	no	—
48	C₂×C₇₃	#1357	1	1	yes	characteristic
49	C₂×C₇₃	#1360	1	2	yes	—
50	#1709	—	73	1	no	—
51	#1367	—	73	2	no	—
52	C₇₃	#7092	1	1	yes	characteristic, complemented, Sylow
53	#494	—	73	1	no	—
54	#493	#64408	1	1	yes	characteristic
55	#486	—	146	2	no	—
56	C₄⋊C₁₆	—	146	2	no	—
57	C₈²	—	73	1	no	—
58	C₂×Q₁₆	—	292	1	no	—
59	C₂×D₈	—	4	1	no	—
60	C₄⋊Q₈	—	146	1	no	—
61	#127	—	2	1	no	—
62	C₂×C₁₆	—	292	2	no	—
63	#102	—	292	2	no	—
64	#102	—	292	2	no	—
65	C₄×C₈	—	73	2	no	—
66	C₄×C₈	C₂×#64408	1	1	yes	characteristic
67	C₂×Q₈	—	292	1	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	4	2	no	—
70	Q₁₆	—	584	2	no	—
71	D₈	—	8	2	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	73	2	no	—
74	C₂×C₈	—	73	2	no	—
75	C₄⋊C₄	—	292	2	no	—
76	C₄²	C₂²⋊#64408	1	1	yes	characteristic
77	C₁₆	—	584	2	no	—
78	C₂³	—	8	1	no	—
79	Q₈	—	584	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	2	no	—
82	D₄	—	8	2	no	—
83	C₂×C₄	—	584	1	no	—
84	C₂×C₄	#14556651	1	1	yes	characteristic
85	C₂×C₄	#14556653	1	2	yes	—
86	C₈	—	2	2	no	—
87	C₈	—	146	2	no	—
88	C₈	—	146	2	no	—
89	C₂²	—	16	1	no	—
90	C₂²	—	16	2	no	—
91	C₂²	#17989445	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	1168	1	no	—
94	C₄	#17989501	1	2	yes	—
95	C₄	#17989503	1	2	yes	—
96	C₂	—	32	1	no	—
97	C₂	#22624939	1	1	yes	characteristic
98	C₂	#22624943	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22624954

C₂

yes

maximal, complemented

#22609536

C₂

yes

maximal, characteristic

C₄⋊#11924031

—

#17988967

C₄

yes

characteristic

#17988933

—

C₈×#10596831

C₂²

yes

characteristic

#493×C₇₃

C₄

yes

characteristic

C₂×#11924031

—

C₂×Dic₂₉₂

—

—

—

—

D₄

yes

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924031

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

C₂×#10596831

D₈

yes

—

C₂×#10596831

SD₁₆

yes

—

C₄⋊Dic₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#100

yes

characteristic

C₂×C₄×C₇₃

#102

yes

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#331

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#359

yes

—

C₄×C₇₃

#360

yes

—

Dic₇₃

—

#7092

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1357

yes

characteristic

C₂×C₇₃

#1360

yes

—

#1709

—

#1367

—

C₇₃

#7092

yes

characteristic, complemented, Sylow

—

yes

—

146

—

C₄⋊C₁₆

—

146

—

C₈²

—

C₂×Q₁₆

—

292

—

C₂×D₈

—

C₄⋊Q₈

—

146

—

#127

—

C₂×C₁₆

—

292

—

#102

—

292

—

#102

—

292

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂×Q₈

—

292

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

584

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

292

—

C₄²

C₂²⋊#64408

yes

characteristic

C₁₆

—

584

—

C₂³

—

Q₈

—

584

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

584

—

C₂×C₄

#14556651

yes

characteristic

C₂×C₄

#14556653

yes

—

C₈

—

C₈

—

146

—

C₈

—

146

—

C₂²

—

C₂²

—

C₂²

#17989445

yes

characteristic

C₄

—

C₄

—

1168

—

C₄

#17989501

yes

—

C₄

#17989503

yes

—

C₂

—

C₂

#22624939

yes

characteristic

C₂

#22624943

yes

—

	subgroup	G/H
center	C₂²	#17989445
commutator	C₄×C₈×C₇₃	C₂×C₄
Frattini	C₄×C₈	C₂×#64408
Fitting, prime radical	#493×C₇₃	C₄
socle	C₂²×C₇₃	#331
cosocle	C₈×#10596831	C₂²
hypercenter	#493	#64408
hypocenter	C₇₃	#7092
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, prime radical

#493×C₇₃

C₄

C₂²×C₇₃

C₂²

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #22624954	C₂	semidirect
52. C₇₃	#7092	semidirect

H ≅ G/N

type

1. #22624954

C₂

semidirect

52. C₇₃

#7092

semidirect

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

subgroup	index
#22624954	2
#22609536	2
#7092	73

subgroup

index

#22624954

#22609536

#7092

Subgroups — #34000060

Conjugacy classes of subgroups

Complements

Maximal subgroups