Group #34001252 — subgroups

Subgroups — #34001252

Order 18688.

186 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 22 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	#22625063	C₂	1	2	yes	maximal, complemented
2	#22609545	C₂	1	1	yes	maximal, characteristic
3	#17989492	—	2	2	no	—
4	#17988964	C₂²	1	1	yes	characteristic
5	#17988963	—	2	2	no	—
6	C₈×#10596831	C₄	1	1	yes	characteristic, complemented
7	#493×C₇₃	C₄	1	1	yes	characteristic, complemented
8	#14556650	—	4	2	no	—
9	C₂×D₅₈₄	—	4	1	no	—
10	C₂×#11924013	—	4	4	no	—
11	#14556588	D₄	1	2	yes	—
12	C₄×#10596831	—	2	1	no	—
13	C₂×D₈×C₇₃	—	4	1	no	—
14	#127×C₇₃	—	2	1	no	—
15	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊#64408	—	8	2	no	—
17	C₂×D₂₉₂	—	2	2	no	—
18	C₂×D₂₉₂	—	4	2	no	—
19	D₅₈₄	—	8	2	no	—
20	#11924013	—	8	8	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×D₄×C₇₃	—	4	2	no	—
23	C₂×D₄×C₇₃	—	4	1	no	—
24	D₈×C₇₃	—	8	2	no	—
25	C₂×C₈×C₇₃	—	2	1	no	—
26	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂³×C₇₃	—	8	1	no	—
28	C₂²×D₇₃	—	4	2	no	—
29	C₂×#64408	—	16	2	no	—
30	D₄×C₇₃	—	8	2	no	—
31	D₄×C₇₃	—	8	4	no	—
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
34	D₂₉₂	—	8	2	no	—
35	D₂₉₂	—	8	4	no	—
36	C₈×C₇₃	—	2	2	no	—
37	#10596831	—	2	4	no	—
38	C₂²×C₇₃	—	16	2	no	—
39	C₂²×C₇₃	—	16	1	no	—
40	C₂²×C₇₃	#353	1	1	yes	characteristic
41	C₂×D₇₃	—	8	2	no	—
42	C₂×D₇₃	—	16	2	no	—
43	#64408	—	32	2	no	—
44	C₄×C₇₃	—	2	2	no	—
45	C₄×C₇₃	—	2	1	no	—
46	#7529	—	73	1	no	maximal, Sylow
47	C₂×C₇₃	—	2	1	no	—
48	C₂×C₇₃	—	32	1	no	—
49	C₂×C₇₃	#1408	1	1	yes	characteristic
50	D₇₃	—	16	2	no	—
51	#1712	—	73	1	no	—
52	#1408	—	73	2	no	—
53	C₇₃	#7529	1	1	yes	characteristic, complemented, Sylow
54	#493	—	73	1	no	—
55	#493	—	146	2	no	—
56	#493	#64408	1	1	yes	characteristic, complemented
57	#353	—	146	2	no	—
58	C₈²	—	73	1	no	—
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	—	292	1	no	—
61	C₂×D₈	—	292	4	no	—
62	#127	—	2	1	no	—
63	#127	—	73	2	no	—
64	#99	—	292	2	no	—
65	C₄×C₈	—	146	1	no	—
66	C₄×C₈	C₂×#64408	1	1	yes	characteristic
67	C₂×D₄	—	4	2	no	—
68	C₂×D₄	—	4	1	no	—
69	C₂×D₄	—	292	2	no	—
70	C₂×D₄	—	146	2	no	—
71	D₈	—	8	2	no	—
72	D₈	—	584	2	no	—
73	D₈	—	584	8	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	146	2	no	—
76	C₂²⋊C₄	—	584	2	no	—
77	C₄²	C₂²⋊#64408	1	1	yes	characteristic
78	C₂³	—	8	1	no	—
79	C₂³	—	292	2	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	584	2	no	—
83	D₄	—	584	4	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	1168	2	no	—
86	C₂×C₄	#14556649	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	—	146	4	no	—
89	C₂²	—	16	2	no	—
90	C₂²	—	16	1	no	—
91	C₂²	—	584	2	no	—
92	C₂²	—	1168	2	no	—
93	C₂²	#17989491	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	2336	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	32	1	no	—
99	C₂	—	1168	2	no	—
100	C₂	#22625062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625063

C₂

yes

maximal, complemented

#22609545

C₂

yes

maximal, characteristic

#17989492

—

#17988964

C₂²

yes

characteristic

#17988963

—

C₈×#10596831

C₄

yes

characteristic, complemented

#493×C₇₃

C₄

yes

characteristic, complemented

#14556650

—

C₂×D₅₈₄

—

C₂×#11924013

—

#14556588

D₄

yes

—

C₄×#10596831

—

C₂×D₈×C₇₃

—

#127×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂²⋊#64408

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

D₅₈₄

—

#11924013

—

C₂×#10596831

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂²⋊C₄

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

C₂×#64408

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#99

yes

characteristic

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#353

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

#7529

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#1408

yes

characteristic

D₇₃

—

#1712

—

#1408

—

C₇₃

#7529

yes

characteristic, complemented, Sylow

#493

—

#493

—

146

—

#493

#64408

yes

characteristic, complemented

#353

—

146

—

C₈²

—

C₂×D₈

—

C₂×D₈

—

292

—

C₂×D₈

—

292

—

#127

—

#127

—

#99

—

292

—

C₄×C₈

—

146

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

292

—

C₂×D₄

—

146

—

D₈

—

D₈

—

584

—

D₈

—

584

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂²⋊C₄

—

584

—

C₄²

C₂²⋊#64408

yes

characteristic

C₂³

—

C₂³

—

292

—

D₄

—

D₄

—

D₄

—

584

—

D₄

—

584

—

C₂×C₄

—

C₂×C₄

—

1168

—

C₂×C₄

#14556649

yes

characteristic

C₈

—

C₈

—

146

—

C₂²

—

C₂²

—

C₂²

—

584

—

C₂²

—

1168

—

C₂²

#17989491

yes

characteristic

C₄

—

C₄

—

C₄

—

2336

—

C₂

—

C₂

—

C₂

—

1168

—

100

C₂

#22625062

yes

characteristic

	subgroup	G/H
center	C₂	#22625062
commutator	C₄×C₈×C₇₃	C₂×C₄
Frattini	C₄×C₈	C₂×#64408
Fitting	#493×C₇₃	C₄
socle	C₂×C₇₃	#1408
prime radical	#22609545	C₂
cosocle	#17988964	C₂²
hypercenter	#493	#64408
hypocenter	C₇₃	#7529
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#22625062

commutator

C₄×C₈×C₇₃

C₂×C₄

Frattini

C₄×C₈

C₂×#64408

Fitting

#493×C₇₃

C₄

C₂×C₇₃

C₂

C₂²

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #22625063	C₂	semidirect
6. C₈×#10596831	C₄	semidirect
7. #493×C₇₃	C₄	semidirect
53. C₇₃	#7529	semidirect
56. #493	#64408	semidirect

H ≅ G/N

type

1. #22625063

C₂

semidirect

6. C₈×#10596831

C₄

semidirect

7. #493×C₇₃

C₄

semidirect

53. C₇₃

#7529

semidirect

56. #493

#64408

semidirect

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

subgroup	index
#22625063	2
#22609545	2
#7529	73

subgroup

index

#22625063

#22609545

#7529

Subgroups — #34001252

Conjugacy classes of subgroups

Complements

Maximal subgroups