Group #34068699 — subgroups

Subgroups — #34068699

Order 19328.

224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 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	#23752617	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752584	C₂	1	2	yes	maximal, complemented
3	#23752558	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558992	—	2	4	no	—
7	#18012739	C₂²	1	2	yes	—
8	#18012673	—	2	2	no	—
9	#18012660	—	2	4	no	—
10	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₁₅₁	—	2	2	no	—
13	#127×C₁₅₁	C₂²	1	2	yes	—
14	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₀₂	—	2	2	no	—
16	#14558992	—	4	8	no	—
17	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
18	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
19	C₄⋊Dic₁₅₁	—	2	2	no	—
20	C₄⋊Dic₁₅₁	—	2	2	no	—
21	C₂×D₄×C₁₅₁	—	2	2	no	—
22	C₂×D₄×C₁₅₁	—	2	2	no	—
23	C₂×D₄×C₁₅₁	—	2	2	no	—
24	D₈×C₁₅₁	—	4	4	no	—
25	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
26	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
27	Dic₃₀₂	—	4	4	no	—
28	C₂×Dic₁₅₁	—	4	2	no	—
29	C₁₅₁⋊C₈	—	2	2	no	—
30	C₁₅₁⋊C₈	D₈	1	4	yes	complemented
31	C₂³×C₁₅₁	—	4	2	no	—
32	D₄×C₁₅₁	—	4	4	no	—
33	D₄×C₁₅₁	—	4	4	no	—
34	D₄×C₁₅₁	—	4	4	no	—
35	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₅₁	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₅₁	—	8	2	no	—
40	C₂²×C₁₅₁	—	8	2	no	—
41	C₂²×C₁₅₁	—	8	2	no	—
42	C₂²×C₁₅₁	#127	1	1	yes	characteristic
43	C₄×C₁₅₁	C₂×D₈	1	2	yes	—
44	C₄×C₁₅₁	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₅₁	#135	1	2	yes	—
46	Dic₁₅₁	—	8	2	no	—
47	C₂×C₁₅₁	—	16	2	no	—
48	C₂×C₁₅₁	#492	1	1	yes	characteristic
49	C₂×C₁₅₁	#493	1	1	yes	characteristic
50	C₂×C₁₅₁	#495	1	1	yes	characteristic
51	C₁₅₁	#1706	1	1	yes	characteristic, complemented, Sylow
52	#1706	—	151	1	no	maximal, Sylow
53	#500	—	151	1	no	—
54	#493	D₁₅₁	1	1	yes	characteristic
55	#492	—	151	2	no	—
56	#486	—	151	2	no	—
57	C₈²	—	151	1	no	—
58	C₂×SD₁₆	—	302	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	151	2	no	—
61	#127	C₂×D₁₅₁	1	2	yes	—
62	#107	—	302	2	no	—
63	#102	—	302	4	no	—
64	C₄×C₈	—	151	1	no	—
65	C₄×C₈	—	151	1	no	—
66	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
67	C₂×Q₈	—	302	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	SD₁₆	—	604	8	no	—
72	D₈	—	4	4	no	—
73	C₂×C₈	—	151	2	no	—
74	C₂×C₈	—	151	2	no	—
75	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
76	C₄⋊C₄	—	302	2	no	—
77	C₄⋊C₄	—	302	2	no	—
78	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
79	C₂³	—	4	2	no	—
80	Q₈	—	604	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	4	4	no	—
84	C₂×C₄	—	604	2	no	—
85	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
86	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
88	C₈	—	302	2	no	—
89	C₈	—	151	4	no	—
90	C₈	#14558992	1	4	yes	complemented
91	C₂²	—	8	2	no	—
92	C₂²	—	8	2	no	—
93	C₂²	—	8	2	no	—
94	C₂²	#18012735	1	1	yes	characteristic
95	C₄	—	1208	2	no	—
96	C₄	D₈×D₁₅₁	1	2	yes	—
97	C₄	C₂×#14558992	1	2	yes	—
98	C₄	#18012758	1	2	yes	—
99	C₂	—	16	2	no	—
100	C₂	#23752584	1	1	yes	characteristic
101	C₂	#23752587	1	1	yes	characteristic
102	C₂	#23752595	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752617

C₂

yes

maximal, characteristic, complemented

#23752584

C₂

yes

maximal, complemented

#23752558

C₂

yes

maximal, complemented

C₈×C₁₅₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#493×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558992

—

#18012739

C₂²

yes

—

#18012673

—

#18012660

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₁₅₁

—

#127×C₁₅₁

C₂²

yes

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

#14558992

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

D₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

D₈

yes

complemented

C₂³×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₈×C₁₅₁

SD₁₆

yes

complemented

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#127

yes

characteristic

C₄×C₁₅₁

C₂×D₈

yes

—

C₄×C₁₅₁

C₂×SD₁₆

yes

—

C₄×C₁₅₁

#135

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1706

—

151

maximal, Sylow

#500

—

151

—

#493

D₁₅₁

yes

characteristic

#492

—

151

—

#486

—

151

—

C₈²

—

151

—

C₂×SD₁₆

—

302

—

C₂×D₈

—

C₄⋊Q₈

—

151

—

#127

C₂×D₁₅₁

yes

—

#107

—

302

—

#102

—

302

—

C₄×C₈

—

151

—

C₄×C₈

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

302

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

604

—

D₈

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

C₂²⋊D₁₅₁

yes

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄²

C₂²×D₁₅₁

yes

characteristic

C₂³

—

Q₈

—

604

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

604

—

C₂×C₄

D₄×D₁₅₁

yes

characteristic

C₂×C₄

D₄×D₁₅₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₈

—

302

—

C₈

—

151

—

C₈

#14558992

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

#18012735

yes

characteristic

C₄

—

1208

—

C₄

D₈×D₁₅₁

yes

—

C₄

C₂×#14558992

yes

—

C₄

#18012758

yes

—

C₂

—

100

C₂

#23752584

yes

characteristic

101

C₂

#23752587

yes

characteristic

102

C₂

#23752595

yes

characteristic

	subgroup	G/H
center	C₂²	#18012735
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting, prime radical	#493×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#127
hypercenter	#493	D₁₅₁
hypocenter	C₁₅₁	#1706
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#18012735

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting, prime radical

#493×C₁₅₁

C₂

C₂²×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #23752617	C₂	semidirect
2. #23752584	C₂	semidirect
3. #23752558	C₂	semidirect
4. C₈×C₁₅₁⋊C₈	C₂	semidirect
30. C₁₅₁⋊C₈	D₈	semidirect
38. C₈×C₁₅₁	SD₁₆	semidirect
51. C₁₅₁	#1706	semidirect
90. C₈	#14558992	semidirect

H ≅ G/N

type

1. #23752617

C₂

semidirect

2. #23752584

C₂

semidirect

3. #23752558

C₂

semidirect

4. C₈×C₁₅₁⋊C₈

C₂

semidirect

30. C₁₅₁⋊C₈

D₈

semidirect

38. C₈×C₁₅₁

SD₁₆

semidirect

51. C₁₅₁

#1706

semidirect

90. C₈

#14558992

semidirect

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

subgroup	index
#23752617	2
#23752584	2
#23752558	2
C₈×C₁₅₁⋊C₈	2
#493×C₁₅₁	2
#1706	151

subgroup

index

#23752617

#23752584

#23752558

C₈×C₁₅₁⋊C₈

#493×C₁₅₁

#1706

151

Subgroups — #34068699

Conjugacy classes of subgroups

Complements

Maximal subgroups