Group #34068774 — subgroups

Subgroups — #34068774

Order 19328.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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	#23752589	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752588	C₂	1	2	yes	maximal, complemented
3	#23752560	C₂	1	2	yes	maximal, complemented
4	#23751933	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₀₈	—	2	2	no	—
7	C₂×#14558989	—	2	4	no	—
8	#18012736	C₂²	1	2	yes	—
9	#18012659	—	2	4	no	—
10	C₄×C₁₅₁⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₅₁	—	2	2	no	—
12	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	#127×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂×D₆₀₄	—	2	2	no	—
16	C₂×D₆₀₄	—	2	2	no	—
17	C₂×D₆₀₄	—	2	2	no	—
18	D₁₂₀₈	—	4	4	no	—
19	#14558989	—	4	8	no	—
20	C₂×C₁₅₁⋊C₈	D₄	1	4	yes	—
21	C₂×Q₈×C₁₅₁	—	2	1	no	—
22	C₂×D₄×C₁₅₁	—	2	2	no	—
23	C₂×D₄×C₁₅₁	—	2	1	no	—
24	SD₁₆×C₁₅₁	—	4	4	no	—
25	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₅₁	—	2	2	no	—
27	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₂²×D₁₅₁	—	4	2	no	—
29	D₆₀₄	—	4	4	no	—
30	D₆₀₄	—	4	4	no	—
31	D₆₀₄	—	4	4	no	—
32	C₁₅₁⋊C₈	—	2	4	no	—
33	C₂³×C₁₅₁	—	4	1	no	—
34	Q₈×C₁₅₁	—	4	2	no	—
35	D₄×C₁₅₁	—	4	2	no	—
36	D₄×C₁₅₁	—	4	4	no	—
37	C₂×C₄×C₁₅₁	—	4	1	no	—
38	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₁₅₁	D₈	1	4	yes	complemented
41	C₂²×C₁₅₁	—	8	1	no	—
42	C₂²×C₁₅₁	—	8	2	no	—
43	C₂²×C₁₅₁	#127	1	1	yes	characteristic
44	C₂×D₁₅₁	—	8	2	no	—
45	C₂×D₁₅₁	—	8	2	no	—
46	C₂×D₁₅₁	—	8	2	no	—
47	C₄×C₁₅₁	—	8	1	no	—
48	C₄×C₁₅₁	C₂×D₈	1	2	yes	—
49	C₄×C₁₅₁	AGL(1,ℤ/8)	1	4	yes	—
50	C₂×C₁₅₁	—	16	1	no	—
51	C₂×C₁₅₁	#493	1	1	yes	characteristic
52	C₂×C₁₅₁	#496	1	2	yes	—
53	D₁₅₁	—	16	2	no	—
54	C₁₅₁	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	151	1	no	maximal, Sylow
56	#493	—	151	2	no	—
57	#493	—	151	1	no	—
58	#492	D₁₅₁	1	1	yes	characteristic, complemented
59	#486	—	151	2	no	—
60	#322	—	151	1	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	302	4	no	—
63	C₂×D₈	—	302	2	no	—
64	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
65	#127	—	151	2	no	—
66	#127	C₂×D₁₅₁	1	1	yes	characteristic
67	#102	—	302	4	no	—
68	C₄×C₈	—	151	2	no	—
69	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	302	2	no	—
74	C₂×D₄	—	302	2	no	—
75	C₂×D₄	—	302	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	604	8	no	—
78	D₈	—	604	4	no	—
79	C₂×C₈	—	151	4	no	—
80	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	604	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	604	4	no	—
89	D₄	—	604	4	no	—
90	D₄	—	604	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
94	C₈	—	302	4	no	—
95	C₈	#14558989	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	1208	2	no	—
99	C₂²	—	1208	2	no	—
100	C₂²	—	1208	2	no	—
101	C₂²	#18012735	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#14558989	1	2	yes	—
104	C₄	#18012766	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	2416	2	no	—
107	C₂	#23752588	1	1	yes	characteristic
108	C₂	#23752598	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752589

C₂

yes

maximal, characteristic, complemented

#23752588

C₂

yes

maximal, complemented

#23752560

C₂

yes

maximal, complemented

#23751933

C₂

yes

maximal, characteristic, complemented

#492×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×D₁₂₀₈

—

C₂×#14558989

—

#18012736

C₂²

yes

—

#18012659

—

C₄×C₁₅₁⋊C₈

C₂²

yes

—

C₂×SD₁₆×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

#127×C₁₅₁

C₂²

yes

characteristic

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

D₁₂₀₈

—

#14558989

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×Q₈×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

SD₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₈×C₁₅₁

D₈

yes

complemented

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#127

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₈

yes

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#493

yes

characteristic

C₂×C₁₅₁

#496

yes

—

D₁₅₁

—

C₁₅₁

#1721

yes

characteristic, complemented, Sylow

#1721

—

151

maximal, Sylow

#493

—

151

—

#493

—

151

—

#492

D₁₅₁

yes

characteristic, complemented

#486

—

151

—

#322

—

151

—

C₂×SD₁₆

—

C₂×D₈

—

302

—

C₂×D₈

—

302

—

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

#127

—

151

—

#127

C₂×D₁₅₁

yes

characteristic

#102

—

302

—

C₄×C₈

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

SD₁₆

—

D₈

—

604

—

D₈

—

604

—

C₂×C₈

—

151

—

C₂×C₈

C₂²⋊D₁₅₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₅₁

yes

characteristic

C₂³

—

C₂³

—

604

—

Q₈

—

D₄

—

D₄

—

D₄

—

604

—

D₄

—

604

—

D₄

—

604

—

C₂×C₄

—

C₂×C₄

D₄×D₁₅₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₈

—

302

—

C₈

#14558989

yes

complemented

C₂²

—

C₂²

—

C₂²

—

1208

—

C₂²

—

1208

—

100

C₂²

—

1208

—

101

C₂²

#18012735

yes

characteristic

102

C₄

—

103

C₄

C₂×#14558989

yes

—

104

C₄

#18012766

yes

—

105

C₂

—

106

C₂

—

2416

—

107

C₂

#23752588

yes

characteristic

108

C₂

#23752598

yes

—

	subgroup	G/H
center	C₂²	#18012735
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#492×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#127
hypercenter	#492	D₁₅₁
hypocenter	C₁₅₁	#1721
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#18012735

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#492×C₁₅₁

C₂

C₂²×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #23752589	C₂	semidirect
2. #23752588	C₂	semidirect
3. #23752560	C₂	semidirect
4. #23751933	C₂	semidirect
5. #492×C₁₅₁	C₂	semidirect
40. C₈×C₁₅₁	D₈	semidirect
54. C₁₅₁	#1721	semidirect
58. #492	D₁₅₁	semidirect
95. C₈	#14558989	semidirect

H ≅ G/N

type

1. #23752589

C₂

semidirect

2. #23752588

C₂

semidirect

3. #23752560

C₂

semidirect

4. #23751933

C₂

semidirect

5. #492×C₁₅₁

C₂

semidirect

40. C₈×C₁₅₁

D₈

semidirect

54. C₁₅₁

#1721

semidirect

58. #492

D₁₅₁

semidirect

95. C₈

#14558989

semidirect

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

subgroup	index
#23752589	2
#23752588	2
#23752560	2
#23751933	2
#492×C₁₅₁	2
#1721	151

subgroup

index

#492×C₁₅₁

151

Subgroups — #34068774

Conjugacy classes of subgroups

Complements

Maximal subgroups