Group #34068776 — subgroups

Subgroups — #34068776

Order 19328.

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

normal subgroups only

This group has 242 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	#23752589	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752589	C₂	1	2	yes	maximal, complemented
3	#23752585	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752559	C₂	1	2	yes	maximal, complemented
5	#322×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14558993	—	2	2	no	—
7	C₂×D₁₂₀₈	—	2	4	no	—
8	C₂×D₁₂₀₈	—	2	2	no	—
9	#18012739	C₂²	1	1	yes	characteristic
10	#18012736	C₂²	1	2	yes	—
11	#18012736	C₂²	1	1	yes	characteristic
12	#18012661	—	2	4	no	—
13	C₄×C₈×C₁₅₁	C₂²	1	2	yes	—
14	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₀₂	—	2	1	no	—
16	C₂×D₆₀₄	—	2	2	no	—
17	C₂×D₆₀₄	—	2	2	no	—
18	C₂×D₆₀₄	—	2	2	no	—
19	C₂×D₆₀₄	—	2	1	no	—
20	C₂×D₆₀₄	—	2	2	no	—
21	#14558993	—	4	4	no	—
22	D₁₂₀₈	—	4	8	no	—
23	D₁₂₀₈	—	4	4	no	—
24	C₄⋊Dic₁₅₁	—	2	2	no	—
25	C₂×C₈×C₁₅₁	D₄	1	4	yes	—
26	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
27	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₂²×D₁₅₁	—	4	1	no	—
29	C₂²×D₁₅₁	—	4	2	no	—
30	Dic₃₀₂	—	4	2	no	—
31	D₆₀₄	—	4	4	no	—
32	D₆₀₄	—	4	2	no	—
33	D₆₀₄	—	4	4	no	—
34	D₆₀₄	—	4	4	no	—
35	D₆₀₄	—	4	4	no	—
36	C₂×Dic₁₅₁	—	4	1	no	—
37	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
39	C₈×C₁₅₁	—	2	4	no	—
40	C₈×C₁₅₁	D₈	1	4	yes	complemented
41	C₂²×C₁₅₁	#127	1	1	yes	characteristic
42	C₂×D₁₅₁	—	8	2	no	—
43	C₂×D₁₅₁	—	8	1	no	—
44	C₂×D₁₅₁	—	8	2	no	—
45	C₂×D₁₅₁	—	8	2	no	—
46	C₂×D₁₅₁	—	8	2	no	—
47	C₄×C₁₅₁	C₂×D₈	1	2	yes	—
48	C₄×C₁₅₁	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₁₅₁	—	8	1	no	—
50	C₂×C₁₅₁	#493	1	1	yes	characteristic
51	C₂×C₁₅₁	#496	1	2	yes	—
52	D₁₅₁	—	16	2	no	—
53	D₁₅₁	—	16	1	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	—	151	1	no	—
59	#486	—	151	2	no	—
60	#322	D₁₅₁	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	302	2	no	—
62	C₂×D₈	—	302	4	no	—
63	C₂×D₈	—	302	2	no	—
64	C₄⋊Q₈	—	151	1	no	—
65	#127	—	151	2	no	—
66	#127	—	151	1	no	—
67	#102	—	302	4	no	—
68	C₄×C₈	C₂×D₁₅₁	1	2	yes	—
69	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
70	C₂×Q₈	—	302	1	no	—
71	C₂×D₄	—	302	2	no	—
72	C₂×D₄	—	302	2	no	—
73	C₂×D₄	—	302	2	no	—
74	C₂×D₄	—	302	2	no	—
75	C₂×D₄	—	302	1	no	—
76	SD₁₆	—	604	4	no	—
77	D₈	—	604	8	no	—
78	D₈	—	604	4	no	—
79	C₂×C₈	D₆₀₄	1	4	yes	—
80	C₂×C₈	D₆₀₄	1	2	yes	—
81	C₄⋊C₄	—	302	2	no	—
82	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
83	C₂³	—	604	2	no	—
84	C₂³	—	604	1	no	—
85	Q₈	—	604	2	no	—
86	D₄	—	604	4	no	—
87	D₄	—	604	2	no	—
88	D₄	—	604	4	no	—
89	D₄	—	604	4	no	—
90	D₄	—	604	4	no	—
91	C₂×C₄	—	604	1	no	—
92	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
93	C₂×C₄	C₂×D₆₀₄	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₁₂₀₈	1	4	yes	complemented
96	C₂²	—	1208	2	no	—
97	C₂²	—	1208	1	no	—
98	C₂²	—	1208	2	no	—
99	C₂²	—	1208	2	no	—
100	C₂²	—	1208	2	no	—
101	C₂²	#18012736	1	1	yes	characteristic
102	C₄	—	1208	1	no	—
103	C₄	C₂×D₁₂₀₈	1	2	yes	—
104	C₄	#18012767	1	4	yes	—
105	C₂	—	2416	2	no	—
106	C₂	—	2416	1	no	—
107	C₂	#23752589	1	1	yes	characteristic
108	C₂	#23752601	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752589

C₂

yes

maximal, characteristic, complemented

#23752589

C₂

yes

maximal, complemented

#23752585

C₂

yes

maximal, characteristic, complemented

#23752559

C₂

yes

maximal, complemented

#322×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14558993

—

C₂×D₁₂₀₈

—

C₂×D₁₂₀₈

—

#18012739

C₂²

yes

characteristic

#18012736

C₂²

yes

—

#18012736

C₂²

yes

characteristic

#18012661

—

C₄×C₈×C₁₅₁

C₂²

yes

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

#14558993

—

D₁₂₀₈

—

D₁₂₀₈

—

C₄⋊Dic₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

Dic₃₀₂

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

C₂×Dic₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₈×C₁₅₁

—

C₈×C₁₅₁

D₈

yes

complemented

C₂²×C₁₅₁

#127

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

C₂×D₈

yes

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

#493

yes

characteristic

C₂×C₁₅₁

#496

yes

—

D₁₅₁

—

D₁₅₁

—

C₁₅₁

#1721

yes

characteristic, complemented, Sylow

#1721

—

151

maximal, Sylow

#493

—

151

—

#493

—

151

—

#492

—

151

—

#486

—

151

—

#322

D₁₅₁

yes

characteristic, complemented

C₂×SD₁₆

—

302

—

C₂×D₈

—

302

—

C₂×D₈

—

302

—

C₄⋊Q₈

—

151

—

#127

—

151

—

#127

—

151

—

#102

—

302

—

C₄×C₈

C₂×D₁₅₁

yes

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

SD₁₆

—

604

—

D₈

—

604

—

D₈

—

604

—

C₂×C₈

D₆₀₄

yes

—

C₂×C₈

D₆₀₄

yes

—

C₄⋊C₄

—

302

—

C₄²

C₂²×D₁₅₁

yes

characteristic

C₂³

—

604

—

C₂³

—

604

—

Q₈

—

604

—

D₄

—

604

—

D₄

—

604

—

D₄

—

604

—

D₄

—

604

—

D₄

—

604

—

C₂×C₄

—

604

—

C₂×C₄

C₂×D₆₀₄

yes

characteristic

C₂×C₄

C₂×D₆₀₄

yes

—

C₈

—

C₈

D₁₂₀₈

yes

complemented

C₂²

—

1208

—

C₂²

—

1208

—

C₂²

—

1208

—

C₂²

—

1208

—

100

C₂²

—

1208

—

101

C₂²

#18012736

yes

characteristic

102

C₄

—

1208

—

103

C₄

C₂×D₁₂₀₈

yes

—

104

C₄

#18012767

yes

—

105

C₂

—

2416

—

106

C₂

—

2416

—

107

C₂

#23752589

yes

characteristic

108

C₂

#23752601

yes

—

	subgroup	G/H
center	C₂²	#18012736
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#322×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#127
hypercenter	#322	D₁₅₁
hypocenter	C₁₅₁	#1721
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#18012736

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#322×C₁₅₁

C₂

C₂²×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #23752589	C₂	semidirect
2. #23752589	C₂	semidirect
3. #23752585	C₂	semidirect
4. #23752559	C₂	semidirect
5. #322×C₁₅₁	C₂	semidirect
40. C₈×C₁₅₁	D₈	semidirect
54. C₁₅₁	#1721	semidirect
60. #322	D₁₅₁	semidirect
95. C₈	D₁₂₀₈	semidirect

H ≅ G/N

type

1. #23752589

C₂

semidirect

2. #23752589

C₂

semidirect

3. #23752585

C₂

semidirect

4. #23752559

C₂

semidirect

5. #322×C₁₅₁

C₂

semidirect

40. C₈×C₁₅₁

D₈

semidirect

54. C₁₅₁

#1721

semidirect

60. #322

D₁₅₁

semidirect

95. C₈

D₁₂₀₈

semidirect

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

subgroup	index
#23752589	2
#23752585	2
#23752559	2
#322×C₁₅₁	2
#1721	151

subgroup

index

#322×C₁₅₁

151

Subgroups — #34068776

Conjugacy classes of subgroups

Complements

Maximal subgroups