Group #34068739 — subgroups

Subgroups — #34068739

Order 19328.

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

normal subgroups only

This group has 220 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
2	#23752586	C₂	1	2	yes	maximal, complemented
3	#23752560	C₂	1	2	yes	maximal, complemented
4	#23751933	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14558991	—	2	4	no	—
7	C₂×D₁₂₀₈	—	2	2	no	—
8	#18012736	C₂²	1	1	yes	characteristic
9	#18012736	C₂²	1	1	yes	characteristic
10	#18012659	—	2	4	no	—
11	C₄×C₁₅₁⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	—
13	#106×C₁₅₁	—	2	2	no	—
14	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂×D₆₀₄	—	2	1	no	—
16	C₂×D₆₀₄	—	2	1	no	—
17	C₂×D₆₀₄	—	2	2	no	—
18	C₂×D₆₀₄	—	2	2	no	—
19	#14558991	—	4	8	no	—
20	D₁₂₀₈	—	4	4	no	—
21	C₂×C₁₅₁⋊C₈	D₄	1	4	yes	—
22	C₂×Q₈×C₁₅₁	—	2	2	no	—
23	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₅₁	—	2	2	no	—
25	C₄⋊C₄×C₁₅₁	—	2	2	no	—
26	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
27	C₂²×D₁₅₁	—	4	1	no	—
28	C₂²×D₁₅₁	—	4	1	no	—
29	D₆₀₄	—	4	2	no	—
30	D₆₀₄	—	4	2	no	—
31	D₆₀₄	—	4	4	no	—
32	D₆₀₄	—	4	4	no	—
33	C₁₅₁⋊C₈	—	2	4	no	—
34	Q₈×C₁₅₁	—	4	4	no	—
35	C₂×C₄×C₁₅₁	—	4	2	no	—
36	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₅₁	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₅₁	#127	1	1	yes	characteristic
40	C₂×D₁₅₁	—	8	1	no	—
41	C₂×D₁₅₁	—	8	1	no	—
42	C₂×D₁₅₁	—	8	2	no	—
43	C₂×D₁₅₁	—	8	2	no	—
44	C₄×C₁₅₁	—	8	2	no	—
45	C₄×C₁₅₁	C₂×SD₁₆	1	2	yes	—
46	C₄×C₁₅₁	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₁₅₁	#492	1	1	yes	characteristic
48	C₂×C₁₅₁	#496	1	2	yes	—
49	D₁₅₁	—	16	1	no	—
50	D₁₅₁	—	16	1	no	—
51	C₁₅₁	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	151	1	no	maximal, Sylow
53	#498	D₁₅₁	1	1	yes	characteristic, complemented
54	#493	—	151	1	no	—
55	#492	—	151	2	no	—
56	#486	—	151	2	no	—
57	#322	—	151	1	no	—
58	C₂×SD₁₆	—	302	4	no	—
59	C₂×D₈	—	302	2	no	—
60	C₄⋊Q₈	C₂×D₁₅₁	1	2	yes	—
61	#127	—	151	1	no	—
62	#127	—	151	1	no	—
63	#106	—	2	2	no	—
64	#102	—	302	4	no	—
65	C₄×C₈	—	151	2	no	—
66	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	302	2	no	—
69	C₂×D₄	—	302	2	no	—
70	C₂×D₄	—	302	1	no	—
71	C₂×D₄	—	302	1	no	—
72	SD₁₆	—	604	8	no	—
73	D₈	—	604	4	no	—
74	C₂×C₈	—	151	4	no	—
75	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
79	C₂³	—	604	1	no	—
80	C₂³	—	604	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	604	2	no	—
83	D₄	—	604	2	no	—
84	D₄	—	604	4	no	—
85	D₄	—	604	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
89	C₈	—	302	4	no	—
90	C₈	#14558991	1	4	yes	complemented
91	C₂²	—	1208	1	no	—
92	C₂²	—	1208	1	no	—
93	C₂²	—	1208	2	no	—
94	C₂²	—	1208	2	no	—
95	C₂²	#18012735	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#14558991	1	2	yes	—
98	C₄	#18012766	1	4	yes	—
99	C₂	—	2416	1	no	—
100	C₂	—	2416	1	no	—
101	C₂	#23752586	1	1	yes	characteristic
102	C₂	#23752598	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752589

C₂

yes

maximal, characteristic

#23752586

C₂

yes

maximal, complemented

#23752560

C₂

yes

maximal, complemented

#23751933

C₂

yes

maximal, characteristic, complemented

#498×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×#14558991

—

C₂×D₁₂₀₈

—

#18012736

C₂²

yes

characteristic

#18012736

C₂²

yes

characteristic

#18012659

—

C₄×C₁₅₁⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

—

#106×C₁₅₁

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

#14558991

—

D₁₂₀₈

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×Q₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

C₁₅₁⋊C₈

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₈×C₁₅₁

SD₁₆

yes

complemented

C₂²×C₁₅₁

#127

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×SD₁₆

yes

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₅₁

#492

yes

characteristic

C₂×C₁₅₁

#496

yes

—

D₁₅₁

—

D₁₅₁

—

C₁₅₁

#1715

yes

characteristic, complemented, Sylow

#1715

—

151

maximal, Sylow

#498

D₁₅₁

yes

characteristic, complemented

#493

—

151

—

#492

—

151

—

#486

—

151

—

#322

—

151

—

C₂×SD₁₆

—

302

—

C₂×D₈

—

302

—

C₄⋊Q₈

C₂×D₁₅₁

yes

—

#127

—

151

—

#127

—

151

—

#106

—

#102

—

302

—

C₄×C₈

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

C₂×D₄

—

302

—

SD₁₆

—

604

—

D₈

—

604

—

C₂×C₈

—

151

—

C₂×C₈

C₂²⋊D₁₅₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₅₁

yes

characteristic

C₂³

—

604

—

C₂³

—

604

—

Q₈

—

D₄

—

604

—

D₄

—

604

—

D₄

—

604

—

D₄

—

604

—

C₂×C₄

—

C₂×C₄

D₄×D₁₅₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₈

—

302

—

C₈

#14558991

yes

complemented

C₂²

—

1208

—

C₂²

—

1208

—

C₂²

—

1208

—

C₂²

—

1208

—

C₂²

#18012735

yes

characteristic

C₄

—

C₄

C₂×#14558991

yes

—

C₄

#18012766

yes

—

C₂

—

2416

—

100

C₂

—

2416

—

101

C₂

#23752586

yes

characteristic

102

C₂

#23752598

yes

—

	subgroup	G/H
center	C₂²	#18012735
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#498×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#127
prime radical	#23752589	C₂
hypercenter	#498	D₁₅₁
hypocenter	C₁₅₁	#1715
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#18012735

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#498×C₁₅₁

C₂

C₂²×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #23752586	C₂	semidirect
3. #23752560	C₂	semidirect
4. #23751933	C₂	semidirect
5. #498×C₁₅₁	C₂	semidirect
38. C₈×C₁₅₁	SD₁₆	semidirect
51. C₁₅₁	#1715	semidirect
53. #498	D₁₅₁	semidirect
90. C₈	#14558991	semidirect

H ≅ G/N

type

2. #23752586

C₂

semidirect

3. #23752560

C₂

semidirect

4. #23751933

C₂

semidirect

5. #498×C₁₅₁

C₂

semidirect

38. C₈×C₁₅₁

SD₁₆

semidirect

51. C₁₅₁

#1715

semidirect

53. #498

D₁₅₁

semidirect

90. C₈

#14558991

semidirect

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

subgroup	index
#23752589	2
#23752586	2
#23752560	2
#23751933	2
#498×C₁₅₁	2
#1715	151

subgroup

index

#498×C₁₅₁

151

Subgroups — #34068739

Conjugacy classes of subgroups

Complements

Maximal subgroups