Group #16845340 — subgroups

Subgroups — #16845340

Order 3968.

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	#14521426	C₂	1	1	yes	maximal, characteristic
2	#14521423	C₂	1	2	yes	maximal, complemented
3	#14521397	C₂	1	2	yes	maximal, complemented
4	#14520770	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×#92481	—	2	4	no	—
8	C₂×D₂₄₈	—	2	2	no	—
9	#11758451	C₂²	1	1	yes	characteristic
10	#11758451	C₂²	1	1	yes	characteristic
11	C₃₁×#106	—	2	2	no	—
12	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	#11758397	—	2	4	no	—
14	C₄×C₃₁⋊C₈	C₂²	1	2	yes	—
15	C₂×Q₈×C₃₁	—	2	2	no	—
16	C₂×D₁₂₄	—	2	2	no	—
17	C₂×D₁₂₄	—	2	1	no	—
18	C₂×D₁₂₄	—	2	1	no	—
19	C₂×D₁₂₄	—	2	2	no	—
20	C₂×C₈×C₃₁	D₄	1	2	yes	—
21	C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₄²×C₃₁	C₂³	1	1	yes	characteristic
24	#92481	—	4	8	no	—
25	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
26	D₂₄₈	—	4	4	no	—
27	C₂²×D₃₁	—	4	1	no	—
28	C₂²×D₃₁	—	4	1	no	—
29	Q₈×C₃₁	—	4	4	no	—
30	C₂×C₄×C₃₁	—	4	2	no	—
31	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
33	D₁₂₄	—	4	2	no	—
34	D₁₂₄	—	4	2	no	—
35	D₁₂₄	—	4	4	no	—
36	D₁₂₄	—	4	4	no	—
37	C₈×C₃₁	SD₁₆	1	4	yes	complemented
38	C₃₁⋊C₈	—	2	4	no	—
39	#1715	—	31	1	no	maximal, Sylow
40	C₂²×C₃₁	#127	1	1	yes	characteristic
41	C₂×D₃₁	—	8	1	no	—
42	C₂×D₃₁	—	8	1	no	—
43	C₂×D₃₁	—	8	2	no	—
44	C₂×D₃₁	—	8	2	no	—
45	C₄×C₃₁	—	8	2	no	—
46	C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
47	C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
48	#498	D₃₁	1	1	yes	characteristic, complemented
49	#493	—	31	1	no	—
50	#492	—	31	2	no	—
51	#486	—	31	2	no	—
52	#322	—	31	1	no	—
53	C₂×C₃₁	#492	1	1	yes	characteristic
54	C₂×C₃₁	#496	1	2	yes	—
55	D₃₁	—	16	1	no	—
56	D₃₁	—	16	1	no	—
57	C₂×SD₁₆	—	62	4	no	—
58	C₂×D₈	—	62	2	no	—
59	C₄⋊Q₈	C₂×D₃₁	1	2	yes	—
60	#127	—	31	1	no	—
61	#127	—	31	1	no	—
62	#106	—	2	2	no	—
63	#102	—	62	4	no	—
64	C₄×C₈	—	31	2	no	—
65	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
66	C₃₁	#1715	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	62	2	no	—
69	C₂×D₄	—	62	2	no	—
70	C₂×D₄	—	62	1	no	—
71	C₂×D₄	—	62	1	no	—
72	SD₁₆	—	124	8	no	—
73	D₈	—	124	4	no	—
74	C₂×C₈	—	31	4	no	—
75	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₃₁	1	1	yes	characteristic
79	C₂³	—	124	1	no	—
80	C₂³	—	124	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	124	4	no	—
83	D₄	—	124	4	no	—
84	D₄	—	124	2	no	—
85	D₄	—	124	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₃₁	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
89	C₈	—	62	4	no	—
90	C₈	#92481	1	4	yes	complemented
91	C₂²	—	248	1	no	—
92	C₂²	—	248	1	no	—
93	C₂²	—	248	2	no	—
94	C₂²	—	248	2	no	—
95	C₂²	#11758517	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	#11758491	1	4	yes	—
98	C₄	C₂×#92481	1	2	yes	—
99	C₂	—	496	1	no	—
100	C₂	—	496	1	no	—
101	C₂	#14521423	1	1	yes	characteristic
102	C₂	#14521435	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521426

C₂

yes

maximal, characteristic

#14521423

C₂

yes

maximal, complemented

#14521397

C₂

yes

maximal, complemented

#14520770

C₂

yes

maximal, characteristic, complemented

C₃₁×#498

C₂

yes

maximal, characteristic, complemented

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₂×#92481

—

C₂×D₂₄₈

—

#11758451

C₂²

yes

characteristic

#11758451

C₂²

yes

characteristic

C₃₁×#106

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758397

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×C₈×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

#92481

—

C₂×C₃₁⋊C₈

D₄

yes

—

D₂₄₈

—

C₂²×D₃₁

—

C₂²×D₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

C₈×C₃₁

SD₁₆

yes

complemented

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×SD₁₆

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

#498

D₃₁

yes

characteristic, complemented

—

—

—

—

C₂×C₃₁

#492

yes

characteristic

C₂×C₃₁

#496

yes

—

D₃₁

—

D₃₁

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₃₁

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1715

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

124

—

D₈

—

124

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

—

124

—

C₂³

—

124

—

Q₈

—

D₄

—

124

—

D₄

—

124

—

D₄

—

124

—

D₄

—

124

—

C₂×C₄

—

C₂×C₄

D₄×D₃₁

yes

—

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

C₈

—

C₈

#92481

yes

complemented

C₂²

—

248

—

C₂²

—

248

—

C₂²

—

248

—

C₂²

—

248

—

C₂²

#11758517

yes

characteristic

C₄

—

C₄

#11758491

yes

—

C₄

C₂×#92481

yes

—

C₂

—

496

—

100

C₂

—

496

—

101

C₂

#14521423

yes

characteristic

102

C₂

#14521435

yes

—

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#498	C₂
socle	C₂²×C₃₁	#127
prime radical	#14521426	C₂
hypercenter	#498	D₃₁
hypocenter	C₃₁	#1715
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#11758517

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#498

C₂

C₂²×C₃₁

C₂

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #14521423	C₂	semidirect
3. #14521397	C₂	semidirect
4. #14520770	C₂	semidirect
5. C₃₁×#498	C₂	semidirect
37. C₈×C₃₁	SD₁₆	semidirect
48. #498	D₃₁	semidirect
66. C₃₁	#1715	semidirect
90. C₈	#92481	semidirect

H ≅ G/N

type

2. #14521423

C₂

semidirect

3. #14521397

C₂

semidirect

4. #14520770

C₂

semidirect

5. C₃₁×#498

C₂

semidirect

37. C₈×C₃₁

SD₁₆

semidirect

48. #498

D₃₁

semidirect

66. C₃₁

#1715

semidirect

90. C₈

#92481

semidirect

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

subgroup	index
#14521426	2
#14521423	2
#14521397	2
#14520770	2
C₃₁×#498	2
#1715	31

subgroup

index

C₃₁×#498

Subgroups — #16845340

Conjugacy classes of subgroups

Complements

Maximal subgroups