Group #16845377 — subgroups

Subgroups — #16845377

Order 3968.

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	#14521426	C₂	1	2	yes	maximal, complemented
2	#14521426	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521422	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521396	C₂	1	2	yes	maximal, complemented
5	C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₄₈	—	2	4	no	—
7	C₂×D₂₄₈	—	2	2	no	—
8	C₂×#92470	—	2	2	no	—
9	#11758451	C₂²	1	2	yes	—
10	#11758451	C₂²	1	1	yes	characteristic
11	#11758446	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	#11758409	—	2	4	no	—
15	C₂×D₁₂₄	—	2	2	no	—
16	C₂×D₁₂₄	—	2	1	no	—
17	C₂×D₁₂₄	—	2	2	no	—
18	C₂×D₁₂₄	—	2	2	no	—
19	C₂×D₁₂₄	—	2	2	no	—
20	C₂×Dic₆₂	—	2	1	no	—
21	C₂×C₈×C₃₁	D₄	1	4	yes	—
22	C₂×C₈×C₃₁	D₄	1	2	yes	—
23	C₄²×C₃₁	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₃₁	—	2	2	no	—
25	D₂₄₈	—	4	8	no	—
26	D₂₄₈	—	4	4	no	—
27	#92470	—	4	4	no	—
28	C₂²×D₃₁	—	4	2	no	—
29	C₂²×D₃₁	—	4	1	no	—
30	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
32	C₂×Dic₃₁	—	4	1	no	—
33	D₁₂₄	—	4	2	no	—
34	D₁₂₄	—	4	4	no	—
35	D₁₂₄	—	4	4	no	—
36	D₁₂₄	—	4	4	no	—
37	D₁₂₄	—	4	4	no	—
38	Dic₆₂	—	4	2	no	—
39	C₈×C₃₁	—	2	4	no	—
40	C₈×C₃₁	D₈	1	4	yes	complemented
41	#1721	—	31	1	no	maximal, Sylow
42	C₂²×C₃₁	#127	1	1	yes	characteristic
43	C₂×D₃₁	—	8	1	no	—
44	C₂×D₃₁	—	8	2	no	—
45	C₂×D₃₁	—	8	2	no	—
46	C₂×D₃₁	—	8	2	no	—
47	C₂×D₃₁	—	8	2	no	—
48	C₄×C₃₁	C₂×D₈	1	2	yes	—
49	C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
50	Dic₃₁	—	8	1	no	—
51	#493	—	31	2	no	—
52	#493	—	31	1	no	—
53	#492	—	31	1	no	—
54	#486	—	31	2	no	—
55	#322	D₃₁	1	1	yes	characteristic, complemented
56	C₂×C₃₁	#493	1	1	yes	characteristic
57	C₂×C₃₁	#496	1	2	yes	—
58	D₃₁	—	16	1	no	—
59	D₃₁	—	16	2	no	—
60	C₂×SD₁₆	—	62	2	no	—
61	C₂×D₈	—	62	4	no	—
62	C₂×D₈	—	62	2	no	—
63	C₄⋊Q₈	—	31	1	no	—
64	#127	—	31	2	no	—
65	#127	—	31	1	no	—
66	#102	—	62	4	no	—
67	C₄×C₈	C₂×D₃₁	1	2	yes	—
68	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
69	C₃₁	#1721	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₈	—	62	1	no	—
71	C₂×D₄	—	62	2	no	—
72	C₂×D₄	—	62	2	no	—
73	C₂×D₄	—	62	2	no	—
74	C₂×D₄	—	62	2	no	—
75	C₂×D₄	—	62	1	no	—
76	SD₁₆	—	124	4	no	—
77	D₈	—	124	8	no	—
78	D₈	—	124	4	no	—
79	C₂×C₈	D₁₂₄	1	4	yes	—
80	C₂×C₈	D₁₂₄	1	2	yes	—
81	C₄⋊C₄	—	62	2	no	—
82	C₄²	C₂²×D₃₁	1	1	yes	characteristic
83	C₂³	—	124	2	no	—
84	C₂³	—	124	1	no	—
85	Q₈	—	124	2	no	—
86	D₄	—	124	4	no	—
87	D₄	—	124	2	no	—
88	D₄	—	124	4	no	—
89	D₄	—	124	4	no	—
90	D₄	—	124	4	no	—
91	C₂×C₄	—	124	1	no	—
92	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
93	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₂₄₈	1	4	yes	complemented
96	C₂²	—	248	2	no	—
97	C₂²	—	248	1	no	—
98	C₂²	—	248	2	no	—
99	C₂²	—	248	2	no	—
100	C₂²	—	248	2	no	—
101	C₂²	#11758451	1	1	yes	characteristic
102	C₄	—	248	1	no	—
103	C₄	C₂×D₂₄₈	1	2	yes	—
104	C₄	#11758485	1	4	yes	—
105	C₂	—	496	2	no	—
106	C₂	—	496	1	no	—
107	C₂	#14521426	1	1	yes	characteristic
108	C₂	#14521438	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521426

C₂

yes

maximal, complemented

#14521426

C₂

yes

maximal, characteristic, complemented

#14521422

C₂

yes

maximal, characteristic, complemented

#14521396

C₂

yes

maximal, complemented

C₃₁×#322

C₂

yes

maximal, characteristic, complemented

C₂×D₂₄₈

—

C₂×D₂₄₈

—

C₂×#92470

—

#11758451

C₂²

yes

—

#11758451

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₄×C₈×C₃₁

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758409

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

D₂₄₈

—

D₂₄₈

—

#92470

—

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×Dic₃₁

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

Dic₆₂

—

C₈×C₃₁

—

C₈×C₃₁

D₈

yes

—

C₂²×C₃₁

yes

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

C₂×D₈

yes

—

C₄×C₃₁

AGL(1,ℤ/8)

yes

—

Dic₃₁

—

—

—

—

—

D₃₁

yes

characteristic, complemented

C₂×C₃₁

#493

yes

characteristic

C₂×C₃₁

#496

yes

—

D₃₁

—

D₃₁

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

#102

—

C₄×C₈

C₂×D₃₁

yes

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1721

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

124

—

D₈

—

124

—

D₈

—

124

—

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

D₁₂₄

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

—

124

—

C₂³

—

124

—

Q₈

—

124

—

D₄

—

124

—

D₄

—

124

—

D₄

—

124

—

D₄

—

124

—

D₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

C₂×D₁₂₄

yes

—

C₈

—

C₈

D₂₄₈

yes

complemented

C₂²

—

248

—

C₂²

—

248

—

C₂²

—

248

—

C₂²

—

248

—

100

C₂²

—

248

—

101

C₂²

#11758451

yes

characteristic

102

C₄

—

248

—

103

C₄

C₂×D₂₄₈

yes

—

104

C₄

#11758485

yes

—

105

C₂

—

496

—

106

C₂

—

496

—

107

C₂

#14521426

yes

characteristic

108

C₂

#14521438

yes

—

	subgroup	G/H
center	C₂²	#11758451
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#322	C₂
socle	C₂²×C₃₁	#127
hypercenter	#322	D₃₁
hypocenter	C₃₁	#1721
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#11758451

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#322

C₂

C₂²×C₃₁

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #14521426	C₂	semidirect
2. #14521426	C₂	semidirect
3. #14521422	C₂	semidirect
4. #14521396	C₂	semidirect
5. C₃₁×#322	C₂	semidirect
40. C₈×C₃₁	D₈	semidirect
55. #322	D₃₁	semidirect
69. C₃₁	#1721	semidirect
95. C₈	D₂₄₈	semidirect

H ≅ G/N

type

1. #14521426

C₂

semidirect

2. #14521426

C₂

semidirect

3. #14521422

C₂

semidirect

4. #14521396

C₂

semidirect

5. C₃₁×#322

C₂

semidirect

40. C₈×C₃₁

D₈

semidirect

55. #322

D₃₁

semidirect

69. C₃₁

#1721

semidirect

95. C₈

D₂₄₈

semidirect

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

subgroup	index
#14521426	2
#14521422	2
#14521396	2
C₃₁×#322	2
#1721	31

subgroup

index

C₃₁×#322

Subgroups — #16845377

Conjugacy classes of subgroups

Complements

Maximal subgroups