Group #34097740 — subgroups

Subgroups — Q₈×Q₁₆×C₁₅₇ #34097740

Order 20096.

336 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 208 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	Q₈²×C₁₅₇	C₂	1	2	yes	maximal
2	#500×C₁₅₇	C₂	1	3	yes	maximal
3	#477×C₁₅₇	C₂	1	6	yes	maximal
4	C₈×Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₁₅₇	C₂	1	3	yes	maximal
6	C₂×Q₁₆×C₁₅₇	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₁₅₇	—	2	6	no	—
8	C₄⋊Q₈×C₁₅₇	C₂²	1	6	yes	—
9	C₄×Q₈×C₁₅₇	—	2	2	no	—
10	C₄×Q₈×C₁₅₇	C₂²	1	6	yes	—
11	C₄×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
12	#107×C₁₅₇	C₂²	1	3	yes	—
13	#107×C₁₅₇	C₂²	1	6	yes	—
14	C₄⋊C₈×C₁₅₇	C₂²	1	3	yes	—
15	#103×C₁₅₇	C₂²	1	6	yes	—
16	C₄×C₈×C₁₅₇	C₂²	1	3	yes	—
17	C₂×Q₈×C₁₅₇	—	2	6	no	—
18	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
19	C₂×Q₈×C₁₅₇	C₂³	1	2	yes	—
20	Q₁₆×C₁₅₇	Q₈	1	4	yes	direct product factor
21	C₂×C₈×C₁₅₇	C₂³	1	3	yes	—
22	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₅₇	—	2	6	no	—
24	C₄⋊C₄×C₁₅₇	—	2	6	no	—
25	C₄⋊C₄×C₁₅₇	D₄	1	3	yes	—
26	C₄⋊C₄×C₁₅₇	C₂³	1	6	yes	—
27	C₄²×C₁₅₇	—	2	6	no	—
28	C₄²×C₁₅₇	C₂³	1	3	yes	—
29	Q₈×C₁₅₇	—	2	2	no	—
30	Q₈×C₁₅₇	—	8	6	no	—
31	Q₈×C₁₅₇	Q₁₆	1	4	yes	direct product factor
32	Q₈×C₁₅₇	C₂×Q₈	1	4	yes	—
33	C₂×C₄×C₁₅₇	—	2	2	no	—
34	C₂×C₄×C₁₅₇	—	2	6	no	—
35	C₂×C₄×C₁₅₇	C₂×D₄	1	3	yes	—
36	C₂×C₄×C₁₅₇	C₂×D₄	1	3	yes	—
37	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
38	C₈×C₁₅₇	—	2	3	no	—
39	C₈×C₁₅₇	C₂×Q₈	1	2	yes	—
40	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₁₅₇	—	2	4	no	—
42	C₄×C₁₅₇	—	2	3	no	—
43	C₄×C₁₅₇	—	4	6	no	—
44	C₄×C₁₅₇	C₂×Q₁₆	1	6	yes	—
45	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₁₅₇	2₋¹⁺⁴	1	1	yes	characteristic
47	C₂×C₁₅₇	D₄×Q₈	1	1	yes	characteristic
48	C₂×C₁₅₇	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂×C₁₅₇	#576	1	1	yes	characteristic
50	C₁₅₇	Q₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
51	Q₈×Q₁₆	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	Q₈²	C₂×C₁₅₇	1	2	yes	—
53	#500	C₂×C₁₅₇	1	3	yes	—
54	#477	C₂×C₁₅₇	1	6	yes	—
55	C₈×Q₈	C₂×C₁₅₇	1	1	yes	characteristic
56	C₄×Q₁₆	C₂×C₁₅₇	1	3	yes	—
57	C₂×Q₁₆	C₂²×C₁₅₇	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	6	no	—
59	C₄⋊Q₈	C₂²×C₁₅₇	1	6	yes	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×Q₈	C₂²×C₁₅₇	1	6	yes	—
62	C₄×Q₈	C₂²×C₁₅₇	1	1	yes	characteristic
63	#107	C₂²×C₁₅₇	1	3	yes	—
64	#107	C₂²×C₁₅₇	1	6	yes	—
65	C₄⋊C₈	C₂²×C₁₅₇	1	3	yes	—
66	#103	C₂²×C₁₅₇	1	6	yes	—
67	C₄×C₈	C₂²×C₁₅₇	1	3	yes	—
68	C₂×Q₈	—	2	6	no	—
69	C₂×Q₈	D₄×C₁₅₇	1	1	yes	characteristic
70	C₂×Q₈	C₂³×C₁₅₇	1	2	yes	—
71	Q₁₆	Q₈×C₁₅₇	1	4	yes	direct product factor
72	C₂×C₈	C₂³×C₁₅₇	1	3	yes	—
73	C₂×C₈	C₂³×C₁₅₇	1	1	yes	characteristic
74	C₄⋊C₄	—	2	6	no	—
75	C₄⋊C₄	—	2	6	no	—
76	C₄⋊C₄	D₄×C₁₅₇	1	3	yes	—
77	C₄⋊C₄	C₂³×C₁₅₇	1	6	yes	—
78	C₄²	—	2	6	no	—
79	C₄²	C₂³×C₁₅₇	1	3	yes	—
80	Q₈	—	2	2	no	—
81	Q₈	—	8	6	no	—
82	Q₈	Q₁₆×C₁₅₇	1	4	yes	direct product factor
83	Q₈	C₂×Q₈×C₁₅₇	1	4	yes	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	6	no	—
86	C₂×C₄	C₂×D₄×C₁₅₇	1	3	yes	—
87	C₂×C₄	C₂×D₄×C₁₅₇	1	3	yes	—
88	C₂×C₄	C₂⁴×C₁₅₇	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	C₂×Q₈×C₁₅₇	1	2	yes	—
91	C₂²	C₂²×D₄×C₁₅₇	1	1	yes	characteristic
92	C₄	—	2	4	no	—
93	C₄	—	2	3	no	—
94	C₄	—	4	6	no	—
95	C₄	C₂×Q₁₆×C₁₅₇	1	6	yes	—
96	C₄	C₂²×Q₈×C₁₅₇	1	1	yes	characteristic
97	C₄	2₋¹⁺⁴×C₁₅₇	1	1	yes	characteristic
98	C₂	D₄×Q₈×C₁₅₇	1	1	yes	characteristic
99	C₂	C₂²×Q₁₆×C₁₅₇	1	1	yes	characteristic
100	C₂	#576×C₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₁₅₇

C₂

yes

maximal

#500×C₁₅₇

C₂

yes

maximal

#477×C₁₅₇

C₂

yes

maximal

C₈×Q₈×C₁₅₇

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₁₅₇

C₂

yes

maximal

C₂×Q₁₆×C₁₅₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

C₄×Q₈×C₁₅₇

—

C₄×Q₈×C₁₅₇

C₂²

yes

—

C₄×Q₈×C₁₅₇

C₂²

yes

characteristic

#107×C₁₅₇

C₂²

yes

—

#107×C₁₅₇

C₂²

yes

—

C₄⋊C₈×C₁₅₇

C₂²

yes

—

#103×C₁₅₇

C₂²

yes

—

C₄×C₈×C₁₅₇

C₂²

yes

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

characteristic

C₂×Q₈×C₁₅₇

C₂³

yes

—

Q₁₆×C₁₅₇

Q₈

yes

direct product factor

C₂×C₈×C₁₅₇

C₂³

yes

—

C₂×C₈×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

D₄

yes

—

C₄⋊C₄×C₁₅₇

C₂³

yes

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

Q₈×C₁₅₇

Q₁₆

yes

direct product factor

Q₈×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×Q₁₆

yes

—

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

C₄×C₁₅₇

2₋¹⁺⁴

yes

characteristic

C₂×C₁₅₇

D₄×Q₈

yes

characteristic

C₂×C₁₅₇

C₂²×Q₁₆

yes

characteristic

C₂×C₁₅₇

#576

yes

characteristic

C₁₅₇

Q₈×Q₁₆

yes

characteristic, direct product factor, Sylow

Q₈×Q₁₆

C₁₅₇

yes

maximal, characteristic, direct product factor, Sylow

Q₈²

C₂×C₁₅₇

yes

—

#500

C₂×C₁₅₇

yes

—

#477

C₂×C₁₅₇

yes

—

C₈×Q₈

C₂×C₁₅₇

yes

characteristic

C₄×Q₁₆

C₂×C₁₅₇

yes

—

C₂×Q₁₆

C₂²×C₁₅₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₅₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₅₇

yes

—

C₄×Q₈

C₂²×C₁₅₇

yes

characteristic

#107

C₂²×C₁₅₇

yes

—

#107

C₂²×C₁₅₇

yes

—

C₄⋊C₈

C₂²×C₁₅₇

yes

—

#103

C₂²×C₁₅₇

yes

—

C₄×C₈

C₂²×C₁₅₇

yes

—

C₂×Q₈

—

C₂×Q₈

D₄×C₁₅₇

yes

characteristic

C₂×Q₈

C₂³×C₁₅₇

yes

—

Q₁₆

Q₈×C₁₅₇

yes

direct product factor

C₂×C₈

C₂³×C₁₅₇

yes

—

C₂×C₈

C₂³×C₁₅₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₅₇

yes

—

C₄⋊C₄

C₂³×C₁₅₇

yes

—

C₄²

—

C₄²

C₂³×C₁₅₇

yes

—

Q₈

—

Q₈

—

Q₈

Q₁₆×C₁₅₇

yes

direct product factor

Q₈

C₂×Q₈×C₁₅₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₅₇

yes

—

C₂×C₄

C₂×D₄×C₁₅₇

yes

—

C₂×C₄

C₂⁴×C₁₅₇

yes

characteristic

C₈

—

C₈

C₂×Q₈×C₁₅₇

yes

—

C₂²

C₂²×D₄×C₁₅₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×Q₁₆×C₁₅₇

yes

—

C₄

C₂²×Q₈×C₁₅₇

yes

characteristic

C₄

2₋¹⁺⁴×C₁₅₇

yes

characteristic

C₂

D₄×Q₈×C₁₅₇

yes

characteristic

C₂

C₂²×Q₁₆×C₁₅₇

yes

characteristic

100

C₂

#576×C₁₅₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₅₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₅₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₇ ◁ C₄×Q₈×C₁₅₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₅₇

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₅₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₇ ◁ C₄×Q₈×C₁₅₇ ◁ G

N	H ≅ G/N	type
20. Q₁₆×C₁₅₇	Q₈	direct (N × H)
31. Q₈×C₁₅₇	Q₁₆	direct (N × H)
50. C₁₅₇	Q₈×Q₁₆	direct (N × H)
51. Q₈×Q₁₆	C₁₅₇	direct (N × H)
71. Q₁₆	Q₈×C₁₅₇	direct (N × H)
82. Q₈	Q₁₆×C₁₅₇	direct (N × H)

H ≅ G/N

type

20. Q₁₆×C₁₅₇

Q₈

direct (N × H)

31. Q₈×C₁₅₇

Q₁₆

direct (N × H)

50. C₁₅₇

Q₈×Q₁₆

direct (N × H)

51. Q₈×Q₁₆

C₁₅₇

direct (N × H)

71. Q₁₆

Q₈×C₁₅₇

direct (N × H)

82. Q₈

Q₁₆×C₁₅₇

direct (N × H)

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

subgroup	index
Q₈²×C₁₅₇	2
#500×C₁₅₇	2
#477×C₁₅₇	2
C₈×Q₈×C₁₅₇	2
C₄×Q₁₆×C₁₅₇	2
Q₈×Q₁₆	157

subgroup

index

Q₈²×C₁₅₇

#500×C₁₅₇

#477×C₁₅₇

C₈×Q₈×C₁₅₇

C₄×Q₁₆×C₁₅₇

Q₈×Q₁₆

157

Subgroups — Q₈×Q₁₆×C₁₅₇ #34097740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×Q16×C157 #34097740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Q₁₆×C₁₅₇ #34097740