Group #38766454 — subgroups

Subgroups — Q₈×Q₁₆×C₁₉₉ #38766454

Order 25472.

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×C₁₉₉	C₂	1	2	yes	maximal
2	#500×C₁₉₉	C₂	1	3	yes	maximal
3	#477×C₁₉₉	C₂	1	6	yes	maximal
4	C₈×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₁₆×C₁₉₉	C₂	1	3	yes	maximal
6	C₂×Q₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₁₉₉	—	2	6	no	—
8	C₄⋊Q₈×C₁₉₉	C₂²	1	6	yes	—
9	C₄×Q₈×C₁₉₉	—	2	2	no	—
10	C₄×Q₈×C₁₉₉	C₂²	1	6	yes	—
11	C₄×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
12	#107×C₁₉₉	C₂²	1	3	yes	—
13	#107×C₁₉₉	C₂²	1	6	yes	—
14	C₄⋊C₈×C₁₉₉	C₂²	1	3	yes	—
15	#103×C₁₉₉	C₂²	1	6	yes	—
16	C₄×C₈×C₁₉₉	C₂²	1	3	yes	—
17	C₂×Q₈×C₁₉₉	—	2	6	no	—
18	C₂×Q₈×C₁₉₉	D₄	1	1	yes	characteristic
19	C₂×Q₈×C₁₉₉	C₂³	1	2	yes	—
20	Q₁₆×C₁₉₉	Q₈	1	4	yes	direct product factor
21	C₂×C₈×C₁₉₉	C₂³	1	3	yes	—
22	C₂×C₈×C₁₉₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₉₉	—	2	6	no	—
24	C₄⋊C₄×C₁₉₉	—	2	6	no	—
25	C₄⋊C₄×C₁₉₉	D₄	1	3	yes	—
26	C₄⋊C₄×C₁₉₉	C₂³	1	6	yes	—
27	C₄²×C₁₉₉	—	2	6	no	—
28	C₄²×C₁₉₉	C₂³	1	3	yes	—
29	Q₈×C₁₉₉	—	2	2	no	—
30	Q₈×C₁₉₉	—	8	6	no	—
31	Q₈×C₁₉₉	Q₁₆	1	4	yes	direct product factor
32	Q₈×C₁₉₉	C₂×Q₈	1	4	yes	—
33	C₂×C₄×C₁₉₉	—	2	2	no	—
34	C₂×C₄×C₁₉₉	—	2	6	no	—
35	C₂×C₄×C₁₉₉	C₂×D₄	1	3	yes	—
36	C₂×C₄×C₁₉₉	C₂×D₄	1	3	yes	—
37	C₂×C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
38	C₈×C₁₉₉	—	2	3	no	—
39	C₈×C₁₉₉	C₂×Q₈	1	2	yes	—
40	C₂²×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₁₉₉	—	2	4	no	—
42	C₄×C₁₉₉	—	2	3	no	—
43	C₄×C₁₉₉	—	4	6	no	—
44	C₄×C₁₉₉	C₂×Q₁₆	1	6	yes	—
45	C₄×C₁₉₉	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
47	C₂×C₁₉₉	D₄×Q₈	1	1	yes	characteristic
48	C₂×C₁₉₉	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂×C₁₉₉	#576	1	1	yes	characteristic
50	C₁₉₉	Q₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
51	Q₈×Q₁₆	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	Q₈²	C₂×C₁₉₉	1	2	yes	—
53	#500	C₂×C₁₉₉	1	3	yes	—
54	#477	C₂×C₁₉₉	1	6	yes	—
55	C₈×Q₈	C₂×C₁₉₉	1	1	yes	characteristic
56	C₄×Q₁₆	C₂×C₁₉₉	1	3	yes	—
57	C₂×Q₁₆	C₂²×C₁₉₉	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	6	no	—
59	C₄⋊Q₈	C₂²×C₁₉₉	1	6	yes	—
60	C₄×Q₈	—	2	2	no	—
61	C₄×Q₈	C₂²×C₁₉₉	1	6	yes	—
62	C₄×Q₈	C₂²×C₁₉₉	1	1	yes	characteristic
63	#107	C₂²×C₁₉₉	1	3	yes	—
64	#107	C₂²×C₁₉₉	1	6	yes	—
65	C₄⋊C₈	C₂²×C₁₉₉	1	3	yes	—
66	#103	C₂²×C₁₉₉	1	6	yes	—
67	C₄×C₈	C₂²×C₁₉₉	1	3	yes	—
68	C₂×Q₈	—	2	6	no	—
69	C₂×Q₈	D₄×C₁₉₉	1	1	yes	characteristic
70	C₂×Q₈	C₂³×C₁₉₉	1	2	yes	—
71	Q₁₆	Q₈×C₁₉₉	1	4	yes	direct product factor
72	C₂×C₈	C₂³×C₁₉₉	1	3	yes	—
73	C₂×C₈	C₂³×C₁₉₉	1	1	yes	characteristic
74	C₄⋊C₄	—	2	6	no	—
75	C₄⋊C₄	—	2	6	no	—
76	C₄⋊C₄	D₄×C₁₉₉	1	3	yes	—
77	C₄⋊C₄	C₂³×C₁₉₉	1	6	yes	—
78	C₄²	—	2	6	no	—
79	C₄²	C₂³×C₁₉₉	1	3	yes	—
80	Q₈	—	2	2	no	—
81	Q₈	—	8	6	no	—
82	Q₈	Q₁₆×C₁₉₉	1	4	yes	direct product factor
83	Q₈	C₂×Q₈×C₁₉₉	1	4	yes	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	6	no	—
86	C₂×C₄	C₂×D₄×C₁₉₉	1	3	yes	—
87	C₂×C₄	C₂×D₄×C₁₉₉	1	3	yes	—
88	C₂×C₄	C₂⁴×C₁₉₉	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	C₂×Q₈×C₁₉₉	1	2	yes	—
91	C₂²	C₂²×D₄×C₁₉₉	1	1	yes	characteristic
92	C₄	—	2	4	no	—
93	C₄	—	2	3	no	—
94	C₄	—	4	6	no	—
95	C₄	C₂×Q₁₆×C₁₉₉	1	6	yes	—
96	C₄	C₂²×Q₈×C₁₉₉	1	1	yes	characteristic
97	C₄	2₋¹⁺⁴×C₁₉₉	1	1	yes	characteristic
98	C₂	D₄×Q₈×C₁₉₉	1	1	yes	characteristic
99	C₂	C₂²×Q₁₆×C₁₉₉	1	1	yes	characteristic
100	C₂	#576×C₁₉₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₁₉₉

C₂

yes

maximal

#500×C₁₉₉

C₂

yes

maximal

#477×C₁₉₉

C₂

yes

maximal

C₈×Q₈×C₁₉₉

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₁₉₉

C₂

yes

maximal

C₂×Q₁₆×C₁₉₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

C₂²

yes

—

C₄×Q₈×C₁₉₉

—

C₄×Q₈×C₁₉₉

C₂²

yes

—

C₄×Q₈×C₁₉₉

C₂²

yes

characteristic

#107×C₁₉₉

C₂²

yes

—

#107×C₁₉₉

C₂²

yes

—

C₄⋊C₈×C₁₉₉

C₂²

yes

—

#103×C₁₉₉

C₂²

yes

—

C₄×C₈×C₁₉₉

C₂²

yes

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

characteristic

C₂×Q₈×C₁₉₉

C₂³

yes

—

Q₁₆×C₁₉₉

Q₈

yes

direct product factor

C₂×C₈×C₁₉₉

C₂³

yes

—

C₂×C₈×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₄⋊C₄×C₁₉₉

C₂³

yes

—

C₄²×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

Q₁₆

yes

direct product factor

Q₈×C₁₉₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂⁴

yes

characteristic

C₈×C₁₉₉

—

C₈×C₁₉₉

C₂×Q₈

yes

—

C₂²×C₁₉₉

C₂²×D₄

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×Q₁₆

yes

—

C₄×C₁₉₉

C₂²×Q₈

yes

characteristic

C₄×C₁₉₉

2₋¹⁺⁴

yes

characteristic

C₂×C₁₉₉

D₄×Q₈

yes

characteristic

C₂×C₁₉₉

C₂²×Q₁₆

yes

characteristic

C₂×C₁₉₉

#576

yes

characteristic

C₁₉₉

Q₈×Q₁₆

yes

characteristic, direct product factor, Sylow

Q₈×Q₁₆

C₁₉₉

yes

maximal, characteristic, direct product factor, Sylow

Q₈²

C₂×C₁₉₉

yes

—

#500

C₂×C₁₉₉

yes

—

#477

C₂×C₁₉₉

yes

—

C₈×Q₈

C₂×C₁₉₉

yes

characteristic

C₄×Q₁₆

C₂×C₁₉₉

yes

—

C₂×Q₁₆

C₂²×C₁₉₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₉₉

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₁₉₉

yes

—

C₄×Q₈

C₂²×C₁₉₉

yes

characteristic

#107

C₂²×C₁₉₉

yes

—

#107

C₂²×C₁₉₉

yes

—

C₄⋊C₈

C₂²×C₁₉₉

yes

—

#103

C₂²×C₁₉₉

yes

—

C₄×C₈

C₂²×C₁₉₉

yes

—

C₂×Q₈

—

C₂×Q₈

D₄×C₁₉₉

yes

characteristic

C₂×Q₈

C₂³×C₁₉₉

yes

—

Q₁₆

Q₈×C₁₉₉

yes

direct product factor

C₂×C₈

C₂³×C₁₉₉

yes

—

C₂×C₈

C₂³×C₁₉₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₁₉₉

yes

—

C₄⋊C₄

C₂³×C₁₉₉

yes

—

C₄²

—

C₄²

C₂³×C₁₉₉

yes

—

Q₈

—

Q₈

—

Q₈

Q₁₆×C₁₉₉

yes

direct product factor

Q₈

C₂×Q₈×C₁₉₉

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₉₉

yes

—

C₂×C₄

C₂×D₄×C₁₉₉

yes

—

C₂×C₄

C₂⁴×C₁₉₉

yes

characteristic

C₈

—

C₈

C₂×Q₈×C₁₉₉

yes

—

C₂²

C₂²×D₄×C₁₉₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×Q₁₆×C₁₉₉

yes

—

C₄

C₂²×Q₈×C₁₉₉

yes

characteristic

C₄

2₋¹⁺⁴×C₁₉₉

yes

characteristic

C₂

D₄×Q₈×C₁₉₉

yes

characteristic

C₂

C₂²×Q₁₆×C₁₉₉

yes

characteristic

100

C₂

#576×C₁₉₉

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₉ ◁ C₄×Q₈×C₁₉₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₉₉

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₉₉

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉₉ ◁ C₄×Q₈×C₁₉₉ ◁ G

N	H ≅ G/N	type
20. Q₁₆×C₁₉₉	Q₈	direct (N × H)
31. Q₈×C₁₉₉	Q₁₆	direct (N × H)
50. C₁₉₉	Q₈×Q₁₆	direct (N × H)
51. Q₈×Q₁₆	C₁₉₉	direct (N × H)
71. Q₁₆	Q₈×C₁₉₉	direct (N × H)
82. Q₈	Q₁₆×C₁₉₉	direct (N × H)

H ≅ G/N

type

20. Q₁₆×C₁₉₉

Q₈

direct (N × H)

31. Q₈×C₁₉₉

Q₁₆

direct (N × H)

50. C₁₉₉

Q₈×Q₁₆

direct (N × H)

51. Q₈×Q₁₆

C₁₉₉

direct (N × H)

71. Q₁₆

Q₈×C₁₉₉

direct (N × H)

82. Q₈

Q₁₆×C₁₉₉

direct (N × H)

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

subgroup	index
Q₈²×C₁₉₉	2
#500×C₁₉₉	2
#477×C₁₉₉	2
C₈×Q₈×C₁₉₉	2
C₄×Q₁₆×C₁₉₉	2
Q₈×Q₁₆	199

subgroup

index

Q₈²×C₁₉₉

#500×C₁₉₉

#477×C₁₉₉

C₈×Q₈×C₁₉₉

C₄×Q₁₆×C₁₉₉

Q₈×Q₁₆

199

Subgroups — Q₈×Q₁₆×C₁₉₉ #38766454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×Q16×C199 #38766454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Q₁₆×C₁₉₉ #38766454