Group #27164682 — subgroups

Subgroups — Q₈×C₂²⋊D₁₉₉ #27164682

Order 12736.

280 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 123 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	C₂×Q₈×D₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263953	C₂	1	3	yes	maximal, complemented
3	#20263934	C₂	1	3	yes	maximal, complemented
4	#20263931	C₂	1	3	yes	maximal, complemented
5	Q₈×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₁₉₉	C₂	1	3	yes	maximal
7	C₂²×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₉₈	C₂²	1	3	yes	—
9	Q₈×D₁₉₉	—	2	4	no	—
10	C₂×C₂²⋊D₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₉₉	C₂²	1	3	yes	—
12	C₄⋊Dic₁₉₉	C₂²	1	3	yes	—
13	Dic₁₉₉⋊C₄	C₂²	1	3	yes	—
14	Dic₁₉₉⋊C₄	C₂²	1	6	yes	—
15	C₂²⋊Dic₁₉₉	C₂²	1	3	yes	—
16	(C₂×D₁₉₉)⋊C₄	C₂²	1	3	yes	—
17	C₄×Dic₁₉₉	C₂²	1	3	yes	—
18	C₂×Q₈×C₁₉₉	—	2	4	no	—
19	C₂×Q₈×C₁₉₉	C₂²	1	3	yes	—
20	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₉	C₂²	1	3	yes	—
22	C₂²×D₁₉₉	C₂³	1	1	yes	characteristic
23	Dic₃₉₈	—	2	6	no	—
24	C₂²⋊D₁₉₉	Q₈	1	4	yes	direct product factor
25	C₂×Dic₁₉₉	C₂³	1	3	yes	—
26	C₂×Dic₁₉₉	C₂³	1	3	yes	—
27	C₂×Dic₁₉₉	C₂³	1	1	yes	characteristic
28	C₄×D₁₉₉	—	2	6	no	—
29	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
30	Q₈×C₁₉₉	—	2	6	no	—
31	Q₈×C₁₉₉	D₄	1	4	yes	complemented
32	C₂×C₄×C₁₉₉	—	2	6	no	—
33	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
34	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
35	C₂²×C₁₉₉	—	2	1	no	—
36	C₂²×C₁₉₉	—	2	1	no	—
37	C₂²×C₁₉₉	C₂×Q₈	1	2	yes	complemented
38	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
39	C₂×D₁₉₉	—	2	1	no	—
40	C₂×D₁₉₉	—	2	1	no	—
41	C₂×D₁₉₉	C₂×Q₈	1	2	yes	complemented
42	C₄×C₁₉₉	—	2	3	no	—
43	C₄×C₁₉₉	C₂×D₄	1	6	yes	—
44	Dic₁₉₉	—	2	3	no	—
45	Dic₁₉₉	—	2	3	no	—
46	Dic₁₉₉	C₂×Q₈	1	2	yes	complemented
47	C₂×C₁₉₉	—	2	2	no	—
48	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₁₉₉	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
51	D₁₉₉	—	2	2	no	—
52	C₁₉₉	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
53	D₄×Q₈	—	199	1	no	maximal, Sylow
54	C₂²×Q₈	—	199	1	no	—
55	C₂²×Q₈	D₁₉₉	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	199	3	no	—
57	C₂²⋊Q₈	—	199	3	no	—
58	C₂²⋊Q₈	—	199	3	no	—
59	C₄×Q₈	—	199	1	no	—
60	C₄×D₄	—	199	3	no	—
61	C₂×Q₈	—	2	4	no	—
62	C₂×Q₈	—	398	4	no	—
63	C₂×Q₈	—	199	3	no	—
64	C₂×Q₈	C₂×D₁₉₉	1	3	yes	—
65	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
66	C₂×D₄	—	199	1	no	—
67	C₂²×C₄	—	199	3	no	—
68	C₂²×C₄	C₂×D₁₉₉	1	3	yes	—
69	C₄⋊C₄	—	199	3	no	—
70	C₄⋊C₄	—	199	6	no	—
71	C₄⋊C₄	—	199	3	no	—
72	C₂²⋊C₄	—	199	3	no	—
73	C₂²⋊C₄	—	199	3	no	—
74	C₄²	—	199	3	no	—
75	C₂³	—	199	1	no	—
76	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
77	Q₈	—	2	6	no	—
78	Q₈	—	398	6	no	—
79	Q₈	C₂²⋊D₁₉₉	1	4	yes	direct product factor
80	D₄	—	199	4	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	398	6	no	—
83	C₂×C₄	—	199	3	no	—
84	C₂×C₄	—	199	3	no	—
85	C₂×C₄	—	199	1	no	—
86	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
87	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	398	1	no	—
91	C₂²	—	398	1	no	—
92	C₂²	—	199	2	no	—
93	C₂²	Q₈×D₁₉₉	1	2	yes	complemented
94	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
95	C₄	—	2	3	no	—
96	C₄	—	398	3	no	—
97	C₄	—	398	3	no	—
98	C₄	—	199	2	no	—
99	C₄	C₂×C₂²⋊D₁₉₉	1	6	yes	—
100	C₂	—	2	2	no	—
101	C₂	—	398	2	no	—
102	C₂	C₂²×C₂²⋊D₁₉₉	1	1	yes	characteristic
103	C₂	C₂×Q₈×D₁₉₉	1	1	yes	characteristic
104	C₂	#20264003	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₉₉

C₂

yes

maximal, characteristic, complemented

#20263953

C₂

yes

maximal, complemented

#20263934

C₂

yes

maximal, complemented

#20263931

C₂

yes

maximal, complemented

Q₈×Dic₁₉₉

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₉₉

C₂

yes

maximal

C₂²×Q₈×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₉₈

C₂²

yes

—

Q₈×D₁₉₉

—

C₂×C₂²⋊D₁₉₉

C₂²

yes

characteristic

C₂×C₄×D₁₉₉

C₂²

yes

—

C₄⋊Dic₁₉₉

C₂²

yes

—

Dic₁₉₉⋊C₄

C₂²

yes

—

Dic₁₉₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₉₉

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

C₄×Dic₁₉₉

C₂²

yes

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

C₂²

yes

—

C₂×Q₈×C₁₉₉

C₂²

yes

characteristic

C₂²×C₄×C₁₉₉

C₂²

yes

—

C₂²×D₁₉₉

C₂³

yes

characteristic

Dic₃₉₈

—

C₂²⋊D₁₉₉

Q₈

yes

direct product factor

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

characteristic

C₄×D₁₉₉

—

C₂³×C₁₉₉

C₂³

yes

characteristic

Q₈×C₁₉₉

—

Q₈×C₁₉₉

D₄

yes

complemented

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×Q₈

yes

complemented

C₂²×C₁₉₉

C₂⁴

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

C₂×Q₈

yes

complemented

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×D₄

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

C₂×Q₈

yes

complemented

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂²×D₄

yes

characteristic

C₂×C₁₉₉

C₂²×Q₈

yes

characteristic

C₂×C₁₉₉

2₋¹⁺⁴

yes

characteristic

D₁₉₉

—

C₁₉₉

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

199

maximal, Sylow

C₂²×Q₈

—

199

—

C₂²×Q₈

D₁₉₉

yes

characteristic, complemented

C₄⋊Q₈

—

199

—

C₂²⋊Q₈

—

199

—

C₂²⋊Q₈

—

199

—

C₄×Q₈

—

199

—

C₄×D₄

—

199

—

C₂×Q₈

—

C₂×Q₈

—

398

—

C₂×Q₈

—

199

—

C₂×Q₈

C₂×D₁₉₉

yes

—

C₂×Q₈

C₂×D₁₉₉

yes

characteristic

C₂×D₄

—

199

—

C₂²×C₄

—

199

—

C₂²×C₄

C₂×D₁₉₉

yes

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₄²

—

199

—

C₂³

—

199

—

C₂³

C₂²×D₁₉₉

yes

characteristic

Q₈

—

Q₈

—

398

—

Q₈

C₂²⋊D₁₉₉

yes

direct product factor

D₄

—

199

—

C₂×C₄

—

C₂×C₄

—

398

—

C₂×C₄

—

199

—

C₂×C₄

—

199

—

C₂×C₄

—

199

—

C₂×C₄

C₂²×D₁₉₉

yes

—

C₂×C₄

C₂²×D₁₉₉

yes

—

C₂²

—

C₂²

—

C₂²

—

398

—

C₂²

—

398

—

C₂²

—

199

—

C₂²

Q₈×D₁₉₉

yes

complemented

C₂²

C₂³×D₁₉₉

yes

characteristic

C₄

—

C₄

—

398

—

C₄

—

398

—

C₄

—

199

—

C₄

C₂×C₂²⋊D₁₉₉

yes

—

100

C₂

—

101

C₂

—

398

—

102

C₂

C₂²×C₂²⋊D₁₉₉

yes

characteristic

103

C₂

C₂×Q₈×D₁₉₉

yes

characteristic

104

C₂

#20264003

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Fitting	C₂²×Q₈×C₁₉₉	C₂
prime radical	C₂×C₂²⋊D₁₉₉	C₂²
hypercenter	C₂²×Q₈	D₁₉₉
hypocenter	C₁₉₉	D₄×Q₈
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉₉

commutator, socle, cosocle

C₂²×C₁₉₉

C₂⁴

Fitting

C₂²×Q₈×C₁₉₉

C₂

prime radical

C₂×C₂²⋊D₁₉₉

C₂²

hypercenter

C₂²×Q₈

D₁₉₉

hypocenter

C₁₉₉

D₄×Q₈

LCS

G ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₁₉₉	C₂	semidirect
2. #20263953	C₂	semidirect
3. #20263934	C₂	semidirect
4. #20263931	C₂	semidirect
5. Q₈×Dic₁₉₉	C₂	semidirect
7. C₂²×Q₈×C₁₉₉	C₂	semidirect
24. C₂²⋊D₁₉₉	Q₈	direct (N × H)
31. Q₈×C₁₉₉	D₄	semidirect
37. C₂²×C₁₉₉	C₂×Q₈	semidirect
41. C₂×D₁₉₉	C₂×Q₈	semidirect
46. Dic₁₉₉	C₂×Q₈	semidirect
52. C₁₉₉	D₄×Q₈	semidirect
55. C₂²×Q₈	D₁₉₉	semidirect
79. Q₈	C₂²⋊D₁₉₉	direct (N × H)
93. C₂²	Q₈×D₁₉₉	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₉₉

C₂

semidirect

2. #20263953

C₂

semidirect

3. #20263934

C₂

semidirect

4. #20263931

C₂

semidirect

5. Q₈×Dic₁₉₉

C₂

semidirect

7. C₂²×Q₈×C₁₉₉

C₂

semidirect

24. C₂²⋊D₁₉₉

Q₈

direct (N × H)

31. Q₈×C₁₉₉

D₄

semidirect

37. C₂²×C₁₉₉

C₂×Q₈

semidirect

41. C₂×D₁₉₉

C₂×Q₈

semidirect

46. Dic₁₉₉

C₂×Q₈

semidirect

52. C₁₉₉

D₄×Q₈

semidirect

55. C₂²×Q₈

D₁₉₉

semidirect

79. Q₈

C₂²⋊D₁₉₉

direct (N × H)

93. C₂²

Q₈×D₁₉₉

semidirect

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

subgroup	index
C₂×Q₈×D₁₉₉	2
#20263953	2
#20263934	2
#20263931	2
Q₈×Dic₁₉₉	2
C₄×C₂²⋊D₁₉₉	2
C₂²×Q₈×C₁₉₉	2
D₄×Q₈	199

subgroup

index

C₂×Q₈×D₁₉₉

#20263953

#20263934

#20263931

Q₈×Dic₁₉₉

C₄×C₂²⋊D₁₉₉

C₂²×Q₈×C₁₉₉

D₄×Q₈

199

Subgroups — Q₈×C₂²⋊D₁₉₉ #27164682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C22⋊D199 #27164682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂²⋊D₁₉₉ #27164682