Group #27133327 — subgroups

Subgroups — Q₈×C₂²⋊D₁₉₁ #27133327

Order 12224.

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×D₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259320	C₂	1	3	yes	maximal, complemented
3	#20259301	C₂	1	3	yes	maximal, complemented
4	#20259298	C₂	1	3	yes	maximal, complemented
5	Q₈×Dic₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₂²⋊D₁₉₁	C₂	1	3	yes	maximal
7	C₂²×Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₈₂	C₂²	1	3	yes	—
9	Q₈×D₁₉₁	—	2	4	no	—
10	C₂×C₂²⋊D₁₉₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₉₁	C₂²	1	3	yes	—
12	C₄⋊Dic₁₉₁	C₂²	1	3	yes	—
13	Dic₁₉₁⋊C₄	C₂²	1	3	yes	—
14	Dic₁₉₁⋊C₄	C₂²	1	6	yes	—
15	C₂²⋊Dic₁₉₁	C₂²	1	3	yes	—
16	(C₂×D₁₉₁)⋊C₄	C₂²	1	3	yes	—
17	C₄×Dic₁₉₁	C₂²	1	3	yes	—
18	C₂×Q₈×C₁₉₁	—	2	4	no	—
19	C₂×Q₈×C₁₉₁	C₂²	1	3	yes	—
20	C₂×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₁	C₂²	1	3	yes	—
22	C₂²×D₁₉₁	C₂³	1	1	yes	characteristic
23	Dic₃₈₂	—	2	6	no	—
24	C₂²⋊D₁₉₁	Q₈	1	4	yes	direct product factor
25	C₂×Dic₁₉₁	C₂³	1	3	yes	—
26	C₂×Dic₁₉₁	C₂³	1	3	yes	—
27	C₂×Dic₁₉₁	C₂³	1	1	yes	characteristic
28	C₄×D₁₉₁	—	2	6	no	—
29	C₂³×C₁₉₁	C₂³	1	1	yes	characteristic
30	Q₈×C₁₉₁	—	2	6	no	—
31	Q₈×C₁₉₁	D₄	1	4	yes	complemented
32	C₂×C₄×C₁₉₁	—	2	6	no	—
33	C₂×C₄×C₁₉₁	C₂³	1	3	yes	—
34	C₂×C₄×C₁₉₁	C₂³	1	3	yes	—
35	C₂²×C₁₉₁	—	2	1	no	—
36	C₂²×C₁₉₁	—	2	1	no	—
37	C₂²×C₁₉₁	C₂×Q₈	1	2	yes	complemented
38	C₂²×C₁₉₁	C₂⁴	1	1	yes	characteristic
39	C₂×D₁₉₁	—	2	1	no	—
40	C₂×D₁₉₁	—	2	1	no	—
41	C₂×D₁₉₁	C₂×Q₈	1	2	yes	complemented
42	C₄×C₁₉₁	—	2	3	no	—
43	C₄×C₁₉₁	C₂×D₄	1	6	yes	—
44	Dic₁₉₁	—	2	3	no	—
45	Dic₁₉₁	—	2	3	no	—
46	Dic₁₉₁	C₂×Q₈	1	2	yes	complemented
47	C₂×C₁₉₁	—	2	2	no	—
48	C₂×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
51	D₁₉₁	—	2	2	no	—
52	C₁₉₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
53	D₄×Q₈	—	191	1	no	maximal, Sylow
54	C₂²×Q₈	—	191	1	no	—
55	C₂²×Q₈	D₁₉₁	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	191	3	no	—
57	C₂²⋊Q₈	—	191	3	no	—
58	C₂²⋊Q₈	—	191	3	no	—
59	C₄×Q₈	—	191	1	no	—
60	C₄×D₄	—	191	3	no	—
61	C₂×Q₈	—	2	4	no	—
62	C₂×Q₈	—	382	4	no	—
63	C₂×Q₈	—	191	3	no	—
64	C₂×Q₈	C₂×D₁₉₁	1	3	yes	—
65	C₂×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
66	C₂×D₄	—	191	1	no	—
67	C₂²×C₄	—	191	3	no	—
68	C₂²×C₄	C₂×D₁₉₁	1	3	yes	—
69	C₄⋊C₄	—	191	3	no	—
70	C₄⋊C₄	—	191	6	no	—
71	C₄⋊C₄	—	191	3	no	—
72	C₂²⋊C₄	—	191	3	no	—
73	C₂²⋊C₄	—	191	3	no	—
74	C₄²	—	191	3	no	—
75	C₂³	—	191	1	no	—
76	C₂³	C₂²×D₁₉₁	1	1	yes	characteristic
77	Q₈	—	2	6	no	—
78	Q₈	—	382	6	no	—
79	Q₈	C₂²⋊D₁₉₁	1	4	yes	direct product factor
80	D₄	—	191	4	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	382	6	no	—
83	C₂×C₄	—	191	3	no	—
84	C₂×C₄	—	191	3	no	—
85	C₂×C₄	—	191	1	no	—
86	C₂×C₄	C₂²×D₁₉₁	1	3	yes	—
87	C₂×C₄	C₂²×D₁₉₁	1	3	yes	—
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	382	1	no	—
91	C₂²	—	382	1	no	—
92	C₂²	—	191	2	no	—
93	C₂²	Q₈×D₁₉₁	1	2	yes	complemented
94	C₂²	C₂³×D₁₉₁	1	1	yes	characteristic
95	C₄	—	2	3	no	—
96	C₄	—	382	3	no	—
97	C₄	—	382	3	no	—
98	C₄	—	191	2	no	—
99	C₄	C₂×C₂²⋊D₁₉₁	1	6	yes	—
100	C₂	—	2	2	no	—
101	C₂	—	382	2	no	—
102	C₂	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
103	C₂	C₂×Q₈×D₁₉₁	1	1	yes	characteristic
104	C₂	#20259370	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₉₁

C₂

yes

maximal, characteristic, complemented

#20259320

C₂

yes

maximal, complemented

#20259301

C₂

yes

maximal, complemented

#20259298

C₂

yes

maximal, complemented

Q₈×Dic₁₉₁

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₉₁

C₂

yes

maximal

C₂²×Q₈×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₈₂

C₂²

yes

—

Q₈×D₁₉₁

—

C₂×C₂²⋊D₁₉₁

C₂²

yes

characteristic

C₂×C₄×D₁₉₁

C₂²

yes

—

C₄⋊Dic₁₉₁

C₂²

yes

—

Dic₁₉₁⋊C₄

C₂²

yes

—

Dic₁₉₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₉₁

C₂²

yes

—

(C₂×D₁₉₁)⋊C₄

C₂²

yes

—

C₄×Dic₁₉₁

C₂²

yes

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

C₂²

yes

—

C₂×Q₈×C₁₉₁

C₂²

yes

characteristic

C₂²×C₄×C₁₉₁

C₂²

yes

—

C₂²×D₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

C₂²⋊D₁₉₁

Q₈

yes

direct product factor

C₂×Dic₁₉₁

C₂³

yes

—

C₂×Dic₁₉₁

C₂³

yes

—

C₂×Dic₁₉₁

C₂³

yes

characteristic

C₄×D₁₉₁

—

C₂³×C₁₉₁

C₂³

yes

characteristic

Q₈×C₁₉₁

—

Q₈×C₁₉₁

D₄

yes

complemented

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂³

yes

—

C₂×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×Q₈

yes

complemented

C₂²×C₁₉₁

C₂⁴

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

C₂×Q₈

yes

complemented

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×D₄

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

C₂×Q₈

yes

complemented

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂²×D₄

yes

characteristic

C₂×C₁₉₁

C₂²×Q₈

yes

characteristic

C₂×C₁₉₁

2₋¹⁺⁴

yes

characteristic

D₁₉₁

—

C₁₉₁

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

191

maximal, Sylow

C₂²×Q₈

—

191

—

C₂²×Q₈

D₁₉₁

yes

characteristic, complemented

C₄⋊Q₈

—

191

—

C₂²⋊Q₈

—

191

—

C₂²⋊Q₈

—

191

—

C₄×Q₈

—

191

—

C₄×D₄

—

191

—

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂×Q₈

C₂×D₁₉₁

yes

—

C₂×Q₈

C₂×D₁₉₁

yes

characteristic

C₂×D₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

C₂×D₁₉₁

yes

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₄⋊C₄

—

191

—

C₂²⋊C₄

—

191

—

C₂²⋊C₄

—

191

—

C₄²

—

191

—

C₂³

—

191

—

C₂³

C₂²×D₁₉₁

yes

characteristic

Q₈

—

Q₈

—

382

—

Q₈

C₂²⋊D₁₉₁

yes

direct product factor

D₄

—

191

—

C₂×C₄

—

C₂×C₄

—

382

—

C₂×C₄

—

191

—

C₂×C₄

—

191

—

C₂×C₄

—

191

—

C₂×C₄

C₂²×D₁₉₁

yes

—

C₂×C₄

C₂²×D₁₉₁

yes

—

C₂²

—

C₂²

—

C₂²

—

382

—

C₂²

—

382

—

C₂²

—

191

—

C₂²

Q₈×D₁₉₁

yes

complemented

C₂²

C₂³×D₁₉₁

yes

characteristic

C₄

—

C₄

—

382

—

C₄

—

382

—

C₄

—

191

—

C₄

C₂×C₂²⋊D₁₉₁

yes

—

100

C₂

—

101

C₂

—

382

—

102

C₂

C₂²×C₂²⋊D₁₉₁

yes

characteristic

103

C₂

C₂×Q₈×D₁₉₁

yes

characteristic

104

C₂

#20259370

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₁
commutator, socle, cosocle	C₂²×C₁₉₁	C₂⁴
Fitting	C₂²×Q₈×C₁₉₁	C₂
prime radical	C₂×C₂²⋊D₁₉₁	C₂²
hypercenter	C₂²×Q₈	D₁₉₁
hypocenter	C₁₉₁	D₄×Q₈
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉₁

commutator, socle, cosocle

C₂²×C₁₉₁

C₂⁴

Fitting

C₂²×Q₈×C₁₉₁

C₂

prime radical

C₂×C₂²⋊D₁₉₁

C₂²

hypercenter

C₂²×Q₈

D₁₉₁

hypocenter

C₁₉₁

D₄×Q₈

LCS

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₁₉₁	C₂	semidirect
2. #20259320	C₂	semidirect
3. #20259301	C₂	semidirect
4. #20259298	C₂	semidirect
5. Q₈×Dic₁₉₁	C₂	semidirect
7. C₂²×Q₈×C₁₉₁	C₂	semidirect
24. C₂²⋊D₁₉₁	Q₈	direct (N × H)
31. Q₈×C₁₉₁	D₄	semidirect
37. C₂²×C₁₉₁	C₂×Q₈	semidirect
41. C₂×D₁₉₁	C₂×Q₈	semidirect
46. Dic₁₉₁	C₂×Q₈	semidirect
52. C₁₉₁	D₄×Q₈	semidirect
55. C₂²×Q₈	D₁₉₁	semidirect
79. Q₈	C₂²⋊D₁₉₁	direct (N × H)
93. C₂²	Q₈×D₁₉₁	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₉₁

C₂

semidirect

2. #20259320

C₂

semidirect

3. #20259301

C₂

semidirect

4. #20259298

C₂

semidirect

5. Q₈×Dic₁₉₁

C₂

semidirect

7. C₂²×Q₈×C₁₉₁

C₂

semidirect

24. C₂²⋊D₁₉₁

Q₈

direct (N × H)

31. Q₈×C₁₉₁

D₄

semidirect

37. C₂²×C₁₉₁

C₂×Q₈

semidirect

41. C₂×D₁₉₁

C₂×Q₈

semidirect

46. Dic₁₉₁

C₂×Q₈

semidirect

52. C₁₉₁

D₄×Q₈

semidirect

55. C₂²×Q₈

D₁₉₁

semidirect

79. Q₈

C₂²⋊D₁₉₁

direct (N × H)

93. C₂²

Q₈×D₁₉₁

semidirect

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

subgroup	index
C₂×Q₈×D₁₉₁	2
#20259320	2
#20259301	2
#20259298	2
Q₈×Dic₁₉₁	2
C₄×C₂²⋊D₁₉₁	2
C₂²×Q₈×C₁₉₁	2
D₄×Q₈	191

subgroup

index

C₂×Q₈×D₁₉₁

#20259320

#20259301

#20259298

Q₈×Dic₁₉₁

C₄×C₂²⋊D₁₉₁

C₂²×Q₈×C₁₉₁

D₄×Q₈

191

Subgroups — Q₈×C₂²⋊D₁₉₁ #27133327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C22⋊D191 #27133327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂²⋊D₁₉₁ #27133327