Group #37568321 — subgroups

Subgroups — #1579×C₁₉₁ #37568321

Order 24448.

160 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 100 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	C₈×Q₈×C₁₉₁	C₂	1	2	yes	maximal
2	C₄×Q₁₆×C₁₉₁	C₂	1	1	yes	maximal, characteristic
3	#334×C₁₉₁	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₈×C₁₉₁	C₂	1	2	yes	maximal
5	#322×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₁₉₁	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₉₁	C₂²	1	2	yes	—
8	#107×C₁₉₁	C₄	1	1	yes	characteristic
9	C₄⋊C₈×C₁₉₁	—	2	2	no	—
10	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
11	#103×C₁₉₁	C₄	1	2	yes	—
12	C₄×C₈×C₁₉₁	—	2	2	no	—
13	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₉₁	C₂×C₄	1	2	yes	—
17	Q₁₆×C₁₉₁	C₈	1	4	yes	complemented
18	C₂×C₈×C₁₉₁	—	2	2	no	—
19	C₂×C₈×C₁₉₁	—	2	1	no	—
20	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₉₁	—	2	2	no	—
24	C₄⋊C₄×C₁₉₁	C₂×C₄	1	2	yes	—
25	C₄²×C₁₉₁	—	2	2	no	—
26	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
27	Q₈×C₁₉₁	—	2	2	no	—
28	Q₈×C₁₉₁	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₁₉₁	—	2	2	no	—
30	C₂×C₄×C₁₉₁	—	2	2	no	—
31	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
34	C₈×C₁₉₁	—	2	1	no	—
35	C₈×C₁₉₁	—	2	2	no	—
36	C₈×C₁₉₁	—	4	1	no	—
37	C₈×C₁₉₁	—	4	2	no	—
38	C₈×C₁₉₁	C₂×C₈	1	2	yes	—
39	C₂²×C₁₉₁	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₉₁	—	2	4	no	—
41	C₄×C₁₉₁	—	2	1	no	—
42	C₄×C₁₉₁	—	4	2	no	—
43	C₄×C₁₉₁	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₁₉₁	#131	1	1	yes	characteristic
45	C₄×C₁₉₁	#137	1	2	yes	—
46	C₂×C₁₉₁	C₈×D₄	1	1	yes	characteristic
47	C₂×C₁₉₁	#441	1	1	yes	characteristic
48	C₂×C₁₉₁	#444	1	1	yes	characteristic
49	C₁₉₁	#1579	1	1	yes	characteristic, direct product factor, Sylow
50	#1579	C₁₉₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₈×Q₈	C₂×C₁₉₁	1	2	yes	—
52	C₄×Q₁₆	C₂×C₁₉₁	1	1	yes	characteristic
53	#334	C₂×C₁₉₁	1	1	yes	characteristic
54	Q₈⋊C₈	C₂×C₁₉₁	1	2	yes	—
55	#322	C₂×C₁₉₁	1	1	yes	characteristic
56	C₂×Q₁₆	C₄×C₁₉₁	1	1	yes	characteristic
57	C₄×Q₈	C₂²×C₁₉₁	1	2	yes	—
58	#107	C₄×C₁₉₁	1	1	yes	characteristic
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	C₂²×C₁₉₁	1	2	yes	—
61	#103	C₄×C₁₉₁	1	2	yes	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	C₂²×C₁₉₁	1	1	yes	characteristic
64	C₄×C₈	C₂²×C₁₉₁	1	1	yes	characteristic
65	C₄×C₈	C₂²×C₁₉₁	1	1	yes	characteristic
66	C₂×Q₈	C₂×C₄×C₁₉₁	1	2	yes	—
67	Q₁₆	C₈×C₁₉₁	1	4	yes	complemented
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	C₂×C₄×C₁₉₁	1	1	yes	characteristic
71	C₂×C₈	C₂×C₄×C₁₉₁	1	1	yes	characteristic
72	C₂×C₈	D₄×C₁₉₁	1	2	yes	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	C₂×C₄×C₁₉₁	1	2	yes	—
75	C₄²	—	2	2	no	—
76	C₄²	C₂³×C₁₉₁	1	1	yes	characteristic
77	Q₈	—	2	2	no	—
78	Q₈	C₂×C₈×C₁₉₁	1	4	yes	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂²×C₄×C₁₉₁	1	1	yes	characteristic
82	C₂×C₄	C₂×D₄×C₁₉₁	1	1	yes	characteristic
83	C₂×C₄	#41×C₁₉₁	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	4	2	no	—
88	C₈	C₂×C₈×C₁₉₁	1	2	yes	—
89	C₂²	C₄×D₄×C₁₉₁	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	2	1	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂²×C₈×C₁₉₁	1	1	yes	characteristic
94	C₄	#131×C₁₉₁	1	1	yes	characteristic
95	C₄	#137×C₁₉₁	1	2	yes	—
96	C₂	C₈×D₄×C₁₉₁	1	1	yes	characteristic
97	C₂	#441×C₁₉₁	1	1	yes	characteristic
98	C₂	#444×C₁₉₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Q₈×C₁₉₁

C₂

yes

maximal

C₄×Q₁₆×C₁₉₁

C₂

yes

maximal, characteristic

#334×C₁₉₁

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₁₉₁

C₂

yes

maximal

#322×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₉₁

C₄

yes

characteristic

C₄×Q₈×C₁₉₁

C₂²

yes

—

#107×C₁₉₁

C₄

yes

characteristic

C₄⋊C₈×C₁₉₁

—

C₄⋊C₈×C₁₉₁

C₂²

yes

—

#103×C₁₉₁

C₄

yes

—

C₄×C₈×C₁₉₁

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Q₈×C₁₉₁

C₂×C₄

yes

—

Q₁₆×C₁₉₁

C₈

yes

complemented

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

D₄

yes

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

Q₈×C₁₉₁

—

Q₈×C₁₉₁

C₂×C₈

yes

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₈×C₁₉₁

C₂×C₈

yes

—

C₂²×C₁₉₁

C₄×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂²×C₈

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

—

C₂×C₁₉₁

C₈×D₄

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, direct product factor, Sylow

#1579

C₁₉₁

yes

maximal, characteristic, direct product factor, Sylow

C₈×Q₈

C₂×C₁₉₁

yes

—

C₄×Q₁₆

C₂×C₁₉₁

yes

characteristic

#334

C₂×C₁₉₁

yes

characteristic

Q₈⋊C₈

C₂×C₁₉₁

yes

—

#322

C₂×C₁₉₁

yes

characteristic

C₂×Q₁₆

C₄×C₁₉₁

yes

characteristic

C₄×Q₈

C₂²×C₁₉₁

yes

—

#107

C₄×C₁₉₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₁₉₁

yes

—

#103

C₄×C₁₉₁

yes

—

C₄×C₈

—

C₄×C₈

C₂²×C₁₉₁

yes

characteristic

C₄×C₈

C₂²×C₁₉₁

yes

characteristic

C₄×C₈

C₂²×C₁₉₁

yes

characteristic

C₂×Q₈

C₂×C₄×C₁₉₁

yes

—

Q₁₆

C₈×C₁₉₁

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₉₁

yes

characteristic

C₂×C₈

C₂×C₄×C₁₉₁

yes

characteristic

C₂×C₈

D₄×C₁₉₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₁₉₁

yes

—

C₄²

—

C₄²

C₂³×C₁₉₁

yes

characteristic

Q₈

—

Q₈

C₂×C₈×C₁₉₁

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₁₉₁

yes

characteristic

C₂×C₄

C₂×D₄×C₁₉₁

yes

characteristic

C₂×C₄

#41×C₁₉₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×C₈×C₁₉₁

yes

—

C₂²

C₄×D₄×C₁₉₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂²×C₈×C₁₉₁

yes

characteristic

C₄

#131×C₁₉₁

yes

characteristic

C₄

#137×C₁₉₁

yes

—

C₂

C₈×D₄×C₁₉₁

yes

characteristic

C₂

#441×C₁₉₁

yes

characteristic

C₂

#444×C₁₉₁

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉₁	C₂×D₄
commutator	C₄	C₂²×C₈×C₁₉₁
Frattini, cosocle	C₄²	C₂³×C₁₉₁
socle, prime radical	C₂²×C₁₉₁	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉₁ ◁ C₄×C₈×C₁₉₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₉₁

C₂×D₄

commutator

C₄

C₂²×C₈×C₁₉₁

Frattini, cosocle

C₄²

C₂³×C₁₉₁

socle, prime radical

C₂²×C₁₉₁

C₄×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₉₁ ◁ C₄×C₈×C₁₉₁ ◁ G

N	H ≅ G/N	type
17. Q₁₆×C₁₉₁	C₈	semidirect
49. C₁₉₁	#1579	direct (N × H)
50. #1579	C₁₉₁	direct (N × H)
67. Q₁₆	C₈×C₁₉₁	semidirect

H ≅ G/N

type

17. Q₁₆×C₁₉₁

C₈

semidirect

49. C₁₉₁

#1579

direct (N × H)

50. #1579

C₁₉₁

direct (N × H)

67. Q₁₆

C₈×C₁₉₁

semidirect

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

subgroup	index
C₈×Q₈×C₁₉₁	2
C₄×Q₁₆×C₁₉₁	2
#334×C₁₉₁	2
Q₈⋊C₈×C₁₉₁	2
#322×C₁₉₁	2
#1579	191

subgroup

index

C₈×Q₈×C₁₉₁

C₄×Q₁₆×C₁₉₁

#334×C₁₉₁

Q₈⋊C₈×C₁₉₁

#322×C₁₉₁

#1579

191

Subgroups — #1579×C₁₉₁ #37568321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1579×C191 #37568321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1579×C₁₉₁ #37568321