Group #38739165 — subgroups

Subgroups — #2231×C₁₉₇ #38739165

Order 25216.

120 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 60 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	#499×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#443×C₁₉₇	C₂	1	1	yes	maximal, characteristic
3	#365×C₁₉₇	C₂	1	2	yes	maximal, complemented
4	#364×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#360×C₁₉₇	C₂	1	2	yes	maximal
6	#135×C₁₉₇	C₂²	1	1	yes	characteristic
7	#131×C₁₉₇	—	2	1	no	—
8	#125×C₁₉₇	—	2	1	no	—
9	M₃₂×C₁₉₇	C₂²	1	2	yes	—
10	#108×C₁₉₇	C₂²	1	1	yes	characteristic
11	#107×C₁₉₇	—	2	1	no	—
12	#106×C₁₉₇	C₂²	1	2	yes	—
13	#104×C₁₉₇	—	2	1	no	—
14	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	#41×C₁₉₇	—	2	1	no	—
16	Q₁₆×C₁₉₇	Q₈	1	1	yes	characteristic
17	SD₁₆×C₁₉₇	—	2	1	no	—
18	D₈×C₁₉₇	Q₈	1	1	yes	characteristic
19	M₁₆×C₁₉₇	—	2	1	no	—
20	M₁₆×C₁₉₇	—	4	1	no	—
21	C₂×C₈×C₁₉₇	—	4	1	no	—
22	C₂×C₈×C₁₉₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₉₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₉₇	—	2	2	no	—
25	C₄⋊C₄×C₁₉₇	—	4	2	no	—
26	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
27	C₁₆×C₁₉₇	—	2	2	no	—
28	Q₈×C₁₉₇	—	2	1	no	—
29	D₄×C₁₉₇	—	2	1	no	—
30	D₄×C₁₉₇	—	4	1	no	—
31	C₂×C₄×C₁₉₇	—	2	1	no	—
32	C₂×C₄×C₁₉₇	—	4	1	no	—
33	C₂×C₄×C₁₉₇	—	4	2	no	—
34	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₉₇	—	4	1	no	—
36	C₈×C₁₉₇	SD₁₆	1	1	yes	characteristic
37	C₈×C₁₉₇	SD₁₆	1	1	yes	characteristic
38	C₈×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₁₉₇	#41	1	1	yes	characteristic
40	C₂²×C₁₉₇	—	4	1	no	—
41	C₂²×C₁₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
42	C₄×C₁₉₇	—	4	1	no	—
43	C₄×C₁₉₇	—	4	1	no	—
44	C₄×C₁₉₇	—	8	2	no	—
45	C₄×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
46	C₄×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
47	C₂×C₁₉₇	—	2	1	no	—
48	C₂×C₁₉₇	—	8	1	no	—
49	C₂×C₁₉₇	#476	1	1	yes	characteristic
50	C₁₉₇	#2231	1	1	yes	characteristic, direct product factor, Sylow
51	#2231	C₁₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	#499	C₂×C₁₉₇	1	1	yes	characteristic, complemented
53	#443	C₂×C₁₉₇	1	1	yes	characteristic
54	#365	C₂×C₁₉₇	1	2	yes	complemented
55	#364	C₂×C₁₉₇	1	1	yes	characteristic, complemented
56	#360	C₂×C₁₉₇	1	2	yes	—
57	#135	C₂²×C₁₉₇	1	1	yes	characteristic
58	#131	—	2	1	no	—
59	#125	—	2	1	no	—
60	M₃₂	C₂²×C₁₉₇	1	2	yes	—
61	#108	C₂²×C₁₉₇	1	1	yes	characteristic
62	#107	—	2	1	no	—
63	#106	C₂²×C₁₉₇	1	2	yes	—
64	#104	—	2	1	no	—
65	C₄×C₈	C₂²×C₁₉₇	1	1	yes	characteristic
66	#41	—	2	1	no	—
67	Q₁₆	Q₈×C₁₉₇	1	1	yes	characteristic
68	SD₁₆	—	2	1	no	—
69	D₈	Q₈×C₁₉₇	1	1	yes	characteristic
70	M₁₆	—	2	1	no	—
71	M₁₆	—	4	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	D₄×C₁₉₇	1	1	yes	characteristic
74	C₂×C₈	C₂³×C₁₉₇	1	1	yes	characteristic
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄²	D₄×C₁₉₇	1	1	yes	characteristic
78	C₁₆	—	2	2	no	—
79	Q₈	—	2	1	no	—
80	D₄	—	2	1	no	—
81	D₄	—	4	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	C₂×D₄×C₁₉₇	1	1	yes	characteristic
86	C₈	—	4	1	no	—
87	C₈	SD₁₆×C₁₉₇	1	1	yes	characteristic
88	C₈	SD₁₆×C₁₉₇	1	1	yes	characteristic
89	C₈	C₂×Q₈×C₁₉₇	1	1	yes	characteristic
90	C₈	#41×C₁₉₇	1	1	yes	characteristic
91	C₂²	—	4	1	no	—
92	C₂²	AGL(1,ℤ/8)×C₁₉₇	1	1	yes	characteristic
93	C₄	—	4	1	no	—
94	C₄	—	4	1	no	—
95	C₄	—	8	2	no	—
96	C₄	C₂²⋊Q₈×C₁₉₇	1	1	yes	characteristic
97	C₄	C₂×SD₁₆×C₁₉₇	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	8	1	no	—
100	C₂	#476×C₁₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#499×C₁₉₇

C₂

yes

maximal, characteristic, complemented

#443×C₁₉₇

C₂

yes

maximal, characteristic

#365×C₁₉₇

C₂

yes

maximal, complemented

#364×C₁₉₇

C₂

yes

maximal, characteristic, complemented

#360×C₁₉₇

C₂

yes

maximal

#135×C₁₉₇

C₂²

yes

characteristic

#131×C₁₉₇

—

#125×C₁₉₇

—

M₃₂×C₁₉₇

C₂²

yes

—

#108×C₁₉₇

C₂²

yes

characteristic

#107×C₁₉₇

—

#106×C₁₉₇

C₂²

yes

—

#104×C₁₉₇

—

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

#41×C₁₉₇

—

Q₁₆×C₁₉₇

Q₈

yes

characteristic

SD₁₆×C₁₉₇

—

D₈×C₁₉₇

Q₈

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

characteristic

C₂×C₈×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

characteristic

C₁₆×C₁₉₇

—

Q₈×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

—

C₈×C₁₉₇

SD₁₆

yes

characteristic

C₈×C₁₉₇

SD₁₆

yes

characteristic

C₈×C₁₉₇

C₂×Q₈

yes

characteristic

C₈×C₁₉₇

#41

yes

characteristic

C₂²×C₁₉₇

—

C₂²×C₁₉₇

AGL(1,ℤ/8)

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₉₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#476

yes

characteristic

C₁₉₇

#2231

yes

characteristic, direct product factor, Sylow

#2231

C₁₉₇

yes

maximal, characteristic, direct product factor, Sylow

#499

C₂×C₁₉₇

yes

characteristic, complemented

#443

C₂×C₁₉₇

yes

characteristic

#365

C₂×C₁₉₇

yes

complemented

#364

C₂×C₁₉₇

yes

characteristic, complemented

#360

C₂×C₁₉₇

yes

—

#135

C₂²×C₁₉₇

yes

characteristic

#131

—

#125

—

M₃₂

C₂²×C₁₉₇

yes

—

#108

C₂²×C₁₉₇

yes

characteristic

#107

—

#106

C₂²×C₁₉₇

yes

—

#104

—

C₄×C₈

C₂²×C₁₉₇

yes

characteristic

#41

—

Q₁₆

Q₈×C₁₉₇

yes

characteristic

SD₁₆

—

D₈

Q₈×C₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

D₄×C₁₉₇

yes

characteristic

C₂×C₈

C₂³×C₁₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₄×C₁₉₇

yes

characteristic

C₁₆

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₉₇

yes

characteristic

C₈

—

C₈

SD₁₆×C₁₉₇

yes

characteristic

C₈

SD₁₆×C₁₉₇

yes

characteristic

C₈

C₂×Q₈×C₁₉₇

yes

characteristic

C₈

#41×C₁₉₇

yes

characteristic

C₂²

—

C₂²

AGL(1,ℤ/8)×C₁₉₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂²⋊Q₈×C₁₉₇

yes

characteristic

C₄

C₂×SD₁₆×C₁₉₇

yes

characteristic

C₂

—

C₂

—

100

C₂

#476×C₁₉₇

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₉₇	#476
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₉₇
prime radical	#135×C₁₉₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉₇ ◁ C₂×C₄×C₁₉₇ ◁ C₄×C₈×C₁₉₇ ◁ G

subgroup

G/H

center, socle

C₂×C₁₉₇

#476

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₁₉₇

prime radical

#135×C₁₉₇

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₉₇ ◁ C₂×C₄×C₁₉₇ ◁ C₄×C₈×C₁₉₇ ◁ G

N	H ≅ G/N	type
1. #499×C₁₉₇	C₂	semidirect
3. #365×C₁₉₇	C₂	semidirect
4. #364×C₁₉₇	C₂	semidirect
50. C₁₉₇	#2231	direct (N × H)
51. #2231	C₁₉₇	direct (N × H)
52. #499	C₂×C₁₉₇	semidirect
54. #365	C₂×C₁₉₇	semidirect
55. #364	C₂×C₁₉₇	semidirect

H ≅ G/N

type

1. #499×C₁₉₇

C₂

semidirect

3. #365×C₁₉₇

C₂

semidirect

4. #364×C₁₉₇

C₂

semidirect

50. C₁₉₇

#2231

direct (N × H)

51. #2231

C₁₉₇

direct (N × H)

52. #499

C₂×C₁₉₇

semidirect

54. #365

C₂×C₁₉₇

semidirect

55. #364

C₂×C₁₉₇

semidirect

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

subgroup	index
#499×C₁₉₇	2
#443×C₁₉₇	2
#365×C₁₉₇	2
#364×C₁₉₇	2
#360×C₁₉₇	2
#2231	197

subgroup

index

#499×C₁₉₇

#443×C₁₉₇

#365×C₁₉₇

#364×C₁₉₇

#360×C₁₉₇

#2231

197

Subgroups — #2231×C₁₉₇ #38739165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2231×C197 #38739165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2231×C₁₉₇ #38739165