Group #38742847 — subgroups

Subgroups — #38742847

Order 25216.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#20263237	C₂	1	1	yes	maximal, characteristic
2	#27160945	C₂	1	1	yes	maximal, characteristic
3	#27160841	C₂	1	4	yes	maximal
4	#384×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13096802	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
7	#20263237	C₄	1	4	yes	complemented
8	#20263233	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₉₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₉₇	—	2	2	no	—
13	C₂×#13096802	—	2	2	no	—
14	C₂×#13096802	—	2	2	no	—
15	C₂×#13096802	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₉₇	—	2	4	no	—
17	C₂²×C₄×C₁₉₇	—	2	2	no	—
18	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₉₇	—	2	4	no	—
21	C₄⋊C₄×C₁₉₇	—	4	4	no	—
22	C₄²×C₁₉₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₉₇	Q₈	1	2	yes	—
24	C₂×Dic₁₉₇	—	4	2	no	—
25	C₂×Dic₁₉₇	—	4	2	no	—
26	C₂×Dic₁₉₇	—	4	2	no	—
27	#13096802	—	2	4	no	—
28	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₇	—	2	2	no	—
30	C₂×C₄×C₁₉₇	—	2	2	no	—
31	C₂×C₄×C₁₉₇	—	4	2	no	—
32	C₂×C₄×C₁₉₇	—	4	4	no	—
33	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₉₇	#41	1	2	yes	—
37	C₂×C₄×C₁₉₇	#41	1	2	yes	—
38	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₉₇	#124	1	1	yes	characteristic
40	C₂²×C₁₉₇	#124	1	1	yes	characteristic
41	C₂²×C₁₉₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₉₇	#137	1	2	yes	—
43	C₄×C₁₉₇	—	2	4	no	—
44	C₄×C₁₉₇	—	8	2	no	—
45	C₄×C₁₉₇	#117	1	2	yes	—
46	C₄×C₁₉₇	#125	1	2	yes	—
47	Dic₁₉₇	—	8	2	no	—
48	C₂×C₁₉₇	#382	1	1	yes	characteristic
49	C₂×C₁₉₇	#428	1	2	yes	—
50	C₂×C₁₉₇	#489	1	2	yes	—
51	C₂×C₁₉₇	#490	1	1	yes	characteristic
52	C₂×C₁₉₇	#491	1	1	yes	characteristic
53	C₁₉₇	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	197	1	no	maximal, Sylow
55	C₂×#97	—	197	1	no	—
56	#384	D₁₉₇	1	1	yes	characteristic
57	#382	—	197	1	no	—
58	#340	—	197	4	no	—
59	C₂²×C₈	—	197	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	197	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
63	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
64	#97	—	197	4	no	—
65	#95	—	394	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	394	2	no	—
68	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
70	C₂×C₈	—	394	2	no	—
71	C₂×C₈	—	394	2	no	—
72	C₂×C₈	—	197	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	394	4	no	—
76	C₄²	C₄×D₁₉₇	1	2	yes	—
77	C₄²	Dic₃₉₄	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	788	2	no	—
84	C₂×C₄	—	788	2	no	—
85	C₂×C₄	—	788	2	no	—
86	C₂×C₄	#15691337	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
89	C₂×C₄	#15691357	1	2	yes	—
90	C₂×C₄	#15691359	1	2	yes	—
91	C₈	—	394	4	no	—
92	C₂²	C₂×#15691337	1	1	yes	characteristic
93	C₂²	#20263313	1	1	yes	characteristic
94	C₂²	#20263315	1	1	yes	characteristic
95	C₂²	#20263352	1	2	yes	—
96	C₂²	#20263357	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	1576	2	no	—
100	C₄	#20263284	1	2	yes	—
101	C₄	#20263317	1	2	yes	—
102	C₂	#27160944	1	1	yes	characteristic
103	C₂	#27161119	1	2	yes	—
104	C₂	#27161441	1	2	yes	—
105	C₂	#27161443	1	1	yes	characteristic
106	C₂	#27161447	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263237

C₂

yes

maximal, characteristic

#27160945

C₂

yes

maximal, characteristic

#27160841

C₂

yes

maximal

#384×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

#20263237

C₄

yes

complemented

#20263233

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

—

C₂×#13096802

—

C₂×#13096802

—

C₂×#13096802

C₂×C₄

yes

—

C₄⋊Dic₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

Q₈

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

—

C₂²×C₁₉₇

#137

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#117

yes

—

C₄×C₁₉₇

#125

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

#382

yes

characteristic

C₂×C₁₉₇

#428

yes

—

C₂×C₁₉₇

#489

yes

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1836

—

197

maximal, Sylow

C₂×#97

—

197

—

#384

D₁₉₇

yes

characteristic

#382

—

197

—

#340

—

197

—

C₂²×C₈

—

197

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#97

—

197

—

#95

—

394

—

C₂²×C₄

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

394

—

C₄²

C₄×D₁₉₇

yes

—

C₄²

Dic₃₉₄

yes

—

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

788

—

C₂×C₄

—

788

—

C₂×C₄

—

788

—

C₂×C₄

#15691337

yes

—

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

C₂×C₄

#15691357

yes

—

C₂×C₄

#15691359

yes

—

C₈

—

394

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#20263357

yes

—

C₄

—

C₄

—

C₄

—

1576

—

100

C₄

#20263284

yes

—

101

C₄

#20263317

yes

—

102

C₂

#27160944

yes

characteristic

103

C₂

#27161119

yes

—

104

C₂

#27161441

yes

—

105

C₂

#27161443

yes

characteristic

106

C₂

#27161447

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₇
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#384×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂×D₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#384	D₁₉₇
hypocenter	C₁₉₇	#1836
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₇

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#384×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂×D₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #20263237	C₄	semidirect
53. C₁₉₇	#1836	semidirect

H ≅ G/N

type

7. #20263237

C₄

semidirect

53. C₁₉₇

#1836

semidirect

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

subgroup	index
C₂×#20263237	2
#27160945	2
#27160841	2
#384×C₁₉₇	2
#1836	197

subgroup

index

#384×C₁₉₇

197

Subgroups — #38742847

Conjugacy classes of subgroups

Complements

Maximal subgroups