Group #38744288 — subgroups

Subgroups — #38744288

Order 25216.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#27161128	C₂	1	1	yes	maximal, characteristic
2	#27161102	C₂	1	1	yes	maximal, characteristic
3	#27160849	C₂	1	2	yes	maximal
4	#27160838	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₂×#15691342	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
9	#20263262	—	2	2	no	—
10	#20263255	—	2	2	no	—
11	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
12	#117×C₁₉₇	C₂²	1	2	yes	—
13	C₂²×Dic₁₉₇	—	2	1	no	—
14	#15691342	—	2	2	no	—
15	#15691342	—	2	2	no	—
16	#15691342	—	2	2	no	—
17	C₂×#13096802	—	2	1	no	—
18	C₂×#13096802	—	2	2	no	—
19	C₄×Dic₁₉₇	—	2	1	no	—
20	C₄×Dic₁₉₇	Q₈	1	1	yes	characteristic
21	C₄×Dic₁₉₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₉₇	—	2	2	no	—
24	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₉₇	—	2	2	no	—
26	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
27	C₄²×C₁₉₇	—	2	2	no	—
28	C₂×Dic₁₉₇	—	2	1	no	—
29	C₂×Dic₁₉₇	—	2	1	no	—
30	C₂×Dic₁₉₇	—	4	2	no	—
31	#13096802	—	4	1	no	—
32	#13096802	—	4	2	no	—
33	C₂³×C₁₉₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₉₇	—	4	2	no	—
35	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	#41	1	2	yes	—
38	C₂×C₄×C₁₉₇	#41	1	2	yes	—
39	C₈×C₁₉₇	—	2	2	no	—
40	C₂²×C₁₉₇	—	4	1	no	—
41	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₉₇	#126	1	2	yes	—
43	C₄×C₁₉₇	—	8	2	no	—
44	C₄×C₁₉₇	#123	1	1	yes	characteristic
45	C₄×C₁₉₇	#124	1	1	yes	characteristic
46	C₄×C₁₉₇	#125	1	2	yes	—
47	Dic₁₉₇	—	2	2	no	—
48	Dic₁₉₇	—	4	1	no	—
49	C₂×C₁₉₇	—	2	1	no	—
50	C₂×C₁₉₇	—	2	2	no	—
51	C₂×C₁₉₇	#400	1	1	yes	characteristic
52	C₁₉₇	#2102	1	1	yes	characteristic, complemented, Sylow
53	#2102	—	197	1	no	maximal, Sylow
54	#430	—	197	1	no	—
55	#423	—	197	1	no	—
56	#344	—	197	2	no	—
57	#339	—	197	2	no	—
58	H(3,ℤ/4)	D₁₉₇	1	1	yes	characteristic
59	C₂×M₁₆	—	197	2	no	—
60	C₂×M₁₆	—	197	1	no	—
61	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
62	#117	C₂×D₁₉₇	1	2	yes	—
63	C₂×C₄²	—	197	1	no	—
64	#108	—	394	2	no	—
65	C₄⋊C₈	—	394	2	no	—
66	C₂²×C₄	—	394	1	no	—
67	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	394	2	no	—
70	M₁₆	—	394	2	no	—
71	M₁₆	—	394	2	no	—
72	C₂×C₈	—	394	1	no	—
73	C₂×C₈	—	394	2	no	—
74	C₂×C₈	D₇₈₈	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	394	1	no	—
79	C₄²	—	197	1	no	—
80	C₄²	—	197	1	no	—
81	C₂³	#15691357	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	788	2	no	—
84	C₂×C₄	—	394	1	no	—
85	C₂×C₄	—	394	1	no	—
86	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
88	C₂×C₄	#15691357	1	2	yes	—
89	C₂×C₄	#15691358	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	788	1	no	—
92	C₈	—	788	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#20263306	1	1	yes	characteristic
95	C₂²	#20263319	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	788	1	no	—
98	C₄	—	394	2	no	—
99	C₄	#20263310	1	1	yes	characteristic
100	C₄	#20263314	1	1	yes	characteristic
101	C₄	#20263316	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#27161019	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161128

C₂

yes

maximal, characteristic

#27161102

C₂

yes

maximal, characteristic

#27160849

C₂

yes

maximal

#27160838

C₂

yes

maximal

H(3,ℤ/4)×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂×#15691342

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

#20263262

—

#20263255

—

C₂×M₁₆×C₁₉₇

C₂²

yes

characteristic

#117×C₁₉₇

C₂²

yes

—

C₂²×Dic₁₉₇

—

—

—

—

—

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

Q₈

yes

characteristic

C₄×Dic₁₉₇

Q₈

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₄⋊C₄×C₁₉₇

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

#13096802

—

C₂³×C₁₉₇

#41

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₇

#126

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#400

yes

characteristic

C₁₉₇

#2102

yes

characteristic, complemented, Sylow

#2102

—

197

maximal, Sylow

#430

—

197

—

#423

—

197

—

#344

—

197

—

#339

—

197

—

H(3,ℤ/4)

D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

characteristic

#117

C₂×D₁₉₇

yes

—

C₂×C₄²

—

197

—

#108

—

394

—

C₄⋊C₈

—

394

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

394

—

M₁₆

—

394

—

M₁₆

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

D₇₈₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

394

—

C₄²

—

197

—

C₄²

—

197

—

C₂³

#15691357

yes

characteristic

C₂×C₄

—

C₂×C₄

—

788

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

C₂×D₇₈₈

yes

characteristic

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

C₂×C₄

#15691357

yes

—

C₂×C₄

#15691358

yes

—

C₈

—

C₈

—

788

—

C₈

—

788

—

C₂²

—

C₂²

#20263306

yes

characteristic

C₂²

#20263319

yes

—

C₄

—

C₄

—

788

—

C₄

—

394

—

C₄

#20263310

yes

characteristic

100

C₄

#20263314

yes

characteristic

101

C₄

#20263316

yes

—

102

C₂

—

103

C₂

—

104

C₂

#27161019

yes

characteristic

	subgroup	G/H
center	C₄	#20263314
commutator, cosocle	C₂²×C₄×C₁₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	H(3,ℤ/4)×C₁₉₇	C₂
socle	C₂×C₁₉₇	#400
prime radical	C₂³×C₁₉₇	#41
hypercenter	H(3,ℤ/4)	D₁₉₇
hypocenter	C₁₉₇	#2102
LCS	G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#20263314

commutator, cosocle

C₂²×C₄×C₁₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

H(3,ℤ/4)×C₁₉₇

C₂

C₂×C₁₉₇

C₂³×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
52. C₁₉₇	#2102	semidirect

H ≅ G/N

type

52. C₁₉₇

#2102

semidirect

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

subgroup	index
#27161128	2
#27161102	2
#27160849	2
#27160838	2
H(3,ℤ/4)×C₁₉₇	2
#2102	197

subgroup

index

H(3,ℤ/4)×C₁₉₇

197

Subgroups — #38744288

Conjugacy classes of subgroups

Complements

Maximal subgroups