Group #38742619 — subgroups

Subgroups — #38742619

Order 25216.

220 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20263261	C₂	1	2	yes	maximal
2	C₄×#15691342	C₂	1	1	yes	maximal, characteristic
3	C₂×#20263237	C₂	1	1	yes	maximal, characteristic
4	#27160843	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₂²×#13096802	C₂²	1	1	yes	characteristic
8	C₂²×#13096802	C₂²	1	1	yes	characteristic
9	#20263261	C₄	1	8	yes	complemented
10	#20263237	—	2	1	no	—
11	#20263237	C₄	1	4	yes	—
12	C₄×#13096802	—	2	1	no	—
13	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
15	#97×C₁₉₇	—	2	1	no	—
16	C₄×C₈×C₁₉₇	—	2	1	no	—
17	#15691342	—	2	2	no	—
18	#15691342	C₂×C₄	1	4	yes	—
19	C₂×#13096802	—	2	2	no	—
20	C₂×#13096802	—	2	2	no	—
21	C₂×#13096802	C₂×C₄	1	2	yes	—
22	C₂×#13096802	C₂×C₄	1	2	yes	—
23	C₂×#13096802	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₉₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₉₇	—	2	2	no	—
27	M₁₆×C₁₉₇	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₉₇	—	2	2	no	—
29	C₄²×C₁₉₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
31	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
32	#13096802	—	2	2	no	—
33	#13096802	—	2	4	no	—
34	#13096802	C₄²	1	4	yes	—
35	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₉₇	—	4	2	no	—
37	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	#41	1	2	yes	—
43	C₈×C₁₉₇	—	2	4	no	—
44	C₂²×C₁₉₇	—	2	2	no	—
45	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₉₇	C₄×Q₈	1	2	yes	—
47	C₄×C₁₉₇	—	2	2	no	—
48	C₄×C₁₉₇	—	2	2	no	—
49	C₄×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₁₉₇	#117	1	1	yes	characteristic
51	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
52	C₂×C₁₉₇	—	2	2	no	—
53	C₂×C₁₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₁₉₇	#430	1	2	yes	—
55	C₁₉₇	#1778	1	1	yes	characteristic, complemented, Sylow
56	#1778	—	197	1	no	maximal, Sylow
57	C₂×#108	—	197	2	no	—
58	C₄×M₁₆	—	197	1	no	—
59	C₄×M₁₆	D₁₉₇	1	1	yes	characteristic
60	C₂×#97	—	197	1	no	—
61	#341	—	197	2	no	—
62	C₂×M₁₆	—	197	2	no	—
63	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
64	C₂²×C₈	—	197	1	no	—
65	C₂²×C₈	—	197	1	no	—
66	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
67	#108	—	197	8	no	—
68	#97	—	2	1	no	—
69	#97	—	394	1	no	—
70	#97	—	197	4	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	394	1	no	—
73	C₂²×C₄	Dic₃₉₄	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	394	2	no	—
77	M₁₆	—	197	4	no	—
78	M₁₆	C₄×D₁₉₇	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	394	2	no	—
81	C₂×C₈	—	394	2	no	—
82	C₂×C₈	—	197	2	no	—
83	C₂×C₈	—	197	2	no	—
84	C₂×C₈	—	197	4	no	—
85	C₄²	C₄×D₁₉₇	1	2	yes	—
86	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
87	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
88	C₂³	C₂×Dic₃₉₄	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	#15691337	1	2	yes	—
91	C₂×C₄	#15691337	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
95	C₂×C₄	#15691358	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	394	2	no	—
98	C₈	—	394	4	no	—
99	C₈	—	197	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#15691337	1	1	yes	characteristic
102	C₂²	C₄×Dic₃₉₄	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₁₉₇	1	1	yes	characteristic
106	C₄	#20263283	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₁₉₇	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#15691337	1	1	yes	characteristic
110	C₂	#27161127	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263261

C₂

yes

maximal

C₄×#15691342

C₂

yes

maximal, characteristic

C₂×#20263237

C₂

yes

maximal, characteristic

#27160843

C₂

yes

maximal

C₄×M₁₆×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂²×#13096802

C₂²

yes

characteristic

C₂²×#13096802

C₂²

yes

characteristic

#20263261

C₄

yes

complemented

#20263237

—

#20263237

C₄

yes

—

C₄×#13096802

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

#97×C₁₉₇

—

C₄×C₈×C₁₉₇

—

#15691342

—

#15691342

C₂×C₄

yes

—

C₂×#13096802

—

C₂×#13096802

—

C₂×#13096802

C₂×C₄

yes

—

C₂×#13096802

C₂×C₄

yes

—

C₂×#13096802

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

C₂×C₄

yes

—

C₂×C₈×C₁₉₇

—

C₄²×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

—

—

C₄²

yes

—

C₂³×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₄×Q₈

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×C₄²

yes

characteristic

C₄×C₁₉₇

#117

yes

characteristic

C₄×C₁₉₇

C₄×D₄

yes

—

C₂×C₁₉₇

—

C₂×C₁₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₉₇

#430

yes

—

C₁₉₇

#1778

yes

characteristic, complemented, Sylow

#1778

—

197

maximal, Sylow

C₂×#108

—

197

—

C₄×M₁₆

—

197

—

C₄×M₁₆

D₁₉₇

yes

characteristic

C₂×#97

—

197

—

#341

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₂²×C₈

—

197

—

C₂²×C₈

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#108

—

197

—

#97

—

#97

—

394

—

#97

—

197

—

C₄×C₈

—

C₄×C₈

—

394

—

C₂²×C₄

Dic₃₉₄

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

394

—

M₁₆

—

197

—

M₁₆

C₄×D₁₉₇

yes

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₄²

C₄×D₁₉₇

yes

—

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₂³

C₂×Dic₃₉₄

yes

characteristic

C₂×C₄

—

C₂×C₄

#15691337

yes

—

C₂×C₄

#15691337

yes

—

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

#15691358

yes

—

C₈

—

C₈

—

394

—

C₈

—

394

—

C₈

—

197

—

100

C₂²

—

101

C₂²

C₂×#15691337

yes

characteristic

102

C₂²

C₄×Dic₃₉₄

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₁₉₇

yes

characteristic

106

C₄

#20263283

yes

characteristic

107

C₄

C₄×C₂²⋊D₁₉₇

yes

—

108

C₂

—

109

C₂

C₄×#15691337

yes

characteristic

110

C₂

#27161127

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉₇
commutator	C₄×C₁₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₄×M₁₆×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₄×M₁₆	D₁₉₇
hypocenter	C₁₉₇	#1778
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₉₇

commutator

C₄×C₁₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₄×M₁₆×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₄×M₁₆

D₁₉₇

hypocenter

C₁₉₇

#1778

LCS

G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #20263261	C₄	semidirect
55. C₁₉₇	#1778	semidirect

H ≅ G/N

type

9. #20263261

C₄

semidirect

55. C₁₉₇

#1778

semidirect

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

subgroup	index
C₂×#20263261	2
C₄×#15691342	2
C₂×#20263237	2
#27160843	2
C₄×M₁₆×C₁₉₇	2
#1778	197

subgroup

index

C₄×M₁₆×C₁₉₇

197

Subgroups — #38742619

Conjugacy classes of subgroups

Complements

Maximal subgroups