Group #38740542 — subgroups

Subgroups — #38740542

Order 25216.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 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	#27161132	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263237	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13096802	C₄	1	2	yes	—
5	C₄⋊#13096802	—	2	1	no	—
6	C₂²⋊#13096802	—	2	1	no	—
7	#20263237	—	2	1	no	—
8	#20263237	—	2	1	no	—
9	C₄×#13096802	C₄	1	2	yes	—
10	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₉₇	—	2	1	no	—
12	C₂²⋊C₈×C₁₉₇	—	2	1	no	—
13	C₄×C₈×C₁₉₇	C₄	1	2	yes	—
14	C₂×#13096802	—	2	2	no	—
15	C₂×#13096802	—	2	2	no	—
16	C₂×#13096802	—	2	2	no	—
17	C₂×#13096802	C₈	1	4	yes	complemented
18	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₇	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₉₇	—	2	2	no	—
22	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
26	#13096802	—	2	4	no	—
27	#13096802	—	4	2	no	—
28	C₂³×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₇	—	4	1	no	—
30	C₂×C₄×C₁₉₇	—	4	1	no	—
31	C₂×C₄×C₁₉₇	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₉₇	M₁₆	1	2	yes	—
39	C₈×C₁₉₇	—	4	2	no	—
40	C₂²×C₁₉₇	—	2	1	no	—
41	C₂²×C₁₉₇	—	2	1	no	—
42	C₂²×C₁₉₇	#95	1	1	yes	characteristic
43	C₂²×C₁₉₇	C₄⋊C₈	1	2	yes	—
44	C₄×C₁₉₇	—	2	1	no	—
45	C₄×C₁₉₇	—	2	1	no	—
46	C₄×C₁₉₇	—	2	1	no	—
47	C₄×C₁₉₇	—	2	1	no	—
48	C₄×C₁₉₇	C₄×C₈	1	1	yes	characteristic
49	C₄×C₁₉₇	#97	1	1	yes	characteristic
50	C₄×C₁₉₇	C₂²⋊C₈	1	2	yes	—
51	C₂×C₁₉₇	—	2	2	no	—
52	C₂×C₁₉₇	#336	1	1	yes	characteristic
53	C₂×C₁₉₇	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₁₉₇	#338	1	1	yes	characteristic
55	C₁₉₇	#1275	1	1	yes	characteristic, complemented, Sylow
56	#1275	—	197	1	no	maximal, Sylow
57	#431	—	197	1	no	—
58	#431	D₁₉₇	1	1	yes	characteristic
59	C₂×#97	—	197	1	no	—
60	C₂²×C₈	—	197	2	no	—
61	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	394	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	394	1	no	—
66	#97	—	394	1	no	—
67	#97	—	394	1	no	—
68	C₄×C₈	—	197	2	no	—
69	C₄×C₈	Dic₁₉₇	1	2	yes	—
70	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
71	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
72	C₂²×C₄	Dic₃₉₄	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	394	2	no	—
75	C₂×C₈	—	394	2	no	—
76	C₂×C₈	—	394	2	no	—
77	C₂×C₈	—	197	4	no	—
78	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
79	C₄²	C₂×Dic₁₉₇	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
81	C₄²	D₇₈₈	1	1	yes	characteristic
82	C₂³	#15691337	1	1	yes	characteristic
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	C₄×Dic₁₉₇	1	1	yes	characteristic
86	C₂×C₄	#15691335	1	1	yes	characteristic
87	C₂×C₄	#15691335	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₁₉₇	1	1	yes	characteristic
89	C₂×C₄	#15691337	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₁₉₇	1	1	yes	characteristic
91	C₂×C₄	C₈×D₁₉₇	1	2	yes	—
92	C₂×C₄	#15691341	1	2	yes	—
93	C₈	—	4	2	no	—
94	C₈	—	788	2	no	—
95	C₈	—	394	4	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#20263233	1	1	yes	characteristic
99	C₂²	#20263255	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₁₉₇	1	1	yes	characteristic
105	C₄	#20263236	1	1	yes	characteristic
106	C₄	#20263238	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	#27160833	1	1	yes	characteristic
109	C₂	#27160835	1	1	yes	characteristic
110	C₂	#27160837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161132

C₂

yes

maximal, characteristic

C₂×#20263237

C₂

yes

maximal, characteristic

#431×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

—

C₂²⋊C₈×C₁₉₇

—

C₄×C₈×C₁₉₇

C₄

yes

—

—

—

—

C₈

yes

complemented

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

Q₈

yes

characteristic

C₂×C₈×C₁₉₇

—

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

#13096802

—

#13096802

—

C₂³×C₁₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄²

yes

characteristic

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×C₈

yes

—

C₂×C₄×C₁₉₇

M₁₆

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#95

yes

characteristic

C₂²×C₁₉₇

C₄⋊C₈

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄×C₈

yes

characteristic

C₄×C₁₉₇

#97

yes

characteristic

C₄×C₁₉₇

C₂²⋊C₈

yes

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1275

—

197

maximal, Sylow

#431

—

197

—

#431

D₁₉₇

yes

characteristic

C₂×#97

—

197

—

C₂²×C₈

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

394

—

C₂²⋊C₈

—

C₂²⋊C₈

—

394

—

#97

—

394

—

#97

—

394

—

C₄×C₈

—

197

—

C₄×C₈

Dic₁₉₇

yes

—

C₂²×C₄

C₄×D₁₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

Dic₃₉₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₄²

C₄×D₁₉₇

yes

characteristic

C₄²

C₂×Dic₁₉₇

yes

characteristic

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₄²

D₇₈₈

yes

characteristic

C₂³

#15691337

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₉₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₉₇

yes

characteristic

C₂×C₄

#15691337

yes

characteristic

C₂×C₄

C₄⋊Dic₁₉₇

yes

characteristic

C₂×C₄

C₈×D₁₉₇

yes

—

C₂×C₄

#15691341

yes

—

C₈

—

C₈

—

788

—

C₈

—

394

—

C₂²

—

C₂²

—

C₂²

#20263233

yes

characteristic

C₂²

#20263255

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₁₉₇

yes

characteristic

105

C₄

#20263236

yes

characteristic

106

C₄

#20263238

yes

—

107

C₂

—

108

C₂

#27160833

yes

characteristic

109

C₂

#27160835

yes

characteristic

110

C₂

#27160837

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₉₇
commutator	C₄×C₁₉₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₉₇
Fitting	#431×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#95
prime radical	C₂³×C₁₉₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉₇	C₂²
hypercenter	#431	D₁₉₇
hypocenter	C₁₉₇	#1275
LCS	G ▷ C₄×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₄×Dic₁₉₇

commutator

C₄×C₁₉₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₉₇

Fitting

#431×C₁₉₇

C₂

socle

C₂²×C₁₉₇

#95

prime radical

C₂³×C₁₉₇

C₄⋊C₄

cosocle

C₂×C₄²×C₁₉₇

C₂²

D₁₉₇

C₁₉₇

G ▷ C₄×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
17. C₂×#13096802	C₈	semidirect
55. C₁₉₇	#1275	semidirect

H ≅ G/N

type

17. C₂×#13096802

C₈

semidirect

55. C₁₉₇

#1275

semidirect

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

subgroup	index
#27161132	2
C₂×#20263237	2
#431×C₁₉₇	2
#1275	197

subgroup

index

#27161132

C₂×#20263237

#431×C₁₉₇

#1275

197

Subgroups — #38740542

Conjugacy classes of subgroups

Complements

Maximal subgroups