Group #38741852 — subgroups

Subgroups — #38741852

Order 25216.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#27161458	C₂	1	1	yes	maximal, characteristic
2	#27161374	C₂	1	2	yes	maximal
3	#27160828	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#13096802	C₂	1	2	yes	maximal
5	Q₈²×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691348	—	2	2	no	—
7	#20263326	C₂²	1	1	yes	characteristic
8	#20263259	—	2	2	no	—
9	C₄⋊#13096802	C₂²	1	2	yes	—
10	#20263250	—	2	2	no	—
11	C₄×#13096802	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₉₇	—	2	2	no	—
13	C₄⋊Q₈×C₁₉₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₉₇	—	2	2	no	—
16	C₄×Q₈×C₁₉₇	C₂²	1	2	yes	—
17	C₂×Dic₃₉₄	—	2	1	no	—
18	#15691348	—	4	4	no	—
19	C₂×#13096802	—	2	1	no	—
20	C₂×#13096802	—	2	2	no	—
21	C₄⋊Dic₁₉₇	—	2	2	no	—
22	C₂×Q₈×C₁₉₇	—	2	2	no	—
23	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
24	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₉₇	—	2	4	no	—
26	C₄⋊C₄×C₁₉₇	—	2	2	no	—
27	C₄⋊C₄×C₁₉₇	—	2	2	no	—
28	C₄⋊C₄×C₁₉₇	D₄	1	2	yes	—
29	C₄²×C₁₉₇	—	2	2	no	—
30	C₄²×C₁₉₇	—	2	2	no	—
31	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
32	Dic₃₉₄	—	4	2	no	—
33	C₂×Dic₁₉₇	—	4	1	no	—
34	#13096802	—	2	2	no	—
35	#13096802	—	4	2	no	—
36	Q₈×C₁₉₇	—	2	2	no	—
37	Q₈×C₁₉₇	—	4	2	no	—
38	Q₈×C₁₉₇	—	8	2	no	—
39	Q₈×C₁₉₇	Q₁₆	1	4	yes	complemented
40	C₂×C₄×C₁₉₇	—	2	2	no	—
41	C₂×C₄×C₁₉₇	—	2	2	no	—
42	C₂×C₄×C₁₉₇	—	2	2	no	—
43	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
45	C₂²×C₁₉₇	#120	1	1	yes	characteristic
46	C₄×C₁₉₇	—	2	4	no	—
47	C₄×C₁₉₇	—	2	1	no	—
48	C₄×C₁₉₇	—	4	2	no	—
49	C₄×C₁₉₇	—	4	2	no	—
50	C₄×C₁₉₇	C₂×Q₁₆	1	2	yes	—
51	C₄×C₁₉₇	#137	1	2	yes	—
52	Dic₁₉₇	—	8	1	no	—
53	C₂×C₁₉₇	#451	1	2	yes	—
54	C₂×C₁₉₇	#453	1	1	yes	characteristic
55	C₁₉₇	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	197	1	no	maximal, Sylow
57	Q₈²	D₁₉₇	1	1	yes	characteristic
58	#494	—	197	1	no	—
59	#477	—	197	2	no	—
60	#333	—	197	1	no	—
61	Q₈⋊C₈	—	197	2	no	—
62	C₂×Q₁₆	—	394	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	197	1	no	—
66	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₁₉₇	1	2	yes	—
69	#107	—	394	2	no	—
70	C₄⋊C₈	—	197	2	no	—
71	#103	—	394	2	no	—
72	C₄×C₈	—	197	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	394	1	no	—
75	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
77	Q₁₆	—	788	4	no	—
78	C₂×C₈	—	394	2	no	—
79	C₂×C₈	—	394	1	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	394	2	no	—
84	C₄⋊C₄	C₂²⋊D₁₉₇	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	788	2	no	—
92	Q₈	#15691348	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	788	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
99	C₈	—	788	2	no	—
100	C₈	—	394	2	no	—
101	C₂²	#20263295	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	1576	1	no	—
107	C₄	C₂×#15691348	1	2	yes	—
108	C₄	#20263357	1	2	yes	—
109	C₂	#27161235	1	2	yes	—
110	C₂	#27161247	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161458

C₂

yes

maximal, characteristic

#27161374

C₂

yes

maximal

#27160828

C₂

yes

maximal, characteristic

Q₈⋊#13096802

C₂

yes

maximal

Q₈²×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691348

—

#20263326

C₂²

yes

characteristic

#20263259

—

C₄⋊#13096802

C₂²

yes

—

#20263250

—

C₄×#13096802

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

C₄×Q₈×C₁₉₇

—

C₄×Q₈×C₁₉₇

C₂²

yes

—

C₂×Dic₃₉₄

—

#15691348

—

C₂×#13096802

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

C₂×Dic₁₉₇

—

#13096802

—

#13096802

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

Q₁₆

yes

complemented

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂²×C₁₉₇

#120

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×Q₁₆

yes

—

C₄×C₁₉₇

#137

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

#451

yes

—

C₂×C₁₉₇

#453

yes

characteristic

C₁₉₇

#1633

yes

characteristic, complemented, Sylow

#1633

—

197

maximal, Sylow

Q₈²

D₁₉₇

yes

characteristic

#494

—

197

—

#477

—

197

—

#333

—

197

—

Q₈⋊C₈

—

197

—

C₂×Q₁₆

—

394

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₁₉₇

yes

—

#107

—

394

—

C₄⋊C₈

—

197

—

#103

—

394

—

C₄×C₈

—

197

—

C₂×Q₈

—

C₂×Q₈

—

394

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

Q₁₆

—

788

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

394

—

C₄⋊C₄

C₂²⋊D₁₉₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₉₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

788

—

Q₈

#15691348

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

788

—

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

C₈

—

788

—

100

C₈

—

394

—

101

C₂²

#20263295

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

1576

—

107

C₄

C₂×#15691348

yes

—

108

C₄

#20263357

yes

—

109

C₂

#27161235

yes

—

110

C₂

#27161247

yes

characteristic

	subgroup	G/H
center	C₂²	#20263295
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	Q₈²×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#120
hypercenter	Q₈²	D₁₉₇
hypocenter	C₁₉₇	#1633
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#20263295

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting

Q₈²×C₁₉₇

C₂

socle, prime radical

C₂²×C₁₉₇

#120

hypercenter

Q₈²

D₁₉₇

hypocenter

C₁₉₇

#1633

LCS

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
39. Q₈×C₁₉₇	Q₁₆	semidirect
55. C₁₉₇	#1633	semidirect
92. Q₈	#15691348	semidirect

H ≅ G/N

type

39. Q₈×C₁₉₇

Q₁₆

semidirect

55. C₁₉₇

#1633

semidirect

92. Q₈

#15691348

semidirect

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

subgroup	index
#27161458	2
#27161374	2
#27160828	2
Q₈⋊#13096802	2
Q₈²×C₁₉₇	2
#1633	197

subgroup

index

Q₈²×C₁₉₇

197

Subgroups — #38741852

Conjugacy classes of subgroups

Complements

Maximal subgroups