Group #38741850 — subgroups

Subgroups — #38741850

Order 25216.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27161809	C₂	1	1	yes	maximal, characteristic
2	#27161371	C₂	1	2	yes	maximal
3	#27160828	C₂	1	1	yes	maximal, characteristic
4	#27160811	C₂	1	2	yes	maximal
5	#494×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	#20263325	—	2	2	no	—
7	#20263324	—	2	2	no	—
8	C₄×Dic₃₉₄	C₂²	1	2	yes	—
9	#20263291	—	2	2	no	—
10	#20263259	—	2	2	no	—
11	C₄⋊#13096802	C₂²	1	2	yes	—
12	#20263249	—	2	2	no	—
13	C₂×Q₁₆×C₁₉₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₉₄	—	2	2	no	—
18	C₂×Dic₃₉₄	D₄	1	2	yes	—
19	C₂×#13096802	—	2	2	no	—
20	C₄⋊Dic₁₉₇	D₄	1	2	yes	—
21	#15691337	—	2	4	no	—
22	#15691337	—	2	2	no	—
23	C₄×Dic₁₉₇	—	2	2	no	—
24	C₄×Dic₁₉₇	—	2	2	no	—
25	C₂×Q₈×C₁₉₇	—	2	1	no	—
26	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
27	Q₁₆×C₁₉₇	—	4	4	no	—
28	C₂×C₈×C₁₉₇	—	2	1	no	—
29	C₄⋊C₄×C₁₉₇	—	2	2	no	—
30	C₄⋊C₄×C₁₉₇	—	2	2	no	—
31	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
32	Dic₃₉₄	—	2	2	no	—
33	Dic₃₉₄	—	8	2	no	—
34	Dic₃₉₄	Q₁₆	1	4	yes	complemented
35	C₂×Dic₁₉₇	—	2	2	no	—
36	C₂×Dic₁₉₇	—	2	2	no	—
37	#13096802	—	4	2	no	—
38	Q₈×C₁₉₇	—	4	2	no	—
39	Q₈×C₁₉₇	—	4	2	no	—
40	C₂×C₄×C₁₉₇	—	2	2	no	—
41	C₂×C₄×C₁₉₇	—	4	1	no	—
42	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₉₇	—	2	2	no	—
45	C₂²×C₁₉₇	#120	1	1	yes	characteristic
46	C₄×C₁₉₇	—	2	1	no	—
47	C₄×C₁₉₇	—	4	2	no	—
48	C₄×C₁₉₇	—	8	1	no	—
49	C₄×C₁₉₇	C₂×Q₁₆	1	2	yes	—
50	C₄×C₁₉₇	#137	1	2	yes	—
51	Dic₁₉₇	—	2	4	no	—
52	Dic₁₉₇	—	4	2	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	—	197	1	no	—
58	#494	D₁₉₇	1	1	yes	characteristic
59	#477	—	197	2	no	—
60	#333	—	197	1	no	—
61	Q₈⋊C₈	—	197	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	394	2	no	—
64	C₄⋊Q₈	—	394	2	no	—
65	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
67	C₄×Q₈	—	394	2	no	—
68	C₄×Q₈	—	197	2	no	—
69	#107	—	394	2	no	—
70	C₄⋊C₈	—	197	2	no	—
71	#103	—	394	2	no	—
72	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	394	2	no	—
75	C₂×Q₈	—	197	2	no	—
76	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	394	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	394	4	no	—
83	C₄⋊C₄	—	394	2	no	—
84	C₄⋊C₄	—	197	2	no	—
85	C₄²	—	394	2	no	—
86	C₄²	—	394	2	no	—
87	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	1576	2	no	—
91	Q₈	—	394	2	no	—
92	Q₈	—	197	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	394	2	no	—
96	C₂×C₄	—	394	2	no	—
97	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	788	2	no	—
101	C₂²	#20263294	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	788	2	no	—
106	C₄	—	394	4	no	—
107	C₄	Q₁₆×D₁₉₇	1	2	yes	—
108	C₄	#20263358	1	2	yes	—
109	C₂	#27161233	1	2	yes	—
110	C₂	#27161246	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161809

C₂

yes

maximal, characteristic

#27161371

C₂

yes

maximal

#27160828

C₂

yes

maximal, characteristic

#27160811

C₂

yes

maximal

#494×C₁₉₇

C₂

yes

maximal, characteristic

#20263325

—

#20263324

—

C₄×Dic₃₉₄

C₂²

yes

—

#20263291

—

#20263259

—

C₄⋊#13096802

C₂²

yes

—

#20263249

—

C₂×Q₁₆×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

D₄

yes

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

D₄

yes

—

#15691337

—

#15691337

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

Q₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

Dic₃₉₄

—

Dic₃₉₄

Q₁₆

yes

complemented

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

#120

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×Q₁₆

yes

—

C₄×C₁₉₇

#137

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

#451

yes

—

C₂×C₁₉₇

#453

yes

characteristic

C₁₉₇

#1633

yes

characteristic, complemented, Sylow

#1633

—

197

maximal, Sylow

Q₈²

—

197

—

#494

D₁₉₇

yes

characteristic

#477

—

197

—

#333

—

197

—

Q₈⋊C₈

—

197

—

C₂×Q₁₆

—

C₄⋊Q₈

—

394

—

C₄⋊Q₈

—

394

—

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

C₄×Q₈

—

394

—

C₄×Q₈

—

197

—

#107

—

394

—

C₄⋊C₈

—

197

—

#103

—

394

—

C₄×C₈

C₂×D₁₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

394

—

C₂×Q₈

—

197

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

197

—

C₄²

—

394

—

C₄²

—

394

—

C₄²

C₂²×D₁₉₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

1576

—

Q₈

—

394

—

Q₈

—

197

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

D₄×D₁₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₈

—

100

C₈

—

788

—

101

C₂²

#20263294

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

788

—

106

C₄

—

394

—

107

C₄

Q₁₆×D₁₉₇

yes

—

108

C₄

#20263358

yes

—

109

C₂

#27161233

yes

—

110

C₂

#27161246

yes

characteristic

	subgroup	G/H
center	C₂²	#20263294
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#494×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	#120
hypercenter	#494	D₁₉₇
hypocenter	C₁₉₇	#1633
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#20263294

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting

#494×C₁₉₇

C₂

C₂²×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₃₉₄	Q₁₆	semidirect
55. C₁₉₇	#1633	semidirect

H ≅ G/N

type

34. Dic₃₉₄

Q₁₆

semidirect

55. C₁₉₇

#1633

semidirect

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

subgroup	index
#27161809	2
#27161371	2
#27160828	2
#27160811	2
#494×C₁₉₇	2
#1633	197

subgroup

index

#494×C₁₉₇

197

Subgroups — #38741850

Conjugacy classes of subgroups

Complements

Maximal subgroups