Group #49157923 — subgroups

Subgroups — #49157923

Order 37504.

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	#34036859	C₂	1	1	yes	maximal, characteristic
2	#34036421	C₂	1	2	yes	maximal
3	#34035878	C₂	1	1	yes	maximal, characteristic
4	#34035861	C₂	1	2	yes	maximal
5	#494×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	#22627623	—	2	2	no	—
7	#22627622	—	2	2	no	—
8	C₄×Dic₅₈₆	C₂²	1	2	yes	—
9	#22627589	—	2	2	no	—
10	#22627557	—	2	2	no	—
11	C₄⋊#14556873	C₂²	1	2	yes	—
12	#22627547	—	2	2	no	—
13	C₂×Q₁₆×C₂₉₃	—	2	2	no	—
14	C₄⋊Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₈₆	—	2	2	no	—
18	C₂×Dic₅₈₆	D₄	1	2	yes	—
19	C₂×#14556873	—	2	2	no	—
20	C₄⋊Dic₂₉₃	D₄	1	2	yes	—
21	#17989781	—	2	4	no	—
22	#17989781	—	2	2	no	—
23	C₄×Dic₂₉₃	—	2	2	no	—
24	C₄×Dic₂₉₃	—	2	2	no	—
25	C₂×Q₈×C₂₉₃	—	2	1	no	—
26	C₂×Q₈×C₂₉₃	D₄	1	2	yes	—
27	Q₁₆×C₂₉₃	—	4	4	no	—
28	C₂×C₈×C₂₉₃	—	2	1	no	—
29	C₄⋊C₄×C₂₉₃	—	2	2	no	—
30	C₄⋊C₄×C₂₉₃	—	2	2	no	—
31	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
32	Dic₅₈₆	—	2	2	no	—
33	Dic₅₈₆	—	8	2	no	—
34	Dic₅₈₆	Q₁₆	1	4	yes	complemented
35	C₂×Dic₂₉₃	—	2	2	no	—
36	C₂×Dic₂₉₃	—	2	2	no	—
37	#14556873	—	4	2	no	—
38	Q₈×C₂₉₃	—	4	2	no	—
39	Q₈×C₂₉₃	—	4	2	no	—
40	C₂×C₄×C₂₉₃	—	2	2	no	—
41	C₂×C₄×C₂₉₃	—	4	1	no	—
42	C₂×C₄×C₂₉₃	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₉₃	—	2	2	no	—
45	C₂²×C₂₉₃	#120	1	1	yes	characteristic
46	C₄×C₂₉₃	—	2	1	no	—
47	C₄×C₂₉₃	—	4	2	no	—
48	C₄×C₂₉₃	—	8	1	no	—
49	C₄×C₂₉₃	C₂×Q₁₆	1	2	yes	—
50	C₄×C₂₉₃	#137	1	2	yes	—
51	Dic₂₉₃	—	2	4	no	—
52	Dic₂₉₃	—	4	2	no	—
53	C₂×C₂₉₃	#451	1	2	yes	—
54	C₂×C₂₉₃	#453	1	1	yes	characteristic
55	C₂₉₃	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	293	1	no	maximal, Sylow
57	Q₈²	—	293	1	no	—
58	#494	D₂₉₃	1	1	yes	characteristic
59	#477	—	293	2	no	—
60	#333	—	293	1	no	—
61	Q₈⋊C₈	—	293	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	586	2	no	—
64	C₄⋊Q₈	—	586	2	no	—
65	C₄⋊Q₈	C₂×D₂₉₃	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₂₉₃	1	1	yes	characteristic
67	C₄×Q₈	—	293	2	no	—
68	C₄×Q₈	—	586	2	no	—
69	#107	—	586	2	no	—
70	C₄⋊C₈	—	293	2	no	—
71	#103	—	586	2	no	—
72	C₄×C₈	C₂×D₂₉₃	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	293	2	no	—
75	C₂×Q₈	—	586	2	no	—
76	C₂×Q₈	C₂²⋊D₂₉₃	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	586	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	293	2	no	—
83	C₄⋊C₄	—	586	4	no	—
84	C₄⋊C₄	—	586	2	no	—
85	C₄²	—	586	2	no	—
86	C₄²	—	586	2	no	—
87	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	293	4	no	—
91	Q₈	—	2344	2	no	—
92	Q₈	—	586	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	586	2	no	—
96	C₂×C₄	—	586	2	no	—
97	C₂×C₄	D₄×D₂₉₃	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	1172	2	no	—
101	C₂²	#22627592	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	586	4	no	—
106	C₄	—	1172	2	no	—
107	C₄	Q₁₆×D₂₉₃	1	2	yes	—
108	C₄	#22627656	1	2	yes	—
109	C₂	#34036283	1	2	yes	—
110	C₂	#34036296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036859

C₂

yes

maximal, characteristic

#34036421

C₂

yes

maximal

#34035878

C₂

yes

maximal, characteristic

#34035861

C₂

yes

maximal

#494×C₂₉₃

C₂

yes

maximal, characteristic

#22627623

—

#22627622

—

C₄×Dic₅₈₆

C₂²

yes

—

#22627589

—

#22627557

—

C₄⋊#14556873

C₂²

yes

—

#22627547

—

C₂×Q₁₆×C₂₉₃

—

C₄⋊Q₈×C₂₉₃

C₂²

yes

characteristic

C₄⋊Q₈×C₂₉₃

C₂²

yes

characteristic

C₄×C₈×C₂₉₃

C₂²

yes

characteristic

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

D₄

yes

—

C₂×#14556873

—

C₄⋊Dic₂₉₃

D₄

yes

—

#17989781

—

#17989781

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

D₄

yes

—

Q₁₆×C₂₉₃

—

C₂×C₈×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

C₂³

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

Dic₅₈₆

Q₁₆

yes

complemented

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

Q₈×C₂₉₃

—

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

#120

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×Q₁₆

yes

—

C₄×C₂₉₃

#137

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

#451

yes

—

C₂×C₂₉₃

#453

yes

characteristic

C₂₉₃

#1633

yes

characteristic, complemented, Sylow

#1633

—

293

maximal, Sylow

Q₈²

—

293

—

#494

D₂₉₃

yes

characteristic

#477

—

293

—

#333

—

293

—

Q₈⋊C₈

—

293

—

C₂×Q₁₆

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

C₂×D₂₉₃

yes

characteristic

C₄⋊Q₈

C₂×D₂₉₃

yes

characteristic

C₄×Q₈

—

293

—

C₄×Q₈

—

586

—

#107

—

586

—

C₄⋊C₈

—

293

—

#103

—

586

—

C₄×C₈

C₂×D₂₉₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×Q₈

C₂²⋊D₂₉₃

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

586

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄²

—

586

—

C₄²

—

586

—

C₄²

C₂²×D₂₉₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

293

—

Q₈

—

2344

—

Q₈

—

586

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

586

—

C₂×C₄

—

586

—

C₂×C₄

D₄×D₂₉₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

C₈

—

100

C₈

—

1172

—

101

C₂²

#22627592

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

586

—

106

C₄

—

1172

—

107

C₄

Q₁₆×D₂₉₃

yes

—

108

C₄

#22627656

yes

—

109

C₂

#34036283

yes

—

110

C₂

#34036296

yes

characteristic

	subgroup	G/H
center	C₂²	#22627592
commutator, cosocle	C₄²×C₂₉₃	C₂³
Frattini	C₄²	C₂²×D₂₉₃
Fitting	#494×C₂₉₃	C₂
socle, prime radical	C₂²×C₂₉₃	#120
hypercenter	#494	D₂₉₃
hypocenter	C₂₉₃	#1633
LCS	G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#22627592

commutator, cosocle

C₄²×C₂₉₃

C₂³

Frattini

C₄²

C₂²×D₂₉₃

Fitting

#494×C₂₉₃

C₂

C₂²×C₂₉₃

D₂₉₃

C₂₉₃

G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₅₈₆	Q₁₆	semidirect
55. C₂₉₃	#1633	semidirect

H ≅ G/N

type

34. Dic₅₈₆

Q₁₆

semidirect

55. C₂₉₃

#1633

semidirect

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

subgroup	index
#34036859	2
#34036421	2
#34035878	2
#34035861	2
#494×C₂₉₃	2
#1633	293

subgroup

index

#494×C₂₉₃

293

Subgroups — #49157923

Conjugacy classes of subgroups

Complements

Maximal subgroups