Group #49157861 — subgroups

Subgroups — #49157861

Order 37504.

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

normal subgroups only

This group has 228 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, complemented
2	#34036326	C₂	1	2	yes	maximal
3	#34035878	C₂	1	1	yes	maximal, characteristic, complemented
4	#34035861	C₂	1	2	yes	maximal, complemented
5	#492×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#17989790	—	2	2	no	—
7	#22627623	—	2	2	no	—
8	#22627622	—	2	2	no	—
9	C₄×Dic₅₈₆	C₂²	1	2	yes	—
10	#22627589	—	2	2	no	—
11	C₄⋊#14556873	C₂²	1	2	yes	—
12	#22627545	—	2	2	no	—
13	C₂×SD₁₆×C₂₉₃	—	2	2	no	—
14	C₄⋊Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
15	#127×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	#17989790	—	4	4	no	—
20	C₂×#14556873	—	2	2	no	—
21	C₄⋊Dic₂₉₃	D₄	1	2	yes	—
22	#17989781	—	2	4	no	—
23	#17989781	—	2	2	no	—
24	C₄×Dic₂₉₃	—	2	2	no	—
25	C₄×Dic₂₉₃	—	2	2	no	—
26	C₂×Q₈×C₂₉₃	D₄	1	2	yes	—
27	C₂×D₄×C₂₉₃	—	2	2	no	—
28	C₂×D₄×C₂₉₃	—	2	1	no	—
29	SD₁₆×C₂₉₃	—	4	4	no	—
30	C₂×C₈×C₂₉₃	—	2	1	no	—
31	C₄⋊C₄×C₂₉₃	—	2	2	no	—
32	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
33	Dic₅₈₆	—	2	2	no	—
34	Dic₅₈₆	—	8	2	no	—
35	Dic₅₈₆	SD₁₆	1	4	yes	complemented
36	C₂×Dic₂₉₃	—	2	2	no	—
37	C₂×Dic₂₉₃	—	2	2	no	—
38	#14556873	—	4	2	no	—
39	C₂³×C₂₉₃	—	4	1	no	—
40	Q₈×C₂₉₃	—	4	2	no	—
41	D₄×C₂₉₃	—	4	2	no	—
42	D₄×C₂₉₃	—	4	2	no	—
43	D₄×C₂₉₃	—	4	2	no	—
44	C₂×C₄×C₂₉₃	—	2	2	no	—
45	C₂×C₄×C₂₉₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₉₃	—	2	2	no	—
48	C₂²×C₂₉₃	—	8	2	no	—
49	C₂²×C₂₉₃	—	8	1	no	—
50	C₂²×C₂₉₃	#120	1	1	yes	characteristic
51	C₄×C₂₉₃	—	2	1	no	—
52	C₄×C₂₉₃	—	4	2	no	—
53	C₄×C₂₉₃	C₂×SD₁₆	1	2	yes	—
54	C₄×C₂₉₃	#137	1	2	yes	—
55	Dic₂₉₃	—	2	4	no	—
56	Dic₂₉₃	—	4	2	no	—
57	C₂×C₂₉₃	—	16	1	no	—
58	C₂×C₂₉₃	#448	1	2	yes	—
59	C₂×C₂₉₃	#453	1	1	yes	characteristic
60	C₂₉₃	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	293	1	no	maximal, Sylow
62	Q₈²	—	293	1	no	—
63	#492	D₂₉₃	1	1	yes	characteristic
64	#460	—	293	2	no	—
65	#333	—	293	1	no	—
66	Q₈⋊C₈	—	293	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	586	2	no	—
69	C₄⋊Q₈	—	586	2	no	—
70	C₄⋊Q₈	—	586	2	no	—
71	C₄⋊Q₈	C₂×D₂₉₃	1	1	yes	characteristic
72	#127	C₂×D₂₉₃	1	1	yes	characteristic
73	C₄×Q₈	—	293	2	no	—
74	C₄×Q₈	—	586	2	no	—
75	C₄⋊C₈	—	293	2	no	—
76	#102	—	586	2	no	—
77	C₄×C₈	C₂×D₂₉₃	1	1	yes	characteristic
78	C₂×Q₈	—	293	2	no	—
79	C₂×Q₈	—	586	2	no	—
80	C₂×Q₈	C₂²⋊D₂₉₃	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	1172	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	586	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	293	2	no	—
89	C₄⋊C₄	—	586	4	no	—
90	C₄⋊C₄	—	586	2	no	—
91	C₄²	—	586	2	no	—
92	C₄²	—	586	2	no	—
93	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	293	4	no	—
97	Q₈	—	2344	2	no	—
98	Q₈	—	586	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	586	2	no	—
104	C₂×C₄	—	586	2	no	—
105	C₂×C₄	D₄×D₂₉₃	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	1172	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#22627592	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	586	4	no	—
115	C₄	—	1172	2	no	—
116	C₄	SD₁₆×D₂₉₃	1	2	yes	—
117	C₄	#22627653	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#34036266	1	2	yes	—
120	C₂	#34036296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036859

C₂

yes

maximal, characteristic, complemented

#34036326

C₂

yes

maximal

#34035878

C₂

yes

maximal, characteristic, complemented

#34035861

C₂

yes

maximal, complemented

#492×C₂₉₃

C₂

yes

maximal, characteristic

C₂×#17989790

—

#22627623

—

#22627622

—

C₄×Dic₅₈₆

C₂²

yes

—

#22627589

—

C₄⋊#14556873

C₂²

yes

—

#22627545

—

C₂×SD₁₆×C₂₉₃

—

C₄⋊Q₈×C₂₉₃

C₂²

yes

characteristic

#127×C₂₉₃

C₂²

yes

characteristic

C₄×C₈×C₂₉₃

C₂²

yes

characteristic

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

D₄

yes

—

#17989790

—

C₂×#14556873

—

C₄⋊Dic₂₉₃

D₄

yes

—

#17989781

—

#17989781

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

—

C₂×Q₈×C₂₉₃

D₄

yes

—

C₂×D₄×C₂₉₃

—

C₂×D₄×C₂₉₃

—

SD₁₆×C₂₉₃

—

C₂×C₈×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

C₂³

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

Dic₅₈₆

SD₁₆

yes

complemented

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

C₂³×C₂₉₃

—

Q₈×C₂₉₃

—

D₄×C₂₉₃

—

D₄×C₂₉₃

—

D₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

#120

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×SD₁₆

yes

—

C₄×C₂₉₃

#137

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

#448

yes

—

C₂×C₂₉₃

#453

yes

characteristic

C₂₉₃

#1623

yes

characteristic, complemented, Sylow

#1623

—

293

maximal, Sylow

Q₈²

—

293

—

#492

D₂₉₃

yes

characteristic

#460

—

293

—

#333

—

293

—

Q₈⋊C₈

—

293

—

C₂×SD₁₆

—

C₂×SD₁₆

—

586

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

C₂×D₂₉₃

yes

characteristic

#127

C₂×D₂₉₃

yes

characteristic

C₄×Q₈

—

293

—

C₄×Q₈

—

586

—

C₄⋊C₈

—

293

—

#102

—

586

—

C₄×C₈

C₂×D₂₉₃

yes

characteristic

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×Q₈

C₂²⋊D₂₉₃

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

1172

—

C₂×C₈

—

C₂×C₈

—

586

—

C₄⋊C₄

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄²

—

586

—

C₄²

—

586

—

C₄²

C₂²×D₂₉₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

293

—

Q₈

—

2344

—

Q₈

—

586

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

586

—

104

C₂×C₄

—

586

—

105

C₂×C₄

D₄×D₂₉₃

yes

—

106

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

107

C₈

—

108

C₈

—

1172

—

109

C₂²

—

110

C₂²

—

111

C₂²

#22627592

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

586

—

115

C₄

—

1172

—

116

C₄

SD₁₆×D₂₉₃

yes

—

117

C₄

#22627653

yes

—

118

C₂

—

119

C₂

#34036266

yes

—

120

C₂

#34036296

yes

characteristic

	subgroup	G/H
center	C₂²	#22627592
commutator, cosocle	C₄²×C₂₉₃	C₂³
Frattini	C₄²	C₂²×D₂₉₃
Fitting	#492×C₂₉₃	C₂
socle	C₂²×C₂₉₃	#120
prime radical	#127×C₂₉₃	C₂²
hypercenter	#492	D₂₉₃
hypocenter	C₂₉₃	#1623
LCS	G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22627592

commutator, cosocle

C₄²×C₂₉₃

C₂³

Frattini

C₄²

C₂²×D₂₉₃

Fitting

#492×C₂₉₃

C₂

socle

C₂²×C₂₉₃

#120

prime radical

#127×C₂₉₃

C₂²

D₂₉₃

C₂₉₃

G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #34036859	C₂	semidirect
3. #34035878	C₂	semidirect
4. #34035861	C₂	semidirect
35. Dic₅₈₆	SD₁₆	semidirect
60. C₂₉₃	#1623	semidirect

H ≅ G/N

type

1. #34036859

C₂

semidirect

3. #34035878

C₂

semidirect

4. #34035861

C₂

semidirect

35. Dic₅₈₆

SD₁₆

semidirect

60. C₂₉₃

#1623

semidirect

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

subgroup	index
#34036859	2
#34036326	2
#34035878	2
#34035861	2
#492×C₂₉₃	2
#1623	293

subgroup

index

#492×C₂₉₃

293

Subgroups — #49157861

Conjugacy classes of subgroups

Complements

Maximal subgroups