Group #49170589 — subgroups

Subgroups — #49170589

Order 37504.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#34036942	C₂	1	4	yes	maximal, complemented
2	#34036401	C₂	1	4	yes	maximal, complemented
3	#34036311	C₂	1	4	yes	maximal, complemented
4	#34036247	C₂	1	2	yes	maximal, complemented
5	#494×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#22627672	C₂²	1	4	yes	—
7	#22627656	—	2	8	no	—
8	#22627655	C₂²	1	8	yes	—
9	#22627647	—	2	4	no	—
10	#22627646	C₂²	1	2	yes	—
11	Q₁₆×D₂₉₃	—	2	4	no	—
12	#22627630	C₂²	1	4	yes	—
13	#22627560	C₂²	1	2	yes	—
14	#22627552	C₂²	1	4	yes	—
15	#22627543	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₂₉₃	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₂₉₃	C₂²	1	2	yes	—
18	C₄×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
19	#17989803	—	2	8	no	—
20	#17989802	—	4	4	no	—
21	#17989802	C₂³	1	4	yes	—
22	Q₈×D₂₉₃	—	2	8	no	—
23	Dic₁₁₇₂	D₄	1	2	yes	—
24	#17989792	—	2	8	no	—
25	#17989791	D₄	1	4	yes	—
26	#17989789	—	2	8	no	—
27	D₂₃₄₄	D₄	1	2	yes	—
28	#17989786	C₂³	1	4	yes	—
29	#17989785	—	2	4	no	—
30	C₈×D₂₉₃	—	2	4	no	—
31	C₂×Q₈×C₂₉₃	C₂³	1	4	yes	—
32	Q₁₆×C₂₉₃	—	2	4	no	—
33	Q₁₆×C₂₉₃	—	2	4	no	—
34	C₂×C₈×C₂₉₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₂₉₃	—	4	2	no	—
36	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
37	Dic₅₈₆	—	4	4	no	—
38	Dic₅₈₆	C₂×D₄	1	4	yes	—
39	D₁₁₇₂	—	4	4	no	—
40	D₁₁₇₂	C₂×D₄	1	4	yes	—
41	C₂²⋊D₂₉₃	—	2	4	no	—
42	C₄×D₂₉₃	—	2	4	no	—
43	C₄×D₂₉₃	—	4	8	no	—
44	#14556873	—	2	4	no	—
45	Q₈×C₂₉₃	—	2	8	no	—
46	C₂×C₄×C₂₉₃	—	2	1	no	—
47	C₂×C₄×C₂₉₃	—	2	4	no	—
48	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
49	C₈×C₂₉₃	C₂×D₄	1	2	yes	—
50	C₈×C₂₉₃	C₂×D₄	1	2	yes	—
51	C₂²×C₂₉₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₂₉₃	—	2	4	no	—
53	C₄×C₂₉₃	—	2	2	no	—
54	C₄×C₂₉₃	—	4	4	no	—
55	C₄×C₂₉₃	C₂²×D₄	1	2	yes	—
56	Dic₂₉₃	—	2	4	no	—
57	C₂×C₂₉₃	—	2	1	no	—
58	C₂×C₂₉₃	D₄²	1	1	yes	characteristic
59	D₂₉₃	—	4	4	no	—
60	C₂₉₃	#3293	1	1	yes	characteristic, complemented, Sylow
61	#3293	—	293	1	no	maximal, Sylow
62	#578	—	293	4	no	—
63	#494	D₂₉₃	1	1	yes	characteristic, complemented
64	#473	—	293	4	no	—
65	#456	—	293	4	no	—
66	#443	—	293	2	no	—
67	2₋¹⁺⁴	—	293	4	no	—
68	#137	—	293	8	no	—
69	#137	—	586	8	no	—
70	#135	—	293	2	no	—
71	#135	—	586	4	no	—
72	C₂×Q₁₆	—	586	4	no	—
73	C₂×Q₁₆	C₂×D₂₉₃	1	4	yes	—
74	#131	—	293	4	no	—
75	C₄⋊Q₈	C₂×D₂₉₃	1	2	yes	—
76	#108	—	293	2	no	—
77	#104	—	293	4	no	—
78	#101	—	293	4	no	—
79	C₄×C₈	C₂×D₂₉₃	1	1	yes	characteristic
80	#41	—	293	4	no	—
81	#41	—	586	8	no	—
82	#41	—	1172	4	no	—
83	C₂×Q₈	—	586	8	no	—
84	C₂×Q₈	C₂²×D₂₉₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	293	2	no	—
88	Q₁₆	—	586	8	no	—
89	SD₁₆	—	293	4	no	—
90	SD₁₆	—	586	8	no	—
91	D₈	—	293	2	no	—
92	M₁₆	—	293	4	no	—
93	M₁₆	—	586	4	no	—
94	C₂×C₈	—	586	4	no	—
95	C₂×C₈	C₂²×D₂₉₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	293	4	no	—
100	Q₈	—	1172	4	no	—
101	D₄	—	293	4	no	—
102	D₄	—	586	4	no	—
103	D₄	—	1172	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	586	4	no	—
107	C₂×C₄	—	1172	8	no	—
108	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
109	C₈	—	586	4	no	—
110	C₈	D₄×D₂₉₃	1	2	yes	—
111	C₈	D₄×D₂₉₃	1	2	yes	—
112	C₂²	—	586	4	no	—
113	C₂²	#22627669	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	586	4	no	—
117	C₄	C₂×D₄×D₂₉₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	1172	4	no	—
120	C₂	#34036764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036942

C₂

yes

maximal, complemented

#34036401

C₂

yes

maximal, complemented

#34036311

C₂

yes

maximal, complemented

#34036247

C₂

yes

maximal, complemented

#494×C₂₉₃

C₂

yes

maximal, characteristic, complemented

#22627672

C₂²

yes

—

#22627656

—

#22627655

C₂²

yes

—

#22627647

—

#22627646

C₂²

yes

—

Q₁₆×D₂₉₃

—

#22627630

C₂²

yes

—

#22627560

C₂²

yes

—

#22627552

C₂²

yes

—

#22627543

C₂²

yes

—

C₂×Q₁₆×C₂₉₃

C₂²

yes

—

C₄⋊Q₈×C₂₉₃

C₂²

yes

—

C₄×C₈×C₂₉₃

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₂₉₃

—

Dic₁₁₇₂

D₄

yes

—

#17989792

—

#17989791

D₄

yes

—

#17989789

—

D₂₃₄₄

D₄

yes

—

#17989786

C₂³

yes

—

#17989785

—

C₈×D₂₉₃

—

C₂×Q₈×C₂₉₃

C₂³

yes

—

Q₁₆×C₂₉₃

—

Q₁₆×C₂₉₃

—

C₂×C₈×C₂₉₃

C₂³

yes

—

C₄⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

C₂³

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

C₂×D₄

yes

—

D₁₁₇₂

—

D₁₁₇₂

C₂×D₄

yes

—

C₂²⋊D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

—

#14556873

—

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂⁴

yes

characteristic

C₈×C₂₉₃

C₂×D₄

yes

—

C₈×C₂₉₃

C₂×D₄

yes

—

C₂²×C₂₉₃

H(5,𝔽₂)

yes

characteristic

C₂×D₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂²×D₄

yes

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

D₄²

yes

characteristic

D₂₉₃

—

C₂₉₃

#3293

yes

characteristic, complemented, Sylow

#3293

—

293

maximal, Sylow

#578

—

293

—

#494

D₂₉₃

yes

characteristic, complemented

#473

—

293

—

#456

—

293

—

#443

—

293

—

2₋¹⁺⁴

—

293

—

#137

—

293

—

#137

—

586

—

#135

—

293

—

#135

—

586

—

C₂×Q₁₆

—

586

—

C₂×Q₁₆

C₂×D₂₉₃

yes

—

#131

—

293

—

C₄⋊Q₈

C₂×D₂₉₃

yes

—

#108

—

293

—

#104

—

293

—

#101

—

293

—

C₄×C₈

C₂×D₂₉₃

yes

characteristic

#41

—

293

—

#41

—

586

—

#41

—

1172

—

C₂×Q₈

—

586

—

C₂×Q₈

C₂²×D₂₉₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

293

—

Q₁₆

—

586

—

SD₁₆

—

293

—

SD₁₆

—

586

—

D₈

—

293

—

M₁₆

—

293

—

M₁₆

—

586

—

C₂×C₈

—

586

—

C₂×C₈

C₂²×D₂₉₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₂₉₃

yes

characteristic

Q₈

—

Q₈

—

293

—

100

Q₈

—

1172

—

101

D₄

—

293

—

102

D₄

—

586

—

103

D₄

—

1172

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

586

—

107

C₂×C₄

—

1172

—

108

C₂×C₄

C₂³×D₂₉₃

yes

characteristic

109

C₈

—

586

—

110

C₈

D₄×D₂₉₃

yes

—

111

C₈

D₄×D₂₉₃

yes

—

112

C₂²

—

586

—

113

C₂²

#22627669

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

586

—

117

C₄

C₂×D₄×D₂₉₃

yes

—

118

C₂

—

119

C₂

—

1172

—

120

C₂

#34036764

yes

characteristic

	subgroup	G/H
center	C₂	#34036764
commutator, cosocle	C₂×C₄×C₂₉₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉₃
Fitting	#494×C₂₉₃	C₂
socle	C₂×C₂₉₃	D₄²
hypercenter	#494	D₂₉₃
hypocenter	C₂₉₃	#3293
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#34036764

commutator, cosocle

C₂×C₄×C₂₉₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉₃

Fitting

#494×C₂₉₃

C₂

socle

C₂×C₂₉₃

D₄²

D₂₉₃

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #34036942	C₂	semidirect
2. #34036401	C₂	semidirect
3. #34036311	C₂	semidirect
4. #34036247	C₂	semidirect
5. #494×C₂₉₃	C₂	semidirect
60. C₂₉₃	#3293	semidirect
63. #494	D₂₉₃	semidirect

H ≅ G/N

type

1. #34036942

C₂

semidirect

2. #34036401

C₂

semidirect

3. #34036311

C₂

semidirect

4. #34036247

C₂

semidirect

5. #494×C₂₉₃

C₂

semidirect

60. C₂₉₃

#3293

semidirect

63. #494

D₂₉₃

semidirect

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

subgroup	index
#34036942	2
#34036401	2
#34036311	2
#34036247	2
#494×C₂₉₃	2
#3293	293

subgroup

index

#494×C₂₉₃

293

Subgroups — #49170589

Conjugacy classes of subgroups

Complements

Maximal subgroups