Group #49159565 — subgroups

Subgroups — #49159565

Order 37504.

196 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#22627560	C₂	1	1	yes	maximal, characteristic
2	#34036172	C₂	1	2	yes	maximal
3	C₂×#22627556	C₂	1	1	yes	maximal, characteristic
4	#34035897	C₂	1	2	yes	maximal
5	#405×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#17989786	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
8	#22627560	C₄	1	4	yes	complemented
9	#22627557	—	2	2	no	—
10	#22627556	—	2	1	no	—
11	#22627556	—	2	1	no	—
12	#22627555	—	2	2	no	—
13	C₂×M₁₆×C₂₉₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
15	#117×C₂₉₃	C₂²	1	1	yes	characteristic
16	#97×C₂₉₃	—	2	1	no	—
17	C₄×C₈×C₂₉₃	—	2	1	no	—
18	C₂²×Dic₂₉₃	—	2	2	no	—
19	#17989786	—	2	2	no	—
20	#17989786	C₂×C₄	1	4	yes	—
21	C₂×#14556873	—	2	2	no	—
22	C₄⋊Dic₂₉₃	—	2	2	no	—
23	C₄⋊Dic₂₉₃	—	2	2	no	—
24	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
25	M₁₆×C₂₉₃	—	2	1	no	—
26	M₁₆×C₂₉₃	Q₈	1	2	yes	—
27	C₂×C₈×C₂₉₃	—	2	1	no	—
28	C₂×C₈×C₂₉₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₂₉₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₂₉₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₂₉₃	Q₈	1	2	yes	—
32	C₄⋊C₄×C₂₉₃	D₄	1	2	yes	—
33	C₂²⋊C₄×C₂₉₃	—	2	1	no	—
34	C₄²×C₂₉₃	—	2	1	no	—
35	C₂×Dic₂₉₃	—	4	2	no	—
36	C₂×Dic₂₉₃	—	8	2	no	—
37	#14556873	—	2	4	no	—
38	C₂³×C₂₉₃	#41	1	1	yes	characteristic
39	C₂×C₄×C₂₉₃	—	2	2	no	—
40	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₉₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₉₃	#41	1	1	yes	characteristic
46	C₈×C₂₉₃	—	2	2	no	—
47	C₈×C₂₉₃	C₄⋊C₄	1	2	yes	—
48	C₈×C₂₉₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₂₉₃	—	2	2	no	—
50	C₂²×C₂₉₃	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₂₉₃	#121	1	1	yes	characteristic
52	C₂²×C₂₉₃	#125	1	1	yes	characteristic
53	C₄×C₂₉₃	—	4	2	no	—
54	C₄×C₂₉₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₂₉₃	C₄×D₄	1	1	yes	characteristic
56	C₄×C₂₉₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₂₉₃	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₂₉₃	—	8	2	no	—
59	C₂×C₂₉₃	—	2	1	no	—
60	C₂×C₂₉₃	—	2	1	no	—
61	C₂×C₂₉₃	#389	1	1	yes	characteristic
62	C₂×C₂₉₃	#471	1	2	yes	—
63	C₂₉₃	#1951	1	1	yes	characteristic, complemented, Sylow
64	#1951	—	293	1	no	maximal, Sylow
65	C₂×#108	—	293	1	no	—
66	#428	—	293	2	no	—
67	C₂×#106	—	293	1	no	—
68	#405	D₂₉₃	1	1	yes	characteristic
69	#343	—	293	2	no	—
70	C₂×M₁₆	—	293	2	no	—
71	C₂×M₁₆	C₂×D₂₉₃	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₂₉₃	1	1	yes	characteristic
73	#117	C₂×D₂₉₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	293	2	no	—
75	#108	—	293	4	no	—
76	#107	—	586	2	no	—
77	#106	—	586	1	no	—
78	#106	—	586	1	no	—
79	#106	—	586	2	no	—
80	#97	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₂²×C₄	—	586	2	no	—
83	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
84	M₁₆	—	2	1	no	—
85	M₁₆	—	293	4	no	—
86	M₁₆	—	586	2	no	—
87	M₁₆	Dic₅₈₆	1	2	yes	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	586	2	no	—
90	C₂×C₈	C₄×D₂₉₃	1	2	yes	—
91	C₂×C₈	C₂²⋊D₂₉₃	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₂₉₃	1	1	yes	characteristic
93	C₂×C₈	Dic₅₈₆	1	2	yes	—
94	C₄⋊C₄	—	586	2	no	—
95	C₄⋊C₄	—	586	2	no	—
96	C₄⋊C₄	D₁₁₇₂	1	2	yes	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₂³	#17989802	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2344	2	no	—
102	C₂×C₄	—	1172	2	no	—
103	C₂×C₄	C₂×C₄×D₂₉₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
105	C₂×C₄	C₂×D₁₁₇₂	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₅₈₆	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₅₈₆	1	1	yes	characteristic
108	C₂×C₄	#17989802	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	586	4	no	—
111	C₈	#17989781	1	2	yes	—
112	C₈	#17989781	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	C₄×Dic₅₈₆	1	1	yes	characteristic
115	C₂²	#22627598	1	1	yes	characteristic
116	C₂²	#22627616	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	2344	2	no	—
119	C₄	C₂×#17989781	1	1	yes	characteristic
120	C₄	C₄×D₁₁₇₂	1	1	yes	characteristic
121	C₄	#22627603	1	1	yes	characteristic
122	C₄	#22627624	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#34036027	1	1	yes	characteristic
126	C₂	#34036392	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627560

C₂

yes

maximal, characteristic

#34036172

C₂

yes

maximal

C₂×#22627556

C₂

yes

maximal, characteristic

#34035897

C₂

yes

maximal

#405×C₂₉₃

C₂

yes

maximal, characteristic

C₂×#17989786

C₂²

yes

—

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

#22627560

C₄

yes

—

—

—

—

C₂×M₁₆×C₂₉₃

C₂²

yes

characteristic

C₂²×C₈×C₂₉₃

C₂²

yes

characteristic

#117×C₂₉₃

C₂²

yes

characteristic

#97×C₂₉₃

—

C₄×C₈×C₂₉₃

—

C₂²×Dic₂₉₃

—

#17989786

—

#17989786

C₂×C₄

yes

—

C₂×#14556873

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

M₁₆×C₂₉₃

—

M₁₆×C₂₉₃

Q₈

yes

—

C₂×C₈×C₂₉₃

—

C₂×C₈×C₂₉₃

C₂×C₄

yes

—

C₂×C₈×C₂₉₃

D₄

yes

characteristic

C₂×C₈×C₂₉₃

D₄

yes

characteristic

C₂×C₈×C₂₉₃

Q₈

yes

—

C₄⋊C₄×C₂₉₃

D₄

yes

—

C₂²⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

C₂³×C₂₉₃

#41

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₉₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₉₃

#41

yes

characteristic

C₈×C₂₉₃

—

C₈×C₂₉₃

C₄⋊C₄

yes

—

C₈×C₂₉₃

C₄⋊C₄

yes

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₄×Q₈

yes

C₂²×C₂₉₃

yes

C₂²×C₂₉₃

yes

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₉₃

C₄×D₄

yes

characteristic

C₄×C₂₉₃

C₂²⋊Q₈

yes

characteristic

C₄×C₂₉₃

C₄⋊Q₈

yes

characteristic

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

#389

yes

characteristic

C₂×C₂₉₃

#471

yes

—

C₂₉₃

#1951

yes

characteristic, complemented, Sylow

#1951

—

293

maximal, Sylow

C₂×#108

—

293

—

#428

—

293

—

C₂×#106

—

293

—

#405

D₂₉₃

yes

characteristic

#343

—

293

—

C₂×M₁₆

—

293

—

C₂×M₁₆

C₂×D₂₉₃

yes

characteristic

C₂²×C₈

C₂×D₂₉₃

yes

characteristic

#117

C₂×D₂₉₃

yes

characteristic

C₂×C₄⋊C₄

—

293

—

#108

—

293

—

#107

—

586

—

#106

—

586

—

#106

—

586

—

#106

—

586

—

#97

—

C₄×C₈

—

C₂²×C₄

—

586

—

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

M₁₆

—

M₁₆

—

293

—

M₁₆

—

586

—

M₁₆

Dic₅₈₆

yes

—

C₂×C₈

—

C₂×C₈

—

586

—

C₂×C₈

C₄×D₂₉₃

yes

—

C₂×C₈

C₂²⋊D₂₉₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₉₃

yes

characteristic

C₂×C₈

Dic₅₈₆

yes

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄⋊C₄

D₁₁₇₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#17989802

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

2344

—

102

C₂×C₄

—

1172

—

103

C₂×C₄

C₂×C₄×D₂₉₃

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

105

C₂×C₄

C₂×D₁₁₇₂

yes

characteristic

106

C₂×C₄

C₂×Dic₅₈₆

yes

characteristic

107

C₂×C₄

C₂×Dic₅₈₆

yes

characteristic

108

C₂×C₄

#17989802

yes

characteristic

109

C₈

—

110

C₈

—

586

—

111

C₈

#17989781

yes

—

112

C₈

#17989781

yes

—

113

C₂²

—

114

C₂²

C₄×Dic₅₈₆

yes

characteristic

115

C₂²

#22627598

yes

characteristic

116

C₂²

#22627616

yes

characteristic

117

C₄

—

118

C₄

—

2344

—

119

C₄

C₂×#17989781

yes

characteristic

120

C₄

C₄×D₁₁₇₂

yes

characteristic

121

C₄

#22627603

yes

characteristic

122

C₄

#22627624

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#34036027

yes

characteristic

126

C₂

#34036392

yes

—

	subgroup	G/H
center	C₂²	#22627598
commutator	C₂×C₄×C₂₉₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉₃
Fitting	#405×C₂₉₃	C₂
socle	C₂²×C₂₉₃	#121
prime radical	C₂³×C₂₉₃	#41
cosocle	C₂²×C₄×C₂₉₃	C₂³
hypercenter	#405	D₂₉₃
hypocenter	C₂₉₃	#1951
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#22627598

commutator

C₂×C₄×C₂₉₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉₃

Fitting

#405×C₂₉₃

C₂

C₂²×C₂₉₃

C₂³×C₂₉₃

C₂²×C₄×C₂₉₃

C₂³

D₂₉₃

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #22627560	C₄	semidirect
63. C₂₉₃	#1951	semidirect

H ≅ G/N

type

8. #22627560

C₄

semidirect

63. C₂₉₃

#1951

semidirect

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

subgroup	index
C₂×#22627560	2
#34036172	2
C₂×#22627556	2
#34035897	2
#405×C₂₉₃	2
#1951	293

subgroup

index

#405×C₂₉₃

293

Subgroups — #49159565

Conjugacy classes of subgroups

Complements

Maximal subgroups