Group #49175015 — subgroups

Subgroups — #49175015

Order 37504.

228 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 90 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×#22627698	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#17989805	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#17989805	C₂	1	4	yes	maximal, complemented
4	C₂×#22627631	C₂	1	1	yes	maximal, characteristic
5	#22627698	—	2	2	no	—
6	C₂×#17989805	—	2	2	no	—
7	C₂×#17989805	C₂²	1	2	yes	—
8	C₂×#17989805	C₂²	1	2	yes	—
9	#22627631	—	2	2	no	—
10	#22627631	—	2	2	no	—
11	C₂×#17989786	C₄	1	2	yes	—
12	C₂×#17989786	C₄	1	1	yes	characteristic
13	C₂²×#14556873	C₂²	1	2	yes	—
14	C₂²×#14556873	C₂²	1	1	yes	characteristic
15	C₂×#41×C₂₉₃	C₄	1	1	yes	characteristic
16	#17989805	—	2	2	no	—
17	#17989805	—	2	2	no	—
18	#17989786	—	2	4	no	—
19	#17989786	—	2	2	no	—
20	C₂×#14556873	—	2	2	no	—
21	C₂×#14556873	—	2	2	no	—
22	C₂×#14556873	C₂×C₄	1	2	yes	—
23	C₂×#14556873	C₂×C₄	1	1	yes	characteristic
24	C₂×#14556873	D₄	1	2	yes	—
25	C₂×#14556873	D₄	1	2	yes	—
26	C₂×#14556873	C₂³	1	1	yes	characteristic
27	#41×C₂₉₃	—	2	2	no	—
28	#41×C₂₉₃	—	2	2	no	—
29	C₂×Q₈×C₂₉₃	C₈	1	1	yes	characteristic
30	C₂×D₄×C₂₉₃	C₈	1	2	yes	—
31	C₂×D₄×C₂₉₃	C₈	1	1	yes	characteristic
32	C₂²×C₄×C₂₉₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₂₉₃	C₂×C₄	1	1	yes	characteristic
34	#14556873	—	2	2	no	—
35	#14556873	C₂²⋊C₄	1	2	yes	—
36	#14556873	C₂×D₄	1	1	yes	characteristic
37	#14556873	C₂×D₄	1	1	yes	characteristic
38	C₂³×C₂₉₃	C₂×C₈	1	2	yes	—
39	C₂³×C₂₉₃	C₂×C₈	1	1	yes	characteristic
40	Q₈×C₂₉₃	—	2	2	no	—
41	D₄×C₂₉₃	—	2	4	no	—
42	D₄×C₂₉₃	—	2	2	no	—
43	C₂×C₄×C₂₉₃	—	2	2	no	—
44	C₂×C₄×C₂₉₃	—	2	2	no	—
45	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₉₃	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₂₉₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×C₂₉₃	M₁₆	1	2	yes	—
49	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₂₉₃	—	2	2	no	—
51	C₂²×C₂₉₃	—	2	2	no	—
52	C₂²×C₂₉₃	—	2	4	no	—
53	C₂²×C₂₉₃	C₂²⋊C₈	1	2	yes	—
54	C₂²×C₂₉₃	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₂₉₃	—	2	2	no	—
56	C₄×C₂₉₃	C₂²⋊C₈	1	2	yes	—
57	C₄×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₂₉₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₂₉₃	—	2	2	no	—
60	C₂×C₂₉₃	—	4	2	no	—
61	C₂×C₂₉₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₂₉₃	#504	1	2	yes	—
63	C₂₉₃	#2113	1	1	yes	characteristic, complemented, Sylow
64	#2113	—	293	1	no	maximal, Sylow
65	C₂×#131	—	293	1	no	—
66	C₂×M₃₂	—	293	1	no	—
67	C₂²×C₁₆	—	293	1	no	—
68	C₂²⋊C₁₆	—	293	4	no	—
69	C₂×#41	#11924078	1	1	yes	characteristic
70	#131	—	586	2	no	—
71	#131	—	586	2	no	—
72	C₂×M₁₆	—	293	2	no	—
73	C₂×M₁₆	—	293	1	no	—
74	C₂²×C₈	—	293	1	no	—
75	C₂²×C₈	—	293	2	no	—
76	M₃₂	—	586	2	no	—
77	C₂×C₁₆	—	293	2	no	—
78	C₂×C₁₆	—	293	2	no	—
79	C₂×C₁₆	—	586	2	no	—
80	#41	—	2	2	no	—
81	#41	—	2	2	no	—
82	C₂×Q₈	#14556880	1	1	yes	characteristic
83	C₂×D₄	#14556880	1	2	yes	—
84	C₂×D₄	#14556880	1	1	yes	characteristic
85	C₂²×C₄	C₂×#11924078	1	2	yes	—
86	C₂²×C₄	C₂×#11924078	1	1	yes	characteristic
87	M₁₆	—	586	4	no	—
88	M₁₆	—	586	2	no	—
89	C₂×C₈	—	293	2	no	—
90	C₂×C₈	—	293	2	no	—
91	C₂×C₈	—	293	1	no	—
92	C₂×C₈	—	293	1	no	—
93	C₂×C₈	—	293	2	no	—
94	C₂×C₈	—	586	2	no	—
95	C₂×C₈	—	586	2	no	—
96	C₁₆	—	586	2	no	—
97	C₁₆	—	586	2	no	—
98	C₂³	C₂×#14556880	1	2	yes	—
99	C₂³	C₂×#14556880	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	D₄	—	2	4	no	—
102	D₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂²⋊#11924078	1	2	yes	—
106	C₂×C₄	C₂×#14556880	1	2	yes	—
107	C₂×C₄	C₂×#14556880	1	1	yes	characteristic
108	C₂×C₄	#17989811	1	2	yes	—
109	C₂×C₄	C₂²×#11924078	1	1	yes	characteristic
110	C₈	—	293	1	no	—
111	C₈	—	293	1	no	—
112	C₈	—	293	2	no	—
113	C₈	—	586	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	2	4	no	—
117	C₂²	C₂²⋊#14556880	1	2	yes	—
118	C₂²	C₂²×#14556880	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	C₂²⋊#14556880	1	2	yes	—
121	C₄	C₂×C₂²⋊#11924078	1	1	yes	characteristic
122	C₄	C₂×#17989811	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	4	2	no	—
125	C₂	C₂×C₂²⋊#14556880	1	1	yes	characteristic
126	C₂	#34037187	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627698

C₂

yes

maximal, characteristic, complemented

C₂²×#17989805

C₂

yes

maximal, characteristic, complemented

C₂²⋊#17989805

C₂

yes

maximal, complemented

C₂×#22627631

C₂

yes

maximal, characteristic

#22627698

—

C₂×#17989805

—

C₂×#17989805

C₂²

yes

—

C₂×#17989805

C₂²

yes

—

#22627631

—

#22627631

—

C₂×#17989786

C₄

yes

—

C₂×#17989786

C₄

yes

characteristic

C₂²×#14556873

C₂²

yes

—

C₂²×#14556873

C₂²

yes

characteristic

C₂×#41×C₂₉₃

C₄

yes

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#14556873

C₂×C₄

yes

characteristic

C₂×#14556873

D₄

yes

—

C₂×#14556873

D₄

yes

—

C₂×#14556873

C₂³

yes

characteristic

#41×C₂₉₃

—

#41×C₂₉₃

—

C₂×Q₈×C₂₉₃

C₈

yes

characteristic

C₂×D₄×C₂₉₃

C₈

yes

—

C₂×D₄×C₂₉₃

C₈

yes

characteristic

C₂²×C₄×C₂₉₃

C₂×C₄

yes

—

C₂²×C₄×C₂₉₃

C₂×C₄

yes

characteristic

#14556873

—

#14556873

C₂²⋊C₄

yes

—

#14556873

C₂×D₄

yes

characteristic

#14556873

C₂×D₄

yes

characteristic

C₂³×C₂₉₃

C₂×C₈

yes

—

C₂³×C₂₉₃

C₂×C₈

yes

characteristic

Q₈×C₂₉₃

—

D₄×C₂₉₃

—

D₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉₃

C₂×C₈

yes

—

C₂×C₄×C₂₉₃

C₂×C₈

yes

characteristic

C₂×C₄×C₂₉₃

M₁₆

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₂²⋊C₈

yes

—

C₂²×C₂₉₃

C₂²×C₈

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂²⋊C₈

yes

—

C₄×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉₃

C₂×M₁₆

yes

characteristic

C₂×C₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₉₃

#504

yes

—

C₂₉₃

#2113

yes

characteristic, complemented, Sylow

#2113

—

293

maximal, Sylow

C₂×#131

—

293

—

C₂×M₃₂

—

293

—

C₂²×C₁₆

—

293

—

C₂²⋊C₁₆

—

293

—

C₂×#41

#11924078

yes

characteristic

#131

—

586

—

#131

—

586

—

C₂×M₁₆

—

293

—

C₂×M₁₆

—

293

—

C₂²×C₈

—

293

—

C₂²×C₈

—

293

—

M₃₂

—

586

—

C₂×C₁₆

—

293

—

C₂×C₁₆

—

293

—

C₂×C₁₆

—

586

—

#41

—

#41

—

C₂×Q₈

#14556880

yes

characteristic

C₂×D₄

#14556880

yes

—

C₂×D₄

#14556880

yes

characteristic

C₂²×C₄

C₂×#11924078

yes

—

C₂²×C₄

C₂×#11924078

yes

characteristic

M₁₆

—

586

—

M₁₆

—

586

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

586

—

C₂×C₈

—

586

—

C₁₆

—

586

—

C₁₆

—

586

—

C₂³

C₂×#14556880

yes

—

C₂³

C₂×#14556880

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊#11924078

yes

—

106

C₂×C₄

C₂×#14556880

yes

—

107

C₂×C₄

C₂×#14556880

yes

characteristic

108

C₂×C₄

#17989811

yes

—

109

C₂×C₄

C₂²×#11924078

yes

characteristic

110

C₈

—

293

—

111

C₈

—

293

—

112

C₈

—

293

—

113

C₈

—

586

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂²⋊#14556880

yes

—

118

C₂²

C₂²×#14556880

yes

characteristic

119

C₄

—

120

C₄

C₂²⋊#14556880

yes

—

121

C₄

C₂×C₂²⋊#11924078

yes

characteristic

122

C₄

C₂×#17989811

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂×C₂²⋊#14556880

yes

characteristic

126

C₂

#34037187

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#11924078
commutator, socle	C₂²×C₂₉₃	C₂²×C₈
Fitting, prime radical	C₂×#41×C₂₉₃	C₄
cosocle	C₂×#14556873	C₂³
hypercenter	C₂×#41	#11924078
hypocenter	C₂₉₃	#2113
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#11924078

commutator, socle

C₂²×C₂₉₃

C₂²×C₈

Fitting, prime radical

C₂×#41×C₂₉₃

C₄

C₂³

C₂×#41

C₂₉₃

G ▷ C₂²×C₂₉₃ ▷ C₂₉₃

C₁ ◁ C₂×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#22627698	C₂	semidirect
2. C₂²×#17989805	C₂	semidirect
3. C₂²⋊#17989805	C₂	semidirect
63. C₂₉₃	#2113	semidirect

H ≅ G/N

type

1. C₂×#22627698

C₂

semidirect

2. C₂²×#17989805

C₂

semidirect

3. C₂²⋊#17989805

C₂

semidirect

63. C₂₉₃

#2113

semidirect

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

subgroup	index
C₂×#22627698	2
C₂²×#17989805	2
C₂²⋊#17989805	2
C₂×#22627631	2
#2113	293

subgroup

index

293

Subgroups — #49175015

Conjugacy classes of subgroups

Complements

Maximal subgroups