Group #50317449 — subgroups

Subgroups — C₁₅₁×#12524 #50317449

Order 38656.

246 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 110 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	#2254×C₁₅₁	C₂	1	2	yes	maximal
2	#2252×C₁₅₁	C₂	1	2	yes	maximal
3	#1713×C₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#1710×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	#1311×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	#500×C₁₅₁	—	2	1	no	—
7	#500×C₁₅₁	C₂²	1	1	yes	characteristic
8	#500×C₁₅₁	C₂²	1	1	yes	characteristic
9	#500×C₁₅₁	C₂²	1	2	yes	—
10	#494×C₁₅₁	—	2	1	no	—
11	#487×C₁₅₁	—	2	1	no	—
12	#487×C₁₅₁	—	2	1	no	—
13	#367×C₁₅₁	—	2	4	no	—
14	#366×C₁₅₁	—	2	4	no	—
15	C₄×C₁₆×C₁₅₁	C₂²	1	2	yes	—
16	C₈²×C₁₅₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₅₁	—	4	2	no	—
18	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
19	C₄⋊Q₈×C₁₅₁	—	2	1	no	—
20	C₄⋊Q₈×C₁₅₁	—	2	2	no	—
21	C₂×C₁₆×C₁₅₁	D₄	1	4	yes	—
22	#107×C₁₅₁	—	2	2	no	—
23	#107×C₁₅₁	—	2	2	no	—
24	#107×C₁₅₁	—	2	2	no	—
25	#107×C₁₅₁	—	2	2	no	—
26	#107×C₁₅₁	—	4	2	no	—
27	#103×C₁₅₁	—	4	2	no	—
28	#103×C₁₅₁	—	4	2	no	—
29	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₅₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₁₅₁	—	4	1	no	—
33	C₂×Q₈×C₁₅₁	—	4	1	no	—
34	C₂×Q₈×C₁₅₁	—	4	2	no	—
35	Q₁₆×C₁₅₁	—	8	4	no	—
36	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
37	C₂×C₈×C₁₅₁	D₈	1	2	yes	—
38	C₂×C₈×C₁₅₁	C₂×D₄	1	2	yes	—
39	C₄⋊C₄×C₁₅₁	—	4	1	no	—
40	C₄⋊C₄×C₁₅₁	—	4	2	no	—
41	C₄⋊C₄×C₁₅₁	—	4	1	no	—
42	C₄⋊C₄×C₁₅₁	—	4	1	no	—
43	C₄⋊C₄×C₁₅₁	—	4	1	no	—
44	C₄⋊C₄×C₁₅₁	—	4	2	no	—
45	C₄²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₁₅₁	Q₁₆	1	8	yes	complemented
47	Q₈×C₁₅₁	—	8	2	no	—
48	Q₈×C₁₅₁	—	8	4	no	—
49	Q₈×C₁₅₁	—	8	2	no	—
50	C₂×C₄×C₁₅₁	—	8	1	no	—
51	C₂×C₄×C₁₅₁	—	8	2	no	—
52	C₂×C₄×C₁₅₁	—	8	1	no	—
53	C₂×C₄×C₁₅₁	#127	1	1	yes	characteristic
54	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₅₁	C₂×D₈	1	1	yes	characteristic
56	C₈×C₁₅₁	—	2	2	no	—
57	C₈×C₁₅₁	—	2	2	no	—
58	C₈×C₁₅₁	C₂×Q₁₆	1	4	yes	—
59	C₂²×C₁₅₁	#493	1	1	yes	characteristic
60	C₄×C₁₅₁	—	16	1	no	—
61	C₄×C₁₅₁	—	16	2	no	—
62	C₄×C₁₅₁	—	16	1	no	—
63	C₄×C₁₅₁	#494	1	2	yes	—
64	C₄×C₁₅₁	#509	1	2	yes	—
65	C₄×C₁₅₁	#510	1	2	yes	—
66	C₂×C₁₅₁	#1713	1	1	yes	characteristic
67	C₂×C₁₅₁	#2250	1	1	yes	characteristic
68	C₂×C₁₅₁	#2251	1	1	yes	characteristic
69	#12524	C₁₅₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₁₅₁	#12524	1	1	yes	characteristic, direct product factor, Sylow
71	#2254	C₂×C₁₅₁	1	2	yes	—
72	#2252	C₂×C₁₅₁	1	2	yes	—
73	#1713	C₂×C₁₅₁	1	1	yes	characteristic
74	#1710	C₂×C₁₅₁	1	1	yes	characteristic
75	#1311	C₂×C₁₅₁	1	1	yes	characteristic
76	#500	—	2	1	no	—
77	#500	C₂²×C₁₅₁	1	1	yes	characteristic
78	#500	C₂²×C₁₅₁	1	1	yes	characteristic
79	#500	C₂²×C₁₅₁	1	2	yes	—
80	#494	—	2	1	no	—
81	#487	—	2	1	no	—
82	#487	—	2	1	no	—
83	#367	—	2	4	no	—
84	#366	—	2	4	no	—
85	C₄×C₁₆	C₂²×C₁₅₁	1	2	yes	—
86	C₈²	C₂²×C₁₅₁	1	1	yes	characteristic
87	C₂×Q₁₆	—	4	2	no	—
88	C₄⋊Q₈	—	2	1	no	—
89	C₄⋊Q₈	—	2	1	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₂×C₁₆	D₄×C₁₅₁	1	4	yes	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#107	—	2	2	no	—
95	#107	—	2	2	no	—
96	#107	—	4	2	no	—
97	#103	—	4	2	no	—
98	#103	—	4	2	no	—
99	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
100	C₄×C₈	D₄×C₁₅₁	1	1	yes	characteristic
101	C₄×C₈	C₂³×C₁₅₁	1	1	yes	characteristic
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	4	2	no	—
105	Q₁₆	—	8	4	no	—
106	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
107	C₂×C₈	D₈×C₁₅₁	1	2	yes	—
108	C₂×C₈	C₂×D₄×C₁₅₁	1	2	yes	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄²	C₂×D₄×C₁₅₁	1	1	yes	characteristic
116	C₁₆	Q₁₆×C₁₅₁	1	8	yes	complemented
117	Q₈	—	8	4	no	—
118	Q₈	—	8	2	no	—
119	Q₈	—	8	2	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	8	2	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	#127×C₁₅₁	1	1	yes	characteristic
124	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
125	C₂×C₄	C₂×D₈×C₁₅₁	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	C₂×Q₁₆×C₁₅₁	1	4	yes	—
129	C₂²	#493×C₁₅₁	1	1	yes	characteristic
130	C₄	—	16	2	no	—
131	C₄	—	16	1	no	—
132	C₄	—	16	1	no	—
133	C₄	#494×C₁₅₁	1	2	yes	—
134	C₄	#509×C₁₅₁	1	2	yes	—
135	C₄	#510×C₁₅₁	1	2	yes	—
136	C₂	#1713×C₁₅₁	1	1	yes	characteristic
137	C₂	#2250×C₁₅₁	1	1	yes	characteristic
138	C₂	#2251×C₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2254×C₁₅₁

C₂

yes

maximal

#2252×C₁₅₁

C₂

yes

maximal

#1713×C₁₅₁

C₂

yes

maximal, characteristic

#1710×C₁₅₁

C₂

yes

maximal, characteristic

#1311×C₁₅₁

C₂

yes

maximal, characteristic

#500×C₁₅₁

—

#500×C₁₅₁

C₂²

yes

characteristic

#500×C₁₅₁

C₂²

yes

characteristic

#500×C₁₅₁

C₂²

yes

—

#494×C₁₅₁

—

#487×C₁₅₁

—

#487×C₁₅₁

—

#367×C₁₅₁

—

#366×C₁₅₁

—

C₄×C₁₆×C₁₅₁

C₂²

yes

—

C₈²×C₁₅₁

C₂²

yes

characteristic

C₂×Q₁₆×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₄⋊Q₈×C₁₅₁

—

C₂×C₁₆×C₁₅₁

D₄

yes

—

#107×C₁₅₁

—

#107×C₁₅₁

—

#107×C₁₅₁

—

#107×C₁₅₁

—

#107×C₁₅₁

—

#103×C₁₅₁

—

#103×C₁₅₁

—

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₄×C₈×C₁₅₁

D₄

yes

characteristic

C₄×C₈×C₁₅₁

C₂³

yes

characteristic

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

—

Q₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

D₈

yes

—

C₂×C₈×C₁₅₁

D₈

yes

—

C₂×C₈×C₁₅₁

C₂×D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×D₄

yes

characteristic

C₁₆×C₁₅₁

Q₁₆

yes

complemented

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

#127

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₈

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₈

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₈×C₁₅₁

C₂×Q₁₆

yes

—

C₂²×C₁₅₁

#493

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#494

yes

—

C₄×C₁₅₁

#509

yes

—

C₄×C₁₅₁

#510

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

maximal, characteristic, direct product factor, Sylow

C₁₅₁

#12524

yes

characteristic, direct product factor, Sylow

#2254

C₂×C₁₅₁

yes

—

#2252

C₂×C₁₅₁

yes

—

#1713

C₂×C₁₅₁

yes

characteristic

#1710

C₂×C₁₅₁

yes

characteristic

#1311

C₂×C₁₅₁

yes

characteristic

#500

—

#500

C₂²×C₁₅₁

yes

characteristic

#500

C₂²×C₁₅₁

yes

characteristic

#500

C₂²×C₁₅₁

yes

—

—

—

—

—

—

C₄×C₁₆

C₂²×C₁₅₁

yes

—

C₈²

C₂²×C₁₅₁

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₄×C₁₅₁

yes

—

—

—

—

—

—

—

—

C₄×C₈

D₄×C₁₅₁

yes

characteristic

100

C₄×C₈

D₄×C₁₅₁

yes

characteristic

101

C₄×C₈

C₂³×C₁₅₁

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

Q₁₆

—

106

C₂×C₈

D₈×C₁₅₁

yes

—

107

C₂×C₈

D₈×C₁₅₁

yes

—

108

C₂×C₈

C₂×D₄×C₁₅₁

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

C₂×D₄×C₁₅₁

yes

characteristic

116

C₁₆

Q₁₆×C₁₅₁

yes

complemented

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

#127×C₁₅₁

yes

characteristic

124

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

125

C₂×C₄

C₂×D₈×C₁₅₁

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

C₂×Q₁₆×C₁₅₁

yes

—

129

C₂²

#493×C₁₅₁

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#494×C₁₅₁

yes

—

134

C₄

#509×C₁₅₁

yes

—

135

C₄

#510×C₁₅₁

yes

—

136

C₂

#1713×C₁₅₁

yes

characteristic

137

C₂

#2250×C₁₅₁

yes

characteristic

138

C₂

#2251×C₁₅₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₅₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₅₁
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ C₈²×C₁₅₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₅₁

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₅₁

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₅₁ ◁ C₄²×C₁₅₁ ◁ C₈²×C₁₅₁ ◁ G

N	H ≅ G/N	type
46. C₁₆×C₁₅₁	Q₁₆	semidirect
69. #12524	C₁₅₁	direct (N × H)
70. C₁₅₁	#12524	direct (N × H)
116. C₁₆	Q₁₆×C₁₅₁	semidirect

H ≅ G/N

type

46. C₁₆×C₁₅₁

Q₁₆

semidirect

69. #12524

C₁₅₁

direct (N × H)

70. C₁₅₁

#12524

direct (N × H)

116. C₁₆

Q₁₆×C₁₅₁

semidirect

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

subgroup	index
#2254×C₁₅₁	2
#2252×C₁₅₁	2
#1713×C₁₅₁	2
#1710×C₁₅₁	2
#1311×C₁₅₁	2
#12524	151

subgroup

index

#2254×C₁₅₁

#2252×C₁₅₁

#1713×C₁₅₁

#1710×C₁₅₁

#1311×C₁₅₁

#12524

151

Subgroups — C₁₅₁×#12524 #50317449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C151×#12524 #50317449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₅₁×#12524 #50317449