Group #16790258 — subgroups

Subgroups — #16790258

Order 3328.

342 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 144 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×#11718233	C₂	1	4	yes	maximal
2	#13131168	C₂	1	2	yes	maximal
3	#13114523	C₂	1	1	yes	maximal, characteristic
4	C₂×#88106	C₂²	1	4	yes	—
5	C₁₃×#376	C₄	1	1	yes	characteristic
6	C₂×#88043	C₄	1	2	yes	—
7	C₂×#88043	C₄	1	1	yes	characteristic
8	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
9	C₂×C₄×#1055	C₂²	1	2	yes	—
10	#11718233	—	2	8	no	—
11	C₄⋊#6092	—	2	4	no	—
12	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
15	C₂²×#1055	C₂×C₄	1	2	yes	—
16	C₂²×#1055	C₂³	1	1	yes	characteristic
17	#88106	—	2	8	no	—
18	C₂×#6092	—	2	4	no	—
19	C₁₃×#95	—	2	2	no	—
20	#88043	—	2	4	no	—
21	#88043	C₈	1	4	yes	—
22	C₄×#1055	D₄	1	2	yes	—
23	C₄×#1055	D₄	1	4	yes	—
24	C₄×#1055	Q₈	1	2	yes	—
25	C₄×#1055	Q₈	1	4	yes	—
26	#12901	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	—	2	2	no	—
28	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₁₃	—	2	2	no	—
31	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
32	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
33	C₄²×C₁₃	C₂×C₈	1	2	yes	—
34	C₂×#1055	—	2	2	no	—
35	C₂×#1055	C₄⋊C₄	1	2	yes	—
36	C₂×#1055	C₄⋊C₄	1	4	yes	—
37	C₂×#1055	C₂×C₈	1	4	yes	—
38	C₂×#1055	M₁₆	1	4	yes	—
39	C₂×#1055	C₂×D₄	1	1	yes	characteristic
40	C₂×#1055	C₂×D₄	1	2	yes	—
41	C₂×#1055	C₂×Q₈	1	2	yes	—
42	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
43	#6092	—	4	4	no	—
44	C₂×#364	—	13	4	no	—
45	#2150	—	13	2	no	—
46	#1726	—	13	1	no	—
47	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	—	4	2	no	—
50	C₂×C₄×C₁₃	—	4	4	no	—
51	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
56	#1055	—	2	2	no	—
57	#1055	C₄⋊C₈	1	8	yes	—
58	#1055	#127	1	1	yes	characteristic
59	#1055	C₄⋊Q₈	1	1	yes	characteristic
60	#1055	C₄⋊Q₈	1	2	yes	—
61	C₂×M₃₂	—	13	4	no	—
62	C₂×#97	—	13	1	no	—
63	C₂×#97	—	13	2	no	—
64	C₂×C₄×C₈	—	13	2	no	—
65	C₂×C₄×C₈	—	13	1	no	—
66	#376	#261	1	1	yes	characteristic
67	#364	—	26	8	no	—
68	C₄⋊C₁₆	—	26	4	no	—
69	C₂²×C₁₃	—	2	2	no	—
70	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂²×C₁₃	#423	1	2	yes	—
72	C₄×C₁₃	—	2	4	no	—
73	C₄×C₁₃	—	4	4	no	—
74	C₄×C₁₃	#384	1	1	yes	characteristic
75	C₄×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
76	C₄×C₁₃	#423	1	1	yes	characteristic
77	C₂²×C₈	—	13	2	no	—
78	C₂²×C₈	—	13	1	no	—
79	C₂²×C₈	—	13	1	no	—
80	C₂×C₄²	C₂×#261	1	2	yes	—
81	C₂×C₄²	C₂×#261	1	1	yes	characteristic
82	M₃₂	—	26	8	no	—
83	C₂×C₁₆	—	26	4	no	—
84	#97	—	13	4	no	—
85	#97	—	26	4	no	—
86	C₄×C₈	—	13	2	no	—
87	C₄×C₈	—	13	4	no	—
88	C₄×C₈	—	13	4	no	—
89	C₄×C₈	—	13	2	no	—
90	#95	—	2	2	no	—
91	C₂×C₁₃	—	2	2	no	—
92	C₂×C₁₃	#1842	1	1	yes	characteristic
93	C₂×C₁₃	#2153	1	2	yes	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	#6122	1	2	yes	—
96	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
97	C₂×C₈	—	13	4	no	—
98	C₂×C₈	—	13	4	no	—
99	C₂×C₈	—	13	2	no	—
100	C₂×C₈	—	13	4	no	—
101	C₂×C₈	—	13	2	no	—
102	C₂×C₈	—	13	1	no	—
103	C₂×C₈	—	13	1	no	—
104	C₂×C₈	—	13	2	no	—
105	C₂×C₈	—	26	2	no	—
106	C₄²	—	2	2	no	—
107	C₄²	D₁₃⋊C₈	1	2	yes	—
108	C₄²	C₄⋊#261	1	4	yes	—
109	C₄²	C₄⋊#261	1	2	yes	—
110	C₁₆	—	52	4	no	—
111	C₁₃	#12901	1	1	yes	characteristic, complemented, Sylow
112	C₂³	C₂×#6122	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	#88112	1	4	yes	—
117	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
118	C₂×C₄	D₁₃⋊M₁₆	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊#261	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
121	C₈	—	13	8	no	—
122	C₈	—	13	1	no	—
123	C₈	—	13	2	no	—
124	C₈	—	13	1	no	—
125	C₈	—	26	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂×#88112	1	1	yes	characteristic
128	C₂²	#11719129	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	D₁₃⋊(C₄⋊C₈)	1	2	yes	—
132	C₄	#11719053	1	1	yes	characteristic
133	C₄	#11719059	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#13130769	1	1	yes	characteristic
136	C₂	#13131179	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718233

C₂

yes

maximal

#13131168

C₂

yes

maximal

#13114523

C₂

yes

maximal, characteristic

C₂×#88106

C₂²

yes

—

C₁₃×#376

C₄

yes

characteristic

C₂×#88043

C₄

yes

—

C₂×#88043

C₄

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

C₂×C₄×#1055

C₂²

yes

—

#11718233

—

C₄⋊#6092

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

characteristic

C₂²×#1055

C₂×C₄

yes

—

C₂²×#1055

C₂³

yes

characteristic

#88106

—

C₂×#6092

—

C₁₃×#95

—

#88043

—

#88043

C₈

yes

—

C₄×#1055

D₄

yes

—

C₄×#1055

D₄

yes

—

C₄×#1055

Q₈

yes

—

C₄×#1055

Q₈

yes

—

#12901

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

M₁₆

yes

—

C₂²×C₄×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂×C₈

yes

—

C₂×#1055

—

C₂×#1055

C₄⋊C₄

yes

—

C₂×#1055

C₄⋊C₄

yes

—

C₂×#1055

C₂×C₈

yes

—

C₂×#1055

M₁₆

yes

—

C₂×#1055

C₂×D₄

yes

characteristic

C₂×#1055

C₂×D₄

yes

—

C₂×#1055

C₂×Q₈

yes

—

C₂×#1055

C₂×Q₈

yes

—

—

—

—

C₂³×C₁₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

C₂×M₁₆

yes

—

#1055

—

#1055

C₄⋊C₈

yes

—

yes

C₄⋊Q₈

yes

characteristic

#1055

C₄⋊Q₈

yes

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

yes

—

C₄⋊C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₁₃

#423

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#384

yes

characteristic

C₄×C₁₃

C₂×C₄⋊C₈

yes

—

C₄×C₁₃

#423

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#261

yes

—

C₂×C₄²

C₂×#261

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

#1842

yes

characteristic

C₂×C₁₃

#2153

yes

—

C₂²×C₄

—

C₂²×C₄

#6122

yes

—

C₂²×C₄

C₂²×#261

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄²

—

107

C₄²

D₁₃⋊C₈

yes

—

108

C₄²

C₄⋊#261

yes

—

109

C₄²

C₄⋊#261

yes

—

110

C₁₆

—

111

C₁₃

#12901

yes

characteristic, complemented, Sylow

112

C₂³

C₂×#6122

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#88112

yes

—

117

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

118

C₂×C₄

D₁₃⋊M₁₆

yes

—

119

C₂×C₄

C₂×C₄⋊#261

yes

—

120

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₂²

—

127

C₂²

C₂×#88112

yes

characteristic

128

C₂²

#11719129

yes

—

129

C₄

—

130

C₄

—

131

C₄

D₁₃⋊(C₄⋊C₈)

yes

—

132

C₄

#11719053

yes

characteristic

133

C₄

#11719059

yes

characteristic

134

C₂

—

135

C₂

#13130769

yes

characteristic

136

C₂

#13131179

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₁₃×#376	C₄
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₂×M₁₆
cosocle	C₂²×#1055	C₂³
hypercenter	#376	#261
hypocenter	C₁₃	#12901
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#261

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#261

Fitting

C₁₃×#376

C₄

socle

C₂²×C₁₃

C₂×C₄⋊C₈

prime radical

C₂³×C₁₃

C₂×M₁₆

cosocle

C₂²×#1055

C₂³

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
111. C₁₃	#12901	semidirect

H ≅ G/N

type

111. C₁₃

#12901

semidirect

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

subgroup	index
C₂×#11718233	2
#13131168	2
#13114523	2
#12901	13

subgroup

index

Subgroups — #16790258

Conjugacy classes of subgroups

Complements

Maximal subgroups