Group #16782371 — subgroups

Subgroups — #16782371

Order 3328.

262 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 80 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	#13130761	C₂	1	1	yes	maximal, characteristic
2	#13130758	C₂	1	1	yes	maximal, characteristic
3	#13129690	C₂	1	4	yes	maximal
4	#13113121	C₂	1	1	yes	maximal, characteristic
5	#11719060	C₂²	1	2	yes	—
6	#11719053	C₂²	1	2	yes	—
7	#11718346	C₂²	1	2	yes	—
8	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
9	#11718268	—	2	2	no	—
10	#11718265	—	2	2	no	—
11	C₁₃×#322	C₄	1	1	yes	characteristic, complemented
12	#11718047	C₄	1	1	yes	characteristic, complemented
13	C₈×#1055	C₄	1	2	yes	complemented
14	C₂×C₄⋊#261	—	2	2	no	—
15	C₂×#6118	—	2	2	no	—
16	C₂×C₈×D₁₃	D₄	1	2	yes	—
17	C₄²×D₁₃	C₂³	1	1	yes	characteristic
18	(C₂×#261)⋊C₄	—	2	4	no	—
19	#88112	—	2	4	no	—
20	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	#88060	—	2	2	no	—
23	#88054	—	2	2	no	—
24	C₈×Dic₁₃	Q₈	1	2	yes	—
25	C₄×#1055	C₂×C₄	1	2	yes	—
26	C₄×#1055	C₂×C₄	1	1	yes	characteristic
27	#8791	—	13	1	no	maximal, Sylow
28	C₂²×#261	—	4	2	no	—
29	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₃	#41	1	2	yes	—
31	C₂×#1057	—	4	2	no	—
32	C₄⋊#261	—	4	4	no	—
33	#6118	—	4	4	no	—
34	C₂×C₈×C₁₃	—	2	2	no	—
35	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
36	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₄×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
38	C₄×Dic₁₃	#41	1	2	yes	—
39	C₂×#1055	—	2	2	no	—
40	C₂×#1055	C₄⋊C₄	1	2	yes	—
41	C₈×D₁₃	—	2	4	no	—
42	#1837	—	13	1	no	—
43	#1835	—	13	1	no	—
44	#1450	—	13	1	no	—
45	#1286	—	13	4	no	—
46	C₂²×D₁₃	#124	1	1	yes	characteristic
47	C₂×#261	—	8	4	no	—
48	C₂×#261	—	8	2	no	—
49	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₃	#117	1	2	yes	—
51	C₂×Dic₁₃	#124	1	1	yes	characteristic
52	C₂×Dic₁₃	#125	1	2	yes	—
53	C₄×D₁₃	—	4	2	no	—
54	C₄×D₁₃	#135	1	4	yes	—
55	#1057	—	8	2	no	—
56	C₈×C₁₃	—	2	2	no	—
57	C₈×C₁₃	—	4	2	no	—
58	#1055	—	2	2	no	—
59	#1055	—	4	2	no	—
60	#431	—	13	2	no	—
61	#423	—	13	2	no	—
62	C₂×C₄×C₈	—	13	1	no	—
63	#382	—	13	2	no	—
64	#341	—	26	2	no	—
65	#340	—	26	2	no	—
66	#322	—	13	1	no	—
67	#322	#261	1	1	yes	characteristic, complemented
68	C₈²	—	13	2	no	—
69	C₂²×C₁₃	#382	1	1	yes	characteristic
70	C₂×D₁₃	—	2	2	no	—
71	C₂×D₁₃	#488	1	2	yes	—
72	#261	—	16	2	no	—
73	C₄×C₁₃	—	2	2	no	—
74	C₄×C₁₃	#427	1	2	yes	—
75	Dic₁₃	—	2	2	no	—
76	Dic₁₃	#488	1	2	yes	—
77	C₂×M₁₆	—	26	2	no	—
78	C₂²×C₈	—	13	2	no	—
79	C₂×C₄⋊C₄	—	26	2	no	—
80	C₂×C₄²	—	13	1	no	—
81	C₄⋊C₈	—	26	2	no	—
82	C₄⋊C₈	—	26	4	no	—
83	C₂²⋊C₈	—	26	2	no	—
84	C₄×C₈	—	13	2	no	—
85	C₄×C₈	—	13	1	no	—
86	C₄×C₈	—	13	2	no	—
87	C₄×C₈	C₂×#261	1	2	yes	—
88	C₄×C₈	C₂×#261	1	1	yes	characteristic
89	#95	—	26	4	no	—
90	C₂×C₁₃	#1833	1	2	yes	—
91	C₂×C₁₃	#1835	1	1	yes	characteristic
92	D₁₃	—	2	2	no	—
93	C₂²×C₄	—	13	1	no	—
94	C₂²×C₄	—	13	2	no	—
95	C₂²×C₄	—	52	2	no	—
96	M₁₆	—	52	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	13	2	no	—
99	C₂×C₈	—	26	4	no	—
100	C₂×C₈	—	26	2	no	—
101	C₂×C₈	—	52	2	no	—
102	C₂×C₈	C₄⋊#261	1	2	yes	—
103	C₄⋊C₄	—	52	4	no	—
104	C₄²	—	13	1	no	—
105	C₄²	—	13	2	no	—
106	C₄²	C₂²×#261	1	1	yes	characteristic
107	C₁₃	#8791	1	1	yes	characteristic, complemented, Sylow
108	C₂³	—	13	1	no	—
109	C₂×C₄	—	13	1	no	—
110	C₂×C₄	—	13	4	no	—
111	C₂×C₄	—	13	2	no	—
112	C₂×C₄	—	52	2	no	—
113	C₂×C₄	—	104	2	no	—
114	C₂×C₄	—	104	4	no	—
115	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
116	C₂×C₄	#88237	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	26	2	no	—
120	C₈	—	52	2	no	—
121	C₈	—	104	2	no	—
122	C₂²	—	13	2	no	—
123	C₂²	—	26	2	no	—
124	C₂²	#11719060	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	13	2	no	—
127	C₄	—	26	2	no	—
128	C₄	—	208	2	no	—
129	C₄	#11719092	1	2	yes	—
130	C₂	—	26	2	no	—
131	C₂	#13130756	1	2	yes	—
132	C₂	#13130758	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130761

C₂

yes

maximal, characteristic

#13130758

C₂

yes

maximal, characteristic

#13129690

C₂

yes

maximal

#13113121

C₂

yes

maximal, characteristic

#11719060

C₂²

yes

—

#11719053

C₂²

yes

—

#11718346

C₂²

yes

—

C₄×C₈×D₁₃

C₂²

yes

characteristic

#11718268

—

#11718265

—

C₁₃×#322

C₄

yes

characteristic, complemented

#11718047

C₄

yes

characteristic, complemented

C₈×#1055

C₄

yes

complemented

C₂×C₄⋊#261

—

C₂×#6118

—

C₂×C₈×D₁₃

D₄

yes

—

C₄²×D₁₃

C₂³

yes

characteristic

(C₂×#261)⋊C₄

—

#88112

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

#88060

—

#88054

—

C₈×Dic₁₃

Q₈

yes

—

C₄×#1055

C₂×C₄

yes

—

C₄×#1055

C₂×C₄

yes

characteristic

#8791

—

maximal, Sylow

C₂²×#261

—

C₂×C₄×D₁₃

C₂×D₄

yes

characteristic

C₂×C₄×D₁₃

#41

yes

—

C₂×#1057

—

C₄⋊#261

—

#6118

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄×Dic₁₃

C₂×Q₈

yes

characteristic

C₄×Dic₁₃

#41

yes

—

C₂×#1055

—

C₂×#1055

C₄⋊C₄

yes

—

C₈×D₁₃

—

—

—

—

—

C₂²×D₁₃

yes

—

—

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

#117

yes

—

C₂×Dic₁₃

#124

yes

characteristic

C₂×Dic₁₃

#125

yes

—

C₄×D₁₃

—

C₄×D₁₃

#135

yes

—

#1057

—

C₈×C₁₃

—

C₈×C₁₃

—

—

—

—

—

C₂×C₄×C₈

—

—

—

—

—

yes

characteristic, complemented

C₈²

—

C₂²×C₁₃

#382

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

#488

yes

—

#261

—

C₄×C₁₃

—

C₄×C₁₃

#427

yes

—

Dic₁₃

—

Dic₁₃

#488

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#261

yes

—

C₄×C₈

C₂×#261

yes

characteristic

#95

—

C₂×C₁₃

#1833

yes

—

C₂×C₁₃

#1835

yes

characteristic

D₁₃

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₄⋊#261

yes

—

103

C₄⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂²×#261

yes

characteristic

107

C₁₃

#8791

yes

characteristic, complemented, Sylow

108

C₂³

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

104

—

114

C₂×C₄

—

104

—

115

C₂×C₄

C₂×C₄⋊#261

yes

characteristic

116

C₂×C₄

#88237

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

104

—

122

C₂²

—

123

C₂²

—

124

C₂²

#11719060

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

208

—

129

C₄

#11719092

yes

—

130

C₂

—

131

C₂

#13130756

yes

—

132

C₂

#13130758

yes

characteristic

	subgroup	G/H
center	C₂²	#11719060
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₄²	C₂²×#261
Fitting	C₁₃×#322	C₄
socle	C₂²×C₁₃	#382
prime radical	C₂²×D₁₃	#124
cosocle	C₄²×D₁₃	C₂³
hypercenter	#322	#261
hypocenter	C₁₃	#8791
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#11719060

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₄²

C₂²×#261

Fitting

C₁₃×#322

C₄

C₂²×C₁₃

C₂²×D₁₃

C₄²×D₁₃

C₂³

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
11. C₁₃×#322	C₄	semidirect
12. #11718047	C₄	semidirect
13. C₈×#1055	C₄	semidirect
67. #322	#261	semidirect
107. C₁₃	#8791	semidirect

H ≅ G/N

type

11. C₁₃×#322

C₄

semidirect

12. #11718047

C₄

semidirect

13. C₈×#1055

C₄

semidirect

67. #322

#261

semidirect

107. C₁₃

#8791

semidirect

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

subgroup	index
#13130761	2
#13130758	2
#13129690	2
#13113121	2
#8791	13

subgroup

index

Subgroups — #16782371

Conjugacy classes of subgroups

Complements

Maximal subgroups