Group #16788866 — subgroups

Subgroups — #16788866

Order 3328.

200 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 86 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	#13131251	C₂	1	2	yes	maximal
2	#13129858	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#6092	C₂	1	2	yes	maximal
4	#13129749	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#1055	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
7	Q₈×#1055	C₂²	1	2	yes	—
8	#11718231	—	2	2	no	—
9	C₄⋊#6092	—	2	2	no	—
10	C₄⋊#6092	C₂²	1	2	yes	—
11	C₄×#6092	C₂²	1	1	yes	characteristic
12	C₄×#6092	C₂²	1	1	yes	characteristic
13	Q₈⋊#1055	C₄	1	2	yes	—
14	#11718050	C₄	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₁₆	C₈	1	1	yes	characteristic
17	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×#6092	—	2	2	no	—
19	C₂×#6092	—	2	1	no	—
20	C₂×#6092	D₄	1	2	yes	—
21	C₁₃×#107	C₈	1	1	yes	characteristic
22	C₁₃×#103	C₈	1	2	yes	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄⋊#1055	—	2	2	no	—
25	C₄⋊#1055	C₂×C₄	1	2	yes	—
26	C₄×#1055	—	2	2	no	—
27	C₄×#1055	C₂×C₄	1	1	yes	characteristic
28	C₄×#1055	C₂³	1	1	yes	characteristic
29	#12061	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
31	C₁₃×Q₁₆	—	2	2	no	—
32	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
36	C₄²×C₁₃	—	2	2	no	—
37	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×#1055	—	2	2	no	—
39	C₂×#1055	—	2	1	no	—
40	C₂×#1055	C₂×D₄	1	1	yes	characteristic
41	C₂×#1055	#41	1	1	yes	characteristic
42	#6092	—	2	2	no	—
43	#6092	—	4	1	no	—
44	#6092	—	4	2	no	—
45	#2183	—	13	2	no	—
46	C₈×Q₁₆	—	13	1	no	—
47	#1367	—	13	1	no	—
48	Q₈⋊C₁₆	—	13	2	no	—
49	#1312	—	13	1	no	—
50	Q₈×C₁₃	—	2	2	no	—
51	Q₈×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	2	2	no	—
54	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
57	C₈×C₁₃	—	2	1	no	—
58	C₈×C₁₃	M₃₂	1	2	yes	—
59	#1055	—	2	1	no	—
60	#1055	—	4	1	no	—
61	#1055	—	4	2	no	—
62	#1055	#137	1	2	yes	—
63	C₈×Q₈	—	13	2	no	—
64	C₄×Q₁₆	#261	1	1	yes	characteristic
65	C₄⋊C₁₆	—	13	2	no	—
66	C₄⋊C₁₆	—	26	2	no	—
67	#346	—	26	2	no	—
68	C₄×C₁₆	—	13	1	no	—
69	C₄×C₁₆	—	13	1	no	—
70	#335	—	13	1	no	—
71	Q₈⋊C₈	—	13	2	no	—
72	C₈²	—	13	1	no	—
73	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
74	C₄×C₁₃	—	2	1	no	—
75	C₄×C₁₃	—	4	2	no	—
76	C₄×C₁₃	—	4	2	no	—
77	C₄×C₁₃	#441	1	1	yes	characteristic
78	C₄×C₁₃	#444	1	1	yes	characteristic
79	C₄×C₁₃	C₂×M₃₂	1	1	yes	characteristic
80	C₄×C₁₃	#504	1	1	yes	characteristic
81	C₂×Q₁₆	#1057	1	1	yes	characteristic
82	C₄×Q₈	C₂×#261	1	2	yes	—
83	C₂×C₁₆	—	13	2	no	—
84	C₂×C₁₆	—	26	1	no	—
85	C₂×C₁₆	—	26	2	no	—
86	#107	#1057	1	1	yes	characteristic
87	C₄⋊C₈	—	13	2	no	—
88	C₄⋊C₈	—	26	2	no	—
89	#103	#1057	1	2	yes	—
90	C₄×C₈	—	13	1	no	—
91	C₄×C₈	—	13	1	no	—
92	C₄×C₈	—	26	2	no	—
93	C₄×C₈	C₂×#261	1	1	yes	characteristic
94	C₂×C₁₃	#1579	1	1	yes	characteristic
95	C₂×C₁₃	#2169	1	1	yes	characteristic
96	C₂×C₁₃	#2171	1	1	yes	characteristic
97	C₂×Q₈	C₂×#1057	1	2	yes	—
98	Q₁₆	—	2	2	no	—
99	C₂×C₈	—	13	1	no	—
100	C₂×C₈	—	13	1	no	—
101	C₂×C₈	—	26	1	no	—
102	C₂×C₈	—	26	2	no	—
103	C₂×C₈	C₂×#1057	1	1	yes	characteristic
104	C₂×C₈	C₂×#1057	1	1	yes	characteristic
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂×#1057	1	2	yes	—
107	C₄²	—	2	2	no	—
108	C₄²	C₂²×#261	1	1	yes	characteristic
109	C₁₆	—	26	2	no	—
110	C₁₆	—	52	2	no	—
111	C₁₆	—	52	1	no	—
112	C₁₃	#12061	1	1	yes	characteristic, complemented, Sylow
113	Q₈	—	2	2	no	—
114	Q₈	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	#88238	1	1	yes	characteristic
118	C₂×C₄	D₄×#261	1	1	yes	characteristic
119	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	13	2	no	—
122	C₈	—	26	1	no	—
123	C₈	—	52	1	no	—
124	C₈	—	52	2	no	—
125	C₈	#88106	1	2	yes	—
126	C₂²	D₄×#1057	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	#11719110	1	1	yes	characteristic
131	C₄	#11719112	1	1	yes	characteristic
132	C₄	C₂×#88106	1	1	yes	characteristic
133	C₄	#11719162	1	1	yes	characteristic
134	C₂	#13130429	1	1	yes	characteristic
135	C₂	#13131220	1	1	yes	characteristic
136	C₂	#13131226	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13131251

C₂

yes

maximal

#13129858

C₂

yes

maximal, characteristic

Q₈⋊#6092

C₂

yes

maximal

#13129749

C₂

yes

maximal, characteristic

Q₁₆×#1055

C₂

yes

maximal, characteristic

C₄×C₁₃×Q₁₆

C₄

yes

characteristic

Q₈×#1055

C₂²

yes

—

#11718231

—

C₄⋊#6092

—

C₄⋊#6092

C₂²

yes

—

C₄×#6092

C₂²

yes

characteristic

C₄×#6092

C₂²

yes

characteristic

Q₈⋊#1055

C₄

yes

—

#11718050

C₄

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₂×C₁₃×Q₁₆

C₈

yes

characteristic

C₄×Q₈×C₁₃

C₂×C₄

yes

—

C₂×#6092

—

C₂×#6092

—

C₂×#6092

D₄

yes

—

C₁₃×#107

C₈

yes

characteristic

C₁₃×#103

C₈

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄⋊#1055

—

C₄⋊#1055

C₂×C₄

yes

—

C₄×#1055

—

C₄×#1055

C₂×C₄

yes

characteristic

C₄×#1055

C₂³

yes

characteristic

#12061

—

maximal, Sylow

C₂×Q₈×C₁₃

C₂×C₈

yes

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

C₂×C₈

yes

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

—

—

—

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

characteristic

C₈×C₁₃

—

C₈×C₁₃

M₃₂

yes

—

—

—

—

yes

—

C₈×Q₈

—

C₄×Q₁₆

#261

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂²×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×M₃₂

yes

C₄×C₁₃

yes

C₂×Q₁₆

yes

C₄×Q₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

#1057

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#103

#1057

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

yes

—

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂×#1057

yes

characteristic

104

C₂×C₈

C₂×#1057

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×#1057

yes

—

107

C₄²

—

108

C₄²

C₂²×#261

yes

characteristic

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₃

#12061

yes

characteristic, complemented, Sylow

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#88238

yes

characteristic

118

C₂×C₄

D₄×#261

yes

characteristic

119

C₂×C₄

C₂²×#1057

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

#88106

yes

—

126

C₂²

D₄×#1057

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#11719110

yes

characteristic

131

C₄

#11719112

yes

characteristic

132

C₄

C₂×#88106

yes

characteristic

133

C₄

#11719162

yes

characteristic

134

C₂

#13130429

yes

characteristic

135

C₂

#13131220

yes

characteristic

136

C₂

#13131226

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#261
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄²	C₂²×#261
Fitting	C₄×C₁₃×Q₁₆	C₄
socle, prime radical	C₂²×C₁₃	C₈×D₄
cosocle	C₄×#1055	C₂³
hypercenter	C₄×Q₁₆	#261
hypocenter	C₁₃	#12061
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×#261

commutator

C₂×C₄×C₁₃

C₂²×C₈

Frattini

C₄²

C₂²×#261

Fitting

C₄×C₁₃×Q₁₆

C₄

socle, prime radical

C₂²×C₁₃

C₈×D₄

C₂³

C₄×Q₁₆

C₁₃

G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
112. C₁₃	#12061	semidirect

H ≅ G/N

type

112. C₁₃

#12061

semidirect

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

subgroup	index
#13131251	2
#13129858	2
Q₈⋊#6092	2
#13129749	2
Q₁₆×#1055	2
#12061	13

subgroup

index

Q₁₆×#1055

Subgroups — #16788866

Conjugacy classes of subgroups

Complements

Maximal subgroups