Group #16776999 — subgroups

Subgroups — #16776999

Order 3328.

158 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 44 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	#13129866	C₂	1	1	yes	maximal, characteristic, complemented
2	#13129859	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114429	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#500	C₄	1	1	yes	characteristic
5	#11718749	—	2	1	no	—
6	#11718746	—	2	1	no	—
7	#11718356	C₄	1	1	yes	characteristic
8	C₄⋊#6092	—	2	1	no	—
9	C₄⋊#6092	—	2	1	no	—
10	C₈×#1055	C₂²	1	1	yes	characteristic
11	C₁₃×C₄⋊Q₈	—	2	1	no	—
12	C₂×#6106	—	4	1	no	—
13	C₂×#6106	—	4	1	no	—
14	C₂×D₁₀₄	—	4	1	no	—
15	#88128	—	2	1	no	—
16	C₂×#6092	—	4	1	no	—
17	C₂×#6092	—	4	1	no	—
18	C₁₃×#107	—	4	1	no	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	#88049	—	4	1	no	—
21	#88049	—	4	1	no	—
22	C₄×#1055	D₄	1	1	yes	characteristic
23	C₄×#1055	D₄	1	1	yes	characteristic
24	#7094	—	13	1	no	maximal, Sylow
25	C₂×Q₈×C₁₃	—	4	1	no	—
26	C₂×D₅₂	—	4	1	no	—
27	C₂×D₅₂	—	4	1	no	—
28	C₂×D₅₂	—	4	1	no	—
29	C₂×C₈×C₁₃	—	2	1	no	—
30	C₁₃×C₄⋊C₄	—	4	1	no	—
31	C₁₃×C₄⋊C₄	—	4	1	no	—
32	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	#6106	—	8	2	no	—
34	#6106	—	8	1	no	—
35	#6106	—	8	1	no	—
36	C₂×#1055	D₈	1	1	yes	characteristic
37	C₂×#1055	D₈	1	1	yes	characteristic
38	C₂×#1055	SD₁₆	1	1	yes	characteristic
39	C₂×#1055	SD₁₆	1	1	yes	characteristic
40	D₁₀₄	—	8	2	no	—
41	#6092	—	8	1	no	—
42	#6092	—	8	1	no	—
43	#1706	—	13	1	no	—
44	#1370	—	13	1	no	—
45	#1367	—	13	1	no	—
46	C₂²×D₁₃	—	8	1	no	—
47	Q₈×C₁₃	—	8	1	no	—
48	Q₈×C₁₃	—	8	1	no	—
49	C₂×C₄×C₁₃	—	8	1	no	—
50	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
51	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
53	D₅₂	—	8	2	no	—
54	D₅₂	—	8	1	no	—
55	D₅₂	—	8	1	no	—
56	D₅₂	—	8	1	no	—
57	D₅₂	—	8	1	no	—
58	C₈×C₁₃	—	2	2	no	—
59	#1055	—	2	1	no	—
60	#1055	—	2	1	no	—
61	#1055	—	2	1	no	—
62	#1055	D₁₆	1	1	yes	characteristic
63	#1055	SD₃₂	1	1	yes	characteristic
64	#500	#261	1	1	yes	characteristic
65	#493	—	13	1	no	—
66	#492	—	26	1	no	—
67	#486	—	26	1	no	—
68	C₄⋊C₁₆	—	26	1	no	—
69	C₄⋊C₁₆	—	26	1	no	—
70	C₈²	—	13	1	no	—
71	C₂²×C₁₃	#331	1	1	yes	characteristic
72	C₂×D₁₃	—	16	1	no	—
73	C₂×D₁₃	—	16	1	no	—
74	C₂×D₁₃	—	16	1	no	—
75	C₄×C₁₃	—	2	1	no	—
76	C₄×C₁₃	—	16	1	no	—
77	C₄×C₁₃	#357	1	1	yes	characteristic
78	C₄×C₁₃	#359	1	1	yes	characteristic
79	C₄×C₁₃	#360	1	1	yes	characteristic
80	C₄×C₁₃	#361	1	1	yes	characteristic
81	C₂×SD₁₆	—	52	1	no	—
82	C₂×SD₁₆	—	52	1	no	—
83	C₂×D₈	—	52	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	#127	—	26	1	no	—
86	C₂×C₁₆	—	52	1	no	—
87	C₂×C₁₆	—	52	1	no	—
88	#107	—	4	1	no	—
89	#102	—	52	1	no	—
90	#102	—	52	1	no	—
91	C₄×C₈	—	13	1	no	—
92	C₄×C₈	—	13	1	no	—
93	C₄×C₈	C₂×#261	1	1	yes	characteristic
94	C₂×C₁₃	#1359	1	1	yes	characteristic
95	C₂×C₁₃	#1360	1	1	yes	characteristic
96	C₂×C₁₃	#1361	1	1	yes	characteristic
97	D₁₃	—	32	1	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×D₄	—	52	1	no	—
100	C₂×D₄	—	52	1	no	—
101	C₂×D₄	—	52	1	no	—
102	SD₁₆	—	104	1	no	—
103	SD₁₆	—	104	1	no	—
104	SD₁₆	—	104	2	no	—
105	D₈	—	104	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	13	1	no	—
108	C₂×C₈	—	13	1	no	—
109	C₂×C₈	—	13	1	no	—
110	C₂×C₈	—	13	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄²	C₂²⋊#261	1	1	yes	characteristic
114	C₁₆	—	104	1	no	—
115	C₁₆	—	104	1	no	—
116	C₁₃	#7094	1	1	yes	characteristic, complemented, Sylow
117	C₂³	—	104	1	no	—
118	Q₈	—	8	1	no	—
119	Q₈	—	8	1	no	—
120	D₄	—	104	2	no	—
121	D₄	—	104	1	no	—
122	D₄	—	104	1	no	—
123	D₄	—	104	1	no	—
124	D₄	—	104	1	no	—
125	C₂×C₄	—	8	1	no	—
126	C₂×C₄	#88115	1	1	yes	characteristic
127	C₂×C₄	#88115	1	1	yes	characteristic
128	C₂×C₄	#88121	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	13	1	no	—
131	C₈	—	13	1	no	—
132	C₈	—	26	1	no	—
133	C₈	—	26	1	no	—
134	C₈	—	26	1	no	—
135	C₂²	—	208	1	no	—
136	C₂²	—	208	1	no	—
137	C₂²	—	208	1	no	—
138	C₂²	#11718297	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	16	1	no	—
141	C₄	#11718276	1	1	yes	characteristic
142	C₄	#11718278	1	1	yes	characteristic
143	C₄	#11718279	1	1	yes	characteristic
144	C₄	#11718281	1	1	yes	characteristic
145	C₂	—	416	1	no	—
146	C₂	#13129847	1	1	yes	characteristic
147	C₂	#13129849	1	1	yes	characteristic
148	C₂	#13129850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129866

C₂

yes

maximal, characteristic, complemented

#13129859

C₂

yes

maximal, characteristic, complemented

#13114429

C₂

yes

maximal, characteristic

C₁₃×#500

C₄

yes

—

—

C₄

yes

—

—

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×#6106

—

C₂×#6106

—

C₂×D₁₀₄

—

#88128

—

C₂×#6092

—

C₂×#6092

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

—

D₄

yes

characteristic

C₄×#1055

D₄

yes

characteristic

#7094

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

—

—

D₈

yes

characteristic

C₂×#1055

D₈

yes

characteristic

C₂×#1055

SD₁₆

yes

characteristic

C₂×#1055

SD₁₆

yes

characteristic

D₁₀₄

—

—

—

—

—

—

C₂²×D₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

—

—

—

D₁₆

yes

characteristic

#1055

SD₃₂

yes

yes

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₈²

—

C₂²×C₁₃

#331

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#102

—

#102

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

SD₁₆

—

104

—

103

SD₁₆

—

104

—

104

SD₁₆

—

104

—

105

D₈

—

104

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂²⋊#261

yes

characteristic

114

C₁₆

—

104

—

115

C₁₆

—

104

—

116

C₁₃

#7094

yes

characteristic, complemented, Sylow

117

C₂³

—

104

—

118

Q₈

—

119

Q₈

—

120

D₄

—

104

—

121

D₄

—

104

—

122

D₄

—

104

—

123

D₄

—

104

—

124

D₄

—

104

—

125

C₂×C₄

—

126

C₂×C₄

#88115

yes

characteristic

127

C₂×C₄

#88115

yes

characteristic

128

C₂×C₄

#88121

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₂²

—

208

—

136

C₂²

—

208

—

137

C₂²

—

208

—

138

C₂²

#11718297

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#11718276

yes

characteristic

142

C₄

#11718278

yes

characteristic

143

C₄

#11718279

yes

characteristic

144

C₄

#11718281

yes

characteristic

145

C₂

—

416

—

146

C₂

#13129847

yes

characteristic

147

C₂

#13129849

yes

characteristic

148

C₂

#13129850

yes

characteristic

	subgroup	G/H
center	C₂²	#11718297
commutator	C₄×C₈×C₁₃	C₂×C₄
Frattini	C₄×C₈	C₂×#261
Fitting	C₁₃×#500	C₄
socle	C₂²×C₁₃	#331
prime radical	#11718356	C₄
cosocle	C₈×#1055	C₂²
hypercenter	#500	#261
hypocenter	C₁₃	#7094
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

#11718297

commutator

C₄×C₈×C₁₃

C₂×C₄

Frattini

C₄×C₈

C₂×#261

Fitting

C₁₃×#500

C₄

C₂²×C₁₃

C₄

C₂²

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
1. #13129866	C₂	semidirect
2. #13129859	C₂	semidirect
116. C₁₃	#7094	semidirect

H ≅ G/N

type

1. #13129866

C₂

semidirect

2. #13129859

C₂

semidirect

116. C₁₃

#7094

semidirect

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

subgroup	index
#13129866	2
#13129859	2
#13114429	2
#7094	13

subgroup

index

Subgroups — #16776999

Conjugacy classes of subgroups

Complements

Maximal subgroups