Finite Group Databasebeta

← #20276283

Subgroups — #20276283

Order 6784.

198 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810489C211yesmaximal, characteristic
2#16810385C211yesmaximal, characteristic
3#16810385C211yesmaximal, characteristic
4#16810385C211yesmaximal, characteristic
5#16810385C211yesmaximal, characteristic
6#16810378C211yesmaximal, characteristic
7C53×#389C211yesmaximal, characteristic
8C22×#88322C2211yescharacteristic
9C22×#88322C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12#13133124—21no—
13#13133124—21no—
14C2×C4⋊C4×C53—21no—
15C2×C4⋊C4×C53C2211yescharacteristic
16C2×C4⋊C4×C53C2211yescharacteristic
17C2×C42×C53C2211yescharacteristic
18#95×C53—21no—
19C22×C4×C53—21no—
20C22×C4×C53—21no—
21C22×C4×C53D411yescharacteristic
22C22×C4×C53D411yescharacteristic
23C22×C4×C53C2311yescharacteristic
24C22×Dic53—21no—
25C22×Dic53—21no—
26C4⋊C4×C53—21no—
27C4⋊C4×C53—21no—
28C4⋊C4×C53—21no—
29C4⋊C4×C53—42no—
30C4⋊C4×C53Q812yes—
31C42×C53—21no—
32C42×C53—21no—
33C4⋊Dic53—21no—
34C4⋊Dic53—21no—
35C4⋊Dic53—21no—
36C4⋊Dic53—21no—
37C2×#88322—21no—
38C2×#88322—21no—
39C2×#88322—21no—
40C2×#88322—21no—
41C2×#88322—21no—
42C2×#88322—21no—
43C23×C53C2×D411yescharacteristic
44C2×C4×C53—22no—
45C2×C4×C53—21no—
46C2×C4×C53—21no—
47C2×C4×C53—21no—
48C2×C4×C53—21no—
49C2×C4×C53—21no—
50C2×C4×C53—21no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53C2×Q811yescharacteristic
57C2×C4×C53#4111yescharacteristic
58C2×C4×C53#4111yescharacteristic
59C2×C4×C53#4111yescharacteristic
60C2×C4×C53#4111yescharacteristic
61C2×C4×C53#4111yescharacteristic
62C2×Dic53—41no—
63C2×Dic53—41no—
64C2×Dic53—41no—
65C2×Dic53—41no—
66C2×Dic53—41no—
67C2×Dic53—41no—
68#88322—41no—
69#88322—41no—
70C22×C53C22⋊Q811yescharacteristic
71C22×C53#12311yescharacteristic
72C22×C53#12411yescharacteristic
73C22×C53#13511yescharacteristic
74C22×C53#13511yescharacteristic
75C22×C53AGL(1,ℤ/8)11yescharacteristic
76C22×C53#13711yescharacteristic
77C4×C53—42no—
78C4×C53—41no—
79C4×C53—41no—
80C4×C53—81no—
81C4×C53#12511yescharacteristic
82C4×C53#12511yescharacteristic
83C4×C53#12611yescharacteristic
84C4×C53#12611yescharacteristic
85Dic53—81no—
86Dic53—81no—
87#2101—531nomaximal, Sylow
88C2×C53#40011yescharacteristic
89C2×C53#47811yescharacteristic
90C2×C53#47911yescharacteristic
91C2×C53#48211yescharacteristic
92C2×C53#48511yescharacteristic
93C2×C53#48811yescharacteristic
94C2×C53#48911yescharacteristic
95#389D5311yescharacteristic
96#382—531no—
97#340—531no—
98#340—531no—
99#340—531no—
100#340—531no—
101#336—531no—
102C53#210111yescharacteristic, complemented, Sylow
103C22×C8—531no—
104C22×C8—531no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—531no—
107C2×C4⋊C4—531no—
108C2×C4⋊C4C2×D5311yescharacteristic
109C2×C4⋊C4C2×D5311yescharacteristic
110C2×C42C2×D5311yescharacteristic
111#95—21no—
112#95—1061no—
113#95—1061no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—1061no—
117C22×C4—1061no—
118C22×C4C22⋊D5311yescharacteristic
119C22×C4C22⋊D5311yescharacteristic
120C22×C4C22×D5311yescharacteristic
121C2×C8—1061no—
122C2×C8—1061no—
123C2×C8—1061no—
124C2×C8—1061no—
125C2×C8—1061no—
126C2×C8—1061no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—42no—
131C4⋊C4—1061no—
132C4⋊C4—1061no—
133C4⋊C4—1061no—
134C4⋊C4—1061no—
135C4⋊C4Dic10612yes—
136C42—21no—
137C42—21no—
138C23C2×C22⋊D5311yescharacteristic
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—2121no—
152C2×C4—2121no—
153C2×C4—2121no—
154C2×C4—2121no—
155C2×C4—2121no—
156C2×C4—2121no—
157C2×C4C2×Dic10611yescharacteristic
158C2×C4#1171994511yescharacteristic
159C2×C4#1171994711yescharacteristic
160C2×C4#1171994711yescharacteristic
161C2×C4#1171994911yescharacteristic
162C2×C4#1171994911yescharacteristic
163C8—2121no—
164C8—2121no—
165C22#1313319611yescharacteristic
166C22#1313319811yescharacteristic
167C22#1313320611yescharacteristic
168C22#1313323711yescharacteristic
169C22#1313323711yescharacteristic
170C22#1313324311yescharacteristic
171C22#1313324811yescharacteristic
172C4—42no—
173C4—41no—
174C4—41no—
175C4—81no—
176C4—4241no—
177C4—4241no—
178C4#1313320811yescharacteristic
179C4#1313320811yescharacteristic
180C4#1313321211yescharacteristic
181C4#1313321211yescharacteristic
182C2#1681056211yescharacteristic
183C2#1681092311yescharacteristic
184C2#1681093011yescharacteristic
185C2#1681094411yescharacteristic
186C2#1681096111yescharacteristic
187C2#1681098011yescharacteristic
188C2#1681098511yescharacteristic
subgroupG/H
centerC23C2×C22⋊D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#389C2
socleC23×C53C2×D4
hypercenter#389D53
hypocenterC53#2101
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
102. C53#2101semidirect