Finite Group Databasebeta

← #20276705

Subgroups — #20276705

Order 6784.

258 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 45 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#16810812C211yesmaximal, characteristic, complemented
2#16810809C211yesmaximal, characteristic
3#16810800C211yesmaximal, characteristic, complemented
4#16810418C211yesmaximal, characteristic, complemented
5#16810413C211yesmaximal, characteristic, complemented
6#16810410C211yesmaximal, characteristic, complemented
7C53×#551C211yesmaximal, characteristic, complemented
8#13133262C2211yescharacteristic
9#13133248—21no—
10#13133200—21no—
11#13133186—21no—
12#13133145—21no—
13#13133134C2211yescharacteristic
14#13133132C2211yescharacteristic
15#13133132C2211yescharacteristic
16#13133131—21no—
17#13133131—21no—
18#126×C53—21no—
19#124×C53C2211yescharacteristic
20#123×C53—21no—
21#123×C53C2211yescharacteristic
22C22⋊Q8×C53—21no—
23#120×C53C2211yescharacteristic
24C4×D4×C53—21no—
25#117×C53—21no—
26C2×C22⋊C4×C53—21no—
27C24×C53D411yescharacteristic
28C2×Q8×C53D411yescharacteristic, complemented
29C2×D4×C53—21no—
30C2×D4×C53C2311yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53D411yescharacteristic, complemented
34C2×C22⋊D53—21no—
35C2×C22⋊D53—21no—
36C2×C22⋊D53—41no—
37#11719947—41no—
38D4×D53—41no—
39#11719945—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53—21no—
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53—21no—
50C22⋊C4×C53D411yescharacteristic
51C22⋊C4×C53D411yescharacteristic, complemented
52C42×C53—21no—
53C42×C53D411yescharacteristic, complemented
54C22⋊Dic53—21no—
55C22⋊Dic53—21no—
56C22⋊Dic53—41no—
57#11719925—41no—
58#11719924—41no—
59#11719921—41no—
60#11719919—41no—
61#11719917—21no—
62C23×C53—21no—
63C23×C53—41no—
64C23×C53—42no—
65C23×C53C2×D411yescharacteristic
66C23×C53C2×D411yescharacteristic
67C22×D53—21no—
68C22×D53—21no—
69Q8×C53—41no—
70D4×C53—41no—
71D4×C53—42no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—42no—
78C2×C4×C53—41no—
79C2×C4×C53—81no—
80C2×C4×C53C2×D411yescharacteristic
81C22⋊D53—41no—
82C22⋊D53—41no—
83C22⋊D53—41no—
84C22⋊D53—82no—
85C2×Dic53—21no—
86C2×Dic53—21no—
87C2×Dic53—41no—
88C2×Dic53—41no—
89D212—21no—
90C4×D53—41no—
91Dic106—21no—
92#88322—41no—
93C22×C53—22no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—42no—
100C22×C53—41no—
101C22×C53—81no—
102C22×C53#12011yescharacteristic
103C2×D53—41no—
104C2×D53—81no—
105C2×D53—81no—
106C4×C53—21no—
107C4×C53—41no—
108C4×C53—41no—
109C4×C53—41no—
110C4×C53—41no—
111C4×C53—41no—
112Dic53—41no—
113Dic53—81no—
114Dic53—81no—
115#2197—531nomaximal, Sylow
116C2×C53—21no—
117C2×C53—41no—
118C2×C53—41no—
119C2×C53—42no—
120C2×C53B(4,𝔽2)11yescharacteristic
121D53—81no—
122#551D5311yescharacteristic, complemented
123#458—531no—
124B(4,𝔽2)—531no—
125#455—531no—
126#354—531no—
127#352—531no—
128AΣL(1,𝔽16)—531no—
129C53#219711yescharacteristic, complemented, Sylow
130H(5,𝔽2)—531no—
131#137—1061no—
132#126—21no—
133#124C2×D5311yescharacteristic
134#123—21no—
135#123—1061no—
136#123C2×D5311yescharacteristic
137C22⋊Q8—21no—
138#120—1061no—
139#120C2×D5311yescharacteristic
140C4×D4—21no—
141#117—21no—
142C2×C22⋊C4—21no—
143#104—1061no—
144#100—531no—
145#99—531no—
146#99—531no—
147#99—1061no—
148#99—1061no—
149C24C22⋊D5311yescharacteristic
150#41—1061no—
151#41—2121no—
152C2×Q8C22⋊D5311yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—1061no—
155C2×D4—1061no—
156C2×D4—2121no—
157C2×D4—2121no—
158C2×D4C22×D5311yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C22⋊D5311yescharacteristic, complemented
162Q16—2121no—
163SD16—2121no—
164M16—1061no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—2121no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C22⋊C4—2121no—
179C22⋊C4—2121no—
180C22⋊C4C22⋊D5311yescharacteristic
181C22⋊C4C22⋊D5311yescharacteristic, complemented
182C42—21no—
183C42C22⋊D5311yescharacteristic, complemented
184C23—21no—
185C23—41no—
186C23—42no—
187C23—1061no—
188C23—1061no—
189C23C2×C22⋊D5311yescharacteristic
190C23C2×C22⋊D5311yescharacteristic
191Q8—41no—
192Q8—1061no—
193D4—41no—
194D4—42no—
195D4—1061no—
196D4—4242no—
197D4—2121no—
198D4—2121no—
199D4—2121no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—2121no—
211C2×C4—2121no—
212C2×C4—2121no—
213C2×C4C2×C22⋊D5311yescharacteristic
214C8—2121no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—4241no—
225C22—4241no—
226C22—2121no—
227C22#1313318611yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—4241no—
235C4—4241no—
236C4—2121no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—4241no—
242C2#1681081011yescharacteristic
subgroupG/H
centerC2#16810810
commutatorC2×D4×C53C23
FrattiniC2×D4C22×D53
FittingC53×#551C2
socleC2×C53B(4,𝔽2)
hypercenter#551D53
hypocenterC53#2197
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #551

Complements

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

NH ≅ G/Ntype
1. #16810812C2semidirect
3. #16810800C2semidirect
4. #16810418C2semidirect
5. #16810413C2semidirect
6. #16810410C2semidirect
7. C53×#551C2semidirect
28. C2×Q8×C53D4semidirect
33. C22×C4×C53D4semidirect
51. C22⋊C4×C53D4semidirect
53. C42×C53D4semidirect
122. #551D53semidirect
129. C53#2197semidirect
152. C2×Q8C22⋊D53semidirect
161. C22×C4C22⋊D53semidirect
181. C22⋊C4C22⋊D53semidirect
183. C42C22⋊D53semidirect