Finite Group Databasebeta

← #20274662

Subgroups — #20274662

Order 6784.

670 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#13133262C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic, complemented
3C2×#13133131C212yesmaximal, complemented
4#16810502C212yesmaximal, complemented
5C53×#386C211yesmaximal, characteristic, complemented
6#13133262—22no—
7#13133262—24no—
8#13133262C414yescomplemented
9C2×#11719945—21no—
10C2×#11719947—22no—
11C22×C22⋊D53—21no—
12C22×C22⋊D53C2212yes—
13C2×D4×D53—22no—
14C2×C22⋊Dic53C2212yescomplemented
15C2×#11719921—22no—
16#13133132—22no—
17#13133131—24no—
18#13133124—22no—
19C22×D4×C53C2211yescharacteristic
20C23×C4×C53C2211yescharacteristic, complemented
21C2×C22⋊C4×C53—21no—
22C2×C22⋊C4×C53C2211yescharacteristic, complemented
23#95×C53—21no—
24C24×C53D411yescharacteristic
25C24×C53C2311yescharacteristic
26C23×D53—21no—
27C23×D53D412yes—
28C2×D4×C53—22no—
29C2×D4×C53—22no—
30C2×D4×C53C2×C412yes—
31C2×D4×C53D412yes—
32C22×C4×C53—21no—
33C22×C4×C53—22no—
34C22×C4×C53—21no—
35C22×C4×C53—42no—
36C22×C4×C53D411yescharacteristic
37C2×C22⋊D53—22no—
38C2×C22⋊D53—24no—
39C2×C22⋊D53—24no—
40C2×C22⋊D53—22no—
41C2×C22⋊D53—22no—
42C2×C22⋊D53—22no—
43C2×C22⋊D53—22no—
44C2×C22⋊D53C2×C414yes—
45C2×C22⋊D53D414yes—
46C22×Dic53—21no—
47C22×Dic53—22no—
48C22×Dic53D412yes—
49#11719947—28no—
50#11719947—28no—
51D4×D53—28no—
52D4×D53—28no—
53#11719945—24no—
54#11719945—24no—
55C2×D212—21no—
56C2×C4×D53—22no—
57C2×Dic106—21no—
58C22⋊C4×C53—22no—
59C22⋊C4×C53—42no—
60C22⋊Dic53—22no—
61C22⋊Dic53—22no—
62#11719921—44no—
63C23×C53—21no—
64C23×C53—21no—
65C23×C53—41no—
66C23×C53—41no—
67C23×C53—41no—
68C23×C53—41no—
69C23×C53C22⋊C412yescomplemented
70C23×C53C22×C411yescharacteristic
71C23×C53C2×D411yescharacteristic
72C23×C53C2×D412yes—
73C23×C53C2×D412yes—
74C23×C53C2×D411yescharacteristic
75C22×D53—22no—
76C22×D53—24no—
77C22×D53—22no—
78C22×D53—42no—
79C22×D53—42no—
80C22×D53—81no—
81C22×D53—81no—
82C22×D53C22⋊C414yescomplemented
83D4×C53—24no—
84D4×C53—24no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—24no—
88C2×C4×C53—22no—
89C2×C4×C53—42no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53C22⋊C412yescomplemented
96C22⋊D53—28no—
97C22⋊D53—24no—
98C22⋊D53—24no—
99C22⋊D53—24no—
100C22⋊D53—24no—
101C2×Dic53—24no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—42no—
105C2×Dic53—42no—
106C2×Dic53—42no—
107C2×Dic53C22⋊C414yescomplemented
108D212—24no—
109C4×D53—28no—
110Dic106—24no—
111C22×C53—22no—
112C22×C53—21no—
113C22×C53—22no—
114C22×C53—21no—
115C22×C53—22no—
116C22×C53—41no—
117C22×C53—41no—
118C22×C53—42no—
119C22×C53—42no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—81no—
123C22×C53C2×C22⋊C412yes—
124C22×C53C2×C22⋊C411yescharacteristic
125C22×C53#12011yescharacteristic
126C22×C53#12011yescharacteristic
127C22×C53#12012yes—
128C2×D53—24no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—44no—
132C2×D53—82no—
133C2×D53—84no—
134C4×C53—22no—
135C4×C53—42no—
136C4×C53—81no—
137Dic53—24no—
138Dic53—82no—
139#1790—531nomaximal, Sylow
140C2×C53—22no—
141C2×C53—21no—
142C2×C53—22no—
143C2×C53—21no—
144C2×C53—42no—
145C2×C53#37911yescharacteristic
146C2×C53B(4,𝔽2)11yescharacteristic
147C2×C53#45811yescharacteristic
148D53—44no—
149C2×H(5,𝔽2)—531no—
150C2×#99—531no—
151C2×#99—532no—
152#386—532no—
153#386D5311yescharacteristic, complemented
154C53#179011yescharacteristic, complemented, Sylow
155H(5,𝔽2)—534no—
156H(5,𝔽2)—1064no—
157H(5,𝔽2)—1062no—
158C2×#41—1062no—
159C2×#41—1061no—
160C22×D4—532no—
161C22×D4—1062no—
162C22×D4—1061no—
163C22×D4C2×D5311yescharacteristic
164C23×C4C2×D5311yescharacteristic, complemented
165C2×C22⋊C4—21no—
166C2×C22⋊C4—532no—
167C2×C22⋊C4—1062no—
168C2×C22⋊C4C2×D5311yescharacteristic, complemented
169#99—1064no—
170#99—1062no—
171#95—21no—
172#95—1062no—
173C24—532no—
174C24—1061no—
175C24C22⋊D5311yescharacteristic
176C24C22×D5311yescharacteristic
177#41—1064no—
178#41—1068no—
179#41—1068no—
180#41—1064no—
181C2×Q8—1061no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—534no—
185C2×D4—534no—
186C2×D4—1061no—
187C2×D4—1068no—
188C2×D4—1068no—
189C2×D4—1062no—
190C2×D4—1064no—
191C2×D4—1062no—
192C2×D4—1062no—
193C2×D4—1062no—
194C2×D4—1062no—
195C2×D4—1064no—
196C2×D4C4×D5312yes—
197C2×D4D21212yes—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—42no—
202C22×C4—532no—
203C22×C4—1062no—
204C22×C4—1061no—
205C22×C4—1062no—
206C22×C4C22⋊D5311yescharacteristic
207C22⋊C4—22no—
208C22⋊C4—42no—
209C22⋊C4—1062no—
210C22⋊C4—1062no—
211C22⋊C4—2124no—
212C23—21no—
213C23—21no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—534no—
219C23—1064no—
220C23—1062no—
221C23—1062no—
222C23—4241no—
223C23—4241no—
224C23—2122no—
225C23—2122no—
226C23#1171992112yescomplemented
227C23C2×C4×D5311yescharacteristic
228C23C2×D21211yescharacteristic
229C23D4×D5312yes—
230C23D4×D5312yes—
231C23C2×C22⋊D5311yescharacteristic
232Q8—1064no—
233D4—24no—
234D4—24no—
235D4—1064no—
236D4—1068no—
237D4—1064no—
238D4—1064no—
239D4—1064no—
240D4—1064no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—534no—
252C2×C4—1068no—
253C2×C4—1064no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4#1171992112yescomplemented
260C22—21no—
261C22—22no—
262C22—21no—
263C22—22no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—42no—
268C22—42no—
269C22—41no—
270C22—41no—
271C22—81no—
272C22—1064no—
273C22—4242no—
274C22—4244no—
275C22—2124no—
276C22—2122no—
277C22—2122no—
278C22C22⋊C4×D5312yes—
279C22C2×#1171992111yescharacteristic
280C22#1313318511yescharacteristic
281C22#1313318511yescharacteristic
282C22#1313318712yes—
283C4—22no—
284C4—42no—
285C4—81no—
286C4—1064no—
287C4—4242no—
288C2—21no—
289C2—22no—
290C2—21no—
291C2—22no—
292C2—42no—
293C2—2124no—
294C2#1681047811yescharacteristic
295C2#1681080911yescharacteristic
296C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#386C2
socleC22×C53#120
hypercenter#386D53
hypocenterC53#1790
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×#13133132C2semidirect
3. C2×#13133131C2semidirect
4. #16810502C2semidirect
5. C53×#386C2semidirect
8. #13133262C4semidirect
14. C2×C22⋊Dic53C22semidirect
20. C23×C4×C53C22semidirect
22. C2×C22⋊C4×C53C22semidirect
69. C23×C53C22⋊C4semidirect
82. C22×D53C22⋊C4semidirect
95. C2×C4×C53C22⋊C4semidirect
107. C2×Dic53C22⋊C4semidirect
153. #386D53semidirect
154. C53#1790semidirect
164. C23×C4C2×D53semidirect
168. C2×C22⋊C4C2×D53semidirect
226. C23#11719921semidirect
259. C2×C4#11719921semidirect