Finite Group Databasebeta

← #20279713

Subgroups — #20279713

Order 6784.

676 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13133213C211yesmaximal, characteristic
2C2×#13133203C212yesmaximal, complemented
3C2×#13133182C211yesmaximal, characteristic, complemented
4#16810541C211yesmaximal, characteristic, complemented
5#16810540C211yesmaximal, characteristic, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810523C212yesmaximal, complemented
8#16810523C212yesmaximal, complemented
9#16810523C212yesmaximal, complemented
10#16810474C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C22×Dic106C2211yescharacteristic
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16#13133213—24no—
17#13133203—28no—
18#13133182—24no—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×C22⋊Dic53C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29C2×C4×Dic53C2211yescharacteristic
30C2×C4×Dic53C2212yes—
31C2×C4×Dic53C2211yescharacteristic
32#13133124C2211yescharacteristic
33C22×D4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2312yes—
39C23×D53C2312yes—
40C2×D4×C53—22no—
41C2×D4×C53—42no—
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×C4×C53C2311yescharacteristic
46C2×C22⋊D53—24no—
47C2×C22⋊D53—24no—
48C2×C22⋊D53—24no—
49C2×C22⋊D53—24no—
50C2×C22⋊D53—24no—
51C2×C22⋊D53—24no—
52C22×Dic53C2312yes—
53C22×Dic53C2311yescharacteristic
54C22×Dic53C2311yescharacteristic
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2311yescharacteristic
57C2×D212—22no—
58C2×D212—42no—
59C2×Dic106—22no—
60C2×Dic106—22no—
61C2×Dic106D414yescomplemented
62C4⋊C4×C53—22no—
63C22⋊C4×C53—24no—
64C22⋊C4×C53—42no—
65C22⋊Dic53—24no—
66C22⋊Dic53—42no—
67#11719921—24no—
68#11719921—24no—
69#11719921—42no—
70#11719921—42no—
71#11719919—22no—
72#11719919—22no—
73#11719919—24no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—24no—
77C4×Dic53D414yes—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53—22no—
84C23×C53—22no—
85C23×C53C2411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—22no—
89C22×D53—22no—
90C22×D53—22no—
91C22×D53—22no—
92C22×D53—22no—
93D4×C53—44no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53C2×D412yes—
103C2×C4×C53C2×D412yes—
104C22⋊D53—48no—
105C22⋊D53—48no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53C2×D412yes—
114C2×Dic53C2×D412yes—
115C2×Dic53C2×D412yes—
116C2×Dic53C2×D412yes—
117C2×Dic53#4112yes—
118C2×Dic53#4114yes—
119D212—44no—
120Dic106—28no—
121C22×C53—42no—
122C22×C53—42no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—42no—
128C22×C53C22×D411yescharacteristic
129C22×C53C22×D411yescharacteristic
130C22×C53C2×#4111yescharacteristic
131C22×C53C2×#4112yes—
132C22×C53H(5,𝔽2)12yes—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C4×C53—22no—
141C4×C53—22no—
142C4×C53—41no—
143C4×C53—42no—
144Dic53—22no—
145Dic53—41no—
146Dic53—42no—
147Dic53#12114yes—
148Dic53#12414yescomplemented
149#2548—531nomaximal, Sylow
150C2×C53—82no—
151C2×C53C2×#12111yescharacteristic
152C2×C53C2×#12411yescharacteristic
153C2×C53#53511yescharacteristic
154C2×C53#55012yes—
155C2×C53#55912yes—
156D53—82no—
157C2×#127—531no—
158C2×#124—532no—
159C2×C4×Q8—531no—
160#394—531no—
161#394—531no—
162#394—531no—
163#394D5311yescharacteristic, complemented
164#390—532no—
165#390—532no—
166#390—532no—
167C4×C4⋊C4—531no—
168C53#254811yescharacteristic, complemented, Sylow
169C22×Q8—531no—
170C22×D4—532no—
171C22×D4—532no—
172C22×D4—531no—
173C22×D4C2×D5311yescharacteristic
174#127—1064no—
175#124—1068no—
176C4×Q8—1064no—
177C2×C4⋊C4—531no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—532no—
180C2×C4⋊C4C2×D5311yescharacteristic
181C2×C22⋊C4—532no—
182C2×C22⋊C4—532no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—532no—
185C2×C22⋊C4—532no—
186C2×C22⋊C4—532no—
187C2×C22⋊C4C2×D5312yes—
188C2×C22⋊C4C2×D5312yes—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—532no—
193#95—531no—
194#95C2×D5311yescharacteristic
195C24—532no—
196C24C22×D5312yes—
197C2×Q8—534no—
198C2×Q8—1062no—
199C2×Q8—1062no—
200C2×D4—22no—
201C2×D4—42no—
202C2×D4—1064no—
203C2×D4—1064no—
204C2×D4—1064no—
205C2×D4—1064no—
206C2×D4—1064no—
207C2×D4—1064no—
208C2×D4—1062no—
209C2×D4—2122no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—532no—
214C22×C4—531no—
215C22×C4C22×D5312yes—
216C22×C4C22×D5311yescharacteristic
217C22×C4C22×D5311yescharacteristic
218C22×C4C22×D5311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—1062no—
221C4⋊C4—1064no—
222C4⋊C4—1062no—
223C22⋊C4—24no—
224C22⋊C4—42no—
225C22⋊C4—1064no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C22⋊C4—2122no—
231C42—534no—
232C42—1062no—
233C42—1062no—
234C42—1064no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—1062no—
243C23—1062no—
244C23—1062no—
245C23—1062no—
246C23—1062no—
247C23—1062no—
248C23—1062no—
249C23C23×D5311yescharacteristic
250Q8—1068no—
251D4—44no—
252D4—2124no—
253D4—2128no—
254D4—2128no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—532no—
264C2×C4—532no—
265C2×C4—532no—
266C2×C4—534no—
267C2×C4—532no—
268C2×C4—532no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1062no—
272C2×C4—1061no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4D4×D5312yes—
277C2×C4D4×D5312yes—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—2122no—
286C22—2122no—
287C22—2122no—
288C22—2122no—
289C22—2122no—
290C22—2122no—
291C22—2122no—
292C22C2×D4×D5311yescharacteristic
293C22C2×D4×D5311yescharacteristic
294C22#41×D5312yes—
295C22#41×D5311yescharacteristic
296C22#1313326212yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—41no—
301C4—534no—
302C4—534no—
303C4—1062no—
304C4—2122no—
305C4—2121no—
306C2—82no—
307C2—4242no—
308C2#121×D5311yescharacteristic
309C2#124×D5311yescharacteristic
310C2#1681118811yescharacteristic
311C2#1681129112yes—
312C2#1681135712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2548
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133203C2semidirect
3. C2×#13133182C2semidirect
4. #16810541C2semidirect
5. #16810540C2semidirect
6. #16810537C2semidirect
7. #16810523C2semidirect
8. #16810523C2semidirect
9. #16810523C2semidirect
10. #16810474C2semidirect
11. C53×#394C2semidirect
61. C2×Dic106D4semidirect
148. Dic53#124semidirect
163. #394D53semidirect
168. C53#2548semidirect