Finite Group Databasebeta

← #20279645

Subgroups — #20279645

Order 6784.

900 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 169 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×#13133200C211yesmaximal, characteristic, complemented
2C2×#13133189C211yesmaximal, characteristic, complemented
3C2×#13133187C211yesmaximal, characteristic, complemented
4C2×#13133185C211yesmaximal, characteristic, complemented
5C22×#11719921C211yesmaximal, characteristic, complemented
6#16810558C212yesmaximal, complemented
7#16810532C212yesmaximal, complemented
8#16810524C212yesmaximal, complemented
9#16810523C212yesmaximal, complemented
10C22⋊C4×Dic53C211yesmaximal, characteristic, complemented
11C2×#123×C53C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C22×D212C2212yes—
14C22×C22⋊D53C2211yescharacteristic
15C22×C22⋊D53C2212yes—
16C22×C22⋊D53C2211yescharacteristic, complemented
17C23×Dic53C2211yescharacteristic
18#13133200—24no—
19#13133189—24no—
20#13133187—24no—
21#13133185—24no—
22C2×#11719919C2212yescomplemented
23C2×#11719919C2211yescharacteristic, complemented
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic, complemented
26C2×#11719921—24no—
27C2×#11719921C2212yes—
28C2×#11719921C2212yes—
29C2×#11719921C2212yes—
30C2×#11719921C2212yes—
31C2×#11719921C2211yescharacteristic, complemented
32C2×#11719921C2212yescomplemented
33C2×C4×Dic53C2212yescomplemented
34#13133124C2212yescomplemented
35C22×D4×C53C2211yescharacteristic
36C23×C4×C53C2211yescharacteristic
37#123×C53—24no—
38C2×C4⋊C4×C53C2212yescomplemented
39C2×C22⋊C4×C53C2211yescharacteristic
40C2×C22⋊C4×C53C2212yescomplemented
41C24×C53C2311yescharacteristic
42C24×C53C2311yescharacteristic
43C23×D53—22no—
44C23×D53—24no—
45C23×D53—22no—
46C23×D53—22no—
47C23×D53—42no—
48C23×D53C2311yescharacteristic
49C23×D53C2312yes—
50C2×D4×C53—22no—
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53C2312yes—
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2312yes—
59C2×C22⋊D53—22no—
60C2×C22⋊D53—22no—
61C2×C22⋊D53—22no—
62C2×C22⋊D53—24no—
63C2×C22⋊D53—24no—
64C2×C22⋊D53—22no—
65C2×C22⋊D53—42no—
66C2×C22⋊D53—42no—
67C22×Dic53—42no—
68C22×Dic53D414yescomplemented
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2312yes—
72C22×Dic53C2311yescharacteristic
73C2×D212—24no—
74C2×D212—24no—
75C2×D212—42no—
76C4⋊C4×C53—24no—
77C22⋊C4×C53—24no—
78C22⋊C4×C53D414yescomplemented
79C22⋊Dic53—22no—
80C22⋊Dic53D414yescomplemented
81#11719921—24no—
82#11719921—24no—
83#11719921—24no—
84#11719921—24no—
85#11719921—22no—
86#11719921—24no—
87#11719919—22no—
88#11719919—24no—
89C4×Dic53—42no—
90C23×C53—22no—
91C23×C53—22no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53C2×D412yes—
98C23×C53C2×D412yes—
99C23×C53C2×D412yes—
100C23×C53C2411yescharacteristic
101C22×D53—28no—
102C22×D53—22no—
103C22×D53—22no—
104C22×D53—22no—
105C22×D53—22no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53—22no—
112C22×D53—22no—
113C22×D53—48no—
114C22×D53—48no—
115C22×D53—44no—
116C22×D53—44no—
117C22×D53#4114yes—
118D4×C53—44no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53—22no—
122C2×C4×C53—22no—
123C2×C4×C53—22no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—44no—
128C2×C4×C53C2×D414yes—
129C22⋊D53—48no—
130C22⋊D53—44no—
131C22⋊D53—44no—
132C2×Dic53—22no—
133C2×Dic53—22no—
134C2×Dic53—22no—
135C2×Dic53—22no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—44no—
140C2×Dic53C2×D412yes—
141C2×Dic53C2×D412yes—
142C2×Dic53C2×D414yes—
143D212—48no—
144C22×C53—22no—
145C22×C53—22no—
146C22×C53—24no—
147C22×C53—24no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53#12714yes—
154C22×C53C22×D411yescharacteristic
155C22×C53C22×D411yescharacteristic
156C22×C53C22×D411yescharacteristic
157C22×C53C2×#4112yes—
158C22×C53H(5,𝔽2)12yes—
159C2×D53—24no—
160C2×D53—24no—
161C2×D53—24no—
162C2×D53—416no—
163C2×D53—42no—
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—42no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—42no—
171C2×D53—42no—
172C4×C53—24no—
173C4×C53—42no—
174C4×C53—41no—
175Dic53—22no—
176Dic53—22no—
177Dic53—24no—
178Dic53—41no—
179#2539—531nomaximal, Sylow
180C2×C53—24no—
181C2×C53—81no—
182C2×C53#52511yescharacteristic
183C2×C53C2×#12711yescharacteristic
184C2×C53#53411yescharacteristic
185C2×C53#54612yes—
186C2×C53#54612yes—
187D53—44no—
188D53—81no—
189C2×#123—531no—
190C2×#123D5311yescharacteristic, complemented
191C2×#121—531no—
192C2×#120—531no—
193C2×#120—531no—
194C22×C22⋊C4—531no—
195#399—532no—
196#392—532no—
197#390—532no—
198#390—532no—
199C4×C22⋊C4—531no—
200C53#253911yescharacteristic, complemented, Sylow
201C25—531no—
202C22×D4—531no—
203C22×D4—532no—
204C22×D4—531no—
205C22×D4—532no—
206C22×D4C2×D5311yescharacteristic
207C23×C4—531no—
208C23×C4C2×D5311yescharacteristic
209#123—24no—
210#123—1064no—
211#121—1064no—
212#120—1064no—
213#120—1064no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—532no—
216C2×C4⋊C4C2×D5312yescomplemented
217C2×C22⋊C4—531no—
218C2×C22⋊C4—531no—
219C2×C22⋊C4—532no—
220C2×C22⋊C4—532no—
221C2×C22⋊C4—532no—
222C2×C22⋊C4—532no—
223C2×C22⋊C4—531no—
224C2×C22⋊C4—532no—
225C2×C22⋊C4—1064no—
226C2×C22⋊C4C2×D5311yescharacteristic
227C2×C22⋊C4C2×D5312yescomplemented
228C2×C42—532no—
229#95—532no—
230C24—532no—
231C24—531no—
232C24—1064no—
233C24—1062no—
234C24—1062no—
235C24—1062no—
236C24—2122no—
237C24C22×D5311yescharacteristic
238C24C22×D5311yescharacteristic
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—1064no—
243C2×D4—1064no—
244C2×D4—1062no—
245C2×D4—1064no—
246C2×D4—1062no—
247C2×D4—1062no—
248C2×D4—1062no—
249C2×D4—1064no—
250C2×D4—2122no—
251C2×D4—2122no—
252C2×D4—2122no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—531no—
257C22×C4—532no—
258C22×C4—531no—
259C22×C4—534no—
260C22×C4—531no—
261C22×C4—2122no—
262C22×C4C22×D5312yes—
263C22×C4C22×D5311yescharacteristic
264C22×C4C22×D5312yes—
265C4⋊C4—24no—
266C4⋊C4—1062no—
267C4⋊C4—1064no—
268C22⋊C4—24no—
269C22⋊C4—534no—
270C22⋊C4—1064no—
271C22⋊C4—1064no—
272C22⋊C4—1064no—
273C22⋊C4—1062no—
274C22⋊C4—1064no—
275C22⋊C4—1062no—
276C22⋊C4—1064no—
277C22⋊C4C22⋊D5314yescomplemented
278C42—2122no—
279C23—22no—
280C23—22no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—534no—
287C23—1068no—
288C23—1062no—
289C23—1062no—
290C23—1062no—
291C23—1062no—
292C23—1062no—
293C23—1062no—
294C23—1061no—
295C23—1061no—
296C23—1061no—
297C23—1062no—
298C23—1062no—
299C23—2128no—
300C23—2128no—
301C23—2124no—
302C23—2124no—
303C23D4×D5312yes—
304C23D4×D5312yes—
305C23C2×C22⋊D5312yes—
306C23C23×D5311yescharacteristic
307D4—44no—
308D4—2128no—
309D4—2128no—
310D4—2124no—
311D4—2124no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—534no—
322C2×C4—532no—
323C2×C4—532no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—1062no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—2124no—
332C2×C4C2×C22⋊D5314yes—
333C22—24no—
334C22—22no—
335C22—22no—
336C22—24no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—1064no—
343C22—1064no—
344C22—1064no—
345C22—21216no—
346C22—2122no—
347C22—2121no—
348C22—2121no—
349C22—2121no—
350C22—2122no—
351C22—2122no—
352C22—2122no—
353C22—2122no—
354C22—2122no—
355C22#1313321314yes—
356C22C2×D4×D5311yescharacteristic
357C22C2×D4×D5311yescharacteristic
358C22C22×C22⋊D5311yescharacteristic
359C22#41×D5312yes—
360C22#1313326312yes—
361C4—24no—
362C4—42no—
363C4—41no—
364C4—1064no—
365C4—1062no—
366C4—1062no—
367C4—2121no—
368C2—24no—
369C2—81no—
370C2—4241no—
371C2—2124no—
372C2#1681113111yescharacteristic
373C2C2×#1313321311yescharacteristic
374C2#1681118411yescharacteristic
375C2#1681126112yes—
376C2#1681126512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2539
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#13133200C2semidirect
2. C2×#13133189C2semidirect
3. C2×#13133187C2semidirect
4. C2×#13133185C2semidirect
5. C22×#11719921C2semidirect
6. #16810558C2semidirect
7. #16810532C2semidirect
8. #16810524C2semidirect
9. #16810523C2semidirect
10. C22⋊C4×Dic53C2semidirect
11. C2×#123×C53C2semidirect
16. C22×C22⋊D53C22semidirect
22. C2×#11719919C22semidirect
23. C2×#11719919C22semidirect
25. C2×C22⋊Dic53C22semidirect
31. C2×#11719921C22semidirect
32. C2×#11719921C22semidirect
33. C2×C4×Dic53C22semidirect
34. #13133124C22semidirect
38. C2×C4⋊C4×C53C22semidirect
40. C2×C22⋊C4×C53C22semidirect
68. C22×Dic53D4semidirect
78. C22⋊C4×C53D4semidirect
80. C22⋊Dic53D4semidirect
190. C2×#123D53semidirect
200. C53#2539semidirect
216. C2×C4⋊C4C2×D53semidirect
227. C2×C22⋊C4C2×D53semidirect
277. C22⋊C4C22⋊D53semidirect