Finite Group Databasebeta

← #27294866

Subgroups — #27294866

Order 13568.

446 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 83 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#20277999C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272735C212yesmaximal, complemented
4#20272714C212yesmaximal, complemented
5C53×#1799C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic, complemented
7#16810587—21no—
8#16810561—21no—
9#16810548C411yescharacteristic, complemented
10#16810548C2211yescharacteristic
11#16810537—21no—
12#16810535C411yescharacteristic, complemented
13#16810535C2211yescharacteristic
14C4×C22⋊Dic53—21no—
15C4×C22⋊Dic53—21no—
16#16810469C411yescharacteristic, complemented
17#16810349—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C53×#380—21no—
22C53×#342C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—41no—
25C4×Dic106—41no—
26C4×Dic106—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C22⋊#88322—22no—
36C22⋊#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C4×D4×C53—41no—
48C4×D4×C53—41no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×C53C2×C411yescharacteristic
55#99×C53—42no—
56#95×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C2×Dic106—41no—
80C2×Dic106—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—82no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C42×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—41no—
93C22⋊Dic53—41no—
94C4⋊Dic53—41no—
95#11719919—42no—
96C4×Dic53—42no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99#11719917—82no—
100C2×#88322—42no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111C23×C53AGL(1,ℤ/8)11yescharacteristic
112C23×C53#13711yescharacteristic
113D4×C53—42no—
114D4×C53—42no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—81no—
128C2×C4×C53—81no—
129C2×C4×C53—82no—
130C2×C4×C53—82no—
131C2×C4×C53—82no—
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#9911yescharacteristic
134C2×Dic53—22no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145C2×Dic53—41no—
146Dic106—42no—
147Dic106—42no—
148#88322—82no—
149#11823—531nomaximal, Sylow
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—42no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53C2×#9911yescharacteristic
165C22×C53#41711yescharacteristic
166C22×C53#42111yescharacteristic
167C4×C53—21no—
168C4×C53—41no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—162no—
173C4×C53#35412yes—
174Dic53—42no—
175Dic53—81no—
176Dic53—81no—
177Dic53—81no—
178#2398—531no—
179#1799D5311yescharacteristic
180#1465—531no—
181#1351—532no—
182#1344—532no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—41no—
186C2×C53—41no—
187C2×C53—81no—
188C2×C53#151311yescharacteristic
189C2×C53#212011yescharacteristic
190C2×C53C2×#35411yescharacteristic
191C2×C4×Q8—531no—
192C2×C4×D4C2×D5311yescharacteristic
193C2×#99—21no—
194#407—1061no—
195#400—1061no—
196#397—531no—
197#397—531no—
198#394—1061no—
199#393—531no—
200#393—531no—
201#380—21no—
202C4×C22⋊C4—1061no—
203C4×C22⋊C4—1061no—
204#376—531no—
205#342C2×D5312yes—
206#324—1061no—
207#323—532no—
208C53#1182311yescharacteristic, complemented, Sylow
209C22×Q8—531no—
210C22×D4C22⋊D5311yescharacteristic
211C23×C4D21211yescharacteristic
212C23×C4C22⋊D5311yescharacteristic
213C2×M16—2121no—
214C4×Q8—2121no—
215C4×Q8—2121no—
216C4×D4—41no—
217C4×D4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4D21211yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4—1061no—
225C2×C22⋊C4—1061no—
226C2×C22⋊C4—1061no—
227C2×C22⋊C4C4×D5311yescharacteristic
228C2×C22⋊C4C22×D5311yescharacteristic
229C2×C42—1061no—
230C2×C42—1061no—
231C2×C42C4×D5311yescharacteristic
232#99—42no—
233C22⋊C8—1062no—
234C22⋊C8—2121no—
235#95—41no—
236#95—531no—
237#95—531no—
238#95—1061no—
239#95—1061no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243C24#1171992111yescharacteristic
244C24C2×C22⋊D5311yescharacteristic
245C2×Q8—2121no—
246C2×Q8—2122no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4#1171992111yescharacteristic
263C22×C4#1171992111yescharacteristic
264C22×C4#1171992111yescharacteristic
265C22×C4C2×C4×D5311yescharacteristic
266C22×C4C2×D21211yescharacteristic
267M16—4242no—
268C2×C8—2122no—
269C4⋊C4—41no—
270C4⋊C4—82no—
271C4⋊C4—2121no—
272C4⋊C4—2122no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—2122no—
278C22⋊C4—2122no—
279C22⋊C4—2122no—
280C22⋊C4—2121no—
281C22⋊C4—2121no—
282C42—21no—
283C42—21no—
284C42—2122no—
285C42—2121no—
286C42—2121no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23C2×#1171992111yescharacteristic
297C23#1313324311yescharacteristic
298C23#1313324811yescharacteristic
299Q8—2122no—
300Q8—2122no—
301D4—42no—
302D4—42no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—1062no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4—2121no—
329C2×C4—2121no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4#1313313111yescharacteristic
333C2×C4#1313313111yescharacteristic
334C8—4242no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22C2×#1313313111yescharacteristic
350C22#1681062211yescharacteristic
351C22#1681064111yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—162no—
358C4—4241no—
359C4—4241no—
360C4—4241no—
361C4—2122no—
362C4#1681041712yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2—81no—
368C2#2027325611yescharacteristic
369C2#2027635411yescharacteristic
370C2C2×#1681041711yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1799C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1799D53
hypocenterC53#11823
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20272735C2semidirect
4. #20272714C2semidirect
6. C2×C4×Dic106C4semidirect
9. #16810548C4semidirect
12. #16810535C4semidirect
16. #16810469C4semidirect
208. C53#11823semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202779992
#202730532
#202727352
#202727142
C53×#17992
#1182353