Finite Group Databasebeta

← #27295213

Subgroups — #27295213

Order 13568.

452 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 95 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#20280069C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20272740C212yesmaximal, complemented
4#20272737C212yesmaximal, complemented
5C53×#1459C211yesmaximal, characteristic
6C2×C4×Dic106C411yescharacteristic, complemented
7C2×#13133132—21no—
8#16810559C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810535C411yescharacteristic, complemented
11#16810535C2211yescharacteristic
12#16810526—21no—
13#16810519—21no—
14#16810496—21no—
15#16810496—21no—
16#16810489C411yescharacteristic, complemented
17#16810483—21no—
18#16810390C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×C22⋊C8×C53—21no—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C4×Dic106—41no—
25C4×Dic106—41no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—41no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—22no—
37C2×C4×Dic53—21no—
38#13133132—42no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—41no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×D4×C53D411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C23×C4×C53D411yescharacteristic
48C22×C8×C53—41no—
49C4×D4×C53—41no—
50C4×D4×C53—41no—
51C2×C4⋊C4×C53D411yescharacteristic
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×C53C2×C411yescharacteristic
55C22⋊C8×C53—22no—
56C22⋊C8×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—41no—
60C2×D4×C53—41no—
61C2×D4×C53SD1612yes—
62C2×D4×C53Q1612yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—81no—
68C22×C4×C53—81no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—42no—
79C2×Dic106—41no—
80C2×Dic106—42no—
81C2×C8×C53—42no—
82C2×C8×C53—82no—
83C4⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C42×C53—21no—
87C42×C53—21no—
88C22⋊Dic53—42no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—81no—
93C22⋊Dic53—81no—
94C4⋊Dic53—42no—
95C4⋊Dic53—41no—
96C4⋊Dic53—82no—
97#11719919—41no—
98#11719919—41no—
99#11719919—81no—
100#11719919—81no—
101C4×Dic53—42no—
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×C53#10312yes—
111C23×C53C2×C22⋊C411yescharacteristic
112C23×C53C2×SD1611yescharacteristic
113C23×C53C2×Q1611yescharacteristic
114D4×C53—42no—
115D4×C53—42no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
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#9911yescharacteristic
130C2×C4×C53#9911yescharacteristic
131C2×C4×C53#10312yes—
132C2×Dic53—22no—
133C2×Dic53—41no—
134C2×Dic53—41no—
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—82no—
145C2×Dic53—82no—
146C2×Dic53—82no—
147Dic106—42no—
148Dic106—42no—
149C8×C53—82no—
150#11897—531nomaximal, Sylow
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—21no—
156C22×C53—42no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53—81no—
165C22×C53C2×#9911yescharacteristic
166C22×C53C2×#10311yescharacteristic
167C22×C53#42111yescharacteristic
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—22no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174Dic53—42no—
175Dic53—81no—
176Dic53—81no—
177Dic53—81no—
178Dic53—162no—
179#2580—531no—
180#1799—531no—
181#1459D5311yescharacteristic
182#1352—532no—
183#1351—532no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—41no—
187C2×C53—42no—
188C2×C53—41no—
189C2×C53#152011yescharacteristic
190C2×C53#212011yescharacteristic
191C2×C53#212711yescharacteristic
192C2×C4×Q8—531no—
193C2×C4×D4C2×D5311yescharacteristic
194C2×#99—1061no—
195C2×C22⋊C8—21no—
196#399—531no—
197#399—531no—
198#393—531no—
199#393—531no—
200#390—1061no—
201#389—1061no—
202#385—1061no—
203#385—1061no—
204#382—531no—
205#380—1061no—
206#342—532no—
207#324—21no—
208#323C2×D5312yes—
209C53#1189711yescharacteristic, complemented, Sylow
210C22×Q8—531no—
211C22×D4D21211yescharacteristic
212C23×C4D21211yescharacteristic
213C23×C4C22⋊D5311yescharacteristic
214C22×C8—41no—
215C4×Q8—2121no—
216C4×Q8—2121no—
217C4×D4—41no—
218C4×D4—41no—
219C2×C4⋊C4—531no—
220C2×C4⋊C4—1061no—
221C2×C4⋊C4—1061no—
222C2×C4⋊C4—1061no—
223C2×C4⋊C4—1061no—
224C2×C4⋊C4—2121no—
225C2×C4⋊C4C22⋊D5311yescharacteristic
226C2×C22⋊C4—1061no—
227C2×C22⋊C4—1061no—
228C2×C22⋊C4—1061no—
229C2×C22⋊C4—1061no—
230C2×C22⋊C4—1062no—
231C2×C22⋊C4C4×D5311yescharacteristic
232C2×C22⋊C4C22×D5311yescharacteristic
233C2×C42—1061no—
234C2×C42C4×D5311yescharacteristic
235#99—2122no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238#95—531no—
239#95—531no—
240#95—1061no—
241#95—1061no—
242#95—1061no—
243#95—2121no—
244C24#1171992111yescharacteristic
245C24C2×D21211yescharacteristic
246C2×Q8—2121no—
247C2×Q8—2122no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4#1171991312yes—
251C2×D4Dic21212yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—1061no—
259C22×C4—1061no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—2122no—
263C22×C4#1171992111yescharacteristic
264C22×C4#1171992111yescharacteristic
265C22×C4#1171992111yescharacteristic
266C22×C4C2×C4×D5311yescharacteristic
267C22×C4C2×C22⋊D5311yescharacteristic
268C2×C8—42no—
269C2×C8—82no—
270C4⋊C4—41no—
271C4⋊C4—4242no—
272C4⋊C4—4241no—
273C4⋊C4—4241no—
274C4⋊C4—2121no—
275C4⋊C4—2122no—
276C4⋊C4—2121no—
277C4⋊C4—2121no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—4241no—
281C22⋊C4—4241no—
282C22⋊C4—2122no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C22⋊C4—2122no—
286C42—21no—
287C42—21no—
288C42—2122no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1313314212yes—
298C23C2×#1171992111yescharacteristic
299C23C2×#1171991311yescharacteristic
300C23C2×Dic21211yescharacteristic
301Q8—2122no—
302Q8—2122no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—1062no—
319C2×C4—4242no—
320C2×C4—4242no—
321C2×C4—4242no—
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—2121no—
333C2×C4#1313313111yescharacteristic
334C2×C4#1313313111yescharacteristic
335C2×C4#1313314212yes—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22—81no—
351C22C2×#1313313111yescharacteristic
352C22C2×#1313314211yescharacteristic
353C22#1681063911yescharacteristic
354C4—22no—
355C4—21no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—8482no—
361C4—4241no—
362C4—4241no—
363C4—4241no—
364C4—2122no—
365C2—21no—
366C2—21no—
367C2—41no—
368C2—42no—
369C2—41no—
370C2#2027328811yescharacteristic
371C2#2027635711yescharacteristic
372C2#2027638211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1459C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1459D53
hypocenterC53#11897
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20272740C2semidirect
4. #20272737C2semidirect
6. C2×C4×Dic106C4semidirect
8. #16810559C4semidirect
10. #16810535C4semidirect
16. #16810489C4semidirect
209. C53#11897semidirect

Maximal subgroups

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

subgroupindex
#202800692
#202747032
#202727402
#202727372
C53×#14592
#1189753