Finite Group Databasebeta

← #27276642

Subgroups — #27276642

Order 13568.

648 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 131 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#20275681C211yesmaximal, characteristic, complemented
2#20273342C211yesmaximal, characteristic, complemented
3#20273253C212yesmaximal, complemented
4#20273175C212yesmaximal, complemented
5C53×#3553C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×C4×#88322C2211yescharacteristic
8#16810489C2211yescharacteristic
9#16810385—22no—
10#16810365—22no—
11#16810360—22no—
12#16810357—22no—
13Q8⋊#88322—22no—
14D4⋊#88322—22no—
15C53×#564C411yescharacteristic, complemented
16C53×#563C412yescomplemented
17C53×#562C411yescharacteristic, complemented
18C53×#554—22no—
19C53×#553—22no—
20C53×#552—22no—
21C53×#541—22no—
22C53×#539—22no—
23C53×#539—22no—
24C53×#529C2212yes—
25C2×#125×C53C2211yescharacteristic
26C22×#88322—21no—
27C2×C4⋊Dic53—21no—
28C4⋊#88322—22no—
29C22⋊#88322—22no—
30#13133128—22no—
31C4×#88322—22no—
32#13133124—22no—
33C4⋊Q8×C53—22no—
34#126×C53—22no—
35#126×C53—22no—
36#126×C53—22no—
37#126×C53—22no—
38#126×C53—22no—
39#125×C53—22no—
40#125×C53—22no—
41#125×C53—22no—
42#125×C53—22no—
43#125×C53—22no—
44#125×C53—22no—
45#125×C53C2×C411yescharacteristic
46#125×C53C2×C411yescharacteristic
47#125×C53C2×C412yes—
48#125×C53D412yes—
49#124×C53C2×C412yes—
50#123×C53—22no—
51#123×C53—22no—
52#123×C53—22no—
53#123×C53—22no—
54#123×C53—22no—
55C22⋊Q8×C53—22no—
56C22⋊Q8×C53—22no—
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53—22no—
59C22⋊Q8×C53—22no—
60C22⋊Q8×C53—22no—
61C22⋊Q8×C53D412yes—
62#121×C53—22no—
63#121×C53—22no—
64#121×C53D412yes—
65C4×Q8×C53—22no—
66C4×Q8×C53D412yes—
67C4×D4×C53—22no—
68C4×D4×C53—22no—
69C4×D4×C53—22no—
70C4×D4×C53—22no—
71C4×D4×C53D412yes—
72#117×C53—22no—
73#117×C53—22no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53D412yes—
77C2×C42×C53C2311yescharacteristic
78C2×Q8×C53—22no—
79C2×Q8×C53C22⋊C412yes—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53C22⋊C412yes—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D412yes—
89C22×Dic53—41no—
90C4⋊C4×C53—22no—
91C4⋊C4×C53—22no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53C22⋊C412yes—
106C4⋊C4×C53C22⋊C412yes—
107C4⋊C4×C53C22⋊C412yes—
108C4⋊C4×C53C22⋊C412yes—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C22⋊C4×C53—22no—
114C22⋊C4×C53—22no—
115C22⋊C4×C53—22no—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C42×C53—22no—
119C42×C53—22no—
120C42×C53—22no—
121C42×C53C22×C411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53C2×D412yes—
124C4⋊Dic53—42no—
125C2×#88322—22no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—42no—
129C23×C53—22no—
130C23×C53#12011yescharacteristic
131Q8×C53—42no—
132Q8×C53—82no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—82no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—42no—
145C2×C4×C53—82no—
146C2×C4×C53—82no—
147C2×C4×C53—82no—
148C2×C4×C53—82no—
149C2×C4×C53C2×C22⋊C412yes—
150C2×C4×C53C2×C22⋊C411yescharacteristic
151C2×C4×C53#12012yes—
152C2×C4×C53#12011yescharacteristic
153C2×Dic53—82no—
154C2×Dic53—81no—
155#88322—42no—
156#88322—82no—
157#8529—531nomaximal, Sylow
158C22×C53—22no—
159C22×C53—42no—
160C22×C53—42no—
161C22×C53—42no—
162C22×C53#37911yescharacteristic
163C22×C53#45411yescharacteristic
164C22×C53#45611yescharacteristic
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—42no—
170C4×C53—42no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53—42no—
174Dic53—161no—
175#3553D5311yescharacteristic
176#1991—531no—
177#1534—531no—
178#1512—532no—
179#1494—532no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—82no—
183C2×C53#179311yescharacteristic
184C2×C53#179412yes—
185#564Dic5311yescharacteristic, complemented
186#563Dic5312yescomplemented
187#562Dic5311yescharacteristic, complemented
188#554—22no—
189#553—22no—
190#552—22no—
191#541—22no—
192#539—22no—
193#539—22no—
194#529C2×D5312yes—
195C2×#125C2×D5311yescharacteristic
196#432—532no—
197C2×C4×C8—531no—
198#382—531no—
199#340—1062no—
200#330—1062no—
201#328—1062no—
202#327—1062no—
203Q8⋊C8—1062no—
204D4⋊C8—1062no—
205C53#852911yescharacteristic, complemented, Sylow
206C22×C8—1061no—
207C4⋊Q8—22no—
208#126—22no—
209#126—22no—
210#126—22no—
211#126—22no—
212#126—22no—
213#125—22no—
214#125—22no—
215#125—22no—
216#125—22no—
217#125—22no—
218#125—22no—
219#125C2×Dic5311yescharacteristic
220#125C2×Dic5311yescharacteristic
221#125C2×Dic5312yes—
222#125C22⋊D5312yes—
223#124C2×Dic5312yes—
224#123—22no—
225#123—22no—
226#123—22no—
227#123—22no—
228#123—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8C22⋊D5312yes—
236#121—22no—
237#121—22no—
238#121C22⋊D5312yes—
239C4×Q8—22no—
240C4×Q8C22⋊D5312yes—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4C22⋊D5312yes—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—1061no—
251C2×C4⋊C4C22⋊D5312yes—
252C2×C42C22×D5311yescharacteristic
253C4⋊C8—1062no—
254C22⋊C8—1062no—
255#97—1062no—
256C4×C8—1062no—
257#95—1062no—
258C2×Q8—22no—
259C2×Q8C22⋊Dic5312yes—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4C22⋊Dic5312yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—2121no—
268C22×C4C2×C22⋊D5311yescharacteristic
269C22×C4C2×C22⋊D5312yes—
270C2×C8—1062no—
271C2×C8—2122no—
272C2×C8—2121no—
273C2×C8—2121no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—2122no—
290C4⋊C4C22⋊Dic5312yes—
291C4⋊C4C22⋊Dic5312yes—
292C4⋊C4C22⋊Dic5312yes—
293C4⋊C4C22⋊Dic5312yes—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42C22×Dic5311yescharacteristic
307C42C2×C22⋊D5312yes—
308C42C2×C22⋊D5311yescharacteristic
309C23—22no—
310C23#1313318611yescharacteristic
311Q8—42no—
312Q8—82no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—82no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—4242no—
330C2×C4—4241no—
331C2×C4C2×C22⋊Dic5311yescharacteristic
332C2×C4C2×C22⋊Dic5312yes—
333C2×C4#1313318612yes—
334C2×C4#1313318611yescharacteristic
335C8—4242no—
336C8—2122no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22#1681047911yescharacteristic
342C22#1681079611yescharacteristic
343C22#1681080611yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—8481no—
354C2—21no—
355C2—21no—
356C2—82no—
357C2#2027467411yescharacteristic
358C2#2027467912yes—
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3553C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3553D53
hypocenterC53#8529
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3553

Complements

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

NH ≅ G/Ntype
1. #20275681C2semidirect
2. #20273342C2semidirect
3. #20273253C2semidirect
4. #20273175C2semidirect
15. C53×#564C4semidirect
16. C53×#563C4semidirect
17. C53×#562C4semidirect
185. #564Dic53semidirect
186. #563Dic53semidirect
187. #562Dic53semidirect
205. C53#8529semidirect

Maximal subgroups

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

subgroupindex
#202756812
#202733422
#202732532
#202731752
C53×#35532
#852953