Finite Group Databasebeta

← #27299689

Subgroups — #27299689

Order 13568.

694 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 127 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#20276410C211yesmaximal, characteristic, complemented
2#20276390C211yesmaximal, characteristic, complemented
3#20276365C212yesmaximal
4#20274819C212yesmaximal, complemented
5C53×#3510C211yesmaximal, characteristic
6#16810607C2212yes—
7#16810597C2212yes—
8#16810422—22no—
9#16810418—22no—
10#16810410—24no—
11#16810383—22no—
12#16810379—22no—
13C53×#559—22no—
14C53×#556—22no—
15C53×#551—24no—
16C53×#542—22no—
17C53×#536C2211yescharacteristic
18C53×#535C2211yescharacteristic
19C53×#533C411yescharacteristic, complemented
20C53×#529—22no—
21C2×#124×C53C411yescharacteristic, complemented
22C53×#519C411yescharacteristic, complemented
23C53×#518C411yescharacteristic, complemented
24C4×#41×C53C2211yescharacteristic
25C2×#11719917—22no—
26#13133176—22no—
27#13133136—22no—
28#13133134—22no—
29#13133132—22no—
30#13133132—22no—
31C2×#41×C53C2311yescharacteristic
32C22×Q8×C53C2×C411yescharacteristic
33C22×D4×C53C2×C411yescharacteristic
34C4⋊Q8×C53—21no—
35C4⋊Q8×C53D412yes—
36#127×C53—21no—
37#126×C53—24no—
38#126×C53—24no—
39#125×C53—21no—
40#124×C53—24no—
41#124×C53—21no—
42#124×C53—21no—
43#124×C53—22no—
44#124×C53—21no—
45#124×C53D412yes—
46#123×C53—24no—
47#123×C53—24no—
48#123×C53—22no—
49C22⋊Q8×C53—24no—
50C22⋊Q8×C53—22no—
51C22⋊Q8×C53—22no—
52#121×C53—22no—
53#120×C53D414yes—
54C4×Q8×C53—22no—
55C4×Q8×C53—22no—
56C4×Q8×C53—22no—
57C4×Q8×C53—21no—
58C4×D4×C53—24no—
59C4×D4×C53—22no—
60C4×D4×C53—22no—
61C4×D4×C53—22no—
62C4×D4×C53—21no—
63#117×C53—24no—
64#117×C53—22no—
65#117×C53C2×C411yescharacteristic
66#117×C53C2×C411yescharacteristic
67#117×C53C2×C411yescharacteristic
68#117×C53D412yes—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—22no—
71C2×C42×C53C2×C411yescharacteristic
72C2×C42×C53D412yes—
73C24×C53C22⋊C412yes—
74#41×C53—42no—
75C2×Q8×C53—22no—
76C2×Q8×C53—21no—
77C2×Q8×C53—42no—
78C2×Q8×C53C22⋊C412yes—
79C2×Q8×C53C2×D411yescharacteristic
80C2×D4×C53—24no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—42no—
84C2×D4×C53C2×D411yescharacteristic
85C2×D4×C53C2×D412yes—
86C22×C4×C53—21no—
87C22×C4×C53—24no—
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53C22×C411yescharacteristic
91C22×C4×C53C2×D412yes—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—24no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53—22no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—24no—
108C22⋊C4×C53—24no—
109C22⋊C4×C53C22⋊C412yes—
110C42×C53—24no—
111C42×C53—21no—
112C42×C53—21no—
113C42×C53—22no—
114C42×C53C22⋊C411yescharacteristic
115C42×C53C22⋊C411yescharacteristic
116C42×C53C22⋊C412yes—
117C42×C53C22⋊C412yes—
118C22⋊Dic53—42no—
119C4⋊Dic53—42no—
120C4×Dic53—42no—
121#11719917—42no—
122#11719917—42no—
123C2×#88322—42no—
124C23×C53—22no—
125C23×C53—42no—
126C23×C53—82no—
127C23×C53C2×C22⋊C411yescharacteristic
128C23×C53#12012yes—
129Q8×C53—41no—
130Q8×C53—42no—
131Q8×C53—42no—
132Q8×C53—82no—
133D4×C53—42no—
134D4×C53—41no—
135D4×C53—42no—
136D4×C53—42no—
137D4×C53—84no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—42no—
148C2×C4×C53—82no—
149C2×C4×C53—84no—
150C2×C4×C53—81no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53#12012yes—
154C2×Dic53—82no—
155#88322—82no—
156#12721—531nomaximal, Sylow
157C22×C53—22no—
158C22×C53—21no—
159C22×C53—42no—
160C22×C53—41no—
161C22×C53—44no—
162C22×C53—42no—
163C22×C53—162no—
164C22×C53#37911yescharacteristic
165C4×C53—22no—
166C4×C53—21no—
167C4×C53—42no—
168C4×C53—41no—
169C4×C53—42no—
170C4×C53—41no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53#41012yes—
174Dic53—162no—
175#3510D5311yescharacteristic
176#2135—531no—
177#2129—531no—
178#2122—532no—
179#1827—532no—
180C2×C53—21no—
181C2×C53—42no—
182C2×C53—41no—
183C2×C53—82no—
184C2×C53#178211yescharacteristic
185#559—22no—
186#556—22no—
187#551—24no—
188#542—22no—
189#536C2×D5311yescharacteristic
190#535C2×D5311yescharacteristic
191#533Dic5311yescharacteristic, complemented
192#529—22no—
193C2×#124Dic5311yescharacteristic, complemented
194#519Dic5311yescharacteristic, complemented
195#518Dic5311yescharacteristic, complemented
196C4×#41C2×D5311yescharacteristic
197#413—532no—
198#410—532no—
199#356—1062no—
200#354—1062no—
201AΣL(1,𝔽16)—1064no—
202#339—1062no—
203H(3,ℤ/4)—1062no—
204C53#1272111yescharacteristic, complemented, Sylow
205C2×#41C22×D5311yescharacteristic
206C22×Q8C2×Dic5311yescharacteristic
207C22×D4C2×Dic5311yescharacteristic
208C2×M16—1062no—
209C4⋊Q8—21no—
210C4⋊Q8C22⋊D5312yes—
211#127—21no—
212#126—24no—
213#126—24no—
214#125—21no—
215#124—24no—
216#124—21no—
217#124—22no—
218#124—21no—
219#124—21no—
220#124C22⋊D5312yes—
221#123—24no—
222#123—24no—
223#123—22no—
224C22⋊Q8—24no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227#121—22no—
228#120C22⋊D5314yes—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×D4—24no—
234C4×D4—22no—
235C4×D4—21no—
236C4×D4—22no—
237C4×D4—22no—
238#117—24no—
239#117—22no—
240#117—1062no—
241#117C2×Dic5311yescharacteristic
242#117C2×Dic5311yescharacteristic
243#117C2×Dic5311yescharacteristic
244#117C22⋊D5312yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42C2×Dic5311yescharacteristic
248C2×C42C22⋊D5312yes—
249#101—1062no—
250#100—1062no—
251#99—1062no—
252#99—1062no—
253C24C22⋊Dic5312yes—
254#41—42no—
255C2×Q8—22no—
256C2×Q8—21no—
257C2×Q8—42no—
258C2×Q8C22⋊Dic5312yes—
259C2×Q8C2×C22⋊D5311yescharacteristic
260C2×D4—24no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4C2×C22⋊D5311yescharacteristic
265C2×D4C2×C22⋊D5312yes—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4C22×Dic5311yescharacteristic
271C22×C4C2×C22⋊D5312yes—
272M16—2122no—
273M16—2122no—
274C2×C8—2122no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—2122no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—22no—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—2122no—
294C22⋊C4C22⋊Dic5312yes—
295C42—24no—
296C42—21no—
297C42—21no—
298C42—22no—
299C42—2122no—
300C42C22⋊Dic5312yes—
301C42C22⋊Dic5311yescharacteristic
302C42C22⋊Dic5311yescharacteristic
303C42C22⋊Dic5312yes—
304C23—22no—
305C23—42no—
306C23—82no—
307C23C2×C22⋊Dic5311yescharacteristic
308C23#1313318612yes—
309Q8—41no—
310Q8—42no—
311Q8—42no—
312Q8—82no—
313D4—42no—
314D4—41no—
315D4—42no—
316D4—42no—
317D4—84no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—81no—
331C2×C4—4242no—
332C2×C4C2×C22⋊Dic5311yescharacteristic
333C2×C4C2×C22⋊Dic5311yescharacteristic
334C2×C4#1313318612yes—
335C8—4242no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—41no—
340C22—44no—
341C22—42no—
342C22—162no—
343C22#1681047911yescharacteristic
344C4—22no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4—42no—
352C4—8482no—
353C4#1681059712yes—
354C2—21no—
355C2—42no—
356C2—41no—
357C2—82no—
358C2#2027463311yescharacteristic
subgroupG/H
centerC2#20274633
commutatorC22×C4×C53C22×C4
FrattiniC2×#41C22×D53
FittingC53×#3510C2
socleC2×C53#1782
prime radicalC53×#535C22
cosocleC2×#41×C53C23
hypercenter#3510D53
hypocenterC53#12721
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×#41 ◁ #3510

Complements

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

NH ≅ G/Ntype
1. #20276410C2semidirect
2. #20276390C2semidirect
4. #20274819C2semidirect
19. C53×#533C4semidirect
21. C2×#124×C53C4semidirect
22. C53×#519C4semidirect
23. C53×#518C4semidirect
191. #533Dic53semidirect
193. C2×#124Dic53semidirect
194. #519Dic53semidirect
195. #518Dic53semidirect
204. C53#12721semidirect

Maximal subgroups

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

subgroupindex
#202764102
#202763902
#202763652
#202748192
C53×#35102
#1272153