Finite Group Databasebeta

← #27356885

Subgroups — #27356885

Order 13568.

970 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 183 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#20284841C212yesmaximal, complemented
2#20277999C211yesmaximal, characteristic
3#20273706C214yesmaximal, complemented
4#20273672C214yesmaximal, complemented
5#20273227C212yesmaximal, complemented
6#20273056C211yesmaximal, characteristic
7C53×#3501C211yesmaximal, characteristic
8C2×C4×Dic106C2211yescharacteristic
9#16810951C2214yes—
10#16810905—22no—
11#16810830—22no—
12#16810778C2214yes—
13#16810728—24no—
14#16810680C2212yes—
15#16810561—21no—
16#16810548C2212yes—
17#16810537—21no—
18#16810535C2212yes—
19C4×C22⋊Dic53—22no—
20#16810469C2211yescharacteristic
21#16810360C2218yes—
22#16810347C2214yes—
23C53×#554—22no—
24C53×#554—24no—
25D4×Q8×C53—22no—
26C53×#544—21no—
27C53×#543C2214yes—
28C53×#533C2212yes—
29C2×C22⋊Q8×C53—22no—
30C53×#518—21no—
31C2×C4×D4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic
33C2×#11719925—24no—
34C4×Dic106—22no—
35C4×Dic106—22no—
36C2×C4⋊Dic53C2311yescharacteristic
37C2×#11719919—21no—
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53—22no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53—21no—
42#13133148—24no—
43C4⋊#88322—22no—
44#13133141—24no—
45#13133140—24no—
46C22⋊#88322C2314yes—
47#13133128—22no—
48#13133124—22no—
49#13133124—21no—
50#13133124—22no—
51#13133124C2312yes—
52C22×Q8×C53—22no—
53C22×D4×C53D411yescharacteristic
54C23×C4×C53D412yes—
55C4⋊Q8×C53—24no—
56C4⋊Q8×C53D412yes—
57C4⋊Q8×C53D412yes—
58#125×C53—24no—
59#125×C53—22no—
60C22⋊Q8×C53—24no—
61C22⋊Q8×C53—24no—
62C22⋊Q8×C53—28no—
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53C2314yes—
67C4×Q8×C53—22no—
68C4×Q8×C53D414yes—
69C4×D4×C53—24no—
70C4×D4×C53—24no—
71C4×D4×C53—22no—
72C4×D4×C53—22no—
73#117×C53—22no—
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53—24no—
76C2×C4⋊C4×C53D411yescharacteristic
77C2×C22⋊C4×C53—24no—
78C2×C22⋊C4×C53C2312yes—
79C2×C42×C53C2311yescharacteristic
80C24×C53C2×D412yes—
81C2×Q8×C53—24no—
82C2×Q8×C53—22no—
83C2×Q8×C53—84no—
84C2×Q8×C53C2×D414yes—
85C2×D4×C53—24no—
86C2×D4×C53—22no—
87C22×C4×C53—24no—
88C22×C4×C53—24no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—44no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D412yes—
95C22×C4×C53C2411yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—22no—
98C22×Dic53—21no—
99C2×Dic106—22no—
100C2×Dic106—42no—
101C4⋊C4×C53—24no—
102C4⋊C4×C53—28no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—24no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53C2×D414yes—
110C22⋊C4×C53—28no—
111C22⋊C4×C53—24no—
112C22⋊C4×C53—24no—
113C42×C53—24no—
114C42×C53C2×D412yes—
115C42×C53C2×D412yes—
116C22⋊Dic53—44no—
117C22⋊Dic53—44no—
118#11719925—48no—
119C4⋊Dic53—22no—
120#11719919—42no—
121C4×Dic53—42no—
122C4×Dic53—41no—
123C4×Dic53—41no—
124C2×#88322—24no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53#13714yes—
130C23×C53C22×D411yescharacteristic
131Q8×C53—44no—
132Q8×C53—44no—
133Q8×C53—84no—
134D4×C53—28no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—24no—
139C2×C4×C53—24no—
140C2×C4×C53—41no—
141C2×C4×C53—44no—
142C2×C4×C53—42no—
143C2×C4×C53—82no—
144C2×C4×C53—84no—
145C2×C4×C53—84no—
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12012yes—
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53C22×D412yes—
150C2×Dic53—22no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—41no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158Dic106—24no—
159Dic106—42no—
160#88322—44no—
161#18181—531nomaximal, Sylow
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—24no—
165C22×C53—44no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—82no—
169C22×C53C2×#12011yescharacteristic
170C22×C53C2×#13712yes—
171C4×C53—22no—
172C4×C53—21no—
173C4×C53—22no—
174C4×C53—42no—
175C4×C53—44no—
176C4×C53—44no—
177C4×C53#45512yes—
178Dic53—42no—
179Dic53—82no—
180Dic53—81no—
181#3501D5311yescharacteristic
182#3113—532no—
183#2398—531no—
184#1608—534no—
185#1602—534no—
186#1506—532no—
187#1466—531no—
188C2×C53—22no—
189C2×C53—44no—
190C2×C53#300411yescharacteristic
191C2×C53C2×#45511yescharacteristic
192C2×C53#302511yescharacteristic
193#554—22no—
194#554—24no—
195D4×Q8—22no—
196#544—21no—
197#543C2×D5314yes—
198#533C2×D5312yes—
199C2×C22⋊Q8—22no—
200#518—21no—
201C2×C4×Q8—531no—
202C2×C4×D4C2×D5311yescharacteristic
203#483—534no—
204#475—1062no—
205#462—1062no—
206#451—534no—
207#441—1064no—
208#432—532no—
209#400—1061no—
210#397—532no—
211#394—1061no—
212#393—532no—
213C4×C22⋊C4—1062no—
214#376—531no—
215#328—538no—
216#323—534no—
217C53#1818111yescharacteristic, complemented, Sylow
218C22×Q8—22no—
219C22×Q8—531no—
220C22×D4C22⋊D5311yescharacteristic
221C23×C4C22⋊D5312yes—
222C2×Q16—1064no—
223C4⋊Q8—24no—
224C4⋊Q8C22⋊D5312yes—
225C4⋊Q8C22⋊D5312yes—
226#125—24no—
227#125—22no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—28no—
231C22⋊Q8—22no—
232C22⋊Q8—24no—
233C22⋊Q8—28no—
234C22⋊Q8C22×D5314yes—
235C4×Q8—22no—
236C4×Q8—1062no—
237C4×Q8—1062no—
238C4×Q8C22⋊D5314yes—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—22no—
242C4×D4—22no—
243#117—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4C22⋊D5311yescharacteristic
249C2×C22⋊C4—24no—
250C2×C22⋊C4—1062no—
251C2×C22⋊C4—1062no—
252C2×C22⋊C4C22×D5312yes—
253C2×C42—1061no—
254C2×C42—1061no—
255C2×C42C22×D5311yescharacteristic
256#106—1064no—
257C4⋊C8—1062no—
258#103—1064no—
259#103—1064no—
260C22⋊C8—534no—
261#97—1062no—
262#95—532no—
263#95—1061no—
264#95—1062no—
265#95—1062no—
266C24C2×C22⋊D5312yes—
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—84no—
270C2×Q8—1062no—
271C2×Q8—2122no—
272C2×Q8C2×C22⋊D5314yes—
273C2×D4—24no—
274C2×D4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—44no—
280C22×C4—1061no—
281C22×C4—1062no—
282C22×C4—1061no—
283C22×C4C2×C22⋊D5311yescharacteristic
284C22×C4C2×C22⋊D5311yescharacteristic
285C22×C4C2×C22⋊D5312yes—
286C22×C4C23×D5311yescharacteristic
287Q16—2128no—
288C2×C8—1064no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—24no—
297C4⋊C4—1062no—
298C4⋊C4—2122no—
299C4⋊C4C2×C22⋊D5314yes—
300C22⋊C4—28no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—2124no—
304C22⋊C4—2124no—
305C42—24no—
306C42—2122no—
307C42—2121no—
308C42—2121no—
309C42C2×C22⋊D5312yes—
310C42C2×C22⋊D5312yes—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23#1313324814yes—
316C23C22×C22⋊D5311yescharacteristic
317Q8—44no—
318Q8—44no—
319Q8—84no—
320Q8—1064no—
321Q8—2122no—
322D4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—1062no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4—2121no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4#1313318612yes—
343C2×C4#1313318611yescharacteristic
344C2×C4#1313318611yescharacteristic
345C2×C4C22×C22⋊D5312yes—
346C8—2124no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22C2×#1313318611yescharacteristic
355C22C2×#1313324812yes—
356C4—22no—
357C4—21no—
358C4—22no—
359C4—42no—
360C4—44no—
361C4—44no—
362C4—4242no—
363C4—4241no—
364C4—2122no—
365C4#1681080112yes—
366C2—22no—
367C2—44no—
368C2#2028394911yescharacteristic
369C2C2×#1681080111yescharacteristic
370C2#2028410711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3501C2
socleC22×C53C2×#120
prime radicalC22×D4×C53D4
hypercenter#3501D53
hypocenterC53#18181
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #20284841C2semidirect
3. #20273706C2semidirect
4. #20273672C2semidirect
5. #20273227C2semidirect
217. C53#18181semidirect

Maximal subgroups

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

subgroupindex
#202848412
#202779992
#202737062
#202736722
#202732272
#202730562
C53×#35012
#1818153