Finite Group Databasebeta

← #27281795

Subgroups — #27281795

Order 13568.

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

normal subgroups only

This group has 432 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#20283799C211yesmaximal, characteristic
2#20279930C211yesmaximal, characteristic
3#20273433C211yesmaximal, characteristic
4#20273431C211yesmaximal, characteristic
5#20273058C211yesmaximal, characteristic
6#20272571C211yesmaximal, characteristic
7C53×#1285C211yesmaximal, characteristic
8C2×#13133217C2211yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10#16810752—22no—
11#16810751—22no—
12#16810680C2211yescharacteristic
13#16810680C2211yescharacteristic
14#16810677—21no—
15#16810673—21no—
16#16810563—21no—
17#16810563—21no—
18#16810529C412yes—
19#16810528—21no—
20#16810528—21no—
21#16810489C412yes—
22#16810370—22no—
23#16810368—22no—
24#16810348—21no—
25#16810348—21no—
26C53×#432C2211yescharacteristic
27C53×#432C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×Dic106C2×C411yescharacteristic
31#13133217—21no—
32#13133217—21no—
33#13133217—21no—
34#13133217—21no—
35C4×Dic106—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38C4×Dic106—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2×C412yes—
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×#11719919D411yescharacteristic
43C2×#11719919D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C4⋊#88322—21no—
47C4⋊#88322D412yes—
48#13133146—22no—
49#13133146—22no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52#13133128—21no—
53#13133128—21no—
54#13133127—22no—
55C8×Dic53—22no—
56#13133124—22no—
57#13133124—22no—
58#13133124—22no—
59#13133124—22no—
60C2×C4⋊C4×C53—21no—
61C2×C42×C53C2×C412yes—
62C2×C42×C53C2311yescharacteristic
63C4⋊C8×C53—21no—
64C4⋊C8×C53D412yes—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#97×C53—21no—
68#97×C53—21no—
69#95×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—22no—
75C22×Dic53C22⋊C411yescharacteristic
76C22×Dic53C22⋊C411yescharacteristic
77C22×Dic53C22⋊C411yescharacteristic
78C22×Dic53C22⋊C411yescharacteristic
79C2×Dic106—21no—
80C2×Dic106—21no—
81C2×Dic106—22no—
82C2×Dic106—41no—
83C2×Dic106—44no—
84C2×Dic106—41no—
85C2×Dic106—41no—
86C2×C8×C53—22no—
87C2×C8×C53—41no—
88C4⋊C4×C53—42no—
89C42×C53—42no—
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53#4111yescharacteristic
93C42×C53#4111yescharacteristic
94C4⋊Dic53—21no—
95C4⋊Dic53—21no—
96C4⋊Dic53—22no—
97C4⋊Dic53—22no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104C4×Dic53—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C2×#88322—22no—
109C2×#88322—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53C4×D411yescharacteristic
119C2×C4×C53C4×D411yescharacteristic
120C2×C4×C53#12011yescharacteristic
121C2×C4×C53#12111yescharacteristic
122C2×C4×C53#12111yescharacteristic
123C2×C4×C53#12311yescharacteristic
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—44no—
134C2×Dic53—42no—
135C2×Dic53#10112yes—
136Dic106—42no—
137Dic106—42no—
138Dic106—82no—
139Dic106—82no—
140C8×C53—42no—
141C8×C53—81no—
142#88322—42no—
143#88322—81no—
144#9456—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53#38611yescharacteristic
148C22×C53C2×#10111yescharacteristic
149C22×C53#41311yescharacteristic
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—82no—
156C4×C53#45412yes—
157Dic53—22no—
158Dic53—41no—
159Dic53—41no—
160Dic53—41no—
161Dic53—82no—
162#2983—531no—
163#2568—531no—
164#1556—531no—
165#1556—531no—
166#1467—531no—
167#1285—531no—
168#1285D5311yescharacteristic
169C2×C53—21no—
170C2×C53—21no—
171C2×C53#188811yescharacteristic
172C2×C53#190111yescharacteristic
173C2×C53#190911yescharacteristic
174C2×C4⋊Q8—531no—
175C2×C4×Q8—531no—
176#446—1062no—
177#446—1062no—
178#432—531no—
179#432—531no—
180#432—1061no—
181#432C2×D5311yescharacteristic
182#432C2×D5311yescharacteristic
183#431—1061no—
184#400—1061no—
185#400—1061no—
186#391—532no—
187#391—1061no—
188#391—1061no—
189#382—532no—
190C4×C4⋊C4C2×D5311yescharacteristic
191#332—1062no—
192#332—1062no—
193#324—1061no—
194#324—1061no—
195C53#945611yescharacteristic, complemented, Sylow
196C22×Q8—531no—
197C22×Q8—1061no—
198C4⋊Q8—1061no—
199C4⋊Q8—1061no—
200C4⋊Q8—1061no—
201C4⋊Q8—1061no—
202C4×Q8—1061no—
203C4×Q8—1061no—
204C4×Q8—1061no—
205C4×Q8—1061no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4—531no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4—1061no—
212C2×C42—531no—
213C2×C42—531no—
214C2×C42C4×D5312yes—
215C2×C42C22×D5311yescharacteristic
216C4⋊C8—21no—
217C4⋊C8—532no—
218C4⋊C8—1062no—
219C4⋊C8—1062no—
220C4⋊C8—1061no—
221C4⋊C8D21212yes—
222C22⋊C8—21no—
223C22⋊C8—21no—
224C22⋊C8—1061no—
225C22⋊C8—1061no—
226#97—21no—
227#97—21no—
228#97—1062no—
229#97—1061no—
230#97—1061no—
231C4×C8—1062no—
232#95—21no—
233#95—1062no—
234#95—1062no—
235#95—1062no—
236#95—1062no—
237C2×Q8—1062no—
238C2×Q8—1061no—
239C2×Q8—1061no—
240C2×Q8—2121no—
241C2×Q8—2124no—
242C2×Q8—2121no—
243C2×Q8—2121no—
244C22×C4—22no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—1062no—
250C22×C4C2×C4×D5311yescharacteristic
251C22×C4D4×D5311yescharacteristic
252C22×C4D4×D5311yescharacteristic
253C2×C8—22no—
254C2×C8—41no—
255C2×C8—1062no—
256C2×C8—2121no—
257C4⋊C4—42no—
258C4⋊C4—1061no—
259C4⋊C4—1061no—
260C4⋊C4—1061no—
261C4⋊C4—1061no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4—1062no—
265C4⋊C4—1062no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C42—42no—
269C42—1061no—
270C42—1061no—
271C42—1061no—
272C42—1061no—
273C42C2×D21211yescharacteristic
274C42#1171994511yescharacteristic
275C42D4×D5311yescharacteristic
276C42#1171994911yescharacteristic
277C23C22⋊C4×D5311yescharacteristic
278Q8—4242no—
279Q8—4242no—
280Q8—2122no—
281Q8—2122no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—532no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—2124no—
298C2×C4—2122no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4#1313317911yescharacteristic
302C2×C4C4×D21211yescharacteristic
303C2×C4#1313318711yescharacteristic
304C2×C4#1313319011yescharacteristic
305C2×C4#1313319211yescharacteristic
306C2×C4#1313320211yescharacteristic
307C8—42no—
308C8—81no—
309C8—4241no—
310C8—2122no—
311C22—21no—
312C22—21no—
313C22#1681050111yescharacteristic
314C22#101×D5311yescharacteristic
315C22#1681060411yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—82no—
322C4—1062no—
323C4—4242no—
324C4—2121no—
325C4—2121no—
326C4—2121no—
327C4#1681079712yes—
328C2—21no—
329C2—21no—
330C2#2027511711yescharacteristic
331C2#2027518211yescharacteristic
332C2#2027523611yescharacteristic
subgroupG/H
centerC22#16810501
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1285C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1285D53
hypocenterC53#9456
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1285

Complements

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

NH ≅ G/Ntype
195. C53#9456semidirect

Maximal subgroups

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

subgroupindex
#202837992
#202799302
#202734332
#202734312
#202730582
#202725712
C53×#12852
#945653