Finite Group Databasebeta

← #27294921

Subgroups — #27294921

Order 13568.

400 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 79 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#20280632C211yesmaximal, characteristic
2#20273053C211yesmaximal, characteristic
3#20272744C212yesmaximal, complemented
4#20272720C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7#16810587—21no—
8#16810553C2211yescharacteristic
9#16810551—21no—
10#16810548C411yescharacteristic
11#16810543C2211yescharacteristic
12#16810537—21no—
13#16810535C411yescharacteristic
14#16810507—21no—
15#16810496—21no—
16#16810486C411yescharacteristic
17#16810349C2212yes—
18#16810347—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#423—21no—
21C2×#100×C53—21no—
22C53×#343C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C2×#11719917—41no—
25#13133196—41no—
26#13133196—41no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C22⋊#88322—22no—
36C22⋊#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C22×D4×C53D411yescharacteristic
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—22no—
48C4×D4×C53—41no—
49C4×D4×C53—41no—
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C22⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C4⋊C8×C53—41no—
55C4⋊C8×C53—41no—
56#100×C53—42no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
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×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C2×Dic106—41no—
79C2×Dic106—81no—
80M16×C53—42no—
81M16×C53—42no—
82C2×C8×C53—42no—
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—81no—
91C22⋊Dic53—81no—
92C4⋊Dic53—41no—
93#11719919—41no—
94#11719919—41no—
95#11719919—41no—
96#11719919—41no—
97#11719919—81no—
98#11719917—82no—
99C2×#88322—42no—
100C23×C53—21no—
101C23×C53—21no—
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×C53C2×C22⋊C411yescharacteristic
111D4×C53—42no—
112D4×C53—42no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C53#13711yescharacteristic
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
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—
142Dic106—82no—
143C8×C53—82no—
144#88322—82no—
145#11834—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53—42no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53C2×#9911yescharacteristic
159C22×C53#41711yescharacteristic
160C22×C53#42111yescharacteristic
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168Dic53—81no—
169Dic53—81no—
170Dic53—81no—
171Dic53—81no—
172#2638—531no—
173#1800D5311yescharacteristic
174#1465—531no—
175#1354—532no—
176#1346—532no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—81no—
180C2×C53—81no—
181C2×C53#151311yescharacteristic
182C2×C53#212111yescharacteristic
183C2×C53#213511yescharacteristic
184C2×C22⋊Q8—531no—
185C2×C4×D4C2×D5311yescharacteristic
186#423—21no—
187C2×#100—21no—
188#407—1061no—
189#398—531no—
190#397—531no—
191#397—1061no—
192#395—531no—
193#394—1061no—
194#393—531no—
195#387—1061no—
196#385—1061no—
197#381—531no—
198#343C2×D5312yes—
199#324—532no—
200#323—1061no—
201C53#1183411yescharacteristic, complemented, Sylow
202C22×Q8—531no—
203C22×D4C22⋊D5311yescharacteristic
204C23×C4C4×D5311yescharacteristic
205C23×C4D21211yescharacteristic
206C2×M16—22no—
207C2×M16—2121no—
208C22⋊Q8—2121no—
209C22⋊Q8—2121no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4C22×D5311yescharacteristic
217C2×C22⋊C4—1061no—
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4C4×D5311yescharacteristic
222C2×C22⋊C4D21211yescharacteristic
223C2×C42C22⋊D5311yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—41no—
226#100—42no—
227C22⋊C8—1062no—
228C22⋊C8—2121no—
229#95—531no—
230#95—531no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235#95—1061no—
236C24#1171992111yescharacteristic
237C24#1171992111yescharacteristic
238C2×Q8—4241no—
239C2×Q8—2121no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4C2×C4×D5311yescharacteristic
257C22×C4C2×D21211yescharacteristic
258C22×C4C2×C22⋊D5311yescharacteristic
259M16—42no—
260M16—42no—
261M16—4242no—
262C2×C8—42no—
263C2×C8—2122no—
264C4⋊C4—41no—
265C4⋊C4—4241no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—4241no—
274C22⋊C4—4241no—
275C22⋊C4—2122no—
276C22⋊C4—2122no—
277C42—21no—
278C42—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×#1171992111yescharacteristic
290Q8—4242no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4#1313313111yescharacteristic
319C2×C4#1313313111yescharacteristic
320C2×C4#1313324311yescharacteristic
321C2×C4#1313324811yescharacteristic
322C8—82no—
323C8—4242no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C2×#1313313111yescharacteristic
337C22#1681062211yescharacteristic
338C22#1681064111yescharacteristic
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—4241no—
347C4—4241no—
348C4—4241no—
349C4—4241no—
350C2—21no—
351C2—21no—
352C2—81no—
353C2—81no—
354C2#2027325611yescharacteristic
355C2#2027636211yescharacteristic
356C2#2027640911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1800C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C4⋊C4×C53C23
hypercenter#1800D53
hypocenterC53#11834
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20272744C2semidirect
4. #20272720C2semidirect
201. C53#11834semidirect

Maximal subgroups

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

subgroupindex
#202806322
#202730532
#202727442
#202727202
C53×#18002
#1183453