Finite Group Databasebeta

← #27294861

Subgroups — #27294861

Order 13568.

434 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 83 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#20277974C211yesmaximal, characteristic
2#20273056C211yesmaximal, characteristic
3#20272738C212yesmaximal, complemented
4#20272714C212yesmaximal, complemented
5C53×#1799C211yesmaximal, characteristic
6C2×#13133176C411yescharacteristic, complemented
7#16810680—21no—
8#16810562C411yescharacteristic, complemented
9#16810559C2211yescharacteristic
10#16810553C411yescharacteristic, complemented
11#16810551—21no—
12#16810548C2211yescharacteristic
13#16810537—21no—
14C4×#11719919—21no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53C411yescharacteristic, complemented
17#16810347—21no—
18#16810347C2212yes—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C53×#380—21no—
22C53×#342C2212yes—
23#13133176—41no—
24#13133176—41no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C4⋊#88322—41no—
36C22⋊#88322—22no—
37#13133128—41no—
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
47C4×D4×C53—41no—
48C4×D4×C53—41no—
49C2×C4⋊C4×C53—41no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54C2×C42×C53D411yescharacteristic
55#99×C53—42no—
56#95×C53—41no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
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—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—82no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C42×C53—21no—
86C42×C53—21no—
87C22⋊Dic53—42no—
88C22⋊Dic53—42no—
89C22⋊Dic53—42no—
90C22⋊Dic53—41no—
91C22⋊Dic53—41no—
92C4⋊Dic53—41no—
93C4⋊Dic53—41no—
94#11719919—42no—
95#11719919—42no—
96C4×Dic53—42no—
97C4×Dic53—41no—
98C4×Dic53—41no—
99C2×#88322—42no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53AGL(1,ℤ/8)11yescharacteristic
111C23×C53#13711yescharacteristic
112D4×C53—42no—
113D4×C53—42no—
114C2×C4×C53—22no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—81no—
128C2×C4×C53—82no—
129C2×C4×C53—82no—
130C2×C4×C53—82no—
131C2×C4×C53#9911yescharacteristic
132C2×C4×C53#9911yescharacteristic
133C2×Dic53—22no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53—41no—
145#88322—82no—
146#11822—531nomaximal, Sylow
147C22×C53—21no—
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—81no—
159C22×C53—81no—
160C22×C53—81no—
161C22×C53C2×#9911yescharacteristic
162C22×C53#41711yescharacteristic
163C22×C53#42111yescharacteristic
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—162no—
170C4×C53#35212yes—
171Dic53—42no—
172Dic53—81no—
173Dic53—81no—
174Dic53—81no—
175#2395—531no—
176#1799D5311yescharacteristic
177#1466—531no—
178#1352—532no—
179#1344—532no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—41no—
183C2×C53—41no—
184C2×C53—81no—
185C2×C53#151311yescharacteristic
186C2×C53C2×#35211yescharacteristic
187C2×C53#212611yescharacteristic
188C2×C4×D4C2×D5311yescharacteristic
189C2×#117—531no—
190#432—1061no—
191C2×#99—21no—
192#400—531no—
193#399—531no—
194#398—531no—
195#397—531no—
196#397—1061no—
197#394—1061no—
198#380—21no—
199C4×C4⋊C4—1061no—
200C4×C22⋊C4—531no—
201C4×C22⋊C4—1061no—
202#342C2×D5312yes—
203#323—532no—
204#323—1061no—
205C53#1182211yescharacteristic, complemented, Sylow
206C22×D4C22⋊D5311yescharacteristic
207C23×C4C4×D5311yescharacteristic
208C23×C4C22⋊D5311yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211#117—2121no—
212#117—2121no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4C4×D5311yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4D21211yescharacteristic
225C2×C22⋊C4C22×D5311yescharacteristic
226C2×C42—1061no—
227C2×C42—1061no—
228C2×C42D21211yescharacteristic
229C4⋊C8—2121no—
230#99—42no—
231C22⋊C8—1062no—
232#97—2121no—
233#95—41no—
234#95—531no—
235#95—531no—
236#95—1061no—
237#95—1061no—
238#95—1061no—
239#95—1061no—
240C24#1171992111yescharacteristic
241C24C2×C22⋊D5311yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4#1171992111yescharacteristic
258C22×C4#1171992111yescharacteristic
259C22×C4#1171992111yescharacteristic
260C22×C4C2×C4×D5311yescharacteristic
261C22×C4C2×D21211yescharacteristic
262C2×C8—2122no—
263C4⋊C4—41no—
264C4⋊C4—82no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2122no—
268C4⋊C4—2122no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—2122no—
274C22⋊C4—2122no—
275C22⋊C4—2122no—
276C22⋊C4—2121no—
277C22⋊C4—2121no—
278C42—21no—
279C42—21no—
280C42—2122no—
281C42—2121no—
282C42—2121no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×#1171992111yescharacteristic
293C23#1313324311yescharacteristic
294C23#1313324811yescharacteristic
295D4—42no—
296D4—42no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—1062no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4#1313313111yescharacteristic
327C2×C4#1313313111yescharacteristic
328C8—4242no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×#1313313111yescharacteristic
344C22#1681062211yescharacteristic
345C22#1681064111yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—162no—
352C4—4241no—
353C4—4241no—
354C4—4241no—
355C4—2122no—
356C4#1681041212yes—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—41no—
361C2—81no—
362C2#2027325611yescharacteristic
363C2C2×#1681041211yescharacteristic
364C2#2027637811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1799C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#1799D53
hypocenterC53#11822
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20272738C2semidirect
4. #20272714C2semidirect
6. C2×#13133176C4semidirect
8. #16810562C4semidirect
10. #16810553C4semidirect
16. C4×C22⋊Dic53C4semidirect
205. C53#11822semidirect

Maximal subgroups

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

subgroupindex
#202779742
#202730562
#202727382
#202727142
C53×#17992
#1182253