Finite Group Databasebeta

← #27295249

Subgroups — #27295249

Order 13568.

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

normal subgroups only

This group has 440 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#20279891C211yesmaximal, characteristic
2#20273033C211yesmaximal, characteristic
3#20272744C212yesmaximal, complemented
4#20272741C212yesmaximal, complemented
5C53×#1800C211yesmaximal, characteristic
6C2×#13133196C411yescharacteristic
7C2×C22⋊#88322—21no—
8#16810562C2211yescharacteristic
9#16810559C411yescharacteristic
10#16810543C2211yescharacteristic
11#16810535C411yescharacteristic
12#16810507—21no—
13#16810503—21no—
14#16810496—21no—
15#16810481—21no—
16C2×#13133124C411yescharacteristic
17#16810349C2212yes—
18#16810347—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#423—21no—
21C2×#100×C53—21no—
22C53×#343C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C23×Dic53—21no—
25C22×#88322—41no—
26#13133196—41no—
27#13133196—41no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—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—41no—
60C2×D4×C53—41no—
61C2×D4×C53SD1612yes—
62C2×D4×C53Q1612yes—
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—
78C22×Dic53—42no—
79C22×Dic53—42no—
80C22×Dic53—81no—
81C2×Dic106—41no—
82C2×Dic106—81no—
83M16×C53—42no—
84M16×C53—42no—
85C2×C8×C53—42no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C42×C53—21no—
90C42×C53—21no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—81no—
94C4⋊Dic53—41no—
95#11719919—42no—
96#11719919—41no—
97#11719919—41no—
98#11719919—81no—
99C2×#88322—42no—
100C2×#88322—82no—
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×C53#10312yes—
110C23×C53#10312yes—
111C23×C53C2×C22⋊C411yescharacteristic
112D4×C53—42no—
113D4×C53—42no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53—81no—
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53C2×SD1611yescharacteristic
130C2×C4×C53C2×Q1611yescharacteristic
131C2×Dic53—22no—
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—
142C2×Dic53—41no—
143C2×Dic53—84no—
144Dic106—82no—
145C8×C53—82no—
146#88322—82no—
147#11906—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—22no—
151C22×C53—22no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—81no—
160C22×C53C2×#9911yescharacteristic
161C22×C53C2×#10311yescharacteristic
162C22×C53#42111yescharacteristic
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—21no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—41no—
170Dic53—42no—
171Dic53—81no—
172Dic53—81no—
173Dic53—81no—
174#2564—531no—
175#1800D5311yescharacteristic
176#1459—531no—
177#1354—532no—
178#1353—532no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—42no—
182C2×C53—42no—
183C2×C53#152011yescharacteristic
184C2×C53#212111yescharacteristic
185C2×C53#213211yescharacteristic
186C2×C22⋊Q8—531no—
187C2×C4×D4C2×D5311yescharacteristic
188#423—21no—
189C2×#100—21no—
190C2×C22⋊C8—1061no—
191#400—531no—
192#399—531no—
193#395—531no—
194#393—531no—
195#387—1061no—
196#386—1061no—
197#385—1061no—
198#380—1061no—
199C2×#95—531no—
200#343C2×D5312yes—
201#324—532no—
202#323—1061no—
203C53#1190611yescharacteristic, complemented, Sylow
204C22×Q8—531no—
205C22×D4C22⋊D5311yescharacteristic
206C23×C4—1061no—
207C23×C4C4×D5311yescharacteristic
208C23×C4D21211yescharacteristic
209C2×M16—22no—
210C22×C8—2121no—
211C22⋊Q8—2121no—
212C22⋊Q8—2121no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4C22×D5311yescharacteristic
220C2×C22⋊C4—1061no—
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223C2×C22⋊C4C4×D5311yescharacteristic
224C2×C22⋊C4D21211yescharacteristic
225C2×C42C22⋊D5311yescharacteristic
226C4⋊C8—41no—
227C4⋊C8—41no—
228#100—42no—
229C22⋊C8—1062no—
230C22⋊C8—2121no—
231#95—531no—
232#95—531no—
233#95—1061no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237#95—1061no—
238C24#1171992111yescharacteristic
239C24#1171992111yescharacteristic
240C2×Q8—4241no—
241C2×Q8—2121no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4#1171992312yes—
245C2×D4#1171992512yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4—4241no—
257C22×C4—2122no—
258C22×C4—2122no—
259C22×C4#1171992111yescharacteristic
260C22×C4#1171992111yescharacteristic
261C22×C4C2×C4×D5311yescharacteristic
262C22×C4C2×D21211yescharacteristic
263C22×C4C2×C22⋊D5311yescharacteristic
264M16—42no—
265M16—42no—
266C2×C8—42no—
267C2×C8—4242no—
268C2×C8—2122no—
269C4⋊C4—41no—
270C4⋊C4—4241no—
271C4⋊C4—2121no—
272C4⋊C4—2122no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—4241no—
278C22⋊C4—2122no—
279C22⋊C4—2122no—
280C42—21no—
281C42—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#1313314012yes—
291C23#1313314012yes—
292C23C2×#1171992111yescharacteristic
293Q8—4242no—
294D4—42no—
295D4—42no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—1062no—
310C2×C4—4244no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4#1313313111yescharacteristic
323C2×C4#1313313111yescharacteristic
324C2×C4C2×#1171992311yescharacteristic
325C2×C4C2×#1171992511yescharacteristic
326C8—82no—
327C8—4242no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—81no—
340C22C2×#1313313111yescharacteristic
341C22C2×#1313314011yescharacteristic
342C22#1681064111yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—4241no—
351C4—4241no—
352C4—4241no—
353C4—2122no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2#2027328611yescharacteristic
359C2#2027636211yescharacteristic
360C2#2027639811yescharacteristic
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#11906
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. #20272741C2semidirect
203. C53#11906semidirect

Maximal subgroups

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

subgroupindex
#202798912
#202730332
#202727442
#202727412
C53×#18002
#1190653