Finite Group Databasebeta

← #27294919

Subgroups — #27294919

Order 13568.

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

normal subgroups only

This group has 382 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#20280575C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic, complemented
3#20272746C212yesmaximal, complemented
4#20272719C212yesmaximal
5C53×#1467C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic
7#16810647—21no—
8C2×#13133136—21no—
9#16810558—21no—
10#16810543C411yescharacteristic
11#16810543C2211yescharacteristic
12#16810540C411yescharacteristic
13#16810540C2211yescharacteristic
14#16810534—21no—
15#16810514—21no—
16#16810514—21no—
17#16810491C411yescharacteristic
18#16810392C2212yes—
19C2×C4×Q8×C53C2211yescharacteristic
20C53×#432—21no—
21C53×#324—21no—
22C53×#324C2212yes—
23C22×Dic106C2×C411yescharacteristic
24C22×D212C2×C411yescharacteristic
25C2×#11719917—22no—
26#13133205—41no—
27#13133205—41no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719921—21no—
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C4⋊#88322—41no—
37C4⋊#88322—41no—
38#13133136—42no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124C2×C411yescharacteristic
44#13133124C2×C411yescharacteristic
45C22×Q8×C53D411yescharacteristic
46C4×Q8×C53—41no—
47C4×Q8×C53—41no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C4⋊C4×C53C2311yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C4⋊C8×C53—41no—
55C22⋊C8×C53—22no—
56#97×C53—41no—
57C23×D53—21no—
58C2×Q8×C53—21no—
59C2×Q8×C53—21no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C2×D212—41no—
73C2×D212—81no—
74C2×Dic106—41no—
75C2×Dic106—81no—
76C2×C8×C53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53—21no—
85C42×C53—21no—
86#11719921—42no—
87#11719921—42no—
88#11719921—41no—
89#11719921—41no—
90#11719921—41no—
91#11719921—41no—
92#11719921—81no—
93#11719921—81no—
94#11719919—81no—
95#11719919—81no—
96#11719917—42no—
97#11719917—42no—
98C2×#88322—42no—
99C23×C53C2×C22⋊C411yescharacteristic
100C22×D53—41no—
101C22×D53—41no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53—41no—
105C22×D53—41no—
106C22×D53—41no—
107Q8×C53—42no—
108Q8×C53—42no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53#9911yescharacteristic
125C2×C4×C53#9911yescharacteristic
126C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C53#13711yescharacteristic
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137D212—82no—
138Dic106—82no—
139C8×C53—82no—
140#88322—82no—
141#11833—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53C2×#9911yescharacteristic
145C22×C53#41711yescharacteristic
146C22×C53#42111yescharacteristic
147C2×D53—81no—
148C2×D53—81no—
149C2×D53—81no—
150C2×D53—81no—
151C2×D53—81no—
152C2×D53—81no—
153C2×D53—81no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163Dic53—81no—
164Dic53—81no—
165Dic53—81no—
166#2632—531no—
167#1801—531no—
168#1467D5311yescharacteristic, complemented
169#1354—532no—
170#1345—532no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53#151311yescharacteristic
174C2×C53#212211yescharacteristic
175C2×C53#213311yescharacteristic
176D53—161no—
177C2×#124—531no—
178C2×C4×Q8C2×D5311yescharacteristic
179#432—21no—
180#423—1061no—
181C2×#101—1061no—
182#399—1061no—
183#395—531no—
184#395—531no—
185#394—531no—
186#394—531no—
187#393—1061no—
188#388—1061no—
189#388—1061no—
190#383—531no—
191#343—532no—
192#324—21no—
193#324C2×D5312yes—
194C53#1183311yescharacteristic, complemented, Sylow
195C22×Q8—531no—
196C22×Q8D21211yescharacteristic
197C22×D4—531no—
198C2×M16—1062no—
199#124—2121no—
200#124—2121no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4C4×D5311yescharacteristic
206C2×C4⋊C4C22⋊D5311yescharacteristic
207C2×C4⋊C4C22×D5311yescharacteristic
208C2×C22⋊C4—1061no—
209C2×C22⋊C4—1061no—
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1061no—
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C42C4×D5311yescharacteristic
215C2×C42D21211yescharacteristic
216C2×C42C22⋊D5311yescharacteristic
217C4⋊C8—41no—
218C4⋊C8—2121no—
219C4⋊C8—2121no—
220#101—2122no—
221C22⋊C8—22no—
222#97—41no—
223#95—531no—
224#95—531no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229C24—1061no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—41no—
233C2×Q8—41no—
234C2×Q8—4241no—
235C2×Q8—2121no—
236C2×D4—4241no—
237C2×D4—2121no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4#1171992111yescharacteristic
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4C2×C4×D5311yescharacteristic
246C22×C4C2×D21211yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248M16—2122no—
249M16—2122no—
250C2×C8—42no—
251C2×C8—2122no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—4241no—
256C4⋊C4—4241no—
257C22⋊C4—4241no—
258C22⋊C4—4241no—
259C22⋊C4—2122no—
260C22⋊C4—2122no—
261C22⋊C4—2121no—
262C22⋊C4—2121no—
263C22⋊C4—2121no—
264C22⋊C4—2121no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C23—2121no—
272C23—2121no—
273C23—2121no—
274C23—2121no—
275C23—2121no—
276C23—2121no—
277C23—2121no—
278C23C2×#1171992111yescharacteristic
279Q8—42no—
280Q8—42no—
281Q8—4242no—
282D4—4242no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4#1313313111yescharacteristic
308C2×C4#1313313111yescharacteristic
309C2×C4#1313324511yescharacteristic
310C2×C4#1313325011yescharacteristic
311C8—82no—
312C8—4242no—
313C22—21no—
314C22—21no—
315C22—4241no—
316C22—4241no—
317C22—4241no—
318C22—4241no—
319C22—4241no—
320C22—4241no—
321C22—4241no—
322C22C2×#1313313111yescharacteristic
323C22#1681062511yescharacteristic
324C22#1681063911yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—4241no—
335C4—4241no—
336C4—4241no—
337C2—21no—
338C2—21no—
339C2—8481no—
340C2#2027325911yescharacteristic
341C2#2027636411yescharacteristic
342C2#2027640211yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1467C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1467D53
hypocenterC53#11833
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #20274711C2semidirect
3. #20272746C2semidirect
5. C53×#1467C2semidirect
168. #1467D53semidirect
194. C53#11833semidirect

Maximal subgroups

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

subgroupindex
#202805752
#202747112
#202727462
#202727192
C53×#14672
#1183353