Finite Group Databasebeta

← #27281979

Subgroups — #27281979

Order 13568.

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

normal subgroups only

This group has 458 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#20283353C211yesmaximal, characteristic
2#20280292C211yesmaximal, characteristic
3#20273161C212yesmaximal, complemented
4#20272581C212yesmaximal, complemented
5C53×#1464C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810684—24no—
8#16810647—21no—
9#16810587—21no—
10C4×#11719917C2212yes—
11#16810553C411yescharacteristic
12#16810548C411yescharacteristic
13#16810543C411yescharacteristic
14#16810535C411yescharacteristic
15#16810511—21no—
16#16810507—21no—
17#16810504—21no—
18#16810496—21no—
19#16810491C2211yescharacteristic
20#16810353—24no—
21C2×#117×C53C2211yescharacteristic
22C53×#432C2212yes—
23C53×#324—21no—
24C53×#323—21no—
25C22×Dic106C2×C411yescharacteristic
26C2×#11719917—22no—
27C4×Dic106—22no—
28C4×Dic106—22no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—24no—
37C22⋊#88322—24no—
38#13133128—22no—
39C4×#88322D414yes—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124C2×C411yescharacteristic
47#13133124C2×C411yescharacteristic
48C23×C4×C53D411yescharacteristic
49#117×C53—22no—
50#117×C53—22no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57C4⋊C8×C53—22no—
58C22⋊C8×C53—22no—
59#97×C53—22no—
60C24×C53C22⋊C411yescharacteristic
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C2×Dic106—22no—
76C2×Dic106—42no—
77C2×C8×C53—42no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—22no—
82C42×C53—21no—
83C42×C53—21no—
84C42×C53C2×D412yes—
85C42×C53#4112yes—
86C22⋊Dic53—42no—
87C22⋊Dic53—42no—
88C4⋊Dic53—22no—
89#11719919—42no—
90#11719919—42no—
91#11719919—41no—
92#11719919—41no—
93C4×Dic53—42no—
94#11719917—42no—
95#11719917—42no—
96C2×#88322—24no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53C4×D412yes—
117C2×C4×C53#12011yescharacteristic
118C2×C4×C53#12112yes—
119C2×C4×C53#12311yescharacteristic
120C2×Dic53—22no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132Dic106—24no—
133Dic106—42no—
134C8×C53—82no—
135#88322—44no—
136#9493—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—42no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—81no—
144C22×C53#38611yescharacteristic
145C22×C53#42111yescharacteristic
146C22×C53#42111yescharacteristic
147C4×C53—22no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—22no—
151C4×C53—41no—
152C4×C53—41no—
153C4×C53—41no—
154Dic53—42no—
155Dic53—81no—
156Dic53—81no—
157Dic53—81no—
158#2921—531no—
159#2599—531no—
160#1491—532no—
161#1464D5311yescharacteristic
162#1288—532no—
163C2×C53—21no—
164C2×C53—21no—
165C2×C53—81no—
166C2×C53#189111yescharacteristic
167C2×C53#189711yescharacteristic
168C2×C53#190911yescharacteristic
169C2×C4×Q8—531no—
170C2×#117C2×D5311yescharacteristic
171#433—1064no—
172#432C2×D5312yes—
173#423—1061no—
174#407—1061no—
175C4×M16—532no—
176#398—531no—
177#397—531no—
178#395—531no—
179#393—531no—
180#388—1061no—
181#387—1061no—
182#387—1061no—
183#385—1061no—
184#383—531no—
185Q8⋊C8—1064no—
186#324—21no—
187#323—21no—
188C53#949311yescharacteristic, complemented, Sylow
189C22×Q8—531no—
190C23×C4D21211yescharacteristic
191C2×M16—1062no—
192C4×Q8—1062no—
193C4×Q8—1062no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4C4×D5311yescharacteristic
201C2×C4⋊C4C22⋊D5311yescharacteristic
202C2×C22⋊C4—1061no—
203C2×C22⋊C4—1061no—
204C2×C22⋊C4C4×D5311yescharacteristic
205C2×C22⋊C4C22⋊D5311yescharacteristic
206C2×C42—1061no—
207C2×C42D21211yescharacteristic
208C2×C42C22×D5311yescharacteristic
209C4⋊C8—22no—
210C4⋊C8—1064no—
211C22⋊C8—22no—
212C22⋊C8—1064no—
213#97—22no—
214#97—1062no—
215C4×C8—534no—
216#95—531no—
217#95—531no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223#95—1061no—
224C24#1171992111yescharacteristic
225C2×Q8—1062no—
226C2×Q8—2122no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4#1171992111yescharacteristic
236C22×C4#1171992111yescharacteristic
237C22×C4#1171992111yescharacteristic
238C22×C4C2×C4×D5311yescharacteristic
239C22×C4C2×D21211yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241M16—2122no—
242M16—2122no—
243C2×C8—42no—
244C2×C8—1064no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—1062no—
248C4⋊C4—2122no—
249C4⋊C4—2122no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—2122no—
255C22⋊C4—2122no—
256C42—21no—
257C42—21no—
258C42—2122no—
259C42D4×D5312yes—
260C42#1171994712yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23C2×#1171992111yescharacteristic
267Q8—1064no—
268Q8—2122no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—1062no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4#1313317712yes—
295C2×C4#1313318711yescharacteristic
296C2×C4#1313318912yes—
297C2×C4#1313320111yescharacteristic
298C8—82no—
299C8—2124no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1681050011yescharacteristic
308C22#1681063911yescharacteristic
309C22#1681063911yescharacteristic
310C4—22no—
311C4—22no—
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—4241no—
318C4—4241no—
319C4—4241no—
320C4—2122no—
321C2—21no—
322C2—21no—
323C2—81no—
324C2#2027513111yescharacteristic
325C2#2027516111yescharacteristic
326C2#2027523511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1464C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1464D53
hypocenterC53#9493
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #20273161C2semidirect
4. #20272581C2semidirect
188. C53#9493semidirect

Maximal subgroups

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

subgroupindex
#202833532
#202802922
#202731612
#202725812
C53×#14642
#949353