Finite Group Databasebeta

← #27293201

Subgroups — #27293201

Order 13568.

594 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 93 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#20277306C211yesmaximal, characteristic
2#324×D53C212yesmaximal, complemented
3#20272649C211yesmaximal, characteristic, complemented
4#20272646C212yesmaximal, complemented
5C53×#1319C211yesmaximal, characteristic, complemented
6C22×#11719949C411yescharacteristic
7C2×#13133174C411yescharacteristic
8C2×C4⋊C4×D53C2211yescharacteristic
9C22⋊C8×D53—22no—
10#16810530C411yescharacteristic
11#16810525C411yescharacteristic
12#16810524C2211yescharacteristic
13#16810501—22no—
14#16810482—22no—
15#16810349C2212yes—
16#16810348—22no—
17C53×#391C2211yescharacteristic
18C53×#324C2212yes—
19C2×#11719949—22no—
20C2×#11719949—42no—
21C2×#11719949—42no—
22C2×#11719949C814yescomplemented
23C22×D212—21no—
24C22×D212C2×C411yescharacteristic
25C22×C4×D53—21no—
26C22×C4×D53C2×C411yescharacteristic
27C22×C4×D53D412yes—
28C2×C8×D53—42no—
29#13133174—42no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719919—21no—
32C2×#11719919D412yes—
33C4⋊C4×D53—42no—
34C2×#11719921—21no—
35C2×#11719921—22no—
36C2×#11719921—22no—
37C2×C4×Dic53C2×C411yescharacteristic
38C22⋊#88322—22no—
39#13133129—42no—
40#13133124—22no—
41C22×Q8×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2311yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C22⋊C8×C53—22no—
45#95×C53—22no—
46C23×D53—21no—
47C23×D53C22⋊C411yescharacteristic
48C2×Q8×C53—21no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53C2×C812yescomplemented
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C22×C411yescharacteristic
55C22×C4×C53C2×D412yes—
56C22×Dic53—21no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C412yes—
59#11719949—48no—
60#11719949—48no—
61C2×D212—24no—
62C2×D212—21no—
63C2×D212—41no—
64C2×D212—41no—
65C2×D212—42no—
66C2×D212—42no—
67C2×D212C2×C812yescomplemented
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—24no—
72C2×C4×D53—22no—
73C2×C4×D53—42no—
74C2×C4×D53—42no—
75C2×C4×D53—82no—
76C2×C4×D53C2×C812yes—
77C2×C4×D53M1612yes—
78C2×C8×C53—42no—
79C4⋊C4×C53—41no—
80C42×C53—41no—
81#11719921—42no—
82#11719921—82no—
83#11719921—82no—
84C4⋊Dic53—41no—
85#11719919—42no—
86#11719919—42no—
87C4×Dic53—41no—
88C2×#88322—42no—
89C8×D53—84no—
90C23×C53C2×C22⋊C411yescharacteristic
91C22×D53—22no—
92C22×D53—41no—
93C22×D53—41no—
94C22×D53—42no—
95C22×D53—42no—
96C22×D53—41no—
97C22×D53—41no—
98C22×D53C22⋊C812yes—
99C22×D53#9912yes—
100C22×D53#10012yes—
101Q8×C53—42no—
102Q8×C53—42no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53C22×C811yescharacteristic
114C2×C4×C53C2×M1611yescharacteristic
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—22no—
118C2×Dic53—42no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53C22⋊C812yes—
123D212—42no—
124D212—42no—
125D212—48no—
126C4×D53—24no—
127C4×D53—24no—
128C4×D53—48no—
129C4×D53—44no—
130C4×D53—82no—
131C8×C53—82no—
132#88322—82no—
133#11432—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53C2×C22⋊C811yescharacteristic
137C22×C53C2×#9911yescharacteristic
138C22×C53C2×#10011yescharacteristic
139C2×D53—22no—
140C2×D53—22no—
141C2×D53—42no—
142C2×D53—42no—
143C2×D53—44no—
144C2×D53—42no—
145C2×D53—82no—
146C2×D53—81no—
147C2×D53—81no—
148C2×D53#32314yes—
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—42no—
152C4×C53—42no—
153C4×C53—81no—
154Dic53—22no—
155Dic53—42no—
156Dic53—81no—
157#2311—531no—
158C2×#324—532no—
159#1319—531no—
160#1319D5311yescharacteristic, complemented
161#1318—532no—
162C2×C53—21no—
163C2×C53—21no—
164C2×C53C2×#32311yescharacteristic
165C2×C53#212711yescharacteristic
166C2×C53#213311yescharacteristic
167D53—24no—
168D53—82no—
169C22×#41—531no—
170C2×#117—531no—
171C22×C4⋊C4—531no—
172C2×C22⋊C8—1062no—
173#391—531no—
174#391C2×D5311yescharacteristic
175#390—531no—
176#390—531no—
177#386—1062no—
178#380—1062no—
179#324—532no—
180#324—1062no—
181#324C2×D5312yes—
182C53#1143211yescharacteristic, complemented, Sylow
183C2×#41—534no—
184C2×#41—1062no—
185C2×#41—2122no—
186C2×#41—2122no—
187C22×Q8C4×D5311yescharacteristic
188C22×D4—531no—
189C22×D4—1061no—
190C23×C4—531no—
191C23×C4—532no—
192C23×C4—1061no—
193C22×C8—2122no—
194#117—2122no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—532no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—2122no—
199C2×C4⋊C4C22×D5311yescharacteristic
200C2×C22⋊C4—1062no—
201C2×C22⋊C4—1062no—
202C2×C22⋊C4—1061no—
203C2×C42—531no—
204C2×C42C4×D5311yescharacteristic
205C22⋊C8—22no—
206C22⋊C8—1062no—
207C22⋊C8—2122no—
208#95—22no—
209#95—1062no—
210C24—531no—
211C24—1061no—
212#41—2128no—
213#41—2128no—
214C2×Q8—21no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8C8×D5312yescomplemented
218C2×D4—532no—
219C2×D4—1064no—
220C2×D4—1061no—
221C2×D4—2121no—
222C2×D4—2121no—
223C2×D4—2122no—
224C2×D4—2122no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—531no—
228C22×C4—532no—
229C22×C4—532no—
230C22×C4—532no—
231C22×C4—1064no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—1062no—
235C22×C4—1062no—
236C22×C4—1061no—
237C22×C4—4242no—
238C22×C4—2122no—
239C22×C4—2122no—
240C22×C4C2×C4×D5311yescharacteristic
241C22×C4D4×D5312yes—
242C2×C8—42no—
243C2×C8—4244no—
244C2×C8—2122no—
245C4⋊C4—41no—
246C4⋊C4—2122no—
247C4⋊C4—2121no—
248C4⋊C4—2122no—
249C22⋊C4—4242no—
250C22⋊C4—4242no—
251C22⋊C4—2122no—
252C42—41no—
253C42—2121no—
254C23—532no—
255C23—532no—
256C23—532no—
257C23—1062no—
258C23—2121no—
259C23—2121no—
260C23—2122no—
261C23—2122no—
262C23—2121no—
263C23—2121no—
264C23C22⋊C4×D5311yescharacteristic
265Q8—42no—
266Q8—42no—
267D4—2122no—
268D4—2122no—
269D4—2128no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—532no—
281C2×C4—1064no—
282C2×C4—1064no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—1062no—
286C2×C4—4242no—
287C2×C4—2128no—
288C2×C4—2124no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2122no—
293C2×C4C2×C8×D5311yescharacteristic
294C2×C4M16×D5311yescharacteristic
295C8—82no—
296C8—4242no—
297C22—21no—
298C22—21no—
299C22—534no—
300C22—1062no—
301C22—1062no—
302C22—4242no—
303C22—4241no—
304C22—4241no—
305C22—2122no—
306C22—2122no—
307C22—2124no—
308C22—2122no—
309C22C22⋊C8×D5311yescharacteristic
310C22#99×D5311yescharacteristic
311C22#100×D5311yescharacteristic
312C4—21no—
313C4—21no—
314C4—42no—
315C4—42no—
316C4—81no—
317C4—1062no—
318C4—4241no—
319C4—2122no—
320C2—21no—
321C2—21no—
322C2—1064no—
323C2—4242no—
324C2#323×D5311yescharacteristic
325C2#2027638311yescharacteristic
326C2#2027640311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1319C2
socleC22×C53C2×#99
prime radicalC22×#11719949C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1319D53
hypocenterC53#11432
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
2. #324×D53C2semidirect
3. #20272649C2semidirect
4. #20272646C2semidirect
5. C53×#1319C2semidirect
22. C2×#11719949C8semidirect
51. C2×Q8×C53C2×C8semidirect
67. C2×D212C2×C8semidirect
160. #1319D53semidirect
182. C53#11432semidirect
217. C2×Q8C8×D53semidirect

Maximal subgroups

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

subgroupindex
#202773062
#324×D532
#202726492
#202726462
C53×#13192
#1143253