Finite Group Databasebeta

← #27302376

Subgroups — #27302376

Order 13568.

536 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#16810413C214yesmaximal
2#20274703C211yesmaximal, characteristic
3C4×#13133134C211yesmaximal, characteristic
4C53×#2486C211yesmaximal, characteristic
5C2×#13133134C2212yes—
6C2×#13133132C2212yes—
7C4×#11719917—21no—
8#16810483—21no—
9#16810413—28no—
10#16810392—21no—
11#16810390—21no—
12C2×#123×C53C2212yes—
13C2×C4×D4×C53C2211yescharacteristic
14C22×C4⋊C4×C53C411yescharacteristic, complemented
15C53×#398—21no—
16C53×#397—21no—
17C53×#389—21no—
18C53×#381C412yescomplemented
19C53×#380—21no—
20C53×#380C411yescharacteristic, complemented
21C2×#11719917—22no—
22C2×C4⋊Dic53—41no—
23C2×C22⋊Dic53—22no—
24#13133134—22no—
25#13133134D414yes—
26#13133132—24no—
27#13133128—41no—
28C4×#88322—41no—
29#13133124—41no—
30C22×D4×C53C2311yescharacteristic
31C23×C4×C53C2×C412yes—
32C23×C4×C53C2×C411yescharacteristic
33C23×C4×C53D411yescharacteristic
34#123×C53—24no—
35#123×C53—24no—
36C4×D4×C53—41no—
37C4×D4×C53—41no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—41no—
43C2×C4⋊C4×C53—41no—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C22⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53D411yescharacteristic
49C2×C42×C53D411yescharacteristic
50#95×C53—22no—
51#95×C53—21no—
52#95×C53—22no—
53C24×C53C22×C411yescharacteristic
54C24×C53C2×D411yescharacteristic
55C2×D4×C53—41no—
56C2×D4×C53—41no—
57C2×D4×C53C2×D412yes—
58C2×D4×C53#4112yes—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—24no—
66C22×C4×C53—42no—
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—42no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—44no—
82C22⋊C4×C53—24no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C42×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C4⋊Dic53—82no—
92#11719917—24no—
93#11719917—42no—
94C2×#88322—42no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C4×D412yes—
105C23×C53#12411yescharacteristic
106C23×C53#12711yescharacteristic
107D4×C53—42no—
108D4×C53—42no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—81no—
128C2×C4×C53—84no—
129C2×C4×C53#9911yescharacteristic
130C2×C4×C53#9911yescharacteristic
131C2×C4×C53#12112yes—
132C2×Dic53—82no—
133C2×Dic53—82no—
134C2×Dic53—82no—
135#88322—44no—
136#13217—531nomaximal, Sylow
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—22no—
141C22×C53—21no—
142C22×C53—42no—
143C22×C53—41no—
144C22×C53—41no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—81no—
151C22×C53#39011yescharacteristic
152C22×C53C2×#9911yescharacteristic
153C22×C53#41011yescharacteristic
154C4×C53—21no—
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—82no—
160C4×C53—82no—
161C4×C53#35212yescomplemented
162Dic53—162no—
163#2486D5311yescharacteristic
164C2×#352—534no—
165#1799—531no—
166C4×#100—531no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53—41no—
170C2×C53—42no—
171C2×C53—41no—
172C2×C53#196411yescharacteristic
173C2×C53C2×#35211yescharacteristic
174C2×C53#212311yescharacteristic
175C2×#123C2×D5312yes—
176C2×C4×D4C2×D5311yescharacteristic
177C22×C4⋊C4Dic5311yescharacteristic, complemented
178C2×#100—532no—
179C2×#99—532no—
180C4×M16—1061no—
181#398—21no—
182#397—21no—
183#389—21no—
184#381Dic5312yescomplemented
185#380—21no—
186#380—1061no—
187#380Dic5311yescharacteristic, complemented
188#352—1068no—
189#343—1061no—
190#342—1061no—
191C53#1321711yescharacteristic, complemented, Sylow
192C22×D4C22×D5311yescharacteristic
193C23×C4C2×Dic5312yes—
194C23×C4C2×Dic5311yescharacteristic
195C23×C4C22⋊D5311yescharacteristic
196C2×M16—1062no—
197#123—24no—
198#123—24no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—2121no—
208C2×C4⋊C4C22⋊D5311yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—1062no—
211C2×C22⋊C4C2×Dic5312yes—
212C2×C22⋊C4C2×Dic5311yescharacteristic
213C2×C22⋊C4C22⋊D5311yescharacteristic
214C2×C42C22⋊D5311yescharacteristic
215#100—534no—
216#100—1062no—
217#99—1064no—
218#97—2121no—
219C4×C8—2121no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—2121no—
224C24C22×Dic5311yescharacteristic
225C24C2×C22⋊D5311yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4D4×D5312yes—
229C2×D4#1171994712yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—2122no—
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C22⋊Dic5311yescharacteristic
245C22×C4C22⋊Dic5311yescharacteristic
246C22×C4C2×C22⋊D5311yescharacteristic
247M16—1064no—
248M16—2122no—
249C2×C8—2122no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—44no—
256C4⋊C4—4242no—
257C22⋊C4—24no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×C22⋊Dic5311yescharacteristic
275C23D4×Dic5312yes—
276C23#1313320411yescharacteristic
277C23#1313321311yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—84no—
300C2×C4—4242no—
301C2×C4—4242no—
302C2×C4—4242no—
303C2×C4#1313313211yescharacteristic
304C2×C4#1313313211yescharacteristic
305C2×C4#1313318812yes—
306C8—2124no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1681052611yescharacteristic
322C22C2×#1313313211yescharacteristic
323C22#1681059711yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—82no—
330C4—82no—
331C4—8482no—
332C4#1681041312yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#2027555211yescharacteristic
339C2C2×#1681041311yescharacteristic
340C2#2027636911yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2486C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#2486D53
hypocenterC53#13217
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
14. C22×C4⋊C4×C53C4semidirect
18. C53×#381C4semidirect
20. C53×#380C4semidirect
161. C4×C53#352semidirect
177. C22×C4⋊C4Dic53semidirect
184. #381Dic53semidirect
187. #380Dic53semidirect
191. C53#13217semidirect
332. C4#16810413semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
#202747032
C4×#131331342
C53×#24862
#1321753