Finite Group Databasebeta

← #27276725

Subgroups — #27276725

Order 13568.

1026 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C22×#13133216C211yesmaximal, characteristic
2#20274638C212yesmaximal
3#20273045C211yesmaximal, characteristic
4#20273044C212yesmaximal
5C53×#1463C211yesmaximal, characteristic
6C23×Dic106C411yescharacteristic
7C23×Dic106C2211yescharacteristic
8C2×#13133216—24no—
9C2×#13133216—24no—
10C2×#13133216—22no—
11C2×#13133216—22no—
12C22×C4⋊Dic53C411yescharacteristic
13C22×#11719919C411yescharacteristic
14C22×#11719919C2211yescharacteristic
15C22×C4×Dic53C411yescharacteristic
16C2×#13133135—24no—
17#16810587C2212yes—
18#16810349—22no—
19#16810348—24no—
20C22×C4⋊C4×C53C2211yescharacteristic
21C53×#407C2212yes—
22C53×#324—22no—
23C22×Dic106—22no—
24C22×Dic106—21no—
25C22×Dic106—22no—
26C22×Dic106—21no—
27C22×Dic106—22no—
28C22×Dic106—22no—
29C22×Dic106—44no—
30C22×Dic106—44no—
31C22×Dic106D414yes—
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2×C411yescharacteristic
35C23×Dic53C2×C411yescharacteristic
36C2×#11719917—22no—
37#13133216—216no—
38#13133216—216no—
39C2×C4⋊Dic53—22no—
40C2×C4⋊Dic53—22no—
41C2×C4⋊Dic53—22no—
42C2×#11719919—22no—
43C2×#11719919—22no—
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×#11719919—22no—
47C2×#11719919—22no—
48C2×#11719919D414yes—
49C2×C4×Dic53—22no—
50C2×C4×Dic53—22no—
51C2×C4×Dic53—22no—
52#13133135—48no—
53C22⋊#88322—22no—
54C22⋊#88322—22no—
55C23×C4×C53C2×C411yescharacteristic
56C23×C4×C53C2×C411yescharacteristic
57C23×C4×C53C2311yescharacteristic
58C2×M16×C53—22no—
59C2×C4⋊C4×C53—22no—
60C2×C4⋊C4×C53—22no—
61C2×C4⋊C4×C53D412yes—
62C2×C4⋊C4×C53D412yes—
63C22⋊C8×C53—22no—
64C22⋊C8×C53—22no—
65C24×C53C22×C411yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—24no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—42no—
72C22×C4×C53—42no—
73C22×C4×C53C22⋊C412yes—
74C22×C4×C53C22⋊C412yes—
75C22×C4×C53C2×D412yes—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D412yes—
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—24no—
84C22×Dic53—24no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—21no—
89C22×Dic53—42no—
90C22×Dic53—42no—
91C22×Dic53C22⋊C412yes—
92C22×Dic53C22⋊C412yes—
93C22×Dic53C22⋊C412yes—
94C22×Dic53C22⋊C412yes—
95C2×Dic106—28no—
96C2×Dic106—24no—
97C2×Dic106—24no—
98C2×Dic106—28no—
99C2×Dic106—44no—
100C2×Dic106—44no—
101C2×Dic106—48no—
102C2×Dic106—44no—
103C2×Dic106—48no—
104C2×Dic106—48no—
105C2×Dic106—44no—
106C2×Dic106—44no—
107M16×C53—44no—
108C2×C8×C53—42no—
109C4⋊C4×C53—24no—
110C4⋊C4×C53—24no—
111C4⋊Dic53—28no—
112#11719919—28no—
113#11719919—24no—
114#11719919—24no—
115C4×Dic53—28no—
116#11719917—44no—
117C2×#88322—42no—
118C23×C53—22no—
119C23×C53—22no—
120C23×C53—22no—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53C2×C22⋊C411yescharacteristic
124C2×C4×C53—24no—
125C2×C4×C53—24no—
126C2×C4×C53—22no—
127C2×C4×C53—24no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53#10114yescomplemented
137C2×C4×C53#12012yes—
138C2×C4×C53#12012yes—
139C2×Dic53—24no—
140C2×Dic53—24no—
141C2×Dic53—24no—
142C2×Dic53—24no—
143C2×Dic53—22no—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—22no—
149C2×Dic53—24no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53#9914yes—
156C2×Dic53#10114yescomplemented
157Dic106—48no—
158Dic106—48no—
159Dic106—416no—
160C8×C53—82no—
161#88322—82no—
162#8551—531nomaximal, Sylow
163C22×C53—28no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53#37911yescharacteristic
171C22×C53C2×#9912yes—
172C22×C53C2×#10112yes—
173C22×C53C2×#10112yes—
174C4×C53—22no—
175C4×C53—22no—
176C4×C53—24no—
177C4×C53—42no—
178Dic53—24no—
179Dic53—24no—
180Dic53—42no—
181Dic53—42no—
182C22×C4⋊Q8—531no—
183#1784—532no—
184#1463—532no—
185#1463—531no—
186#1463D5311yescharacteristic
187C2×C53—22no—
188C2×C53—22no—
189C2×C53—22no—
190C2×C53#178111yescharacteristic
191C2×C53#178411yescharacteristic
192C2×C53#178411yescharacteristic
193C23×Q8—531no—
194C23×Q8—531no—
195C2×C4⋊Q8—1064no—
196C2×C4⋊Q8—1064no—
197C2×C4⋊Q8—1062no—
198C2×C4⋊Q8—1062no—
199C22×C4⋊C4—531no—
200C22×C4⋊C4—531no—
201C22×C4⋊C4—531no—
202C22×C4⋊C4C2×D5311yescharacteristic
203C22×C42—531no—
204C2×#101—1064no—
205#407—532no—
206#407C2×D5312yes—
207#324—22no—
208#324—1064no—
209#324—1062no—
210C53#855111yescharacteristic, complemented, Sylow
211C22×Q8—534no—
212C22×Q8—1062no—
213C22×Q8—1062no—
214C22×Q8—1061no—
215C22×Q8—1062no—
216C22×Q8—1061no—
217C22×Q8—1062no—
218C22×Q8—2124no—
219C22×Q8—2124no—
220C23×C4—531no—
221C23×C4—531no—
222C23×C4—531no—
223C23×C4—531no—
224C23×C4C4×D5311yescharacteristic
225C23×C4C4×D5311yescharacteristic
226C23×C4C22×D5311yescharacteristic
227C2×M16—22no—
228C2×M16—1062no—
229C4⋊Q8—10616no—
230C4⋊Q8—10616no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—534no—
234C2×C4⋊C4—1062no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4—1062no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4—1062no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1062no—
242C2×C4⋊C4—1062no—
243C2×C4⋊C4D21212yes—
244C2×C4⋊C4C22⋊D5312yes—
245C2×C42—1062no—
246C2×C42—1062no—
247C2×C42—1062no—
248#101—2128no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—1062no—
252C22⋊C8—1062no—
253C24C2×C4×D5311yescharacteristic
254C2×Q8—1068no—
255C2×Q8—1068no—
256C2×Q8—1064no—
257C2×Q8—1064no—
258C2×Q8—2124no—
259C2×Q8—2124no—
260C2×Q8—2124no—
261C2×Q8—2128no—
262C2×Q8—2124no—
263C2×Q8—2128no—
264C2×Q8—2124no—
265C2×Q8—2128no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—532no—
274C22×C4—532no—
275C22×C4—532no—
276C22×C4—532no—
277C22×C4—1064no—
278C22×C4—1064no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—2122no—
288C22×C4—2122no—
289C22×C4#1171992112yes—
290C22×C4#1171992112yes—
291C22×C4C2×D21211yescharacteristic
292C22×C4D4×D5312yes—
293C22×C4D4×D5312yes—
294C22×C4C2×C22⋊D5311yescharacteristic
295M16—44no—
296M16—2124no—
297C2×C8—42no—
298C2×C8—2122no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—1068no—
302C4⋊C4—1068no—
303C4⋊C4—1064no—
304C4⋊C4—1064no—
305C42—1068no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23C22⋊C4×D5311yescharacteristic
310C23C22⋊C4×D5311yescharacteristic
311C23C2×#1171992111yescharacteristic
312Q8—2128no—
313Q8—2128no—
314Q8—21216no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—534no—
328C2×C4—534no—
329C2×C4—1064no—
330C2×C4—1064no—
331C2×C4—1064no—
332C2×C4—1064no—
333C2×C4—1062no—
334C2×C4—1062no—
335C2×C4—1062no—
336C2×C4—1062no—
337C2×C4—1064no—
338C2×C4—1062no—
339C2×C4—1062no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2122no—
344C2×C4—2122no—
345C2×C4#1313313514yescomplemented
346C2×C4#1313318512yes—
347C2×C4#1313318712yes—
348C8—82no—
349C8—4242no—
350C22—28no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22#1681047811yescharacteristic
358C22#99×D5312yes—
359C22#101×D5312yes—
360C22C2×#1313313512yes—
361C4—22no—
362C4—22no—
363C4—24no—
364C4—42no—
365C4—1064no—
366C4—1064no—
367C4—2122no—
368C4—2122no—
369C2—22no—
370C2—22no—
371C2—22no—
372C2#2027462811yescharacteristic
373C2#2027463711yescharacteristic
374C2#2027463811yescharacteristic
subgroupG/H
centerC22#16810478
commutator, prime radicalC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1463C2
socleC22×C53#379
cosocleC23×C4×C53C23
hypercenter#1463D53
hypocenterC53#8551
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
136. C2×C4×C53#101semidirect
156. C2×Dic53#101semidirect
210. C53#8551semidirect
345. C2×C4#13133135semidirect

Maximal subgroups

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

subgroupindex
C22×#131332162
#202746382
#202730452
#202730442
C53×#14632
#855153