Finite Group Databasebeta

← #27284658

Subgroups — #27284658

Order 13568.

772 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 119 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×#16810941C212yesmaximal, complemented
2#20280817C211yesmaximal, characteristic
3C2×#16810356C212yesmaximal, complemented
4#20272558C211yesmaximal, characteristic, complemented
5C53×#1445C211yesmaximal, characteristic, complemented
6C2×#13133191—22no—
7C2×#13133191C2212yes—
8#16810941—24no—
9#16810941—24no—
10C2×#13133148—22no—
11C2×#13133137—22no—
12C2×C22⋊#88322C2212yes—
13#16810560—22no—
14#16810533—22no—
15#16810356—24no—
16C22×C4⋊C4×C53C2211yescharacteristic
17C2×#95×C53—22no—
18C2×#95×C53C2211yescharacteristic
19C2×#95×C53C2211yescharacteristic
20C22×D212—22no—
21C22×D212D412yes—
22C22×C22⋊D53—22no—
23C22×#88322—22no—
24#13133191—22no—
25#13133191—22no—
26#13133191—22no—
27#13133191—22no—
28#13133191—44no—
29#13133191—44no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53D412yes—
32C2×#11719921—24no—
33C2×#11719921—22no—
34#13133148—28no—
35#13133137—28no—
36C22⋊#88322—22no—
37C22⋊#88322D414yescomplemented
38C23×C4×C53—21no—
39C23×C4×C53—21no—
40C23×C4×C53D412yes—
41C23×C4×C53C2311yescharacteristic
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53D412yescomplemented
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—44no—
49C24×C53C2×D411yescharacteristic
50C23×D53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—42no—
56C22×C4×C53—42no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53—41no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—42no—
66C22×C4×C53SD1614yes—
67C22×C4×C53C2×D412yes—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D412yes—
70C22×C4×C53#4111yescharacteristic
71C2×C22⋊D53—42no—
72C2×C22⋊D53—42no—
73C2×C22⋊D53—44no—
74C2×C22⋊D53—44no—
75C22×Dic53—22no—
76C2×D212—22no—
77C2×D212—22no—
78C2×D212—44no—
79C2×D212—44no—
80C2×D212—82no—
81C2×D212—82no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—42no—
84#11719921—48no—
85#11719921—42no—
86#11719921—42no—
87C4⋊Dic53—22no—
88C4⋊Dic53—22no—
89C4⋊Dic53—44no—
90C2×#88322—24no—
91C2×#88322—44no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—22no—
95C23×C53#12011yescharacteristic
96C23×C53#12111yescharacteristic
97C23×C53#12711yescharacteristic
98C23×C53C2×SD1612yes—
99C23×C53AGL(1,ℤ/8)12yes—
100C22×D53—42no—
101C22×D53—44no—
102C22×D53—42no—
103C22×D53—42no—
104C22×D53—44no—
105C2×C4×C53—24no—
106C2×C4×C53—24no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—44no—
113C2×C4×C53—44no—
114C2×C4×C53—42no—
115C2×C4×C53—44no—
116C2×C4×C53—42no—
117C2×C4×C53—82no—
118C2×C4×C53—82no—
119C2×C4×C53—81no—
120C2×C4×C53—82no—
121C2×C4×C53—81no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53#12012yes—
125C2×C4×C53#12112yes—
126C22⋊D53—84no—
127C22⋊D53—82no—
128C22⋊D53—82no—
129C2×Dic53—42no—
130C2×Dic53—44no—
131D212—44no—
132D212—82no—
133D212—88no—
134#88322—44no—
135#9968—531nomaximal, Sylow
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53#39211yescharacteristic
145C22×C53#45012yes—
146C22×C53#45311yescharacteristic
147C22×C53#45411yescharacteristic
148C22×C53#45812yes—
149C22×C53#46012yes—
150C22×C53#49211yescharacteristic
151C22×C53#49611yescharacteristic
152C2×D53—84no—
153C2×D53—84no—
154C2×D53—82no—
155C2×D53—82no—
156C2×D53—82no—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—44no—
160C4×C53—82no—
161C4×C53—82no—
162Dic53—82no—
163C2×#481—532no—
164#2656—531no—
165C2×#327—532no—
166#1445D5311yescharacteristic, complemented
167#1280—531no—
168C2×C53—21no—
169C2×C53—22no—
170C2×C53—21no—
171C2×C53#161312yes—
172C2×C53#161612yes—
173C2×C53#200611yescharacteristic
174C2×C53#201411yescharacteristic
175C2×C53#202211yescharacteristic
176D53—162no—
177C2×#121—532no—
178C2×#121—1062no—
179C22×C4⋊C4C2×D5311yescharacteristic
180#481—1064no—
181#481—1064no—
182C2×#106—1062no—
183C2×#102—1062no—
184C2×C22⋊C8—532no—
185#399—1062no—
186#392—1062no—
187C2×#95—22no—
188C2×#95C2×D5311yescharacteristic
189C2×#95C2×D5311yescharacteristic
190#327—1064no—
191C53#996811yescharacteristic, complemented, Sylow
192C22×D4—532no—
193C22×D4—1062no—
194C22×D4—1062no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C22⋊D5312yes—
198C23×C4C22×D5311yescharacteristic
199C22×C8—1062no—
200#121—1062no—
201#121—1062no—
202#121—1062no—
203#121—1062no—
204#121—2124no—
205#121—2124no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—532no—
210C2×C4⋊C4—1062no—
211C2×C4⋊C4D21212yescomplemented
212C2×C22⋊C4—1064no—
213C2×C22⋊C4—1062no—
214#106—1068no—
215#102—1068no—
216C22⋊C8—534no—
217C22⋊C8—1062no—
218#95—22no—
219#95—22no—
220#95—44no—
221C24—1062no—
222C24C2×C22⋊D5311yescharacteristic
223C2×D4—1062no—
224C2×D4—1062no—
225C2×D4—4242no—
226C2×D4—4242no—
227C2×D4—2124no—
228C2×D4—2124no—
229C2×D4—2122no—
230C2×D4—2122no—
231C2×D4—2124no—
232C2×D4—2124no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—1062no—
249C22×C4#1171992414yes—
250C22×C4C2×D21211yescharacteristic
251C22×C4#1171994511yescharacteristic
252C22×C4D4×D5312yes—
253C22×C4D4×D5312yes—
254C2×C8—1064no—
255C2×C8—2124no—
256C4⋊C4—24no—
257C4⋊C4—42no—
258C4⋊C4—1062no—
259C4⋊C4—1062no—
260C4⋊C4—2124no—
261C22⋊C4—2128no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—2124no—
268C23—2122no—
269C23—2122no—
270C23—2122no—
271C23—2124no—
272C23#1313318511yescharacteristic
273C23#1313319111yescharacteristic
274C23#1313321311yescharacteristic
275C23C2×#1171992412yes—
276C23#1313324212yes—
277D4—4242no—
278D4—4248no—
279D4—4242no—
280D4—4244no—
281D4—4242no—
282D4—2124no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—2124no—
303C2×C4—2122no—
304C2×C4#1313318712yes—
305C2×C4#1313319012yes—
306C8—2124no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—21no—
314C22—21no—
315C22—4242no—
316C22—4244no—
317C22—4244no—
318C22—4242no—
319C22—4242no—
320C22#1681053211yescharacteristic
321C22#1681077312yes—
322C22#1681079011yescharacteristic
323C22#1681079511yescharacteristic
324C22#1681081212yes—
325C22#1681082312yes—
326C22#1681099711yescharacteristic
327C22#1681101311yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—82no—
333C4—4242no—
334C2—21no—
335C2—22no—
336C2—21no—
337C2—8482no—
338C2#2027373412yes—
339C2#2027375112yes—
340C2#2027575611yescharacteristic
341C2#2027580211yescharacteristic
342C2#2027584611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1445C2
socleC23×C53#120
prime radical#20280817C2
hypercenter#1445D53
hypocenterC53#9968
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#16810941C2semidirect
3. C2×#16810356C2semidirect
4. #20272558C2semidirect
5. C53×#1445C2semidirect
37. C22⋊#88322D4semidirect
45. C2×C4⋊C4×C53D4semidirect
166. #1445D53semidirect
191. C53#9968semidirect
211. C2×C4⋊C4D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168109412
#202808172
C2×#168103562
#202725582
C53×#14452
#996853