Finite Group Databasebeta

← #27280166

Subgroups — #27280166

Order 13568.

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

normal subgroups only

This group has 690 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
1#20277313C211yesmaximal, characteristic, complemented
2#20273283C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic, complemented
4#20272582C212yesmaximal, complemented
5C53×#2617C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7#16810516—24no—
8#16810483C2212yes—
9#16810479—21no—
10C4×C22⋊Dic53—21no—
11#16810357—24no—
12#16810347—22no—
13C2×#121×C53C2211yescharacteristic
14C2×#120×C53—21no—
15C22×C22⋊C4×C53C2211yescharacteristic
16C53×#397C412yescomplemented
17C53×#394—22no—
18C53×#392C412yescomplemented
19C53×#386—21no—
20C53×#380C2211yescharacteristic
21C22×#88322—22no—
22C2×C4⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—22no—
27C22⋊#88322—22no—
28C22⋊#88322—22no—
29#13133124—22no—
30#13133124D414yescomplemented
31C25×C53D411yescharacteristic
32C22×D4×C53—21no—
33C22×D4×C53—22no—
34C22×D4×C53C2×C411yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53C2311yescharacteristic
37#121×C53—22no—
38#121×C53—22no—
39#120×C53—44no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53C2×C412yes—
49C2×C22⋊C4×C53D412yes—
50#95×C53—22no—
51#95×C53C2×C412yes—
52C24×C53—21no—
53C24×C53—21no—
54C24×C53—22no—
55C24×C53—22no—
56C24×C53—21no—
57C24×C53—81no—
58C24×C53—81no—
59C24×C53C22⋊C412yes—
60C24×C53C2×D411yescharacteristic
61C2×D4×C53—22no—
62C2×D4×C53—44no—
63C2×D4×C53—41no—
64C2×D4×C53—41no—
65C2×D4×C53—42no—
66C2×D4×C53—42no—
67C2×D4×C53—82no—
68C2×D4×C53—82no—
69C2×D4×C53—81no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53—81no—
75C22×C4×C53—81no—
76C22×C4×C53C22⋊C412yes—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D412yes—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4112yes—
81C22×Dic53—24no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—81no—
84C22⋊C4×C53—24no—
85C22⋊C4×C53—24no—
86C22⋊C4×C53—44no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—82no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—82no—
93C22⋊Dic53—82no—
94C4⋊Dic53—82no—
95C4×Dic53—82no—
96C2×#88322—42no—
97C2×#88322—82no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—42no—
109C23×C53—42no—
110C23×C53—42no—
111C23×C53—41no—
112C23×C53—41no—
113C23×C53—41no—
114C23×C53—81no—
115C23×C53—81no—
116C23×C53—84no—
117C23×C53—84no—
118C23×C53C2×C22⋊C411yescharacteristic
119C23×C53#12011yescharacteristic
120C23×C53#12311yescharacteristic
121C23×C53#13512yes—
122C23×C53AGL(1,ℤ/8)12yes—
123D4×C53—82no—
124D4×C53—88no—
125D4×C53—82no—
126D4×C53—82no—
127C2×C4×C53—22no—
128C2×C4×C53—41no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—81no—
141C2×C4×C53C4×D412yes—
142C2×C4×C53#12112yes—
143C2×Dic53—44no—
144C2×Dic53—44no—
145C2×Dic53—44no—
146#88322—82no—
147#9205—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53—44no—
157C22×C53—42no—
158C22×C53—42no—
159C22×C53—42no—
160C22×C53—82no—
161C22×C53—84no—
162C22×C53—84no—
163C22×C53—82no—
164C22×C53—82no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53#38611yescharacteristic
169C22×C53#41611yescharacteristic
170C22×C53#41811yescharacteristic
171C22×C53#44912yes—
172C22×C53#48212yes—
173C4×C53—22no—
174C4×C53—22no—
175C4×C53—42no—
176C4×C53—82no—
177C4×C53—81no—
178Dic53—84no—
179#2617D5311yescharacteristic
180#2312—531no—
181#1519—532no—
182C2×#323—531no—
183#1289—532no—
184C2×C53—22no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—22no—
188C2×C53—82no—
189C2×C53—161no—
190C2×C53#187511yescharacteristic
191C2×C53#191111yescharacteristic
192C2×C53#191311yescharacteristic
193C2×#121C2×D5311yescharacteristic
194C2×#120—21no—
195C22×C22⋊C4C2×D5311yescharacteristic
196C2×C22⋊C8—532no—
197#397Dic5312yescomplemented
198#394—22no—
199#392Dic5312yescomplemented
200#388—1064no—
201#386—21no—
202#380—532no—
203#380C2×D5311yescharacteristic
204#379—1061no—
205C4×C22⋊C4—1061no—
206#327—1064no—
207#323—1062no—
208C53#920511yescharacteristic, complemented, Sylow
209C25C22⋊D5311yescharacteristic
210C22×D4—21no—
211C22×D4—22no—
212C22×D4C2×Dic5311yescharacteristic
213C23×C4C22⋊D5311yescharacteristic
214C23×C4C22×D5311yescharacteristic
215C22×C8—1062no—
216#121—22no—
217#121—22no—
218#120—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—1062no—
221C2×C4⋊C4C2×Dic5311yescharacteristic
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—1062no—
229C2×C22⋊C4—1062no—
230C2×C22⋊C4—1062no—
231C2×C22⋊C4C2×Dic5312yes—
232C2×C22⋊C4C22⋊D5312yes—
233C2×C42—1062no—
234C22⋊C8—1062no—
235C22⋊C8—1062no—
236#95—22no—
237#95—534no—
238#95—1062no—
239#95C2×Dic5312yes—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—21no—
245C24—81no—
246C24—81no—
247C24C22⋊Dic5312yes—
248C24C2×C22⋊D5311yescharacteristic
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—1064no—
265C22×C4C22⋊Dic5312yes—
266C22×C4D4×D5312yes—
267C22×C4#1171994712yes—
268C22×C4C22×Dic5311yescharacteristic
269C22×C4C2×C22⋊D5311yescharacteristic
270C2×C8—4242no—
271C2×C8—2122no—
272C4⋊C4—22no—
273C4⋊C4—81no—
274C4⋊C4—4242no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—44no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—82no—
281C22⋊C4—4242no—
282C22⋊C4—4242no—
283C22⋊C4—2122no—
284C22⋊C4—2122no—
285C42—4242no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—81no—
303C23—81no—
304C23—84no—
305C23—84no—
306C23C2×C22⋊Dic5311yescharacteristic
307C23#1313318511yescharacteristic
308C23#1313319811yescharacteristic
309C23#1313323712yes—
310C23#1313324312yes—
311D4—82no—
312D4—88no—
313D4—82no—
314D4—82no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—2124no—
330C2×C4—2124no—
331C2×C4—2124no—
332C2×C4D4×Dic5312yes—
333C2×C4#1313318912yes—
334C8—4242no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—84no—
349C22—84no—
350C22—82no—
351C22—82no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#1681050311yescharacteristic
356C22#1681061911yescharacteristic
357C22#1681062811yescharacteristic
358C22#1681076612yes—
359C22#1681094412yes—
360C4—22no—
361C4—22no—
362C4—42no—
363C4—82no—
364C4—81no—
365C4—4244no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2—22no—
370C2—82no—
371C2—161no—
372C2#2027503811yescharacteristic
373C2#2027524811yescharacteristic
374C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2617C2
socleC22×C53#386
cosocleC23×C4×C53C23
hypercenter#2617D53
hypocenterC53#9205
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. #20277313C2semidirect
2. #20273283C2semidirect
3. C2×#16810347C2semidirect
4. #20272582C2semidirect
16. C53×#397C4semidirect
18. C53×#392C4semidirect
30. #13133124D4semidirect
197. #397Dic53semidirect
199. #392Dic53semidirect
208. C53#9205semidirect

Maximal subgroups

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

subgroupindex
#202773132
#202732832
C2×#168103472
#202725822
C53×#26172
#920553