Finite Group Databasebeta

← #27279107

Subgroups — #27279107

Order 13568.

928 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 135 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#20277318C211yesmaximal, characteristic, complemented
2#20274646C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic, complemented
4#20272594C212yesmaximal, complemented
5C2×C53×#394C211yesmaximal, characteristic
6C2×#13133138—24no—
7C2×C22⋊#88322C2212yes—
8#16810509—24no—
9#16810486—21no—
10#16810483C2212yes—
11C4×C22⋊Dic53—21no—
12#16810349—22no—
13C23×D4×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C22×C22⋊C4×C53—21no—
16C22×C22⋊C4×C53—21no—
17C53×#394—22no—
18C53×#394C414yescomplemented
19C2×#95×C53C2211yescharacteristic
20C22×#88322—22no—
21C2×C4⋊Dic53D414yes—
22C2×C22⋊Dic53—22no—
23C2×C4×Dic53—22no—
24#13133138—48no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—24no—
29#13133124—22no—
30C25×C53—21no—
31C22×D4×C53—22no—
32C22×D4×C53—21no—
33C22×D4×C53—22no—
34C22×D4×C53—44no—
35C22×D4×C53C2×C412yes—
36C23×C4×C53—21no—
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C23×C4×C53C2311yescharacteristic
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53D412yes—
44C2×C22⋊C4×C53—24no—
45C2×C22⋊C4×C53—24no—
46C2×C22⋊C4×C53—44no—
47C2×C22⋊C4×C53—44no—
48#95×C53—21no—
49#95×C53C2×C412yes—
50C24×C53—24no—
51C24×C53—22no—
52C24×C53—42no—
53C24×C53—42no—
54C24×C53—42no—
55C24×C53—41no—
56C24×C53—41no—
57C24×C53—42no—
58C24×C53—42no—
59C24×C53C2×D411yescharacteristic
60C2×D4×C53—28no—
61C2×D4×C53—24no—
62C2×D4×C53—44no—
63C2×D4×C53—44no—
64C2×D4×C53—48no—
65C2×D4×C53—44no—
66C2×D4×C53—42no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—22no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C412yes—
80C22×C4×C53C22⋊C412yes—
81C22×C4×C53C22×C411yescharacteristic
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4112yes—
85C22×Dic53—24no—
86C4⋊C4×C53—24no—
87C4⋊C4×C53—24no—
88C22⋊C4×C53—48no—
89C22⋊C4×C53—48no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—44no—
93C4×Dic53—82no—
94C2×#88322—44no—
95C2×#88322—42no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—24no—
99C23×C53—42no—
100C23×C53—44no—
101C23×C53—48no—
102C23×C53—48no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—42no—
106C23×C53—42no—
107C23×C53—42no—
108C23×C53—44no—
109C23×C53—84no—
110C23×C53—84no—
111C23×C53—82no—
112C23×C53—82no—
113C23×C53—82no—
114C23×C53—82no—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53#12011yescharacteristic
117C23×C53#12311yescharacteristic
118C23×C53#13512yes—
119C23×C53AGL(1,ℤ/8)12yes—
120D4×C53—28no—
121D4×C53—48no—
122D4×C53—44no—
123C2×C4×C53—24no—
124C2×C4×C53—22no—
125C2×C4×C53—22no—
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—81no—
139C2×C4×C53—84no—
140C2×C4×C53C4×D412yes—
141C2×C4×C53#12112yes—
142C2×Dic53—44no—
143C2×Dic53—44no—
144C2×Dic53—44no—
145#88322—82no—
146#9018—531nomaximal, Sylow
147C22×C53—24no—
148C22×C53—24no—
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—44no—
154C22×C53—44no—
155C22×C53—44no—
156C22×C53—82no—
157C22×C53—82no—
158C22×C53—88no—
159C22×C53—88no—
160C22×C53—82no—
161C22×C53—82no—
162C22×C53#38611yescharacteristic
163C22×C53#41611yescharacteristic
164C22×C53#41811yescharacteristic
165C22×C53#44912yes—
166C22×C53#45412yes—
167C22×C53B(4,𝔽2)12yes—
168C22×C53#48212yes—
169C4×C53—22no—
170C4×C53—22no—
171C4×C53—42no—
172C4×C53—42no—
173C4×C53—82no—
174Dic53—84no—
175C2×#394D5311yescharacteristic
176#2313—531no—
177#1786—532no—
178C2×#324—531no—
179#1294—532no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—22no—
183C2×C53—84no—
184C2×C53#159714yes—
185C2×C53#187511yescharacteristic
186C2×C53#188811yescharacteristic
187C2×C53#188911yescharacteristic
188C23×D4C2×D5311yescharacteristic
189C22×C4⋊C4C2×D5311yescharacteristic
190C22×C22⋊C4—21no—
191C22×C22⋊C4—21no—
192C2×#102—1064no—
193C2×C22⋊C8—532no—
194#394—22no—
195#394Dic5314yescomplemented
196#387—1064no—
197#381—1061no—
198#380—532no—
199C4×C22⋊C4—1061no—
200C2×#95C2×D5311yescharacteristic
201#324—1062no—
202C53#901811yescharacteristic, complemented, Sylow
203C25—21no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—44no—
208C22×D4C2×Dic5312yes—
209C23×C4—21no—
210C23×C4C22⋊D5311yescharacteristic
211C23×C4C22⋊D5311yescharacteristic
212C23×C4C22×D5311yescharacteristic
213C22×C8—1062no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—534no—
217C2×C4⋊C4C2×Dic5312yes—
218C2×C4⋊C4C22⋊D5312yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—1062no—
224C2×C42—1062no—
225#102—2128no—
226C22⋊C8—1062no—
227C22⋊C8—1062no—
228#95—21no—
229#95—1062no—
230#95—1064no—
231#95—1062no—
232#95C2×Dic5312yes—
233C24—24no—
234C24—22no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—41no—
239C24—41no—
240C24—42no—
241C24—42no—
242C24C2×C22⋊D5311yescharacteristic
243C2×D4—28no—
244C2×D4—24no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—1064no—
263C22×C4C22⋊Dic5312yes—
264C22×C4C22⋊Dic5312yes—
265C22×C4D4×D5312yes—
266C22×C4#1171994712yes—
267C22×C4C22×Dic5311yescharacteristic
268C22×C4C2×C22⋊D5311yescharacteristic
269C2×C8—2124no—
270C2×C8—2122no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—2124no—
274C22⋊C4—48no—
275C22⋊C4—48no—
276C22⋊C4—2122no—
277C22⋊C4—2122no—
278C42—4242no—
279C23—22no—
280C23—22no—
281C23—24no—
282C23—42no—
283C23—44no—
284C23—42no—
285C23—42no—
286C23—48no—
287C23—48no—
288C23—42no—
289C23—44no—
290C23—42no—
291C23—42no—
292C23—82no—
293C23—84no—
294C23—84no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23C2×C22⋊Dic5311yescharacteristic
299C23#1313318511yescharacteristic
300C23#1313319811yescharacteristic
301C23#1313323712yes—
302C23#1313324312yes—
303D4—28no—
304D4—48no—
305D4—44no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—84no—
323C2×C4—2124no—
324C2×C4—2124no—
325C2×C4—2124no—
326C2×C4D4×Dic5312yes—
327C2×C4#1313318912yes—
328C8—4242no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—24no—
335C22—44no—
336C22—44no—
337C22—44no—
338C22—82no—
339C22—82no—
340C22—88no—
341C22—82no—
342C22—82no—
343C22—88no—
344C22#1681050311yescharacteristic
345C22#1681061911yescharacteristic
346C22#1681062811yescharacteristic
347C22#1681076612yes—
348C22#1681079512yes—
349C22#1681080912yes—
350C22#1681094412yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—82no—
356C4—4244no—
357C2—21no—
358C2—21no—
359C2—22no—
360C2—84no—
361C2#2027364214yes—
362C2#2027503811yescharacteristic
363C2#2027511911yescharacteristic
364C2#2027512411yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#394C2
socleC23×C53#120
prime radicalC23×D4×C53C22
cosocleC23×C4×C53C23
hypercenterC2×#394D53
hypocenterC53#9018
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
1. #20277318C2semidirect
3. C2×#16810349C2semidirect
4. #20272594C2semidirect
18. C53×#394C4semidirect
195. #394Dic53semidirect
202. C53#9018semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202746462
C2×#168103492
#202725942
C2×C53×#3942
#901853