Finite Group Databasebeta

← #27294481

Subgroups — #27294481

Order 13568.

400 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#13133136C211yesmaximal, characteristic, complemented
2#20273039C211yesmaximal, characteristic, complemented
3#20272745C212yesmaximal, complemented
4#20272718C212yesmaximal
5C53×#2632C211yesmaximal, characteristic
6#16810676—21no—
7C2×#13133136—21no—
8C4×#11719917—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×#124×C53C411yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#399—21no—
15C53×#395C411yescharacteristic
16C53×#395C2211yescharacteristic
17C53×#394C411yescharacteristic
18C53×#394C2211yescharacteristic
19C53×#393—21no—
20C53×#388—21no—
21C53×#388—21no—
22C53×#383C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25#13133136—24no—
26C22⋊#88322—22no—
27#13133128—41no—
28C4×#88322—41no—
29C4×#88322—41no—
30C22×Q8×C53C2×C411yescharacteristic
31C22×Q8×C53D411yescharacteristic
32C22×D4×C53C2×C411yescharacteristic
33#124×C53—41no—
34#124×C53—41no—
35C4×Q8×C53—41no—
36C4×Q8×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53C2×C411yescharacteristic
56#95×C53C2×C411yescharacteristic
57C24×C53—21no—
58C2×Q8×C53—22no—
59C2×Q8×C53—22no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×Q8×C53—81no—
63C2×D4×C53—41no—
64C2×D4×C53—81no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—81no—
79C4⋊C4×C53—81no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—81no—
86C22⋊C4×C53—81no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93#11719917—24no—
94#11719917—42no—
95C2×#88322—42no—
96C2×#88322—42no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105Q8×C53—44no—
106Q8×C53—82no—
107D4×C53—82no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53#9911yescharacteristic
132C2×C4×C53#9911yescharacteristic
133C2×C4×C53#10412yes—
134C2×C4×C53#13512yes—
135#88322—44no—
136#88322—82no—
137#11738—531nomaximal, Sylow
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—81no—
145C22×C53—81no—
146C22×C53—81no—
147C22×C53C2×#9911yescharacteristic
148C22×C53#41611yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—22no—
153C4×C53—22no—
154C4×C53—41no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—81no—
159C4×C53—81no—
160C4×C53—81no—
161#2632D5311yescharacteristic
162C4×#101—531no—
163#1461—531no—
164#1354—532no—
165#1345—532no—
166C2×C53—21no—
167C2×C53—21no—
168C2×C53—161no—
169C2×C53#150111yescharacteristic
170C2×C53#212211yescharacteristic
171C2×C53#213311yescharacteristic
172C2×#124Dic5311yescharacteristic
173C2×C4×Q8C2×D5311yescharacteristic
174#431—1061no—
175C2×#101—1061no—
176C4×M16—1061no—
177#399—21no—
178#395Dic5311yescharacteristic
179#395C2×D5311yescharacteristic
180#394Dic5311yescharacteristic
181#394C2×D5311yescharacteristic
182#393—21no—
183#388—21no—
184#388—21no—
185#383Dic5311yescharacteristic
186#343—532no—
187#324—532no—
188#324—1061no—
189C53#1173811yescharacteristic, complemented, Sylow
190C22×Q8C2×Dic5311yescharacteristic
191C22×Q8C22⋊D5311yescharacteristic
192C22×D4C2×Dic5311yescharacteristic
193C2×M16—1062no—
194#124—41no—
195#124—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C2×Dic5311yescharacteristic
201C2×C4⋊C4C22⋊D5311yescharacteristic
202C2×C4⋊C4C22×D5311yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42C2×Dic5311yescharacteristic
210C2×C42C22⋊D5311yescharacteristic
211C2×C42C22⋊D5311yescharacteristic
212C4⋊C8—2121no—
213#101—1064no—
214C22⋊C8—1062no—
215#97—2121no—
216C4×C8—2121no—
217C4×C8—2121no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C2×Dic5311yescharacteristic
223#95C2×Dic5311yescharacteristic
224C24—21no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—81no—
230C2×D4—41no—
231C2×D4—81no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4C22⋊Dic5311yescharacteristic
236C22×C4C22⋊Dic5311yescharacteristic
237C22×C4C22⋊Dic5311yescharacteristic
238C22×C4C22⋊Dic5311yescharacteristic
239C22×C4C22×Dic5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—1064no—
243M16—2122no—
244C2×C8—2122no—
245C2×C8—2122no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C22⋊C4—42no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23C2×C22⋊Dic5311yescharacteristic
272Q8—44no—
273Q8—82no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4#1313313211yescharacteristic
299C2×C4#1313313211yescharacteristic
300C2×C4#1313314412yes—
301C2×C4#1313323712yes—
302C8—4242no—
303C8—2124no—
304C22—21no—
305C22—21no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22C2×#1313313211yescharacteristic
314C22#1681061911yescharacteristic
315C22C2×#1313314411yescharacteristic
316C4—22no—
317C4—22no—
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C2—21no—
328C2—21no—
329C2—161no—
330C2#2027320611yescharacteristic
331C2#2027636511yescharacteristic
332C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2632C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2632D53
hypocenterC53#11738
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2632

Complements

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

NH ≅ G/Ntype
1. C4×#13133136C2semidirect
2. #20273039C2semidirect
3. #20272745C2semidirect
189. C53#11738semidirect

Maximal subgroups

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

subgroupindex
C4×#131331362
#202730392
#202727452
#202727182
C53×#26322
#1173853