Finite Group Databasebeta

← #27295010

Subgroups — #27295010

Order 13568.

418 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 87 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#20274798C211yesmaximal, characteristic, complemented
2#20273029C211yesmaximal, characteristic, complemented
3#20272745C212yesmaximal, complemented
4#20272718C212yesmaximal, complemented
5C53×#2610C211yesmaximal, characteristic
6#16810676—21no—
7#16810587—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349C2212yes—
11#16810347—21no—
12C2×#124×C53C411yescharacteristic
13C2×#117×C53C2211yescharacteristic
14C53×#395C2211yescharacteristic
15C53×#394C2211yescharacteristic
16C53×#393C411yescharacteristic
17C53×#392C411yescharacteristic
18C53×#390—21no—
19C53×#388—21no—
20C53×#388—21no—
21C53×#386—21no—
22C53×#376C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22⋊#88322—41no—
28C4×#88322—41no—
29C22×Q8×C53C2×C411yescharacteristic
30C22×D4×C53—21no—
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33#124×C53—41no—
34#124×C53—41no—
35#117×C53—41no—
36#117×C53—41no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C4⋊C4×C53C2311yescharacteristic
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56C24×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×Q8×C53—41no—
59C2×Q8×C53—81no—
60C2×D4×C53—42no—
61C2×D4×C53—41no—
62C2×D4×C53—42no—
63C2×D4×C53—81no—
64C2×D4×C53—81no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—81no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—81no—
89C22⋊C4×C53—81no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—21no—
93C42×C53—21no—
94C42×C53—42no—
95#11719917—42no—
96#11719917—42no—
97C2×#88322—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53C2×C22⋊C411yescharacteristic
112Q8×C53—82no—
113D4×C53—84no—
114D4×C53—82no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53#9911yescharacteristic
139C2×C4×C53#9911yescharacteristic
140C2×C4×C53#13512yes—
141#88322—82no—
142#88322—82no—
143#11851—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—42no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53—81no—
158C22×C53C2×#9911yescharacteristic
159C22×C53#41611yescharacteristic
160C22×C53#42111yescharacteristic
161C4×C53—22no—
162C4×C53—21no—
163C4×C53—21no—
164C4×C53—42no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—81no—
170C4×C53—81no—
171#2610D5311yescharacteristic
172#1822—531no—
173#1458—531no—
174#1354—532no—
175#1345—532no—
176C2×C53—21no—
177C2×C53—21no—
178C2×C53—81no—
179C2×C53—161no—
180C2×C53#151611yescharacteristic
181C2×C53#212211yescharacteristic
182C2×C53#213211yescharacteristic
183C2×#124Dic5311yescharacteristic
184C2×#117C2×D5311yescharacteristic
185#431—1061no—
186#407—1061no—
187#395C2×D5311yescharacteristic
188#394C2×D5311yescharacteristic
189#393Dic5311yescharacteristic
190#392Dic5311yescharacteristic
191#390—21no—
192#388—21no—
193#388—21no—
194#386—21no—
195#376Dic5311yescharacteristic
196#343—532no—
197#343—1061no—
198#324—532no—
199#323—1061no—
200C53#1185111yescharacteristic, complemented, Sylow
201C22×Q8C2×Dic5311yescharacteristic
202C22×D4—21no—
203C22×D4C2×Dic5311yescharacteristic
204C23×C4C22⋊D5311yescharacteristic
205C2×M16—1062no—
206#124—41no—
207#124—41no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22⋊D5311yescharacteristic
212C2×C4⋊C4C22×D5311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×Dic5311yescharacteristic
220C2×C22⋊C4C22⋊D5311yescharacteristic
221C2×C42—21no—
222C2×C42C2×Dic5311yescharacteristic
223C2×C42C22⋊D5311yescharacteristic
224C4⋊C8—2121no—
225C22⋊C8—1062no—
226C22⋊C8—2121no—
227C22⋊C8—2121no—
228C4×C8—2121no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×Dic5311yescharacteristic
233#95C2×Dic5311yescharacteristic
234C24—21no—
235C24C22⋊Dic5311yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—81no—
242C2×D4—81no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22⋊Dic5311yescharacteristic
252C22×C4C22⋊Dic5311yescharacteristic
253C22×C4C22×Dic5311yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255C22×C4C2×C22⋊D5311yescharacteristic
256M16—2122no—
257M16—2122no—
258C2×C8—2122no—
259C2×C8—2122no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—42no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊Dic5311yescharacteristic
290Q8—82no—
291D4—84no—
292D4—82no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4#1313313211yescharacteristic
317C2×C4#1313313211yescharacteristic
318C2×C4#1313323712yes—
319C8—4242no—
320C8—4242no—
321C22—21no—
322C22—21no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C2×#1313313211yescharacteristic
336C22#1681061911yescharacteristic
337C22#1681064011yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—161no—
352C2#2027327111yescharacteristic
353C2#2027636511yescharacteristic
354C2#2027640011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2610C2
socleC22×C53C2×#99
prime radicalC53×#392C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2610D53
hypocenterC53#11851
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #20274798C2semidirect
2. #20273029C2semidirect
3. #20272745C2semidirect
4. #20272718C2semidirect
200. C53#11851semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202730292
#202727452
#202727182
C53×#26102
#1185153