Finite Group Databasebeta

← #27295134

Subgroups — #27295134

Order 13568.

440 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 103 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#20274711C211yesmaximal, characteristic, complemented
2#20273039C211yesmaximal, characteristic, complemented
3#20272721C212yesmaximal, complemented
4#20272718C212yesmaximal
5C53×#2536C211yesmaximal, characteristic
6#16810676—21no—
7#16810647—21no—
8C2×#13133136—21no—
9#16810392C2212yes—
10#16810349—21no—
11#16810349C2212yes—
12C2×C22⋊Q8×C53—21no—
13C2×C4×Q8×C53C2211yescharacteristic
14C2×C4×D4×C53C411yescharacteristic
15C53×#398C411yescharacteristic
16C53×#398C2211yescharacteristic
17C53×#394C411yescharacteristic
18C53×#394C2211yescharacteristic
19C53×#388—21no—
20C53×#388—21no—
21C53×#382C411yescharacteristic
22C53×#380—21no—
23C2×#11719917—22no—
24C4⋊#88322—41no—
25C4⋊#88322—41no—
26C4⋊#88322—41no—
27#13133136—42no—
28C22⋊#88322—22no—
29C4×#88322—41no—
30C22×Q8×C53D411yescharacteristic
31C22×D4×C53C2×C411yescharacteristic
32C23×C4×C53—21no—
33C22⋊Q8×C53—42no—
34C22⋊Q8×C53—42no—
35C4×Q8×C53—41no—
36C4×Q8×C53—41no—
37C4×D4×C53—41no—
38C4×D4×C53—41no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53C2311yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53C2×C411yescharacteristic
58#95×C53C2×C411yescharacteristic
59C24×C53—21no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×Q8×C53D812yes—
63C2×Q8×C53SD1612yes—
64C2×D4×C53—41no—
65C2×D4×C53—42no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—42no—
70C22×C4×C53—42no—
71C22×C4×C53—42no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—42no—
84C4⋊C4×C53—42no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—42no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—42no—
92C22⋊C4×C53—41no—
93C22⋊C4×C53—41no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—21no—
99C42×C53—21no—
100#11719917—42no—
101#11719917—42no—
102C2×#88322—42no—
103C2×#88322—42no—
104C23×C53—22no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—41no—
111C23×C53C2×C22⋊C411yescharacteristic
112Q8×C53—42no—
113Q8×C53—42no—
114D4×C53—42no—
115D4×C53—42no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
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—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—84no—
139C2×C4×C53#9911yescharacteristic
140C2×C4×C53#9911yescharacteristic
141C2×C4×C53#10212yes—
142C2×C4×C53#10212yes—
143C2×C4×C53C2×D811yescharacteristic
144C2×C4×C53C2×SD1611yescharacteristic
145#88322—82no—
146#88322—82no—
147#11879—531nomaximal, Sylow
148C22×C53—21no—
149C22×C53—21no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53C2×#9911yescharacteristic
158C22×C53C2×#10211yescharacteristic
159C22×C53#42111yescharacteristic
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—21no—
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53—41no—
167C4×C53—41no—
168C4×C53—41no—
169C4×C53—81no—
170C4×C53—81no—
171C4×C53—81no—
172#2536D5311yescharacteristic
173#1801—531no—
174#1461—531no—
175#1346—532no—
176#1345—532no—
177C2×C53—21no—
178C2×C53—21no—
179C2×C53—82no—
180C2×C53#151911yescharacteristic
181C2×C53#212011yescharacteristic
182C2×C53#213311yescharacteristic
183C2×C22⋊Q8—21no—
184C2×C4×Q8C2×D5311yescharacteristic
185C2×C4×D4Dic5311yescharacteristic
186#431—1061no—
187#423—1061no—
188C2×#101—1061no—
189#398Dic5311yescharacteristic
190#398C2×D5311yescharacteristic
191#394Dic5311yescharacteristic
192#394C2×D5311yescharacteristic
193#388—21no—
194#388—21no—
195#382Dic5311yescharacteristic
196#380—21no—
197#343—532no—
198#324—532no—
199#324—1061no—
200C53#1187911yescharacteristic, complemented, Sylow
201C22×Q8C22⋊D5311yescharacteristic
202C22×D4C2×Dic5311yescharacteristic
203C23×C4—21no—
204C2×M16—1062no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic5311yescharacteristic
215C2×C4⋊C4C2×Dic5311yescharacteristic
216C2×C4⋊C4C22⋊D5311yescharacteristic
217C2×C4⋊C4C22×D5311yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42C2×Dic5311yescharacteristic
224C2×C42C22⋊D5311yescharacteristic
225C2×C42C22⋊D5311yescharacteristic
226C4⋊C8—2121no—
227C4⋊C8—2121no—
228C4⋊C8—2121no—
229#101—2122no—
230C22⋊C8—1062no—
231C4×C8—2121no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C2×Dic5311yescharacteristic
236#95C2×Dic5311yescharacteristic
237C24—21no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8#1171992212yes—
241C2×Q8#1171992312yes—
242C2×D4—41no—
243C2×D4—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22⋊Dic5311yescharacteristic
252C22×C4C22⋊Dic5311yescharacteristic
253C22×C4C22⋊Dic5311yescharacteristic
254C22×C4C22×Dic5311yescharacteristic
255C22×C4C2×C22⋊D5311yescharacteristic
256C22×C4C2×C22⋊D5311yescharacteristic
257M16—2122no—
258M16—2122no—
259C2×C8—2122no—
260C2×C8—2122no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—22no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊Dic5311yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—42no—
293D4—42no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
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—84no—
317C2×C4#1313313211yescharacteristic
318C2×C4#1313313211yescharacteristic
319C2×C4#1313313812yes—
320C2×C4#1313313812yes—
321C2×C4C2×#1171992211yescharacteristic
322C2×C4C2×#1171992311yescharacteristic
323C8—4242no—
324C8—4242no—
325C22—21no—
326C22—21no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×#1313313211yescharacteristic
335C22C2×#1313313811yescharacteristic
336C22#1681064011yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—21no—
350C2—21no—
351C2—82no—
352C2#2027328311yescharacteristic
353C2#2027635611yescharacteristic
354C2#2027640411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2536C2
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2536D53
hypocenterC53#11879
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #20274711C2semidirect
2. #20273039C2semidirect
3. #20272721C2semidirect
200. C53#11879semidirect

Maximal subgroups

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

subgroupindex
#202747112
#202730392
#202727212
#202727182
C53×#25362
#1187953