Finite Group Databasebeta

← #27272348

Subgroups — #27272348

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274477C211yesmaximal, characteristic
2#20274409C211yesmaximal, characteristic
3#20272541C212yesmaximal, complemented
4#20272538C212yesmaximal, complemented
5C4×C22⋊Q8×C53C211yesmaximal, characteristic
6C22×#11719917—21no—
7C2×C22⋊#88322—21no—
8C2×#13133128—21no—
9#16810378—21no—
10#16810378C2212yes—
11#16810378C2212yes—
12C2×C22⋊Q8×C53C411yescharacteristic
13C2×C4×Q8×C53C411yescharacteristic
14C22×C42×C53C2211yescharacteristic
15C53×#391C2211yescharacteristic
16C53×#389C2211yescharacteristic
17C53×#387—21no—
18C53×#385—21no—
19C53×#380C411yescharacteristic
20C4×C4⋊C4×C53—21no—
21C4×C4⋊C4×C53C411yescharacteristic
22C4×C22⋊C4×C53—21no—
23C2×#11719917—24no—
24C2×#11719917—42no—
25C2×#11719917—42no—
26C22×#88322—22no—
27C22×#88322—22no—
28C22⋊#88322—24no—
29#13133128—42no—
30C22×Q8×C53C2×C411yescharacteristic
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C22⋊Q8×C53—22no—
35C22⋊Q8×C53—22no—
36C4×Q8×C53—42no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C42×C53—21no—
46C2×C42×C53—21no—
47C2×C42×C53—42no—
48C2×C42×C53C2×C411yescharacteristic
49C2×C42×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C2×C42×C53C2311yescharacteristic
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57C24×C53C22⋊C411yescharacteristic
58C2×Q8×C53—21no—
59C2×Q8×C53—41no—
60C2×Q8×C53—41no—
61C2×Q8×C53M1612yescomplemented
62C22×C4×C53—22no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
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—22no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22×C411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C4⋊C4×C53—24no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—41no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53M1612yescomplemented
91C22⋊C4×C53—24no—
92C22⋊C4×C53—42no—
93C42×C53—22no—
94C42×C53—22no—
95C42×C53—41no—
96C42×C53—41no—
97C42×C53—42no—
98C42×C53—42no—
99#11719917—48no—
100C2×#88322—24no—
101C2×#88322—24no—
102C2×#88322—42no—
103C2×#88322—42no—
104C2×#88322—42no—
105C2×#88322—42no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53#10412yes—
113C23×C53C2×C22⋊C411yescharacteristic
114Q8×C53—42no—
115Q8×C53—42no—
116C2×C4×C53—24no—
117C2×C4×C53—24no—
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—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—44no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53#9912yes—
147C2×C4×C53#10412yes—
148C2×C4×C53C2×M1611yescharacteristic
149C2×C4×C53C2×M1611yescharacteristic
150C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C2×C4×C53#13712yes—
153#88322—44no—
154#88322—44no—
155#7634—531nomaximal, Sylow
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53#40711yescharacteristic
164C22×C53C2×#9911yescharacteristic
165C22×C53#41311yescharacteristic
166C22×C53#41711yescharacteristic
167C22×C53#41911yescharacteristic
168C22×C53C2×#10411yescharacteristic
169C22×C53C2×#10411yescharacteristic
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—22no—
174C4×C53—21no—
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—41no—
178C4×C53—42no—
179C4×C53—42no—
180C4×C53—81no—
181C4×C53—81no—
182C4×C22⋊Q8D5311yescharacteristic
183#1749—531no—
184#1728—531no—
185#1272—532no—
186#1271—532no—
187C2×C53—42no—
188C2×C53#146511yescharacteristic
189C2×C53#148911yescharacteristic
190C2×C53#149111yescharacteristic
191C2×C53#150411yescharacteristic
192C2×C53#150811yescharacteristic
193C2×C53#152411yescharacteristic
194C2×C53#155911yescharacteristic
195C22×M16—1061no—
196C2×C22⋊Q8Dic5311yescharacteristic
197C2×C4×Q8Dic5311yescharacteristic
198C22×C42C2×D5311yescharacteristic
199C2×C22⋊C8—1061no—
200C2×#97—1061no—
201#391C2×D5311yescharacteristic
202#389C2×D5311yescharacteristic
203#387—21no—
204#385—21no—
205#380Dic5311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4Dic5311yescharacteristic
208C4×C22⋊C4—21no—
209#336—532no—
210#336—532no—
211#336—1061no—
212C53#763411yescharacteristic, complemented, Sylow
213C22×Q8C2×Dic5311yescharacteristic
214C23×C4C2×Dic5311yescharacteristic
215C23×C4C22⋊D5311yescharacteristic
216C23×C4C22⋊D5311yescharacteristic
217C2×M16—1064no—
218C2×M16—2122no—
219C2×M16—2122no—
220C22×C8—1062no—
221C22×C8—1062no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C4×Q8—42no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic5311yescharacteristic
229C2×C4⋊C4C2×Dic5311yescharacteristic
230C2×C4⋊C4C2×Dic5311yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—42no—
236C2×C42C2×Dic5311yescharacteristic
237C2×C42C22⋊D5311yescharacteristic
238C2×C42C22⋊D5311yescharacteristic
239C2×C42C22×D5311yescharacteristic
240C22⋊C8—1064no—
241#97—2122no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247C24C22⋊Dic5311yescharacteristic
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8#1171991712yescomplemented
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C22⋊Dic5311yescharacteristic
268C22×C4C22⋊Dic5311yescharacteristic
269C22×C4C22⋊Dic5311yescharacteristic
270C22×C4C22×Dic5311yescharacteristic
271C22×C4C2×C22⋊D5311yescharacteristic
272C22×C4C2×C22⋊D5311yescharacteristic
273M16—2128no—
274C2×C8—1064no—
275C2×C8—1064no—
276C2×C8—2122no—
277C2×C8—2122no—
278C2×C8—2122no—
279C2×C8—2122no—
280C4⋊C4—24no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4#1171991712yescomplemented
288C22⋊C4—24no—
289C22⋊C4—42no—
290C42—22no—
291C42—22no—
292C42—41no—
293C42—41no—
294C42—42no—
295C42—42no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#1313314412yes—
303C23C2×C22⋊Dic5311yescharacteristic
304Q8—42no—
305Q8—42no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4#1313313212yes—
337C2×C4#1313314412yes—
338C2×C4C2×#1171991711yescharacteristic
339C2×C4C2×#1171991711yescharacteristic
340C2×C4#1313324211yescharacteristic
341C2×C4#1313324711yescharacteristic
342C2×C4#1313324812yes—
343C8—2124no—
344C8—2124no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#1681058711yescharacteristic
353C22C2×#1313313211yescharacteristic
354C22#1681060711yescharacteristic
355C22#1681062411yescharacteristic
356C22#1681063211yescharacteristic
357C22C2×#1313314411yescharacteristic
358C22C2×#1313314411yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—22no—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—42no—
367C4—41no—
368C4—42no—
369C4—81no—
370C4—81no—
371C2—42no—
372C2#2027305311yescharacteristic
373C2#2027315411yescharacteristic
374C2#2027316211yescharacteristic
375C2#2027321911yescharacteristic
376C2#2027323511yescharacteristic
377C2#2027330311yescharacteristic
378C2#2027344511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×C22⋊Q8×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenterC4×C22⋊Q8D53
hypocenterC53#7634
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20272541C2semidirect
4. #20272538C2semidirect
61. C2×Q8×C53M16semidirect
90. C4⋊C4×C53M16semidirect
212. C53#7634semidirect
251. C2×Q8#11719917semidirect
287. C4⋊C4#11719917semidirect

Maximal subgroups

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

subgroupindex
#202744772
#202744092
#202725412
#202725382
C4×C22⋊Q8×C532
#763453