Finite Group Databasebeta

← #27283106

Subgroups — #27283106

Order 13568.

526 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274568C211yesmaximal, characteristic, complemented
2C4×#13133128C211yesmaximal, characteristic, complemented
3C2×#16810365C212yesmaximal, complemented
4C2×#16810363C212yesmaximal
5C53×#2861C211yesmaximal, characteristic
6C2×C4⋊#88322—21no—
7C2×#13133128—21no—
8C2×#13133128C2212yes—
9C2×#13133128C2212yes—
10C2×C4×#88322—21no—
11#16810378—21no—
12#16810365—24no—
13#16810363—24no—
14C2×#125×C53C2211yescharacteristic
15C2×#124×C53C2211yescharacteristic
16C53×#391C411yescharacteristic
17C53×#391C411yescharacteristic
18C53×#390C411yescharacteristic
19C53×#390C411yescharacteristic
20C53×#388—21no—
21C53×#388—21no—
22C53×#388—21no—
23C53×#388—21no—
24C43×C53C2211yescharacteristic
25C22×#88322—22no—
26C22×#88322—22no—
27C4⋊#88322—24no—
28#13133128—24no—
29#13133128—22no—
30#13133128—22no—
31#13133128—22no—
32#13133128D414yes—
33C4×#88322—24no—
34C22×Q8×C53C2×C411yescharacteristic
35C22×D4×C53C2×C411yescharacteristic
36#125×C53—22no—
37#125×C53—22no—
38#124×C53—22no—
39#124×C53—22no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—21no—
47C2×C22⋊C4×C53—21no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C42×C53C2×C411yescharacteristic
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61C24×C53—21no—
62C2×Q8×C53—22no—
63C2×Q8×C53—41no—
64C2×Q8×C53—41no—
65C2×D4×C53—22no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53C22⋊C411yescharacteristic
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—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53—22no—
91C42×C53—22no—
92C42×C53—22no—
93C42×C53—41no—
94C42×C53—41no—
95C42×C53—41no—
96C42×C53C2×D412yes—
97C42×C53#4112yes—
98C2×#88322—24no—
99C2×#88322—24no—
100C2×#88322—24no—
101C2×#88322—24no—
102C2×#88322—42no—
103C2×#88322—42no—
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—82no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
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—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
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#10412yes—
139C2×C4×C53#10412yes—
140C2×C4×C53#10412yes—
141C2×C4×C53C4×D412yes—
142C2×C4×C53#12112yes—
143C2×C4×C53#12411yescharacteristic
144C2×C4×C53#12711yescharacteristic
145#88322—28no—
146#88322—44no—
147#9687—531nomaximal, Sylow
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53#39011yescharacteristic
156C22×C53#41311yescharacteristic
157C22×C53#41311yescharacteristic
158C22×C53C2×#10411yescharacteristic
159C22×C53C2×#10411yescharacteristic
160C22×C53C2×#10411yescharacteristic
161C22×C53#42111yescharacteristic
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—21no—
165C4×C53—21no—
166C4×C53—22no—
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—41no—
170C4×C53—81no—
171C4×C53—81no—
172C4×C53—81no—
173#2861D5311yescharacteristic
174#1767—531no—
175C4×#97—531no—
176C2×#330—532no—
177C2×#329—532no—
178C2×C53—161no—
179C2×C53#152411yescharacteristic
180C2×C53#152411yescharacteristic
181C2×C53#152911yescharacteristic
182C2×C53#153211yescharacteristic
183C2×C53#196211yescharacteristic
184C2×C53#196711yescharacteristic
185C2×C53#198011yescharacteristic
186C2×#125C2×D5311yescharacteristic
187C2×#124C2×D5311yescharacteristic
188C2×C4⋊C8—1061no—
189C2×#97—532no—
190C2×#97—532no—
191C2×#97—1061no—
192C2×C4×C8—1061no—
193#391Dic5311yescharacteristic
194#391Dic5311yescharacteristic
195#390Dic5311yescharacteristic
196#390Dic5311yescharacteristic
197#388—21no—
198#388—21no—
199#388—21no—
200#388—21no—
201C43C2×D5311yescharacteristic
202#336—1061no—
203#330—1064no—
204#329—1064no—
205C53#968711yescharacteristic, complemented, Sylow
206C22×Q8C2×Dic5311yescharacteristic
207C22×D4C2×Dic5311yescharacteristic
208C22×C8—1062no—
209C22×C8—1062no—
210#125—22no—
211#125—22no—
212#124—22no—
213#124—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×Dic5311yescharacteristic
217C2×C4⋊C4C2×Dic5311yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42C2×Dic5311yescharacteristic
225C2×C42C2×Dic5311yescharacteristic
226C2×C42C22⋊D5311yescharacteristic
227C2×C42C22⋊D5311yescharacteristic
228C2×C42C22⋊D5311yescharacteristic
229C2×C42C22⋊D5311yescharacteristic
230C2×C42C22×D5311yescharacteristic
231C4⋊C8—1064no—
232#97—534no—
233#97—1064no—
234#97—1062no—
235#97—1062no—
236#97—1062no—
237C4×C8—1064no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242C24—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×D4—22no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4C22⋊Dic5311yescharacteristic
253C22×C4C22⋊Dic5311yescharacteristic
254C22×C4C22⋊Dic5311yescharacteristic
255C22×C4C22⋊Dic5311yescharacteristic
256C22×C4C22×Dic5311yescharacteristic
257C22×C4C2×C22⋊D5311yescharacteristic
258C22×C4C2×C22⋊D5311yescharacteristic
259C2×C8—1064no—
260C2×C8—1064no—
261C2×C8—1064no—
262C2×C8—1064no—
263C2×C8—2122no—
264C2×C8—2122no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—41no—
281C42—41no—
282C42—41no—
283C42D4×D5312yes—
284C42#1171994712yes—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊Dic5311yescharacteristic
293Q8—82no—
294D4—82no—
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—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4#1313314412yes—
320C2×C4#1313314412yes—
321C2×C4#1313314412yes—
322C2×C4D4×Dic5312yes—
323C2×C4#1313318812yes—
324C2×C4#1313320411yescharacteristic
325C2×C4#1313321311yescharacteristic
326C8—1068no—
327C8—2124no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#1681052611yescharacteristic
336C22#1681060711yescharacteristic
337C22#1681060711yescharacteristic
338C22C2×#1313314411yescharacteristic
339C22C2×#1313314411yescharacteristic
340C22C2×#1313314411yescharacteristic
341C22#1681064011yescharacteristic
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—21no—
347C4—21no—
348C4—22no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C2—161no—
354C2#2027330311yescharacteristic
355C2#2027330311yescharacteristic
356C2#2027332211yescharacteristic
357C2#2027333511yescharacteristic
358C2#2027554411yescharacteristic
359C2#2027556511yescharacteristic
360C2#2027563511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2861C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2861D53
hypocenterC53#9687
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C43 ◁ #2861

Complements

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

NH ≅ G/Ntype
1. #20274568C2semidirect
2. C4×#13133128C2semidirect
3. C2×#16810365C2semidirect
205. C53#9687semidirect

Maximal subgroups

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

subgroupindex
#202745682
C4×#131331282
C2×#168103652
C2×#168103632
C53×#28612
#968753