Finite Group Databasebeta

← #27534429

Subgroups — #27534429

Order 13568.

746 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810993C211yesmaximal, characteristic, complemented
2#20284202C211yesmaximal, characteristic
3#20283143C211yesmaximal, characteristic
4#20275709C211yesmaximal, characteristic, complemented
5#20275558C211yesmaximal, characteristic, complemented
6#20275369C214yesmaximal, complemented
7#20275228C214yesmaximal
8#20274855C211yesmaximal, characteristic, complemented
9C53×#3080C211yesmaximal, characteristic
10C2×#13133247C2212yes—
11C2×#13133224C2212yes—
12C2×#13133217C2211yescharacteristic
13#16810993—21no—
14#16810993—21no—
15#16810993—22no—
16#16810833—22no—
17#16810828—22no—
18#16810647C2211yescharacteristic
19#16810631C2212yes—
20C2×#13133140C2214yes—
21C2×#13133135C2214yes—
22#16810579—22no—
23#16810579C2212yes—
24C4×#11719917—21no—
25C4×#11719917C2211yescharacteristic
26C2×#13133128—21no—
27C2×#13133128C2211yescharacteristic
28#16810392C2214yes—
29#16810387C2214yes—
30C2×#125×C53C2211yescharacteristic
31C4×#41×C53C2211yescharacteristic
32C53×#479—22no—
33C53×#478—22no—
34C53×#428C2212yes—
35C53×#423C2211yescharacteristic
36C53×#417C2212yes—
37C22×Dic106C2312yes—
38#13133247—28no—
39C2×#11719925—22no—
40C2×#11719923—22no—
41#13133224—24no—
42#13133224—24no—
43C2×#11719917—22no—
44C2×#11719917C2312yes—
45C2×#11719917C2312yes—
46C22×#88322—22no—
47C22×#88322C2312yes—
48#13133217—22no—
49#13133217—21no—
50#13133217—21no—
51C2×C4⋊Dic53—22no—
52C4⋊#88322—22no—
53#13133140—24no—
54#13133140—24no—
55#13133140—24no—
56#13133135—28no—
57#13133128—21no—
58#13133128—22no—
59#13133128D411yescharacteristic
60#13133128D411yescharacteristic
61#13133128D412yes—
62C4×#88322—21no—
63C4×#88322—22no—
64C2×#41×C53C2311yescharacteristic
65C2×M16×C53C2312yes—
66#125×C53—21no—
67#125×C53—21no—
68#125×C53—22no—
69C4×Q8×C53D412yes—
70C4×D4×C53—22no—
71C4×D4×C53D412yes—
72#117×C53—21no—
73#117×C53C2311yescharacteristic
74C2×C4⋊C4×C53—22no—
75C2×C4⋊C4×C53C2312yes—
76C2×C42×C53—21no—
77C2×C42×C53C2311yescharacteristic
78#107×C53—22no—
79#106×C53—22no—
80C4⋊C8×C53—22no—
81#103×C53—22no—
82#102×C53—22no—
83#41×C53—22no—
84#41×C53#4114yes—
85C2×Q8×C53C2×D411yescharacteristic
86C2×D4×C53—21no—
87C2×D4×C53C2×D411yescharacteristic
88C22×C4×C53—21no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53#4112yes—
92C22×C4×C53C2411yescharacteristic
93C22×Dic53—22no—
94C2×Dic106—22no—
95C2×Dic106—22no—
96C2×Dic106—24no—
97C2×Dic106—24no—
98M16×C53—24no—
99C2×C8×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—42no—
104C4⋊C4×C53—42no—
105C4⋊C4×C53C2×D412yes—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—22no—
108C42×C53—22no—
109C42×C53—21no—
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D412yes—
113#11719925—44no—
114#11719923—44no—
115C4⋊Dic53—42no—
116C4⋊Dic53—42no—
117#11719917—28no—
118#11719917—22no—
119#11719917—24no—
120#11719917#4114yes—
121C2×#88322—28no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—22no—
126C2×#88322C2×D412yes—
127C2×#88322C2×D412yes—
128C23×C53—21no—
129C23×C53C2×#4111yescharacteristic
130Q8×C53—21no—
131Q8×C53#12412yes—
132D4×C53—24no—
133D4×C53—21no—
134D4×C53#12412yes—
135C2×C4×C53—21no—
136C2×C4×C53—24no—
137C2×C4×C53—22no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—82no—
141C2×C4×C53—82no—
142C2×C4×C53#12112yes—
143C2×C4×C53#12112yes—
144C2×C4×C53C22×D411yescharacteristic
145C2×C4×C53C22×D411yescharacteristic
146C2×C4×C53C2×#4112yes—
147C2×C4×C53C2×#4112yes—
148C2×Dic53—42no—
149C2×Dic53—44no—
150Dic106—44no—
151Dic106—42no—
152Dic106—42no—
153C8×C53—42no—
154#88322—22no—
155#88322—24no—
156#88322—22no—
157#88322—42no—
158#33065—531nomaximal, Sylow
159C22×C53—22no—
160C22×C53—22no—
161C22×C53—42no—
162C22×C53C2×#12111yescharacteristic
163C22×C53#52912yes—
164C4×C53—22no—
165C4×C53—22no—
166C4×C53—22no—
167C4×C53—42no—
168C4×C53—82no—
169C4×C53#52512yes—
170C4×C53C2×#12411yescharacteristic
171C4×C53#53211yescharacteristic
172Dic53—82no—
173C2×#491—531no—
174#3080D5311yescharacteristic
175#3039—531no—
176#2875—531no—
177#1998—531no—
178#1966—531no—
179#1930—534no—
180#1908—534no—
181#1838—531no—
182C2×C53—22no—
183C2×C53—42no—
184C2×C53#285011yescharacteristic
185C2×C53#306912yes—
186C2×#137—532no—
187C2×#131—532no—
188C2×C4⋊Q8—531no—
189C2×#125C2×D5311yescharacteristic
190C4×#41C2×D5311yescharacteristic
191#491—1061no—
192#491—1061no—
193#491—1062no—
194#479—22no—
195#478—22no—
196#462—1062no—
197#461—1062no—
198#428C2×D5312yes—
199#423—531no—
200#423C2×D5311yescharacteristic
201#419—532no—
202#417C2×D5312yes—
203C2×#103—534no—
204C2×#101—534no—
205#405—532no—
206#405—1062no—
207C4×M16—531no—
208C4×M16—1061no—
209C2×#97—531no—
210C2×#97—1061no—
211#343—534no—
212#341—534no—
213C53#3306511yescharacteristic, complemented, Sylow
214C2×#41C22×D5311yescharacteristic
215C22×Q8—532no—
216#137—1068no—
217C2×Q16—1062no—
218C2×SD16—1062no—
219#131—1064no—
220#131—1064no—
221C2×M16—532no—
222C2×M16—532no—
223C2×M16—1062no—
224C2×M16C22×D5312yes—
225C22×C8—532no—
226C22×C8—1062no—
227C4⋊Q8—1061no—
228C4⋊Q8—1061no—
229C4⋊Q8—1062no—
230#125—21no—
231#125—21no—
232#125—22no—
233C4×Q8C22⋊D5312yes—
234C4×D4—22no—
235C4×D4C22⋊D5312yes—
236#117—21no—
237#117C22×D5311yescharacteristic
238C2×C4⋊C4—22no—
239C2×C4⋊C4—1062no—
240C2×C4⋊C4C22×D5312yes—
241C2×C42—21no—
242C2×C42C22×D5311yescharacteristic
243#107—22no—
244#106—22no—
245C4⋊C8—22no—
246C4⋊C8—1062no—
247#103—22no—
248#103—1064no—
249#103—1064no—
250#103—1064no—
251#102—22no—
252#101—1068no—
253#97—531no—
254#97—531no—
255#97—532no—
256#97—1062no—
257#97—1061no—
258C4×C8—1062no—
259C4×C8—1061no—
260#41—22no—
261#41#1171994914yes—
262C2×Q8—1062no—
263C2×Q8—1062no—
264C2×Q8—1064no—
265C2×Q8—1064no—
266C2×Q8C2×C22⋊D5311yescharacteristic
267C2×D4—21no—
268C2×D4C2×C22⋊D5311yescharacteristic
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—1062no—
273C22×C4#1171994712yes—
274C22×C4C23×D5311yescharacteristic
275Q16—2124no—
276SD16—2124no—
277M16—24no—
278M16—534no—
279M16—1068no—
280M16—1062no—
281M16—1064no—
282C2×C8—22no—
283C2×C8—532no—
284C2×C8—532no—
285C2×C8—1068no—
286C2×C8—1062no—
287C2×C8—1062no—
288C2×C8—1062no—
289C2×C8—1062no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—2122no—
296C4⋊C4—2122no—
297C4⋊C4C2×C22⋊D5312yes—
298C22⋊C4—21no—
299C22⋊C4—22no—
300C42—22no—
301C42—21no—
302C42D4×D5311yescharacteristic
303C42D4×D5311yescharacteristic
304C42C2×C22⋊D5312yes—
305C23—21no—
306C23C2×#1171994711yescharacteristic
307Q8—21no—
308Q8—2124no—
309Q8—2122no—
310Q8—2122no—
311Q8#1313320612yes—
312D4—24no—
313D4—21no—
314D4#1313320612yes—
315C2×C4—21no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—2122no—
323C2×C4—2124no—
324C2×C4#1313318912yes—
325C2×C4#1313318912yes—
326C2×C4C2×D4×D5311yescharacteristic
327C2×C4C22×C22⋊D5311yescharacteristic
328C2×C4#41×D5312yes—
329C2×C4C2×#1171994912yes—
330C8—42no—
331C8—1062no—
332C8—1062no—
333C8—1064no—
334C8—2122no—
335C22—22no—
336C22—22no—
337C22—42no—
338C22C2×#1313318911yescharacteristic
339C22#1681115912yes—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—42no—
344C4—82no—
345C4—4242no—
346C4#1681113612yes—
347C4C2×#1313320611yescharacteristic
348C4#1681116911yescharacteristic
349C2—22no—
350C2—42no—
351C2#2028301811yescharacteristic
352C2#2028445412yes—
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3080C2
socleC22×C53C2×#121
prime radicalC2×#41×C53C23
hypercenter#3080D53
hypocenterC53#33065
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3080

Complements

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

NH ≅ G/Ntype
1. C2×#16810993C2semidirect
4. #20275709C2semidirect
5. #20275558C2semidirect
6. #20275369C2semidirect
8. #20274855C2semidirect
213. C53#33065semidirect

Maximal subgroups

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

subgroupindex
C2×#168109932
#202842022
#202831432
#202757092
#202755582
#202753692
#202752282
#202748552
C53×#30802
#3306553