Finite Group Databasebeta

← #27278969

Subgroups — #27278969

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284497C211yesmaximal, characteristic
2#20283383C211yesmaximal, characteristic
3#20273118C211yesmaximal, characteristic, complemented
4#20273098C211yesmaximal, characteristic
5#20272610C211yesmaximal, characteristic, complemented
6#20272550C211yesmaximal, characteristic, complemented
7C53×#1493C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810927—21no—
10#16810920—21no—
11#16810915C411yescharacteristic
12#16810910C411yescharacteristic
13#16810905C411yescharacteristic
14#16810900C411yescharacteristic
15#16810683—22no—
16#16810679—21no—
17#16810676C2211yescharacteristic
18#16810675—21no—
19#16810659—21no—
20C2×C4⋊#88322C2211yescharacteristic
21C2×C4⋊#88322C2211yescharacteristic
22#16810617—21no—
23#16810591—21no—
24#16810581—21no—
25#16810355—21no—
26Q8⋊#88322—21no—
27Q8⋊#88322—21no—
28#16810352—21no—
29C53×#533C2211yescharacteristic
30C53×#423C2211yescharacteristic
31C2×C4×C8×C53C2211yescharacteristic
32C53×Q8⋊C8—22no—
33C2×#11719945D411yescharacteristic
34C22×#88322—21no—
35C22×#88322—21no—
36C4×Dic106C2×C411yescharacteristic
37C4×Dic106D411yescharacteristic
38C4×D212C2×C411yescharacteristic
39C4×D212D411yescharacteristic
40C4×C22⋊D53—21no—
41C4×C22⋊D53—21no—
42#13133176D411yescharacteristic
43#13133174—21no—
44C42×D53—21no—
45#13133150—21no—
46#13133150—21no—
47#13133148—21no—
48#13133148—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322C2×C411yescharacteristic
53C4⋊#88322C2×C411yescharacteristic
54C4⋊#88322D411yescharacteristic
55C4⋊#88322D411yescharacteristic
56#13133140—21no—
57#13133140—21no—
58#13133137—21no—
59#13133137—21no—
60C22⋊#88322—21no—
61#13133129—22no—
62#13133129—22no—
63#13133127—22no—
64C4×#88322—21no—
65C8×Dic53—22no—
66C2×M16×C53—21no—
67C22×C8×C53—21no—
68C4⋊Q8×C53C2×C411yescharacteristic
69C4⋊Q8×C53C2×C411yescharacteristic
70#125×C53—21no—
71C22⋊Q8×C53—21no—
72C22⋊Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53—21no—
75#117×C53—21no—
76C2×C42×C53C2311yescharacteristic
77C4⋊C8×C53—21no—
78C4⋊C8×C53D412yes—
79C4×C8×C53—21no—
80C4×C8×C53—21no—
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C22×C4×C53#4111yescharacteristic
86C2×C22⋊D53—21no—
87#11719945—21no—
88#11719945—21no—
89#11719945—21no—
90#11719945—21no—
91C2×D212C22⋊C411yescharacteristic
92C2×C4×D53—21no—
93C2×C4×D53—21no—
94C2×C4×D53—21no—
95C2×Dic106C22⋊C411yescharacteristic
96M16×C53—41no—
97M16×C53—41no—
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C42×C53—41no—
110C42×C53C22×C411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C22⋊Dic53—21no—
115#11719921—21no—
116#11719921—21no—
117C4⋊Dic53C22⋊C411yescharacteristic
118C4⋊Dic53C22⋊C411yescharacteristic
119#11719919—21no—
120#11719919—21no—
121C4×Dic53—21no—
122C4×Dic53—21no—
123C4×Dic53—21no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C2×#88322—41no—
133C23×C53#12311yescharacteristic
134C22×D53—21no—
135Q8×C53—42no—
136Q8×C53—41no—
137Q8×C53—41no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#13511yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53#13711yescharacteristic
152C2×C4×C53#13711yescharacteristic
153C22⋊D53—41no—
154C22⋊D53—41no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158D212—21no—
159D212—21no—
160C4×D53—41no—
161C4×D53—41no—
162C4×D53—41no—
163C4×D53—41no—
164C4×D53—81no—
165Dic106—21no—
166Dic106—21no—
167C8×C53—42no—
168C8×C53—42no—
169#88322—41no—
170#88322—41no—
171#88322—41no—
172#88322—41no—
173#88322—81no—
174#8994—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53#38611yescharacteristic
178C22×C53#48511yescharacteristic
179C22×C53#48511yescharacteristic
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—81no—
187C4×C53—81no—
188C4×C53#41611yescharacteristic
189C4×C53#41911yescharacteristic
190C4×C53#44311yescharacteristic
191C4×C53#44411yescharacteristic
192C4×C53#45211yescharacteristic
193C4×C53#45211yescharacteristic
194Dic53—41no—
195Dic53—41no—
196Dic53—41no—
197#3076—531no—
198#2925—531no—
199#1493D5311yescharacteristic, complemented
200#1481—531no—
201#1476—531no—
202#1301—531no—
203#1277—531no—
204C2×C53—21no—
205C2×C53—21no—
206C2×C53#187311yescharacteristic
207C2×C53#187611yescharacteristic
208C2×C53#188111yescharacteristic
209D53—81no—
210#533C2×D5311yescharacteristic
211C4×#41—531no—
212#479—1061no—
213#478—1061no—
214#477—531no—
215#476—531no—
216#475—531no—
217#474—531no—
218#433—1062no—
219#432—1061no—
220#431—531no—
221#431—1061no—
222#427—1061no—
223#423C2×D5311yescharacteristic
224C2×C4⋊C8—531no—
225C2×C4⋊C8—531no—
226#416—1061no—
227#408—1061no—
228#405—1061no—
229C2×C4×C8C2×D5311yescharacteristic
230Q8⋊C8—22no—
231Q8⋊C8—1061no—
232Q8⋊C8—1061no—
233Q8⋊C8—1061no—
234D4⋊C8—1061no—
235C53#899411yescharacteristic, complemented, Sylow
236C2×#41—531no—
237C2×M16—21no—
238C22×C8—21no—
239C22×C8—1061no—
240C22×C8—1061no—
241C4⋊Q8C4×D5311yescharacteristic
242C4⋊Q8C4×D5311yescharacteristic
243#125—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—531no—
249C4×Q8—531no—
250C4×D4—531no—
251C4×D4—531no—
252C4×D4—1061no—
253C4×D4—1061no—
254#117—21no—
255#117—531no—
256#117—1061no—
257C2×C42—1061no—
258C2×C42C22×D5311yescharacteristic
259#107—1061no—
260#107—1061no—
261#106—1061no—
262#106—1061no—
263C4⋊C8—21no—
264C4⋊C8—531no—
265C4⋊C8—531no—
266C4⋊C8—531no—
267C4⋊C8—531no—
268C4⋊C8—1061no—
269C4⋊C8—1061no—
270C4⋊C8—1061no—
271C4⋊C8C22⋊D5312yes—
272#103—1061no—
273#103—1061no—
274#102—1061no—
275#102—1061no—
276C22⋊C8—1062no—
277C22⋊C8—1062no—
278C22⋊C8—1061no—
279#97—1062no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—1062no—
283C4×C8—1061no—
284#41—1061no—
285#41—1061no—
286#41—1061no—
287#41—1061no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—531no—
291C2×D4—531no—
292C2×D4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4#1171994511yescharacteristic
297C22×C4D4×D5311yescharacteristic
298C22×C4#1171994711yescharacteristic
299M16—41no—
300M16—41no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—1061no—
308C2×C8—1061no—
309C2×C8—2121no—
310C2×C8—2121no—
311C2×C8—2121no—
312C2×C8—2121no—
313C2×C8—2121no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—531no—
321C4⋊C4—531no—
322C4⋊C4—1061no—
323C4⋊C4—1061no—
324C22⋊C4—41no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C42—41no—
329C42—1061no—
330C42—1061no—
331C42—1061no—
332C42C2×C4×D5311yescharacteristic
333C42D4×D5311yescharacteristic
334C42D4×D5311yescharacteristic
335C42C2×C22⋊D5311yescharacteristic
336C23—1061no—
337C23#1313319911yescharacteristic
338Q8—42no—
339Q8—41no—
340Q8—41no—
341Q8—1061no—
342Q8—1061no—
343D4—1061no—
344D4—1061no—
345D4—2121no—
346D4—2121no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—4241no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4C22⋊C4×D5311yescharacteristic
361C2×C4#1313317711yescharacteristic
362C2×C4C4×C22⋊D5311yescharacteristic
363C2×C4#1313318511yescharacteristic
364C2×C4#1313318811yescharacteristic
365C2×C4#1313319011yescharacteristic
366C2×C4#1313323811yescharacteristic
367C2×C4#1313324011yescharacteristic
368C2×C4#1313324611yescharacteristic
369C2×C4#1313324911yescharacteristic
370C8—42no—
371C8—42no—
372C8—4241no—
373C8—2121no—
374C8—2121no—
375C8—2121no—
376C8—2121no—
377C22—21no—
378C22—21no—
379C22—2121no—
380C22—2121no—
381C22—2121no—
382C22#1681049911yescharacteristic
383C22#1681096211yescharacteristic
384C22#1681096311yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—81no—
390C4—2121no—
391C4—2121no—
392C4—2121no—
393C4#1681061811yescharacteristic
394C4#1681063011yescharacteristic
395C4#1681073711yescharacteristic
396C4#1681074211yescharacteristic
397C4#1681078211yescharacteristic
398C4#1681078411yescharacteristic
399C2—21no—
400C2—21no—
401C2—4241no—
402C2#2027502411yescharacteristic
403C2#2027503911yescharacteristic
404C2#2027508011yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1493C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1493D53
hypocenterC53#8994
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
3. #20273118C2semidirect
5. #20272610C2semidirect
6. #20272550C2semidirect
7. C53×#1493C2semidirect
199. #1493D53semidirect
235. C53#8994semidirect

Maximal subgroups

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

subgroupindex
#202844972
#202833832
#202731182
#202730982
#202726102
#202725502
C53×#14932
#899453