Finite Group Databasebeta

← #27538763

Subgroups — #27538763

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811027C211yesmaximal, characteristic, complemented
2#20283486C211yesmaximal, characteristic
3#20276866C212yesmaximal, complemented
4#20276861C212yesmaximal, complemented
5#20276856C212yesmaximal, complemented
6#20276851C212yesmaximal, complemented
7#20276504C211yesmaximal, characteristic, complemented
8#20276470C211yesmaximal, characteristic, complemented
9#20276428C211yesmaximal, characteristic
10#20276425C211yesmaximal, characteristic
11C53×#2159C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13C2×#13133216—21no—
14#16811091—21no—
15C2×#13133157C2211yescharacteristic
16#16811027—21no—
17#16811027—21no—
18#16811027C2212yes—
19#16811027C2212yes—
20#16810717C2211yescharacteristic
21#16810717C2211yescharacteristic
22#16810710—21no—
23#16810710—21no—
24#16810705C2211yescharacteristic
25#16810705C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133150—21no—
28#16810617—21no—
29C2×C8×Dic53C2211yescharacteristic
30#16810450C2212yes—
31#16810448C2212yes—
32#16810441C2212yes—
33#16810441C2212yes—
34#16810428C2212yes—
35#16810426C2212yes—
36#16810425C2212yes—
37#16810423C2212yes—
38C2×M32×C53C2211yescharacteristic
39C53×#427C2211yescharacteristic
40C2×#107×C53C2211yescharacteristic
41C53×#367C2212yes—
42C53×#366C2212yes—
43C2×#11719945—21no—
44C22×Dic106—41no—
45#13133239—22no—
46#13133239Q814yescomplemented
47C2×Dic212C2311yescharacteristic
48C2×#11719913—21no—
49C2×D424C2311yescharacteristic
50C22×#88322D411yescharacteristic
51#13133216—22no—
52#13133216—22no—
53#13133216—41no—
54#13133216—41no—
55#13133182—21no—
56#13133182—21no—
57#13133179—21no—
58#13133179—21no—
59#13133177—41no—
60#13133177—41no—
61#13133174—41no—
62C2×C4⋊Dic53—21no—
63C2×#11719919—41no—
64C2×C4×Dic53D411yescharacteristic
65C42×D53—41no—
66#13133157—22no—
67C2×#11719908C2312yes—
68#13133151—21no—
69#13133151C2312yes—
70#13133150—22no—
71#13133150—22no—
72#13133140—21no—
73#13133140—21no—
74#13133137—21no—
75#13133137—21no—
76C8×Dic53D411yescharacteristic
77C8×Dic53D411yescharacteristic
78C8×Dic53C2311yescharacteristic
79C8×Dic53C2311yescharacteristic
80C22×C8×C53C2311yescharacteristic
81#117×C53—21no—
82C2×C4⋊C4×C53—21no—
83M32×C53—22no—
84C2×C16×C53C2312yes—
85#107×C53—21no—
86#107×C53C2312yes—
87#107×C53C2311yescharacteristic
88#107×C53C2311yescharacteristic
89#106×C53—21no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C2×C22⋊D53—41no—
93C22×Dic53—41no—
94C22×Dic53D811yescharacteristic
95C22×Dic53D811yescharacteristic
96#11719945—24no—
97#11719945—42no—
98C2×D212—21no—
99C2×C4×D53—41no—
100C2×C4×D53—41no—
101C2×C4×D53—41no—
102C2×Dic106—21no—
103C2×Dic106—42no—
104C2×Dic106—42no—
105C2×Dic106—84no—
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53C2×Q812yes—
108C2×C8×C53#4112yes—
109C2×C8×C53C2411yescharacteristic
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53—41no—
116#11719921—41no—
117#11719921—41no—
118C4⋊Dic53—22no—
119C4⋊Dic53—41no—
120#11719919—42no—
121#11719919—42no—
122#11719919—41no—
123#11719919—41no—
124C4×Dic53—41no—
125C4×Dic53—41no—
126C4×Dic53D811yescharacteristic
127C4×Dic53D811yescharacteristic
128C4×Dic53C2×D411yescharacteristic
129C4×Dic53C2×D411yescharacteristic
130C2×#88322—22no—
131C2×#88322C2×D411yescharacteristic
132C2×#88322C2×D411yescharacteristic
133Dic212—21no—
134Dic212C2×Q812yescomplemented
135D424—21no—
136D424C2×Q812yescomplemented
137#11719913—24no—
138C16×C53—22no—
139#11719908—22no—
140C23×C53C2×D811yescharacteristic
141C22×D53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—82no—
146C2×C4×C53C22⋊Q812yes—
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53#13712yes—
149C2×C4×C53C22×D411yescharacteristic
150C22⋊D53—44no—
151C2×Dic53—22no—
152C2×Dic53—22no—
153C2×Dic53—41no—
154C2×Dic53—42no—
155C2×Dic53—82no—
156C2×Dic53C2×D811yescharacteristic
157C2×Dic53C2×D811yescharacteristic
158C2×Dic53C2×D811yescharacteristic
159C2×Dic53C2×D811yescharacteristic
160D212—22no—
161D212—41no—
162C4×D53—44no—
163C4×D53—82no—
164C4×D53—82no—
165Dic106—22no—
166Dic106—41no—
167Dic106—84no—
168Dic106—84no—
169C8×C53C22⋊Q812yes—
170C8×C53C22×Q811yescharacteristic
171C8×C53C2×#4111yescharacteristic
172#88322—21no—
173#88322—21no—
174#33629—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53#47412yes—
178C22×C53C22×D811yescharacteristic
179C2×D53—42no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—82no—
183C4×C53—81no—
184C4×C53—81no—
185C4×C53#47412yes—
186C4×C53C2×C22⋊Q811yescharacteristic
187C4×C53C2×#13711yescharacteristic
188Dic53—21no—
189Dic53—21no—
190Dic53—42no—
191Dic53—82no—
192Dic53#50912yes—
193Dic53#51012yes—
194C2×#500—531no—
195C4×#135—531no—
196#2227—532no—
197#2226—532no—
198#2225—532no—
199#2224—532no—
200#2159—531no—
201#2159D5311yescharacteristic, complemented
202#2150—531no—
203#2139—531no—
204#2139—531no—
205C2×C53—22no—
206C2×C53C2×#47411yescharacteristic
207C2×C53C2×#50911yescharacteristic
208C2×C53C2×#51011yescharacteristic
209D53—82no—
210C2×#135—531no—
211C2×C4⋊Q8—1061no—
212C4×#41—1061no—
213C2×M32—531no—
214C2×M32C2×D5311yescharacteristic
215#500—532no—
216#500—532no—
217#500—1061no—
218#500—1061no—
219C4×Q16—531no—
220C4×Q16—531no—
221C4×SD16—1061no—
222C4×SD16—1061no—
223C4×D8—531no—
224C4×D8—531no—
225#427C2×D5311yescharacteristic
226C2×#107—531no—
227C2×#107—1061no—
228C2×#107C2×D5311yescharacteristic
229#416—1061no—
230C2×C4×C8—531no—
231#367—532no—
232#367C2×D5312yes—
233#366—532no—
234#366C2×D5312yes—
235C4⋊C16—532no—
236C4⋊C16—532no—
237#358—532no—
238#358—532no—
239#357—532no—
240#357—532no—
241C53#3362911yescharacteristic, complemented, Sylow
242C2×#41—1061no—
243C22×Q8—2121no—
244#135—534no—
245#135—1062no—
246C2×Q16—531no—
247C2×SD16—1061no—
248C2×D8—531no—
249C22×C8—531no—
250C22×C8C22×D5311yescharacteristic
251C4⋊Q8—1062no—
252C4⋊Q8—1062no—
253C4⋊Q8—2121no—
254C4⋊Q8—2121no—
255C4×Q8—1061no—
256C4×Q8—1061no—
257C4×D4—1061no—
258C4×D4—1061no—
259C4×D4—2121no—
260C4×D4—2121no—
261#117—21no—
262#117—2121no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—1061no—
265C2×C4⋊C4—2121no—
266C2×C42—531no—
267C2×C42—2121no—
268M32—22no—
269M32—1062no—
270C2×C16—532no—
271C2×C16C22×D5312yes—
272#107—21no—
273#107—532no—
274#107—1062no—
275#107—1062no—
276#107—1061no—
277#107C22×D5312yes—
278#107C22×D5311yescharacteristic
279#107C22×D5311yescharacteristic
280#106—21no—
281#103—1061no—
282#103—1061no—
283#102—1061no—
284#102—1061no—
285C4×C8—531no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—531no—
289#41—1064no—
290#41—2122no—
291C2×Q8—1061no—
292C2×Q8—4244no—
293C2×Q8—2122no—
294C2×Q8—2122no—
295C2×D4—1061no—
296C2×D4—2121no—
297C22×C4—41no—
298C22×C4—531no—
299C22×C4—531no—
300C22×C4—2121no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4—2121no—
304C22×C4D4×D5311yescharacteristic
305Q16—532no—
306Q16—1061no—
307SD16—1064no—
308D8—532no—
309D8—1061no—
310C2×C8—531no—
311C2×C8—531no—
312C2×C8—1062no—
313C2×C8#1171994512yes—
314C2×C8D4×D5311yescharacteristic
315C2×C8Q8×D5312yes—
316C2×C8C23×D5311yescharacteristic
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—1062no—
322C4⋊C4—2122no—
323C4⋊C4—2122no—
324C4⋊C4—2121no—
325C4⋊C4—2121no—
326C4⋊C4—2121no—
327C22⋊C4—41no—
328C22⋊C4—2121no—
329C22⋊C4—2121no—
330C42—41no—
331C42—531no—
332C42—531no—
333C42—531no—
334C42—531no—
335C42—2121no—
336C42—2121no—
337C16—22no—
338C16—1062no—
339C23—2121no—
340C23D8×D5311yescharacteristic
341Q8—1062no—
342Q8—4244no—
343Q8—4244no—
344Q8—2121no—
345D4—1062no—
346D4—2124no—
347D4—2121no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—531no—
353C2×C4—531no—
354C2×C4—531no—
355C2×C4—531no—
356C2×C4—1062no—
357C2×C4—1062no—
358C2×C4—4242no—
359C2×C4—4242no—
360C2×C4—4242no—
361C2×C4—2122no—
362C2×C4—2121no—
363C2×C4—2124no—
364C2×C4#1313319512yes—
365C2×C4D8×D5311yescharacteristic
366C2×C4#1313324912yes—
367C2×C4C2×D4×D5311yescharacteristic
368C8—1061no—
369C8—1061no—
370C8#1313319512yes—
371C8C2×Q8×D5311yescharacteristic
372C8C2×#1171994511yescharacteristic
373C22—21no—
374C22—21no—
375C22—4241no—
376C22—4241no—
377C22—2122no—
378C22#1681090212yes—
379C22C2×D8×D5311yescharacteristic
380C4—82no—
381C4—81no—
382C4—81no—
383C4—532no—
384C4—532no—
385C4—1061no—
386C4—1061no—
387C4—4242no—
388C4—2122no—
389C4#1681090212yes—
390C4C2×#1313319511yescharacteristic
391C4C2×#1313324911yescharacteristic
392C2—22no—
393C2—4242no—
394C2C2×#1681090211yescharacteristic
395C2#509×D5311yescharacteristic
396C2#510×D5311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2159C2
socleC22×C53C22×D8
prime radicalC2×#13133239C22
hypercenter#2159D53
hypocenterC53#33629
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. C2×#16811027C2semidirect
3. #20276866C2semidirect
4. #20276861C2semidirect
5. #20276856C2semidirect
6. #20276851C2semidirect
7. #20276504C2semidirect
8. #20276470C2semidirect
11. C53×#2159C2semidirect
46. #13133239Q8semidirect
134. Dic212C2×Q8semidirect
136. D424C2×Q8semidirect
201. #2159D53semidirect
241. C53#33629semidirect

Maximal subgroups

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

subgroupindex
C2×#168110272
#202834862
#202768662
#202768612
#202768562
#202768512
#202765042
#202764702
#202764282
#202764252
C53×#21592
#3362953