Finite Group Databasebeta

← #27537420

Subgroups — #27537420

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811066C211yesmaximal, characteristic
2#20283486C211yesmaximal, characteristic
3C2×#16810717C211yesmaximal, characteristic, complemented
4#20276585C212yesmaximal, complemented
5#20276578C212yesmaximal, complemented
6#20276573C212yesmaximal, complemented
7#20276566C212yesmaximal, complemented
8#20276434C211yesmaximal, characteristic, complemented
9#20276425C211yesmaximal, characteristic
10M32×Dic53C211yesmaximal, characteristic, complemented
11C53×#2159C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13C22×Dic212C2211yescharacteristic
14#16811091—21no—
15C2×#13133182—21no—
16#16811066C418yescomplemented
17C2×Dic424C2212yes—
18C2×#13133162C2212yes—
19C2×#13133157C2211yescharacteristic
20#16810717—21no—
21#16810717—21no—
22#16810717C2212yes—
23#16810717C2212yes—
24#16810717C2211yescharacteristic
25#16810717C2211yescharacteristic
26#16810710—21no—
27#16810710—21no—
28#16810705C2211yescharacteristic
29#16810705C2211yescharacteristic
30#16810617—21no—
31C2×#13133140—21no—
32C2×C8×Dic53C2211yescharacteristic
33#16810426C2212yes—
34#16810426C2212yes—
35#16810426C2212yes—
36#16810423C2212yes—
37#16810398C2212yes—
38C16×Dic53C2212yes—
39C2×M32×C53C2211yescharacteristic
40C53×#427C2211yescharacteristic
41C2×#107×C53C2211yescharacteristic
42C53×#367C2212yes—
43C53×#366C2212yes—
44C2×#11719945—21no—
45C22×Dic106—21no—
46#13133239—22no—
47#13133239C2×C414yes—
48C2×Dic212—21no—
49C2×Dic212—22no—
50C2×Dic212C2×C414yescomplemented
51C2×Dic212C2312yes—
52C2×Dic212C2311yescharacteristic
53C2×#11719913—21no—
54C2×D424C2311yescharacteristic
55C22×#88322D411yescharacteristic
56#13133182—22no—
57#13133182—22no—
58#13133182—21no—
59#13133182—21no—
60#13133182—41no—
61#13133182—41no—
62#13133179—21no—
63#13133179—21no—
64#13133177—41no—
65#13133177—41no—
66#13133174—41no—
67C2×#11719919—41no—
68C2×C4×Dic53—41no—
69C2×C4×Dic53D411yescharacteristic
70C42×D53—41no—
71Dic424C2×C418yescomplemented
72#13133162C2×C418yescomplemented
73#13133157—22no—
74C2×#11719908C2312yes—
75#13133140—22no—
76#13133140—22no—
77#13133140—21no—
78#13133140—21no—
79#13133137—21no—
80#13133137—21no—
81C8×Dic53D411yescharacteristic
82C8×Dic53D411yescharacteristic
83C8×Dic53C2311yescharacteristic
84C8×Dic53C2311yescharacteristic
85C22×C8×C53C2311yescharacteristic
86#117×C53—21no—
87C2×C4⋊C4×C53—21no—
88M32×C53C2×C414yescomplemented
89C2×C16×C53C2312yes—
90#107×C53—21no—
91#107×C53C2311yescharacteristic
92#107×C53C2312yes—
93#107×C53C2311yescharacteristic
94#106×C53—21no—
95C22×C4×C53—41no—
96C22×C4×C53C2×D411yescharacteristic
97C2×C22⋊D53—41no—
98C22×Dic53—41no—
99C22×Dic53D811yescharacteristic
100C22×Dic53D811yescharacteristic
101#11719945—24no—
102#11719945—42no—
103C2×D212—21no—
104C2×C4×D53—41no—
105C2×C4×D53—41no—
106C2×C4×D53—41no—
107C2×Dic106—24no—
108C2×Dic106—21no—
109C2×Dic106—22no—
110C2×Dic106—41no—
111C2×Dic106—42no—
112C2×C8×C53C22×C412yes—
113C2×C8×C53C2×D411yescharacteristic
114C2×C8×C53#4112yes—
115C2×C8×C53C2411yescharacteristic
116C4⋊C4×C53—21no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—41no—
120C22⋊C4×C53—41no—
121C42×C53—41no—
122#11719921—41no—
123#11719921—41no—
124#11719919—42no—
125#11719919—42no—
126#11719919—41no—
127#11719919—41no—
128C4×Dic53—41no—
129C4×Dic53—41no—
130C4×Dic53—42no—
131C4×Dic53—42no—
132C4×Dic53D811yescharacteristic
133C4×Dic53D811yescharacteristic
134C4×Dic53C2×D411yescharacteristic
135C4×Dic53C2×D411yescharacteristic
136C2×#88322—22no—
137C2×#88322C2×D411yescharacteristic
138C2×#88322C2×D411yescharacteristic
139Dic212—24no—
140Dic212—22no—
141Dic212—21no—
142Dic212C22×C414yes—
143Dic212C22×C412yes—
144D424—21no—
145D424C22×C412yes—
146#11719913—24no—
147C16×C53C22×C414yes—
148#11719908—22no—
149C23×C53C2×D811yescharacteristic
150C22×D53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—42no—
154C2×C4×C53—82no—
155C2×C4×C53C4×D412yes—
156C2×C4×C53C2×D811yescharacteristic
157C2×C4×C53#13512yes—
158C2×C4×C53C22×D411yescharacteristic
159C22⋊D53—44no—
160C2×Dic53—22no—
161C2×Dic53—41no—
162C2×Dic53—42no—
163C2×Dic53—41no—
164C2×Dic53—44no—
165C2×Dic53C2×D811yescharacteristic
166C2×Dic53C2×D811yescharacteristic
167C2×Dic53C2×D811yescharacteristic
168C2×Dic53C2×D811yescharacteristic
169D212—22no—
170D212—41no—
171C4×D53—44no—
172C4×D53—82no—
173C4×D53—82no—
174Dic106—24no—
175Dic106—22no—
176Dic106—42no—
177Dic106—41no—
178Dic106—44no—
179C8×C53C4×D412yes—
180C8×C53C23×C411yescharacteristic
181C8×C53C2×#4111yescharacteristic
182#88322—21no—
183#88322—21no—
184C4×#510—531nomaximal, Sylow
185C22×C53—21no—
186C22×C53—21no—
187C22×C53C4×D812yes—
188C22×C53C22×D811yescharacteristic
189C2×D53—42no—
190C2×D53—81no—
191C2×D53—81no—
192C4×C53—81no—
193C4×C53—81no—
194C4×C53—82no—
195C4×C53C4×D812yes—
196C4×C53C2×C4×D411yescharacteristic
197C4×C53C2×#13511yescharacteristic
198Dic53—21no—
199Dic53—21no—
200Dic53—21no—
201Dic53—44no—
202Dic53—42no—
203Dic53#51012yescomplemented
204C2×#510—531no—
205C4×#135—531no—
206C2×C4×Q16—531no—
207#2176—532no—
208#2175—532no—
209C4×Q32—532no—
210C4×SD32—532no—
211#2159D5311yescharacteristic, complemented
212#2141—531no—
213#2139—531no—
214C4×M32—531no—
215C2×C53—22no—
216C2×C53C2×C4×D811yescharacteristic
217C2×C53C2×#51011yescharacteristic
218C2×C53#341411yescharacteristic
219D53—82no—
220C2×#135—531no—
221C22×Q16—531no—
222C4×#41—1061no—
223C2×C4×Q8—1061no—
224#510—538no—
225C2×Q32—532no—
226C2×SD32—532no—
227C2×M32—531no—
228C2×M32C2×D5311yescharacteristic
229C4×Q16—532no—
230C4×Q16—532no—
231C4×Q16—531no—
232C4×Q16—531no—
233C4×Q16—1061no—
234C4×Q16—1061no—
235C4×SD16—1061no—
236C4×SD16—1061no—
237C4×D8—531no—
238C4×D8—531no—
239#427C2×D5311yescharacteristic
240C2×#107C2×D5311yescharacteristic
241#416—1061no—
242C2×#103—1061no—
243C2×C4×C8—531no—
244#367C2×D5312yes—
245#366C2×D5312yes—
246#358—532no—
247#358—532no—
248#358—532no—
249#357—532no—
250#346—532no—
251C4×C16—532no—
252C53C4×#51011yescharacteristic, complemented, Sylow
253C2×#41—1061no—
254C22×Q8—1061no—
255#135—534no—
256#135—1062no—
257C2×Q16—534no—
258C2×Q16—532no—
259C2×Q16—531no—
260C2×Q16—1061no—
261C2×Q16—1062no—
262C2×SD16—1061no—
263C2×D8—531no—
264C22×C8—531no—
265C22×C8C22×D5311yescharacteristic
266C4×Q8—1062no—
267C4×Q8—1062no—
268C4×Q8—1061no—
269C4×Q8—1061no—
270C4×Q8—2121no—
271C4×Q8—2121no—
272C4×D4—1061no—
273C4×D4—1061no—
274C4×D4—2121no—
275C4×D4—2121no—
276#117—21no—
277#117—2121no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—2121no—
280C2×C42—531no—
281C2×C42—2121no—
282C2×C42—2121no—
283Q32—538no—
284SD32—538no—
285M32—1062no—
286M32C4×D5314yescomplemented
287C2×C16—532no—
288C2×C16C22×D5312yes—
289#107—21no—
290#107C22×D5312yes—
291#107C22×D5311yescharacteristic
292#107C22×D5311yescharacteristic
293#106—21no—
294#103—1062no—
295#103—1062no—
296#103—1061no—
297#103—1061no—
298#102—1061no—
299#102—1061no—
300C4×C8—531no—
301C4×C8—531no—
302C4×C8—531no—
303C4×C8—531no—
304#41—1064no—
305#41—2122no—
306C2×Q8—1064no—
307C2×Q8—1061no—
308C2×Q8—1062no—
309C2×Q8—2121no—
310C2×Q8—2122no—
311C2×D4—1061no—
312C2×D4—2121no—
313C22×C4—41no—
314C22×C4—531no—
315C22×C4—531no—
316C22×C4—2121no—
317C22×C4—2121no—
318C22×C4—2121no—
319C22×C4—2121no—
320C22×C4D4×D5311yescharacteristic
321Q16—534no—
322Q16—532no—
323Q16—1064no—
324Q16—1062no—
325Q16—1061no—
326SD16—1064no—
327D8—532no—
328D8—1061no—
329C2×C8—531no—
330C2×C8—531no—
331C2×C8—1062no—
332C2×C8C2×C4×D5312yes—
333C2×C8D4×D5311yescharacteristic
334C2×C8#1171994912yes—
335C2×C8C23×D5311yescharacteristic
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—2122no—
341C4⋊C4—2122no—
342C4⋊C4—2121no—
343C4⋊C4—2121no—
344C22⋊C4—41no—
345C22⋊C4—2121no—
346C22⋊C4—2121no—
347C42—41no—
348C42—531no—
349C42—531no—
350C42—531no—
351C42—531no—
352C42—2121no—
353C42—2121no—
354C42—2122no—
355C42—2122no—
356C16—1062no—
357C16C2×C4×D5314yes—
358C23—2121no—
359C23D8×D5311yescharacteristic
360Q8—1062no—
361Q8—1064no—
362Q8—2122no—
363Q8—2121no—
364Q8—2124no—
365D4—1062no—
366D4—2124no—
367D4—2121no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4—531no—
373C2×C4—531no—
374C2×C4—531no—
375C2×C4—531no—
376C2×C4—1062no—
377C2×C4—4242no—
378C2×C4—4242no—
379C2×C4—2121no—
380C2×C4—2121no—
381C2×C4—2122no—
382C2×C4—2124no—
383C2×C4—2124no—
384C2×C4#1313317912yes—
385C2×C4D8×D5311yescharacteristic
386C2×C4#1313324012yes—
387C2×C4C2×D4×D5311yescharacteristic
388C8—1061no—
389C8—1061no—
390C8#1313317912yes—
391C8C22×C4×D5311yescharacteristic
392C8C2×#1171994911yescharacteristic
393C22—21no—
394C22—21no—
395C22—4241no—
396C22—4241no—
397C22—2122no—
398C22#1681070512yes—
399C22C2×D8×D5311yescharacteristic
400C4—81no—
401C4—82no—
402C4—81no—
403C4—532no—
404C4—1061no—
405C4—1061no—
406C4—1061no—
407C4—2122no—
408C4—2124no—
409C4#1681070512yes—
410C4C2×#1313317911yescharacteristic
411C4C2×#1313324011yescharacteristic
412C2—22no—
413C2—4242no—
414C2C2×#1681070511yescharacteristic
415C2#510×D5311yescharacteristic
416C2#2028784711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2159C2
socleC22×C53C22×D8
prime radicalC2×#13133239C22
cosocleC2×C8×C53C24
hypercenter#2159D53
hypocenterC53C4×#510
LCSG ▷ 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
3. C2×#16810717C2semidirect
4. #20276585C2semidirect
5. #20276578C2semidirect
6. #20276573C2semidirect
7. #20276566C2semidirect
8. #20276434C2semidirect
10. M32×Dic53C2semidirect
11. C53×#2159C2semidirect
16. #16811066C4semidirect
50. C2×Dic212C2×C4semidirect
71. Dic424C2×C4semidirect
72. #13133162C2×C4semidirect
88. M32×C53C2×C4semidirect
203. Dic53#510semidirect
211. #2159D53semidirect
252. C53C4×#510semidirect
286. M32C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110662
#202834862
C2×#168107172
#202765852
#202765782
#202765732
#202765662
#202764342
#202764252
M32×Dic532
C53×#21592
C4×#51053