Finite Group Databasebeta

← #27285357

Subgroups — #27285357

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811026C211yesmaximal, characteristic, complemented
2C2×#16810900C211yesmaximal, characteristic
3#20279664C211yesmaximal, characteristic
4C2×#16810370C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1981C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×#13133197—22no—
10C2×C4×D212C2211yescharacteristic
11#16811026—22no—
12#16811026—22no—
13#16810900—22no—
14#16810900—22no—
15C2×#13133150—22no—
16C2×#13133150—21no—
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133137—21no—
19C2×C4×#88322C2211yescharacteristic
20#16810560—22no—
21#16810525—21no—
22#16810493—21no—
23#16810370—22no—
24#16810350—22no—
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×#103×C53—22no—
28C4×C4⋊C4×C53C2211yescharacteristic
29C22×Dic106—21no—
30C22×D212D411yescharacteristic
31C22×C4×D53—21no—
32C22×#88322—21no—
33C22×#88322—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217—21no—
37#13133217—21no—
38#13133197—44no—
39#13133197—44no—
40C4×D212—21no—
41C4×D212—21no—
42C4×D212—21no—
43C4×D212—21no—
44C2×C4⋊Dic53—24no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D412yes—
47C2×C4⋊Dic53D411yescharacteristic
48C2×#11719921—24no—
49C2×#11719921—21no—
50#13133150—28no—
51#13133150—24no—
52C4⋊#88322—21no—
53C4⋊#88322D412yes—
54#13133137—24no—
55C4×#88322—21no—
56C4×#88322D412yes—
57C22×Q8×C53—21no—
58C22×C8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53D412yes—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D412yes—
66C2×C42×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70#103×C53—44no—
71#95×C53—21no—
72C23×D53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—41no—
75C2×Q8×C53—42no—
76C2×Q8×C53—42no—
77C22×C4×C53—22no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—22no—
83C22×Dic53—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212Q1612yes—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—42no—
93C2×C4×D53—42no—
94C2×Dic106—41no—
95C2×Dic106—41no—
96C2×Dic106—42no—
97C2×Dic106—42no—
98C2×C8×C53—41no—
99C2×C8×C53—42no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53D814yes—
106C42×C53—41no—
107C42×C53—41no—
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112#11719921—48no—
113#11719921—41no—
114#11719921—41no—
115C4⋊Dic53—21no—
116C4⋊Dic53—22no—
117C4⋊Dic53—21no—
118C4⋊Dic53—22no—
119C4⋊Dic53—48no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C2×#88322—24no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—41no—
126C2×#88322—41no—
127C23×C53#12011yescharacteristic
128C22×D53—22no—
129C22×D53—41no—
130C22×D53—42no—
131C22×D53—41no—
132C22×D53—42no—
133Q8×C53—44no—
134Q8×C53—81no—
135Q8×C53—81no—
136C2×C4×C53—22no—
137C2×C4×C53—24no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—82no—
142C2×C4×C53#12011yescharacteristic
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12711yescharacteristic
148C2×C4×C53C2×D812yes—
149C2×C4×C53C2×Q1611yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53AGL(1,ℤ/8)12yes—
152C2×C4×C53#13711yescharacteristic
153C2×C4×C53#13711yescharacteristic
154C2×Dic53—42no—
155C2×Dic53—41no—
156C2×Dic53—44no—
157C2×Dic53—42no—
158D212—22no—
159D212—22no—
160D212—41no—
161D212—41no—
162C4×D53—82no—
163C4×D53—81no—
164C4×D53—81no—
165Dic106—82no—
166Dic106—81no—
167Dic106—81no—
168C8×C53—81no—
169#88322—24no—
170#88322—42no—
171#10076—531nomaximal, Sylow
172C22×C53#39211yescharacteristic
173C22×C53#44712yes—
174C22×C53#45111yescharacteristic
175C22×C53#45211yescharacteristic
176C22×C53#45612yes—
177C2×D53—44no—
178C2×D53—42no—
179C2×D53—82no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—44no—
185C4×C53—82no—
186C4×C53#45111yescharacteristic
187C4×C53#45211yescharacteristic
188C4×C53#45912yes—
189C4×C53#46711yescharacteristic
190C4×C53#47011yescharacteristic
191C4×C53#49311yescharacteristic
192C4×C53#49611yescharacteristic
193Dic53—82no—
194Dic53—81no—
195C2×#500—531no—
196C2×#474—531no—
197#2542—531no—
198#1981D5311yescharacteristic, complemented
199C2×#332—531no—
200C2×D4⋊C8—531no—
201#1279—531no—
202C2×C53#164712yes—
203C2×C53#164912yes—
204C2×C53#200111yescharacteristic
205C2×C53#200511yescharacteristic
206C2×C53#201111yescharacteristic
207D53—82no—
208C2×C4⋊Q8—531no—
209C2×C4⋊Q8C2×D5311yescharacteristic
210C2×C22⋊Q8—1062no—
211C2×C4×D4—531no—
212#500—1062no—
213#500—1062no—
214#474—1062no—
215#474—1062no—
216C2×#107—1062no—
217C2×#107—1061no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8C2×D5311yescharacteristic
220C2×#103—22no—
221C2×#102—1061no—
222C2×C4×C8—531no—
223#399—1062no—
224#390—1061no—
225#384—1061no—
226C4×C4⋊C4C2×D5311yescharacteristic
227#332—1062no—
228D4⋊C8—1062no—
229C53#1007611yescharacteristic, complemented, Sylow
230C22×Q8—21no—
231C22×Q8—1061no—
232C22×D4—531no—
233C23×C4—1061no—
234C22×C8—21no—
235C22×C8—1061no—
236C22×C8—1061no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—1061no—
241C4⋊Q8—1061no—
242C4⋊Q8—1061no—
243C4⋊Q8—1061no—
244C4⋊Q8D21212yes—
245C22⋊Q8—2124no—
246C22⋊Q8—2124no—
247C4×D4—1061no—
248C4×D4—1061no—
249C4×D4—1061no—
250C4×D4—1061no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—531no—
254C2×C4⋊C4—532no—
255C2×C4⋊C4—1064no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4C22⋊D5312yes—
258C2×C22⋊C4—1064no—
259C2×C22⋊C4—1061no—
260C2×C42—21no—
261C2×C42C22×D5311yescharacteristic
262#107—1064no—
263#107—1068no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—532no—
267C4⋊C8—1061no—
268#103—44no—
269#102—1064no—
270C4×C8—532no—
271C4×C8—1061no—
272#95—21no—
273C24—1061no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—2121no—
279C2×Q8—2121no—
280C2×Q8—2122no—
281C2×Q8—2122no—
282C2×D4—532no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—1062no—
291C22×C4—1061no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2122no—
295C22×C4—2122no—
296C22×C4D4×D5311yescharacteristic
297C22×C4D4×D5311yescharacteristic
298C22×C4C2×C22⋊D5311yescharacteristic
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—1064no—
302C2×C8—1062no—
303C2×C8—1062no—
304C2×C8—2121no—
305C2×C8—2121no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—1061no—
312C4⋊C4—1062no—
313C4⋊C4—1061no—
314C4⋊C4—1062no—
315C4⋊C4—2128no—
316C4⋊C4—2121no—
317C4⋊C4—2121no—
318C4⋊C4#1171992214yes—
319C22⋊C4—2128no—
320C22⋊C4—2121no—
321C22⋊C4—2121no—
322C42—41no—
323C42—41no—
324C42C2×D21211yescharacteristic
325C42#1171994511yescharacteristic
326C42D4×D5311yescharacteristic
327C42D4×D5311yescharacteristic
328C23—1062no—
329C23—2122no—
330C23—2121no—
331C23—2121no—
332C23—2122no—
333C23#1313318511yescharacteristic
334Q8—44no—
335Q8—81no—
336Q8—81no—
337Q8—4241no—
338Q8—4242no—
339Q8—4241no—
340D4—1062no—
341D4—1062no—
342D4—2121no—
343D4—2121no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—4241no—
351C2×C4—4242no—
352C2×C4—4241no—
353C2×C4—2124no—
354C2×C4—2122no—
355C2×C4—2121no—
356C2×C4—2122no—
357C2×C4#1313318711yescharacteristic
358C2×C4#1313318711yescharacteristic
359C2×C4#1313319011yescharacteristic
360C2×C4#1313319011yescharacteristic
361C2×C4#1313319111yescharacteristic
362C2×C4#1313321311yescharacteristic
363C2×C4C2×#1171992212yes—
364C2×C4Q16×D5311yescharacteristic
365C2×C4#1313323811yescharacteristic
366C2×C4#1313324212yes—
367C2×C4#1313324611yescharacteristic
368C2×C4#1313324911yescharacteristic
369C8—81no—
370C8—1064no—
371C8—2122no—
372C22—4242no—
373C22—4241no—
374C22—4241no—
375C22—2122no—
376C22—2124no—
377C22#1681053211yescharacteristic
378C22#1681075612yes—
379C22#1681077711yescharacteristic
380C22#1681078211yescharacteristic
381C22#1681080412yes—
382C4—21no—
383C4—21no—
384C4—44no—
385C4—82no—
386C4—4241no—
387C4—4242no—
388C4#1681078011yescharacteristic
389C4#1681078811yescharacteristic
390C4#1681081812yes—
391C4#1681086311yescharacteristic
392C4#1681088011yescharacteristic
393C4#1681099911yescharacteristic
394C4#1681101311yescharacteristic
395C2—4242no—
396C2#2027394812yes—
397C2#2027396212yes—
398C2#2027572411yescharacteristic
399C2#2027575211yescharacteristic
400C2#2027578511yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1981C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1981D53
hypocenterC53#10076
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1981

Complements

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

NH ≅ G/Ntype
1. C2×#16811026C2semidirect
4. C2×#16810370C2semidirect
6. #20272556C2semidirect
7. C53×#1981C2semidirect
198. #1981D53semidirect
229. C53#10076semidirect

Maximal subgroups

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

subgroupindex
C2×#168110262
C2×#168109002
#202796642
C2×#168103702
C2×#168103502
#202725562
C53×#19812
#1007653