Finite Group Databasebeta

← #27286636

Subgroups — #27286636

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic, complemented
2C2×#16810818C211yesmaximal, characteristic
3#20275510C211yesmaximal, characteristic, complemented
4C2×#16810372C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic, complemented
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#476C211yesmaximal, characteristic, complemented
8C22×#11719922—21no—
9C2×#13133214C2211yescharacteristic
10#16810971—22no—
11#16810971—22no—
12#16810818—22no—
13#16810818—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133137—22no—
17C2×#13133137—21no—
18C2×C22⋊#88322—22no—
19C2×C4×#88322C2211yescharacteristic
20#16810372—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×D4×C53C2211yescharacteristic
24C53×#476—22no—
25C53×#476—22no—
26C2×#106×C53—21no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#102×C53—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719922—24no—
33C2×#11719922—42no—
34C2×#11719922—42no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322D412yes—
38#13133214—21no—
39#13133214—21no—
40#13133214—21no—
41#13133214D412yes—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D412yes—
46#13133137—24no—
47#13133137—28no—
48C22⋊#88322—44no—
49C4×#88322—21no—
50C4×#88322—21no—
51C22×Q8×C53—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C22×C8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53D412yes—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C4×D4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68#106×C53—24no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53D412yes—
71#102×C53—24no—
72C24×C53—21no—
73C23×D53—41no—
74C2×Q8×C53—22no—
75C2×Q8×C53—41no—
76C2×Q8×C53—82no—
77C2×D4×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53SD1612yes—
83C22×C4×C53—21no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53#4111yescharacteristic
92C2×D212—22no—
93C2×D212—22no—
94C2×D212—22no—
95C2×D212—41no—
96C2×D212—42no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—42no—
100C2×D212—41no—
101C2×D212—41no—
102C2×D212—41no—
103C2×D212—41no—
104C2×D212—82no—
105C2×C8×C53—22no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—22no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—41no—
113C4⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120#11719922—48no—
121C2×#88322—22no—
122C2×#88322—22no—
123C2×#88322—22no—
124C2×#88322—22no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—42no—
129C23×C53—22no—
130C23×C53—41no—
131C23×C53—42no—
132C23×C53—41no—
133C23×C53—42no—
134C23×C53#12111yescharacteristic
135C22×D53—42no—
136C22×D53—82no—
137C22×D53—81no—
138C22×D53—81no—
139C22×D53—82no—
140Q8×C53—82no—
141Q8×C53—82no—
142D4×C53—22no—
143D4×C53—22no—
144D4×C53—41no—
145D4×C53—41no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53—82no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53#12011yescharacteristic
155C2×C4×C53#12011yescharacteristic
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12711yescharacteristic
160C2×C4×C53C2×SD1611yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53#13512yes—
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)12yes—
165C2×C4×C53#13711yescharacteristic
166D212—44no—
167D212—42no—
168D212—42no—
169D212—82no—
170D212—82no—
171D212—81no—
172D212—81no—
173D212—81no—
174D212—81no—
175C8×C53—42no—
176#88322—42no—
177#88322—42no—
178#88322—81no—
179#10267—531nomaximal, Sylow
180C22×C53—44no—
181C22×C53—42no—
182C22×C53—82no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53#39211yescharacteristic
186C22×C53#46312yes—
187C22×C53#46511yescharacteristic
188C22×C53#46811yescharacteristic
189C22×C53#47212yes—
190C2×D53—82no—
191C2×D53—82no—
192C2×D53—82no—
193C2×D53—162no—
194C2×D53—162no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—81no—
198C4×C53—82no—
199C4×C53#44811yescharacteristic
200C4×C53#44912yes—
201C4×C53#44911yescharacteristic
202C4×C53#45011yescharacteristic
203C4×C53#45211yescharacteristic
204C4×C53#49511yescharacteristic
205C4×C53#49611yescharacteristic
206C2×#486—531no—
207C2×#476D5311yescharacteristic, complemented
208C2×#459—531no—
209#1954—531no—
210C2×#333—531no—
211C2×#331—531no—
212#1277—531no—
213C2×C53—82no—
214C2×C53#167712yes—
215C2×C53#169712yes—
216C2×C53#200011yescharacteristic
217C2×C53#201211yescharacteristic
218C2×C53#201811yescharacteristic
219D53—162no—
220C22×D8—1061no—
221C2×C4⋊Q8C2×D5311yescharacteristic
222C2×#127—531no—
223C2×C4×D4C2×D5311yescharacteristic
224#486—1062no—
225#486—1062no—
226#476—22no—
227#476—22no—
228#459—1062no—
229#459—1062no—
230C2×#106—21no—
231C2×C4⋊C8—531no—
232C2×C4⋊C8—531no—
233C2×C4⋊C8C2×D5311yescharacteristic
234C2×#102—21no—
235C2×#102—1062no—
236C2×#102—1061no—
237C2×C22⋊C8—1062no—
238C2×C4×C8—531no—
239#333—1062no—
240#331—1062no—
241C53#1026711yescharacteristic, complemented, Sylow
242C22×Q8—21no—
243C22×D4—1061no—
244C22×D4—1061no—
245C22×D4—1061no—
246C22×D4C22⋊D5311yescharacteristic
247C23×C4—21no—
248C2×D8—1064no—
249C2×D8—2122no—
250C2×D8—2122no—
251C22×C8—21no—
252C22×C8—532no—
253C22×C8—1061no—
254C22×C8—1061no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C22⋊D5312yes—
259#127—532no—
260#127—1061no—
261#127—1061no—
262#127—1061no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C22⋊D5311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C42C22×D5311yescharacteristic
272#106—24no—
273C4⋊C8—21no—
274C4⋊C8—532no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278C4⋊C8C22⋊D5312yes—
279#102—24no—
280#102—1064no—
281#102—1068no—
282C22⋊C8—2124no—
283C4×C8—1061no—
284C4×C8—1061no—
285C24—21no—
286C24—2121no—
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—82no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—1062no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—4242no—
299C2×D4—2121no—
300C2×D4—2121no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2122no—
305C2×D4—2121no—
306C2×D4—2122no—
307C2×D4—2121no—
308C2×D4#1171992412yes—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4D4×D5311yescharacteristic
316C22×C4#1171994711yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318D8—2128no—
319C2×C8—22no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—1062no—
323C2×C8—1062no—
324C2×C8—1062no—
325C2×C8—1062no—
326C2×C8—2121no—
327C2×C8—2122no—
328C2×C8—2121no—
329C2×C8—2121no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42D4×D5311yescharacteristic
339C42D4×D5311yescharacteristic
340C42C2×C22⋊D5311yescharacteristic
341C42C2×C22⋊D5311yescharacteristic
342C23—22no—
343C23—41no—
344C23—42no—
345C23—41no—
346C23—42no—
347C23—4242no—
348C23—4241no—
349C23—4241no—
350C23—4242no—
351C23—2122no—
352C23#1313318911yescharacteristic
353Q8—82no—
354Q8—82no—
355D4—22no—
356D4—22no—
357D4—41no—
358D4—41no—
359D4—4242no—
360D4—4242no—
361D4—4241no—
362D4—4241no—
363D4—4241no—
364D4—4241no—
365D4—2124no—
366D4—2122no—
367D4—2122no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—82no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4#1313318511yescharacteristic
377C2×C4#1313318511yescharacteristic
378C2×C4#1313318611yescharacteristic
379C2×C4#1313318811yescharacteristic
380C2×C4#1313318911yescharacteristic
381C2×C4#1313321311yescharacteristic
382C2×C4C2×#1171992411yescharacteristic
383C2×C4#1313323711yescharacteristic
384C2×C4#1313323812yes—
385C2×C4#1313324211yescharacteristic
386C2×C4#1313324412yes—
387C2×C4#1313324811yescharacteristic
388C8—42no—
389C8—4241no—
390C8—2122no—
391C8—2122no—
392C22—42no—
393C22—44no—
394C22—82no—
395C22—81no—
396C22—81no—
397C22—8482no—
398C22—8482no—
399C22—4242no—
400C22—4242no—
401C22—4242no—
402C22#1681053111yescharacteristic
403C22#1681084012yes—
404C22#1681085411yescharacteristic
405C22#1681087111yescharacteristic
406C22#1681089112yes—
407C4—21no—
408C4—21no—
409C4—81no—
410C4—82no—
411C4#1681076211yescharacteristic
412C4#1681076712yes—
413C4#1681076811yescharacteristic
414C4#1681077311yescharacteristic
415C4#1681078311yescharacteristic
416C4#1681100411yescharacteristic
417C4#1681100911yescharacteristic
418C2—82no—
419C2—8482no—
420C2#2027412612yes—
421C2#2027425012yes—
422C2#2027571611yescharacteristic
423C2#2027578911yescharacteristic
424C2#2027582111yescharacteristic
subgroupG/H
centerC23#13133189
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#476C2
socleC23×C53#121
prime radicalC2×#16810818C2
hypercenterC2×#476D53
hypocenterC53#10267
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#16810971C2semidirect
3. #20275510C2semidirect
4. C2×#16810372C2semidirect
5. C2×#16810366C2semidirect
6. #20272549C2semidirect
7. C2×C53×#476C2semidirect
207. C2×#476D53semidirect
241. C53#10267semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
C2×#168108182
#202755102
C2×#168103722
C2×#168103662
#202725492
C2×C53×#4762
#1026753