Finite Group Databasebeta

← #27286650

Subgroups — #27286650

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810971C211yesmaximal, characteristic
2C2×#16810823C211yesmaximal, characteristic
3#20275652C211yesmaximal, characteristic, complemented
4C2×#16810372C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272549C211yesmaximal, characteristic, complemented
7C2×C53×#477C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×#13133214C2211yescharacteristic
10#16810971—22no—
11#16810971—22no—
12#16810823—22no—
13#16810823—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133137—22no—
17C2×#13133137—21no—
18C2×C4×#88322C2211yescharacteristic
19#16810378—22no—
20#16810372—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4×Q8×C53C2211yescharacteristic
24C53×#477—22no—
25C53×#477—22no—
26C2×#107×C53—21no—
27C2×C4⋊C8×C53C2211yescharacteristic
28C2×#103×C53—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212—21no—
32C2×#11719924—24no—
33C2×#11719924—42no—
34C2×#11719924—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—
48C4×#88322—21no—
49C4×#88322—21no—
50C22×Q8×C53—21no—
51C22×Q8×C53D411yescharacteristic
52C22×C8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53D412yes—
57C4×Q8×C53—21no—
58C4×Q8×C53—21no—
59C4×Q8×C53—21no—
60C4×Q8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67#107×C53—24no—
68C4⋊C8×C53—21no—
69C4⋊C8×C53D412yes—
70#103×C53—24no—
71C23×D53—41no—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—22no—
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—41no—
79C2×Q8×C53—82no—
80C2×Q8×C53SD1612yes—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—41no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C2×D212—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—42no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212—42no—
96C2×D212—41no—
97C2×D212—41no—
98C2×D212—41no—
99C2×D212—82no—
100C2×C8×C53—22no—
101C2×C8×C53—41no—
102C2×C8×C53—41no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C42×C53—41no—
112C42×C53—41no—
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117#11719924—48no—
118C2×#88322—22no—
119C2×#88322—22no—
120C2×#88322—22no—
121C2×#88322—22no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—42no—
126C23×C53#12111yescharacteristic
127C22×D53—42no—
128C22×D53—82no—
129C22×D53—81no—
130C22×D53—81no—
131C22×D53—82no—
132Q8×C53—22no—
133Q8×C53—22no—
134Q8×C53—41no—
135Q8×C53—41no—
136Q8×C53—82no—
137Q8×C53—82no—
138C2×C4×C53—22no—
139C2×C4×C53—22no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—82no—
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12711yescharacteristic
152C2×C4×C53C2×SD1611yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53#13512yes—
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53#13711yescharacteristic
157C2×C4×C53#13712yes—
158D212—42no—
159D212—42no—
160D212—44no—
161D212—82no—
162D212—82no—
163D212—81no—
164D212—81no—
165D212—81no—
166D212—81no—
167C8×C53—42no—
168#88322—42no—
169#88322—42no—
170#88322—81no—
171#10269—531nomaximal, Sylow
172C22×C53#39211yescharacteristic
173C22×C53#46412yes—
174C22×C53#46511yescharacteristic
175C22×C53#46811yescharacteristic
176C22×C53#47212yes—
177C2×D53—82no—
178C2×D53—82no—
179C2×D53—82no—
180C2×D53—162no—
181C2×D53—162no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—42no—
185C4×C53—81no—
186C4×C53—82no—
187C4×C53#44811yescharacteristic
188C4×C53#44911yescharacteristic
189C4×C53#45011yescharacteristic
190C4×C53#45211yescharacteristic
191C4×C53#45212yes—
192C4×C53#49511yescharacteristic
193C4×C53#49711yescharacteristic
194C2×#486—531no—
195C2×#477D5311yescharacteristic, complemented
196C2×#460—531no—
197#1984—531no—
198C2×#333—531no—
199C2×#331—531no—
200#1277—531no—
201C2×C53#167812yes—
202C2×C53#169812yes—
203C2×C53#200011yescharacteristic
204C2×C53#201511yescharacteristic
205C2×C53#201811yescharacteristic
206D53—162no—
207C22×SD16—1061no—
208C2×C4⋊Q8C2×D5311yescharacteristic
209C2×#127—531no—
210C2×C4×Q8C2×D5311yescharacteristic
211#486—1062no—
212#486—1062no—
213#477—22no—
214#477—22no—
215#460—1062no—
216#460—1062no—
217C2×#107—21no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8—531no—
220C2×C4⋊C8C2×D5311yescharacteristic
221C2×#103—21no—
222C2×#102—1062no—
223C2×#102—1061no—
224C2×C4×C8—531no—
225#336—1062no—
226#333—1062no—
227#331—1062no—
228C53#1026911yescharacteristic, complemented, Sylow
229C22×Q8—21no—
230C22×Q8C22⋊D5311yescharacteristic
231C22×D4—1061no—
232C22×D4—1061no—
233C22×D4—1061no—
234C2×SD16—1064no—
235C2×SD16—2122no—
236C2×SD16—2122no—
237C22×C8—21no—
238C22×C8—532no—
239C22×C8—1061no—
240C22×C8—1061no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22⋊D5312yes—
245#127—532no—
246#127—1061no—
247#127—1061no—
248#127—1061no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C42—21no—
258C2×C42C22×D5311yescharacteristic
259#107—24no—
260C4⋊C8—21no—
261C4⋊C8—532no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8C22⋊D5312yes—
266#103—24no—
267#102—1064no—
268#102—1068no—
269C4×C8—1061no—
270C4×C8—1061no—
271C24—2121no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×Q8#1171992412yes—
281C2×D4—1062no—
282C2×D4—1062no—
283C2×D4—1062no—
284C2×D4—4242no—
285C2×D4—2121no—
286C2×D4—2121no—
287C2×D4—2121no—
288C2×D4—2121no—
289C2×D4—2121no—
290C2×D4—2122no—
291C2×D4—2121no—
292C2×D4—2121no—
293C2×D4—2122no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4D4×D5311yescharacteristic
298C22×C4#1171994711yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300SD16—2128no—
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—1062no—
305C2×C8—1062no—
306C2×C8—1062no—
307C2×C8—1062no—
308C2×C8—2121no—
309C2×C8—2122no—
310C2×C8—2121no—
311C2×C8—2121no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C42—41no—
321C42—41no—
322C42D4×D5311yescharacteristic
323C42D4×D5311yescharacteristic
324C42C2×C22⋊D5311yescharacteristic
325C42C2×C22⋊D5311yescharacteristic
326C23—4242no—
327C23—4241no—
328C23—4241no—
329C23—4242no—
330C23—2122no—
331C23#1313318911yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—82no—
337Q8—82no—
338D4—4242no—
339D4—4242no—
340D4—4241no—
341D4—4241no—
342D4—4241no—
343D4—4241no—
344D4—2122no—
345D4—2122no—
346D4—2124no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4#1313318511yescharacteristic
356C2×C4#1313318511yescharacteristic
357C2×C4#1313318611yescharacteristic
358C2×C4#1313318811yescharacteristic
359C2×C4#1313318911yescharacteristic
360C2×C4#1313321311yescharacteristic
361C2×C4C2×#1171992411yescharacteristic
362C2×C4#1313323711yescharacteristic
363C2×C4#1313324012yes—
364C2×C4#1313324211yescharacteristic
365C2×C4#1313324811yescharacteristic
366C2×C4#1313324912yes—
367C8—42no—
368C8—4241no—
369C8—2122no—
370C8—2122no—
371C22—8482no—
372C22—8482no—
373C22—4242no—
374C22—4242no—
375C22—4242no—
376C22#1681053111yescharacteristic
377C22#1681084712yes—
378C22#1681085411yescharacteristic
379C22#1681087111yescharacteristic
380C22#1681089112yes—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—82no—
386C4#1681076211yescharacteristic
387C4#1681076811yescharacteristic
388C4#1681077311yescharacteristic
389C4#1681078311yescharacteristic
390C4#1681078412yes—
391C4#1681100811yescharacteristic
392C4#1681101611yescharacteristic
393C2—8482no—
394C2#2027413312yes—
395C2#2027425712yes—
396C2#2027571611yescharacteristic
397C2#2027580611yescharacteristic
398C2#2027582111yescharacteristic
subgroupG/H
centerC23#13133189
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#477C2
socleC23×C53#121
prime radicalC2×#13133214C22
hypercenterC2×#477D53
hypocenterC53#10269
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #20275652C2semidirect
4. C2×#16810372C2semidirect
6. #20272549C2semidirect
7. C2×C53×#477C2semidirect
195. C2×#477D53semidirect
228. C53#10269semidirect

Maximal subgroups

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

subgroupindex
C2×#168109712
C2×#168108232
#202756522
C2×#168103722
C2×#168103662
#202725492
C2×C53×#4772
#1026953