Finite Group Databasebeta

← #27285271

Subgroups — #27285271

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810825C211yesmaximal, characteristic
2#20279889C211yesmaximal, characteristic
3#20275668C211yesmaximal, characteristic, complemented
4C2×#16810372C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×#487C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×C4×D212C2211yescharacteristic
10#16810825—22no—
11#16810825—22no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133141—21no—
15#16810534—22no—
16#16810524—21no—
17#16810488—21no—
18#16810480—22no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810385—22no—
21#16810372—22no—
22#16810352—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4⋊Q8×C53C2211yescharacteristic
25C53×#487—22no—
26C53×#487—22no—
27C2×#103×C53—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719924—24no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C2×C4⋊Dic53D411yescharacteristic
41C2×C4⋊Dic53D412yes—
42C2×#11719921—24no—
43C2×#11719921—21no—
44C2×C4×Dic53—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322D412yes—
49#13133141—24no—
50#13133124—24no—
51#13133124—21no—
52C22×Q8×C53—21no—
53C22×Q8×C53D412yes—
54C22×C8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×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×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68#103×C53—28no—
69C4×C8×C53—21no—
70C4×C8×C53D412yes—
71C23×D53—21no—
72C2×Q8×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53—22no—
75C2×Q8×C53—44no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C22×C4×C53—22no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×Dic53—21no—
87C22×Dic53—22no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212SD1612yes—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—42no—
97C2×C4×D53—42no—
98C2×C8×C53—22no—
99C2×C8×C53—42no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C42×C53C2×D411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112#11719924—48no—
113#11719921—41no—
114#11719921—41no—
115#11719921—84no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53—22no—
119C4⋊Dic53—22no—
120C4×Dic53—41no—
121C4×Dic53—41no—
122C2×#88322—22no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—42no—
127C23×C53#12011yescharacteristic
128C22×D53—22no—
129C22×D53—41no—
130C22×D53—41no—
131C22×D53—42no—
132C22×D53—42no—
133Q8×C53—44no—
134Q8×C53—42no—
135Q8×C53—42no—
136Q8×C53—82no—
137Q8×C53—81no—
138Q8×C53—81no—
139C2×C4×C53—22no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—44no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12711yescharacteristic
151C2×C4×C53C2×SD1611yescharacteristic
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53#13512yes—
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×C4×C53#13712yes—
157C2×Dic53—41no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162D212—22no—
163D212—22no—
164D212—41no—
165D212—41no—
166C4×D53—82no—
167C4×D53—81no—
168C4×D53—81no—
169C8×C53—42no—
170#88322—42no—
171#88322—81no—
172#10064—531nomaximal, Sylow
173C22×C53#39211yescharacteristic
174C22×C53#44811yescharacteristic
175C22×C53#44911yescharacteristic
176C22×C53#45212yes—
177C22×C53#45512yes—
178C2×D53—44no—
179C2×D53—42no—
180C2×D53—82no—
181C2×D53—81no—
182C2×D53—81no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—82no—
186C4×C53—82no—
187C4×C53#45011yescharacteristic
188C4×C53#45211yescharacteristic
189C4×C53#46412yes—
190C4×C53#46511yescharacteristic
191C4×C53#46811yescharacteristic
192C4×C53#49511yescharacteristic
193C4×C53#49711yescharacteristic
194Dic53—82no—
195Dic53—81no—
196C2×#487D5311yescharacteristic, complemented
197C2×#461—531no—
198#2563—531no—
199#1988—531no—
200C2×#333—531no—
201C2×D4⋊C8—531no—
202#1279—531no—
203C2×C53#164312yes—
204C2×C53#164612yes—
205C2×C53#200011yescharacteristic
206C2×C53#200411yescharacteristic
207C2×C53#201511yescharacteristic
208D53—82no—
209C22×SD16—1061no—
210C2×C4⋊Q8C2×D5311yescharacteristic
211C2×C4⋊Q8C2×D5311yescharacteristic
212C2×C4×D4—531no—
213#487—22no—
214#487—22no—
215#461—1062no—
216#461—1062no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8—531no—
219C2×#103—22no—
220C2×#103—1061no—
221C2×C4×C8C2×D5311yescharacteristic
222#393—1062no—
223#390—1061no—
224#382—1061no—
225#380—1062no—
226C4×C4⋊C4—531no—
227#340—1062no—
228#333—1062no—
229D4⋊C8—1062no—
230C53#1006411yescharacteristic, complemented, Sylow
231C22×Q8—21no—
232C22×Q8C22⋊D5312yes—
233C22×D4—531no—
234C23×C4—1061no—
235C2×SD16—1064no—
236C2×SD16—2122no—
237C2×SD16—2122no—
238C22×C8—21no—
239C22×C8—1061no—
240C22×C8—1061no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8C22⋊D5312yes—
249C4×D4—1061no—
250C4×D4—1061no—
251C4×D4—1061no—
252C4×D4—1061no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—531no—
258C2×C4⋊C4—532no—
259C2×C22⋊C4—1064no—
260C2×C22⋊C4—1061no—
261C2×C42—1061no—
262C2×C42C22×D5311yescharacteristic
263C4⋊C8—532no—
264C4⋊C8—1061no—
265C4⋊C8—1061no—
266C4⋊C8—1061no—
267#103—28no—
268#103—1064no—
269C4×C8—21no—
270C4×C8C22⋊D5312yes—
271#95—1064no—
272#95—1061no—
273C24—1061no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—44no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—532no—
284C2×D4—1061no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—1061no—
292C22×C4—1062no—
293C22×C4—2121no—
294C22×C4—2121no—
295C22×C4—2122no—
296C22×C4—2122no—
297C22×C4D4×D5311yescharacteristic
298C22×C4D4×D5311yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300SD16—2128no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—1062no—
304C2×C8—2121no—
305C2×C8—2122no—
306C2×C8—2121no—
307C2×C8—2121no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—1062no—
319C4⋊C4—1062no—
320C22⋊C4—4244no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C42—2121no—
324C42—2121no—
325C42D4×D5311yescharacteristic
326C42#1171994711yescharacteristic
327C42C2×C22⋊D5311yescharacteristic
328C42C2×C22⋊D5311yescharacteristic
329C23—1062no—
330C23—2121no—
331C23—2122no—
332C23—2121no—
333C23—2122no—
334C23#1313318511yescharacteristic
335Q8—44no—
336Q8—42no—
337Q8—42no—
338Q8—82no—
339Q8—81no—
340Q8—81no—
341D4—1062no—
342D4—1062no—
343D4—2121no—
344D4—2121no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—4241no—
352C2×C4—4241no—
353C2×C4—4242no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2121no—
358C2×C4—2122no—
359C2×C4#1313318511yescharacteristic
360C2×C4#1313318611yescharacteristic
361C2×C4#1313318811yescharacteristic
362C2×C4#1313318911yescharacteristic
363C2×C4#1313318911yescharacteristic
364C2×C4#1313321311yescharacteristic
365C2×C4SD16×D5311yescharacteristic
366C2×C4#1313323811yescharacteristic
367C2×C4#1313324012yes—
368C2×C4#1313324411yescharacteristic
369C2×C4#1313324611yescharacteristic
370C2×C4#1313324912yes—
371C8—42no—
372C8—4241no—
373C8—2122no—
374C22—4242no—
375C22—4241no—
376C22—4241no—
377C22—2122no—
378C22—2124no—
379C22#1681053111yescharacteristic
380C22#1681076011yescharacteristic
381C22#1681076711yescharacteristic
382C22#1681078412yes—
383C22#1681080012yes—
384C4—21no—
385C4—21no—
386C4—82no—
387C4—82no—
388C4—4242no—
389C4—4241no—
390C4#1681077211yescharacteristic
391C4#1681078211yescharacteristic
392C4#1681084712yes—
393C4#1681084911yescharacteristic
394C4#1681086611yescharacteristic
395C4#1681100811yescharacteristic
396C4#1681101611yescharacteristic
397C2—4242no—
398C2#2027391912yes—
399C2#2027394012yes—
400C2#2027571511yescharacteristic
401C2#2027574411yescharacteristic
402C2#2027580611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#487C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenterC2×#487D53
hypocenterC53#10064
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
3. #20275668C2semidirect
4. C2×#16810372C2semidirect
6. #20272555C2semidirect
7. C2×C53×#487C2semidirect
196. C2×#487D53semidirect
230. C53#10064semidirect

Maximal subgroups

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

subgroupindex
C2×#168108252
#202798892
#202756682
C2×#168103722
C2×#168103522
#202725552
C2×C53×#4872
#1006453