Finite Group Databasebeta

← #27285215

Subgroups — #27285215

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic
2#20279889C211yesmaximal, characteristic
3#20275664C211yesmaximal, characteristic, complemented
4C2×#16810372C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C2×C53×#494C211yesmaximal, characteristic, complemented
8C2×C4×D212C2211yescharacteristic
9#16810910—22no—
10#16810910—22no—
11C2×#13133148—21no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—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—
23C22×Q16×C53—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×C4⋊Q8×C53C2211yescharacteristic
26C53×#494—22no—
27C53×#494—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C22×#88322—21no—
32C22×#88322—21no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C4×D212—21no—
37C2×C4⋊Dic53D411yescharacteristic
38C2×C4⋊Dic53D412yes—
39C2×#11719921—24no—
40C2×#11719921—21no—
41C2×C4×Dic53—21no—
42#13133148—24no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322D412yes—
47#13133137—24no—
48#13133124—24no—
49#13133124—21no—
50C22×Q8×C53—21no—
51C22×Q8×C53D412yes—
52C2×Q16×C53—28no—
53C2×Q16×C53—48no—
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
68C4×C8×C53—21no—
69C4×C8×C53D412yes—
70C23×D53—21no—
71C2×Q8×C53—22no—
72C2×Q8×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53—44no—
75C2×Q8×C53—44no—
76C2×Q8×C53—41no—
77C22×C4×C53—22no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—22no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212Q1612yescomplemented
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—42no—
93C2×C4×D53—42no—
94Q16×C53—416no—
95C2×C8×C53—24no—
96C2×C8×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C42×C53C2×D411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109#11719921—41no—
110#11719921—41no—
111#11719921—84no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53—22no—
115C4⋊Dic53—22no—
116C4×Dic53—41no—
117C4×Dic53—41no—
118C2×#88322—22no—
119C2×#88322—41no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—42no—
123C23×C53#12011yescharacteristic
124C22×D53—22no—
125C22×D53—41no—
126C22×D53—42no—
127C22×D53—41no—
128C22×D53—42no—
129Q8×C53—44no—
130Q8×C53—44no—
131Q8×C53—82no—
132Q8×C53—82no—
133C2×C4×C53—22no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—44no—
137C2×C4×C53—82no—
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12711yescharacteristic
144C2×C4×C53C2×Q1611yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53#13512yes—
147C2×C4×C53#13711yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×C4×C53#13712yes—
150C2×Dic53—41no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155D212—22no—
156D212—22no—
157D212—41no—
158D212—41no—
159C4×D53—82no—
160C4×D53—81no—
161C4×D53—81no—
162C8×C53—24no—
163#88322—42no—
164#88322—81no—
165#10056—531nomaximal, Sylow
166C22×C53#39211yescharacteristic
167C22×C53#45111yescharacteristic
168C22×C53#45212yes—
169C22×C53#45211yescharacteristic
170C22×C53#45312yes—
171C2×D53—44no—
172C2×D53—42no—
173C2×D53—82no—
174C2×D53—81no—
175C2×D53—81no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—82no—
179C4×C53—82no—
180C4×C53#45111yescharacteristic
181C4×C53#45211yescharacteristic
182C4×C53#46412yes—
183C4×C53#46711yescharacteristic
184C4×C53#47011yescharacteristic
185C4×C53#49511yescharacteristic
186C4×C53#49711yescharacteristic
187Dic53—82no—
188Dic53—81no—
189C2×#494D5311yescharacteristic, complemented
190C2×#476—531no—
191#2563—531no—
192#1987—531no—
193C2×#333—531no—
194C2×D4⋊C8—531no—
195#1279—531no—
196C2×C53#163212yes—
197C2×C53#163412yes—
198C2×C53#200111yescharacteristic
199C2×C53#200211yescharacteristic
200C2×C53#201511yescharacteristic
201D53—82no—
202C22×Q16—22no—
203C2×C4⋊Q8C2×D5311yescharacteristic
204C2×C4⋊Q8C2×D5311yescharacteristic
205C2×C4×D4—531no—
206#494—22no—
207#494—22no—
208#476—1062no—
209#476—1062no—
210C2×#106—1061no—
211C2×C4⋊C8—531no—
212C2×C4⋊C8—531no—
213C2×#102—1061no—
214C2×C4×C8C2×D5311yescharacteristic
215#393—1062no—
216#390—1061no—
217#382—1061no—
218#380—1062no—
219C4×C4⋊C4—531no—
220#340—1062no—
221#333—1062no—
222D4⋊C8—1062no—
223C53#1005611yescharacteristic, complemented, Sylow
224C22×Q8—21no—
225C22×Q8C22⋊D5312yes—
226C22×D4—531no—
227C23×C4—1061no—
228C2×Q16—28no—
229C2×Q16—48no—
230C22×C8—21no—
231C22×C8—1061no—
232C22×C8—1061no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8C22⋊D5312yes—
241C4×D4—1061no—
242C4×D4—1061no—
243C4×D4—1061no—
244C4×D4—1061no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—531no—
250C2×C4⋊C4—532no—
251C2×C22⋊C4—1064no—
252C2×C22⋊C4—1061no—
253C2×C42—1061no—
254C2×C42C22×D5311yescharacteristic
255#106—1064no—
256C4⋊C8—532no—
257C4⋊C8—1061no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260#102—1064no—
261C4×C8—21no—
262C4×C8C22⋊D5312yes—
263#95—1064no—
264#95—1061no—
265C24—1061no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—41no—
272C2×D4—532no—
273C2×D4—1061no—
274C2×D4—1061no—
275C2×D4—1061no—
276C2×D4—1061no—
277C2×D4—1061no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—1061no—
281C22×C4—1062no—
282C22×C4—2121no—
283C22×C4—2121no—
284C22×C4—2122no—
285C22×C4—2122no—
286C22×C4D4×D5311yescharacteristic
287C22×C4D4×D5311yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—416no—
290C2×C8—24no—
291C2×C8—22no—
292C2×C8—1062no—
293C2×C8—2121no—
294C2×C8—2122no—
295C2×C8—2121no—
296C2×C8—2121no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—1062no—
308C4⋊C4—1062no—
309C22⋊C4—4244no—
310C22⋊C4—2121no—
311C22⋊C4—2121no—
312C42—2121no—
313C42—2121no—
314C42D4×D5311yescharacteristic
315C42#1171994711yescharacteristic
316C42C2×C22⋊D5311yescharacteristic
317C42C2×C22⋊D5311yescharacteristic
318C23—1062no—
319C23—2122no—
320C23—2121no—
321C23—2121no—
322C23—2122no—
323C23#1313318511yescharacteristic
324Q8—44no—
325Q8—44no—
326Q8—82no—
327Q8—82no—
328D4—1062no—
329D4—1062no—
330D4—2121no—
331D4—2121no—
332C2×C4—22no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—82no—
337C2×C4—4241no—
338C2×C4—4241no—
339C2×C4—4242no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4—2121no—
344C2×C4—2122no—
345C2×C4#1313318511yescharacteristic
346C2×C4#1313318611yescharacteristic
347C2×C4#1313318811yescharacteristic
348C2×C4#1313318911yescharacteristic
349C2×C4#1313318911yescharacteristic
350C2×C4#1313321311yescharacteristic
351C2×C4Q16×D5311yescharacteristic
352C2×C4#1313324012yes—
353C2×C4#1313324011yescharacteristic
354C2×C4#1313324912yes—
355C2×C4#1313324911yescharacteristic
356C2×C4#1313324911yescharacteristic
357C8—24no—
358C8—4241no—
359C8—2122no—
360C22—4241no—
361C22—4241no—
362C22—4242no—
363C22—2122no—
364C22—2124no—
365C22#1681053111yescharacteristic
366C22#1681077711yescharacteristic
367C22#1681078411yescharacteristic
368C22#1681078412yes—
369C22#1681079012yes—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—82no—
374C4—4241no—
375C4—4242no—
376C4#1681077711yescharacteristic
377C4#1681078411yescharacteristic
378C4#1681084712yes—
379C4#1681086111yescharacteristic
380C4#1681087811yescharacteristic
381C4#1681100811yescharacteristic
382C4#1681101611yescharacteristic
383C2—4242no—
384C2#2027384612yes—
385C2#2027385812yes—
386C2#2027572211yescharacteristic
387C2#2027573011yescharacteristic
388C2#2027580611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#494C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenterC2×#494D53
hypocenterC53#10056
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#494

Complements

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

NH ≅ G/Ntype
3. #20275664C2semidirect
4. C2×#16810372C2semidirect
6. #20272555C2semidirect
7. C2×C53×#494C2semidirect
89. C2×D212Q16semidirect
189. C2×#494D53semidirect
223. C53#10056semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
#202798892
#202756642
C2×#168103722
C2×#168103522
#202725552
C2×C53×#4942
#1005653