Finite Group Databasebeta

← #27283329

Subgroups — #27283329

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810997C211yesmaximal, characteristic
2#20275642C211yesmaximal, characteristic
3C4⋊C4×#88322C211yesmaximal, characteristic, complemented
4C2×#16810373C211yesmaximal, characteristic, complemented
5C2×#16810369C211yesmaximal, characteristic, complemented
6C2×#16810366C211yesmaximal, characteristic
7C53×#1988C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C22×#11719924—21no—
10C2×#13133214C2211yescharacteristic
11#16810997—22no—
12#16810997C414yescomplemented
13C2×C4⋊#88322C2211yescharacteristic
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133137—22no—
16C2×C4×#88322—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810378—21no—
19#16810373—22no—
20#16810369—22no—
21#16810366—22no—
22C2×C4⋊Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C4×C4⋊C4×C53C2211yescharacteristic
25C53×#340—22no—
26C22×D212—21no—
27C22×D212—21no—
28C22×D212—21no—
29C2×#11719924—24no—
30C2×#11719924—24no—
31C2×#11719924—42no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C2×#11719924—42no—
35C22×#88322—21no—
36C22×#88322—21no—
37C22×#88322D411yescharacteristic
38C22×#88322D411yescharacteristic
39#13133214—21no—
40#13133214—21no—
41#13133214—21no—
42#13133214C2×C412yes—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322D412yes—
47#13133137—44no—
48C4×#88322—42no—
49C4×#88322C2×C412yes—
50C4×#88322D412yes—
51C22×Q8×C53—21no—
52C22×C8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53C2×C412yes—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53D412yes—
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62C4⋊C8×C53—21no—
63C4⋊C8×C53—21no—
64#95×C53—21no—
65C23×D53—41no—
66C2×Q8×C53—22no—
67C2×Q8×C53—41no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C22×C4×C53—22no—
73C22×C4×C53—41no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53#4111yescharacteristic
77C2×D212—22no—
78C2×D212—22no—
79C2×D212—22no—
80C2×D212—41no—
81C2×D212—42no—
82C2×D212—41no—
83C2×D212—42no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—82no—
90C2×C8×C53—41no—
91C2×C8×C53—42no—
92C4⋊C4×C53—22no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53SD1614yescomplemented
98C42×C53—41no—
99C42×C53—41no—
100C42×C53C22×C411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104#11719924—48no—
105#11719924—48no—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322—22no—
109C2×#88322—41no—
110C2×#88322—41no—
111C2×#88322—41no—
112C2×#88322—42no—
113C2×#88322C22⋊C412yes—
114C2×#88322C22⋊C412yes—
115C2×#88322D812yes—
116C2×#88322SD1612yes—
117C23×C53#12111yescharacteristic
118C22×D53—42no—
119C22×D53—81no—
120C22×D53—81no—
121C22×D53—82no—
122C22×D53—82no—
123Q8×C53—42no—
124Q8×C53—42no—
125Q8×C53—81no—
126Q8×C53—81no—
127C2×C4×C53—22no—
128C2×C4×C53—24no—
129C2×C4×C53—42no—
130C2×C4×C53—42no—
131C2×C4×C53—42no—
132C2×C4×C53—81no—
133C2×C4×C53—81no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53#12411yescharacteristic
139C2×C4×C53#12711yescharacteristic
140C2×C4×C53C2×D811yescharacteristic
141C2×C4×C53C2×SD1611yescharacteristic
142C2×C4×C53C2×SD1612yes—
143C2×C4×C53#13512yes—
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146D212—42no—
147D212—42no—
148D212—44no—
149D212—82no—
150D212—82no—
151D212—81no—
152D212—81no—
153D212—81no—
154D212—81no—
155C8×C53—81no—
156#88322—22no—
157#88322—42no—
158#88322—81no—
159#88322#10214yescomplemented
160#9732—531nomaximal, Sylow
161C22×C53#39011yescharacteristic
162C22×C53#45911yescharacteristic
163C22×C53#46011yescharacteristic
164C22×C53#46512yes—
165C22×C53#47112yes—
166C2×D53—82no—
167C2×D53—82no—
168C2×D53—82no—
169C2×D53—162no—
170C2×D53—162no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—44no—
174C4×C53—82no—
175C4×C53C2×#10211yescharacteristic
176C4×C53#41811yescharacteristic
177C4×C53C4×SD1612yes—
178C4×C53#45911yescharacteristic
179C4×C53#46011yescharacteristic
180C4×C53#49211yescharacteristic
181C4×C53#49511yescharacteristic
182C2×#492—531no—
183#1988D5311yescharacteristic, complemented
184#1982—531no—
185C8×C4⋊C4—531no—
186C2×#334—531no—
187C2×#332—531no—
188C2×#331—531no—
189C2×C53#166512yes—
190C2×C53#166612yes—
191C2×C53#196811yescharacteristic
192C2×C53#197711yescharacteristic
193C2×C53#198211yescharacteristic
194D53—162no—
195C22×SD16—1061no—
196C22×SD16—1061no—
197C2×C4⋊Q8C2×D5311yescharacteristic
198C2×#127—531no—
199#492—534no—
200#492—1062no—
201C2×C4⋊C8—531no—
202C2×C4⋊C8—531no—
203C2×C4⋊C8C2×D5311yescharacteristic
204C2×#102—1062no—
205C2×C4×C8—531no—
206C2×C4×C8—1061no—
207C4×C4⋊C4C2×D5311yescharacteristic
208#340—22no—
209#336—1061no—
210#334—1062no—
211#332—1062no—
212#331—1062no—
213C53#973211yescharacteristic, complemented, Sylow
214C22×Q8—21no—
215C22×D4—1061no—
216C22×D4—1061no—
217C22×D4—1061no—
218C2×SD16—1064no—
219C2×SD16—1064no—
220C2×SD16—2122no—
221C2×SD16—2122no—
222C2×SD16—2122no—
223C2×SD16—2122no—
224C22×C8—21no—
225C22×C8—531no—
226C22×C8—531no—
227C22×C8—1061no—
228C22×C8—1061no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8C4×D5312yes—
233#127—532no—
234#127—1061no—
235#127—1061no—
236#127—1061no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22⋊D5312yes—
240C2×C42—21no—
241C2×C42C22×D5311yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—532no—
245C4⋊C8—1061no—
246C4⋊C8—1061no—
247C4⋊C8—1061no—
248#102—2124no—
249C4×C8—532no—
250C4×C8—532no—
251C4×C8—2122no—
252#95—21no—
253C24—2121no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×D4—1062no—
261C2×D4—1062no—
262C2×D4—1062no—
263C2×D4—4242no—
264C2×D4—2121no—
265C2×D4—2121no—
266C2×D4—2121no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2122no—
270C2×D4—2122no—
271C2×D4—2121no—
272C2×D4—2121no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4D4×D5311yescharacteristic
276C22×C4#1171994711yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278SD16—2128no—
279SD16—2128no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—532no—
283C2×C8—532no—
284C2×C8—532no—
285C2×C8—532no—
286C2×C8—1061no—
287C2×C8—1061no—
288C2×C8—1062no—
289C2×C8—2121no—
290C2×C8—2122no—
291C2×C8—2121no—
292C2×C8—2121no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4#1171992414yescomplemented
299C42—41no—
300C42—41no—
301C42C2×C4×D5311yescharacteristic
302C42#1171994511yescharacteristic
303C42D4×D5311yescharacteristic
304C42D4×D5311yescharacteristic
305C23—4241no—
306C23—4241no—
307C23—4242no—
308C23—4242no—
309C23—2122no—
310C23#1313318911yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—81no—
314Q8—81no—
315D4—4242no—
316D4—4242no—
317D4—4241no—
318D4—4241no—
319D4—4241no—
320D4—4241no—
321D4—2122no—
322D4—2122no—
323D4—2124no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4C22⋊C4×D5311yescharacteristic
332C2×C4#1313317711yescharacteristic
333C2×C4C4×C22⋊D5311yescharacteristic
334C2×C4#1313319011yescharacteristic
335C2×C4#1313320311yescharacteristic
336C2×C4#1313321311yescharacteristic
337C2×C4D8×D5311yescharacteristic
338C2×C4SD16×D5311yescharacteristic
339C2×C4C2×#1171992412yes—
340C2×C4#1313323712yes—
341C2×C4#1313324111yescharacteristic
342C2×C4#1313324411yescharacteristic
343C8—81no—
344C8—534no—
345C8—1062no—
346C8—4241no—
347C8—2122no—
348C22—8482no—
349C22—8482no—
350C22—4242no—
351C22—4242no—
352C22—4242no—
353C22#1681052311yescharacteristic
354C22#1681081611yescharacteristic
355C22#1681082111yescharacteristic
356C22#1681085412yes—
357C22#1681088812yes—
358C4—21no—
359C4—21no—
360C4—44no—
361C4—82no—
362C4#102×D5311yescharacteristic
363C4#1681062611yescharacteristic
364C4C4×#1171992412yes—
365C4#1681081711yescharacteristic
366C4#1681082211yescharacteristic
367C4#1681099711yescharacteristic
368C4#1681100511yescharacteristic
369C2—8482no—
370C2#2027405812yes—
371C2#2027406512yes—
372C2#2027557211yescharacteristic
373C2#2027561711yescharacteristic
374C2#2027564011yescharacteristic
subgroupG/H
centerC23#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1988C2
socleC23×C53#121
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1988D53
hypocenterC53#9732
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×#88322C2semidirect
4. C2×#16810373C2semidirect
5. C2×#16810369C2semidirect
7. C53×#1988C2semidirect
12. #16810997C4semidirect
97. C4⋊C4×C53SD16semidirect
159. #88322#102semidirect
183. #1988D53semidirect
213. C53#9732semidirect
298. C4⋊C4#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109972
#202756422
C4⋊C4×#883222
C2×#168103732
C2×#168103692
C2×#168103662
C53×#19882
#973253