Finite Group Databasebeta

← #27369655

Subgroups — #27369655

Order 13568.

630 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
1#20284508C211yesmaximal, characteristic
2C2×#16810900C211yesmaximal, characteristic
3#20283755C211yesmaximal, characteristic
4#20274287C214yesmaximal, complemented
5#20274201C214yesmaximal, complemented
6#20273486C211yesmaximal, characteristic, complemented
7#20273400C211yesmaximal, characteristic, complemented
8#20273372C211yesmaximal, characteristic, complemented
9C53×#3209C211yesmaximal, characteristic, complemented
10C2×C4×D212C2211yescharacteristic
11#16810920—22no—
12#16810910C2212yes—
13#16810900—22no—
14#16810900C2212yes—
15#16810900C2214yes—
16#16810702C2214yes—
17#16810700—22no—
18#16810676C2211yescharacteristic
19#16810666—21no—
20C2×#13133150—21no—
21#16810647C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23#16810627—21no—
24C2×#13133137—21no—
25#16810585—22no—
26#16810374C2214yescomplemented
27#16810372C2212yescomplemented
28#16810369C2212yescomplemented
29#16810369C2214yescomplemented
30C53×#533C2211yescharacteristic
31C2×C4⋊Q8×C53C2211yescharacteristic
32C53×#494C2212yescomplemented
33C53×#487C2212yescomplemented
34C53×#462—22no—
35C53×#451—22no—
36C53×#431C2211yescharacteristic
37C22×D212D411yescharacteristic
38C22×C4×D53—21no—
39C2×#11719917—21no—
40C2×#11719912—24no—
41C22×#88322—21no—
42C4×D212—21no—
43C4×D212—21no—
44C4×D212D412yes—
45C4×D212C2312yes—
46C2×C4⋊Dic53D411yescharacteristic
47C2×#11719921—21no—
48#13133150—24no—
49#13133150—22no—
50#13133148—22no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55C4⋊#88322C2312yes—
56#13133137—24no—
57#13133137—22no—
58#13133137—22no—
59C22⋊#88322—21no—
60#13133129—24no—
61C4×#88322—21no—
62C22×Q8×C53—21no—
63C2×Q16×C53—24no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53—21no—
67C4⋊Q8×C53C2311yescharacteristic
68C4⋊Q8×C53C2312yes—
69C4⋊Q8×C53C2311yescharacteristic
70#125×C53—21no—
71C22⋊Q8×C53—21no—
72C22⋊Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53—21no—
75#117×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—21no—
78C2×C42×C53C2311yescharacteristic
79C4⋊C8×C53D412yes—
80#103×C53—24no—
81C22⋊C8×C53D412yes—
82C4×C8×C53C2312yes—
83C23×D53—21no—
84C2×Q8×C53—22no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—41no—
88C2×Q8×C53—82no—
89C22×C4×C53—41no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4111yescharacteristic
93C22×Dic53—21no—
94C2×D212—22no—
95C2×D212—22no—
96C2×D212—21no—
97C2×D212C2×D412yes—
98C2×C4×D53—22no—
99C2×C4×D53—22no—
100C2×C4×D53—82no—
101Q16×C53—84no—
102C2×C8×C53C2×D414yes—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—41no—
110C4⋊C4×C53—41no—
111C4⋊C4×C53—41no—
112C4⋊C4×C53—41no—
113C22⋊C4×C53—41no—
114C42×C53—41no—
115C42×C53C2×D411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53C2411yescharacteristic
119#11719921—22no—
120#11719921—22no—
121C4⋊Dic53—21no—
122C4⋊Dic53C2×D412yes—
123#11719917—42no—
124C2×#88322—22no—
125C2×#88322—22no—
126C2×#88322—22no—
127C2×#88322—42no—
128#11719912—84no—
129C23×C53#12111yescharacteristic
130C22×D53—22no—
131C22×D53—42no—
132C22×D53—81no—
133C22×D53—81no—
134Q8×C53—44no—
135Q8×C53—42no—
136Q8×C53—82no—
137Q8×C53—81no—
138C2×C4×C53—22no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—82no—
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#13712yes—
148C2×C4×C53#13712yes—
149C2×C4×C53C22×D411yescharacteristic
150C2×C4×C53C22×D411yescharacteristic
151C2×C4×C53C2×#4111yescharacteristic
152C2×Dic53—22no—
153C2×Dic53—81no—
154D212—22no—
155D212—22no—
156D212—42no—
157C4×D53—44no—
158C4×D53—82no—
159C8×C53—24no—
160#88322—42no—
161#88322—42no—
162#88322—42no—
163#19423—531nomaximal, Sylow
164C22×C53—21no—
165C22×C53—21no—
166C22×C53#47012yes—
167C22×C53C2×#12111yescharacteristic
168C2×D53—42no—
169C2×D53—42no—
170C2×D53—42no—
171C2×D53—84no—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—82no—
176C4×C53—81no—
177C4×C53—81no—
178C4×C53#47012yes—
179C4×C53C2×#13711yescharacteristic
180C4×C53C2×#13711yescharacteristic
181C4×C53#57612yes—
182Dic53—42no—
183#3209D5311yescharacteristic, complemented
184#3077—531no—
185C2×#474—531no—
186#2975—531no—
187#1703—534no—
188#1689—534no—
189#1570—531no—
190#1549—531no—
191#1542—531no—
192C2×C53—22no—
193C2×C53#304111yescharacteristic
194C2×C53C2×#47011yescharacteristic
195C2×C53#306911yescharacteristic
196D53—82no—
197#533C2×D5311yescharacteristic
198C2×C4⋊Q8C2×D5311yescharacteristic
199C2×C4×D4—531no—
200#494C2×D5312yescomplemented
201#487C2×D5312yescomplemented
202#478—1062no—
203#476—532no—
204#474—532no—
205#474—534no—
206#474—1062no—
207#462—22no—
208#451—22no—
209#436—534no—
210#436—1062no—
211#431—531no—
212#431C2×D5311yescharacteristic
213#428—1061no—
214C2×#107—1061no—
215#423—531no—
216C2×C4⋊C8—531no—
217#418—1061no—
218C2×#102—1061no—
219#407—1062no—
220#334—534no—
221#333—532no—
222#332—532no—
223#332—534no—
224C53#1942311yescharacteristic, complemented, Sylow
225C22×Q8—21no—
226C22×D4—531no—
227C23×C4—1061no—
228C2×Q16—24no—
229C2×M16—1061no—
230C2×M16—1064no—
231C22×C8—1061no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8C22×D5312yes—
236C4⋊Q8C22×D5311yescharacteristic
237C4⋊Q8C22×D5311yescharacteristic
238#125—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×D4—532no—
244C4×D4—532no—
245C4×D4—1061no—
246C4×D4—1061no—
247#117—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—531no—
251C2×C22⋊C4—1061no—
252C2×C42C22×D5311yescharacteristic
253#107—1064no—
254#107—1062no—
255#106—1062no—
256C4⋊C8—532no—
257C4⋊C8—532no—
258C4⋊C8—532no—
259C4⋊C8—1061no—
260C4⋊C8—1061no—
261C4⋊C8C22⋊D5312yes—
262#103—24no—
263#102—1064no—
264#102—1062no—
265#102—1062no—
266C22⋊C8—1061no—
267C22⋊C8—1064no—
268C22⋊C8C22⋊D5312yes—
269C4×C8—1061no—
270C4×C8C22×D5312yes—
271C24—1061no—
272C2×Q8—22no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—82no—
277C2×D4—532no—
278C2×D4—1061no—
279C2×D4—1062no—
280C2×D4—1062no—
281C22×C4—41no—
282C22×C4—1061no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—4242no—
286C22×C4D4×D5311yescharacteristic
287C22×C4#1171994711yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289Q16—84no—
290M16—4244no—
291M16—2122no—
292C2×C8—1062no—
293C2×C8—1062no—
294C2×C8—1062no—
295C2×C8—2122no—
296C2×C8C2×C22⋊D5314yes—
297C4⋊C4—21no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—532no—
308C4⋊C4—1061no—
309C22⋊C4—41no—
310C22⋊C4—1062no—
311C22⋊C4—1062no—
312C42—41no—
313C42D4×D5311yescharacteristic
314C42#1171994711yescharacteristic
315C42C2×C22⋊D5311yescharacteristic
316C42C23×D5311yescharacteristic
317C23—1062no—
318C23—4241no—
319C23—4241no—
320C23—2122no—
321C23#1313318811yescharacteristic
322Q8—42no—
323Q8—44no—
324Q8—81no—
325Q8—82no—
326D4—1062no—
327D4—1062no—
328D4—2122no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—1062no—
336C2×C4—4242no—
337C2×C4—4241no—
338C2×C4—2124no—
339C2×C4#1313318811yescharacteristic
340C2×C4#1313318811yescharacteristic
341C2×C4#1313318811yescharacteristic
342C2×C4#1313324912yes—
343C2×C4#1313324912yes—
344C2×C4C2×D4×D5311yescharacteristic
345C2×C4C22×C22⋊D5311yescharacteristic
346C2×C4C2×#1171994711yescharacteristic
347C8—24no—
348C8—2122no—
349C8—2122no—
350C8—2122no—
351C22—21no—
352C22—21no—
353C22—4244no—
354C22—2122no—
355C22—2122no—
356C22—2122no—
357C22#1681087812yes—
358C22C2×#1313318811yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—82no—
364C4—81no—
365C4—2122no—
366C4#1681087812yes—
367C4C2×#1313324911yescharacteristic
368C4C2×#1313324911yescharacteristic
369C4#1681142612yes—
370C2—22no—
371C2—4242no—
372C2#2028421411yescharacteristic
373C2C2×#1681087811yescharacteristic
374C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3209C2
socleC22×C53C2×#121
prime radicalC22×D212D4
hypercenter#3209D53
hypocenterC53#19423
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
4. #20274287C2semidirect
5. #20274201C2semidirect
6. #20273486C2semidirect
7. #20273400C2semidirect
8. #20273372C2semidirect
9. C53×#3209C2semidirect
26. #16810374C22semidirect
27. #16810372C22semidirect
28. #16810369C22semidirect
29. #16810369C22semidirect
32. C53×#494C22semidirect
33. C53×#487C22semidirect
183. #3209D53semidirect
200. #494C2×D53semidirect
201. #487C2×D53semidirect
224. C53#19423semidirect

Maximal subgroups

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

subgroupindex
#202845082
C2×#168109002
#202837552
#202742872
#202742012
#202734862
#202734002
#202733722
C53×#32092
#1942353