Finite Group Databasebeta

← #27301219

Subgroups — #27301219

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811058C211yesmaximal, characteristic
2C2×#16810430C211yesmaximal, characteristic, complemented
3C2×#16810423C211yesmaximal, characteristic, complemented
4#20275625C211yesmaximal, characteristic, complemented
5#20272823C211yesmaximal, characteristic, complemented
6#20272804C211yesmaximal, characteristic, complemented
7C53×#2951C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9#16811058—22no—
10#16811058C414yescomplemented
11C2×#13133160—21no—
12C2×#13133158—21no—
13C2×#13133157C2211yescharacteristic
14C22×#11719908C2211yescharacteristic
15C2×#13133153C2211yescharacteristic
16#16810627—21no—
17C2×#13133133—21no—
18#16810430—22no—
19#16810423—22no—
20C2×#135×C53C2211yescharacteristic
21C53×#520—21no—
22C53×#440—22no—
23C4×Q16×C53—21no—
24C4×D8×C53—21no—
25C2×#107×C53C2211yescharacteristic
26C53×#417—21no—
27C53×#405C2211yescharacteristic
28C22×D212—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424C2×C412yescomplemented
35C2×#11719917—21no—
36#13133160—24no—
37#13133158—24no—
38#13133157—21no—
39#13133157C2×C412yescomplemented
40C2×#11719908—21no—
41C2×#11719908—21no—
42C2×#11719908—21no—
43C2×#11719908—21no—
44#13133153—21no—
45#13133153—21no—
46#13133137—41no—
47#13133137—41no—
48#13133133—42no—
49C2×#41×C53—21no—
50#135×C53—21no—
51#135×C53—21no—
52#135×C53C2×C412yes—
53#135×C53D412yes—
54C2×Q16×C53D411yescharacteristic
55C2×SD16×C53—21no—
56C2×D8×C53D411yescharacteristic
57C2×M16×C53D411yescharacteristic
58C22×C8×C53C2311yescharacteristic
59C4×Q8×C53—22no—
60C4×D4×C53—22no—
61C4×D4×C53—42no—
62#117×C53—41no—
63#117×C53D411yescharacteristic
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—41no—
66#107×C53—21no—
67#107×C53D412yes—
68#103×C53—22no—
69#102×C53—22no—
70#97×C53—21no—
71C4×C8×C53—21no—
72C23×D53—41no—
73#41×C53—22no—
74#41×C53—22no—
75#41×C53—41no—
76#41×C53—41no—
77C2×Q8×C53—21no—
78C2×D4×C53—21no—
79C2×D4×C53—41no—
80C22×C4×C53—42no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D411yescharacteristic
84C2×D212—22no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90Q16×C53—21no—
91Q16×C53C22⋊C412yes—
92SD16×C53—22no—
93SD16×C53—22no—
94D8×C53—21no—
95D8×C53C22⋊C412yes—
96M16×C53—21no—
97M16×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53C22×C411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53#4111yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C4⋊C4×C53—22no—
106C4⋊C4×C53—42no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53D811yescharacteristic
109C4⋊C4×C53D811yescharacteristic
110C22⋊C4×C53—21no—
111C22⋊C4×C53—42no—
112C42×C53—21no—
113C42×C53—42no—
114#11719917—41no—
115#11719917—41no—
116C2×#88322—41no—
117D424—22no—
118D424—22no—
119D424—41no—
120D424—41no—
121#11719908—22no—
122#11719908—41no—
123C23×C53—41no—
124C23×C53AGL(1,ℤ/8)11yescharacteristic
125C22×D53—42no—
126C22×D53—81no—
127C22×D53—81no—
128C22×D53—161no—
129C22×D53—161no—
130Q8×C53—22no—
131Q8×C53—41no—
132D4×C53—22no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—41no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—162no—
143C2×C4×C53—161no—
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53C2×D811yescharacteristic
148C2×C4×C53#13511yescharacteristic
149C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
150D212—42no—
151D212—42no—
152D212—81no—
153D212—81no—
154C8×C53—41no—
155C8×C53C2×C22⋊C411yescharacteristic
156C8×C53C4×D411yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12311yescharacteristic
159#88322—81no—
160#13022—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—42no—
164C22×C53—81no—
165C22×C53—81no—
166C22×C53#44211yescharacteristic
167C22×C53#44711yescharacteristic
168C22×C53#46311yescharacteristic
169C2×D53—82no—
170C2×D53—162no—
171C2×D53—161no—
172C2×D53—161no—
173C2×D53—162no—
174C4×C53—42no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—82no—
178C4×C53#38611yescharacteristic
179C4×C53C4×D811yescharacteristic
180C4×C53#44911yescharacteristic
181C4×C53#45911yescharacteristic
182C2×#509—531no—
183#2951D5311yescharacteristic, complemented
184C2×#359—531no—
185C2×#357—531no—
186#1978—531no—
187#1387—531no—
188#1379—531no—
189C2×C53—21no—
190C2×C53—21no—
191C2×C53—82no—
192C2×C53#187911yescharacteristic
193C2×C53#219211yescharacteristic
194C2×C53#219311yescharacteristic
195D53—162no—
196C2×#135C2×D5311yescharacteristic
197C22×D8—531no—
198#520—21no—
199#509—534no—
200#509—1062no—
201C2×SD32—1061no—
202C2×D16—1061no—
203C2×M32—531no—
204C22×C16—531no—
205#440—22no—
206C4×Q16—21no—
207C4×D8—21no—
208C2×#108—531no—
209C2×#107C2×D5311yescharacteristic
210#418—1061no—
211#417—21no—
212C2×#100—1061no—
213#405C2×D5311yescharacteristic
214#359—1062no—
215#357—1062no—
216C53#1302211yescharacteristic, complemented, Sylow
217C2×#41—21no—
218C22×D4—1061no—
219#135—21no—
220#135—21no—
221#135C4×D5312yes—
222#135D21212yes—
223C2×Q16C22⋊D5311yescharacteristic
224C2×SD16—21no—
225C2×D8—532no—
226C2×D8—1061no—
227C2×D8—1061no—
228C2×D8—1061no—
229C2×D8—1061no—
230C2×D8—1061no—
231C2×D8C22⋊D5311yescharacteristic
232C2×M16—1061no—
233C2×M16C22⋊D5311yescharacteristic
234C22×C8C22×D5311yescharacteristic
235C4×Q8—22no—
236C4×D4—22no—
237C4×D4—42no—
238#117—41no—
239#117C22⋊D5311yescharacteristic
240C2×C4⋊C4—21no—
241C2×C4⋊C4—41no—
242SD32—1064no—
243D16—1064no—
244M32—532no—
245M32—1061no—
246C2×C16—1061no—
247C2×C16—1061no—
248C2×C16—1061no—
249C2×C16—1061no—
250#108—1061no—
251#108—1061no—
252#107—21no—
253#107C22⋊D5312yes—
254#103—22no—
255#102—22no—
256#102—2121no—
257#102—2121no—
258#100—2122no—
259#97—21no—
260C4×C8—21no—
261C24—2121no—
262#41—22no—
263#41—22no—
264#41—41no—
265#41—41no—
266C2×Q8—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—1062no—
270C2×D4—2121no—
271C2×D4—2121no—
272C2×D4—2121no—
273C2×D4—2121no—
274C2×D4—2121no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C2×C22⋊D5311yescharacteristic
279Q16—21no—
280Q16#1171992112yes—
281SD16—22no—
282SD16—22no—
283D8—21no—
284D8—1062no—
285D8—1062no—
286D8—2121no—
287D8—2121no—
288D8#1171992112yes—
289M16—21no—
290M16—21no—
291M16—2121no—
292M16—2121no—
293C2×C8—21no—
294C2×C8—2121no—
295C2×C8C2×C4×D5311yescharacteristic
296C2×C8C2×D21211yescharacteristic
297C2×C8#1171994511yescharacteristic
298C2×C8#1171994511yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C2×C8C2×C22⋊D5311yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4#1171992211yescharacteristic
305C4⋊C4#1171992211yescharacteristic
306C22⋊C4—21no—
307C22⋊C4—42no—
308C42—21no—
309C42—42no—
310C16—1062no—
311C16—2121no—
312C23—41no—
313C23—8481no—
314C23—8481no—
315C23—4241no—
316C23—4241no—
317C23—2122no—
318C23#1313324311yescharacteristic
319Q8—22no—
320Q8—41no—
321D4—22no—
322D4—42no—
323D4—42no—
324D4—41no—
325D4—4241no—
326D4—4241no—
327D4—2122no—
328D4—2122no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—162no—
336C2×C4—161no—
337C2×C4C4×C22⋊D5311yescharacteristic
338C2×C4#1313318611yescharacteristic
339C2×C4#1313319111yescharacteristic
340C2×C4C2×#1171992211yescharacteristic
341C2×C4#1313323711yescharacteristic
342C2×C4#1313324211yescharacteristic
343C8—41no—
344C8—4241no—
345C8C2×#1171992111yescharacteristic
346C8C4×C22⋊D5311yescharacteristic
347C8#1313319111yescharacteristic
348C8#1313320011yescharacteristic
349C22—21no—
350C22—21no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—8482no—
355C22—8482no—
356C22—8481no—
357C22—8481no—
358C22—4242no—
359C22#1681073311yescharacteristic
360C22#1681075711yescharacteristic
361C22#1681083811yescharacteristic
362C4—42no—
363C4—41no—
364C4—41no—
365C4—82no—
366C4#1681050211yescharacteristic
367C4C4×#1171992211yescharacteristic
368C4#1681076811yescharacteristic
369C4#1681081811yescharacteristic
370C2—21no—
371C2—21no—
372C2—82no—
373C2—8482no—
374C2#2027506311yescharacteristic
375C2#2027667211yescharacteristic
376C2#2027667911yescharacteristic
subgroupG/H
centerC22#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2951C2
socleC22×C53#447
prime radicalC2×#16811058C2
cosocleC22×C8×C53C23
hypercenter#2951D53
hypocenterC53#13022
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
2. C2×#16810430C2semidirect
3. C2×#16810423C2semidirect
4. #20275625C2semidirect
5. #20272823C2semidirect
6. #20272804C2semidirect
7. C53×#2951C2semidirect
10. #16811058C4semidirect
34. C2×D424C2×C4semidirect
39. #13133157C2×C4semidirect
183. #2951D53semidirect
216. C53#13022semidirect

Maximal subgroups

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

subgroupindex
C2×#168110582
C2×#168104302
C2×#168104232
#202756252
#202728232
#202728042
C53×#29512
#1302253