Finite Group Databasebeta

← #27301254

Subgroups — #27301254

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287849C211yesmaximal, characteristic
2#20276450C211yesmaximal, characteristic, complemented
3#20276443C211yesmaximal, characteristic, complemented
4#20275603C211yesmaximal, characteristic, complemented
5#20272831C211yesmaximal, characteristic, complemented
6#20272797C211yesmaximal, characteristic, complemented
7C4×#135×C53C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9#16811064—21no—
10#16811063C412yescomplemented
11#16811059—21no—
12#16811058C412yescomplemented
13#16811053—22no—
14C2×#13133157C2211yescharacteristic
15C2×#13133157C2211yescharacteristic
16C2×#13133153C2211yescharacteristic
17C2×#13133145—21no—
18#16810589—21no—
19#16810432—22no—
20#16810430—22no—
21C2×#135×C53C2211yescharacteristic
22C4×#41×C53—21no—
23C4×Q16×C53—21no—
24C4×SD16×C53—22no—
25C4×D8×C53—21no—
26C53×#427C2211yescharacteristic
27C53×#416—21no—
28C2×C4×C8×C53C2211yescharacteristic
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239—21no—
33#13133239—21no—
34C2×Dic212C2×C411yescharacteristic, complemented
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic, complemented
37C2×#11719917—21no—
38#13133163—22no—
39#13133161—22no—
40#13133160—22no—
41#13133158—22no—
42#13133157—21no—
43#13133157—21no—
44#13133157—21no—
45#13133157C2×C412yescomplemented
46C2×#11719908—21no—
47C2×#11719908—21no—
48#13133153—21no—
49#13133153—21no—
50#13133145—41no—
51#13133145—41no—
52#13133129—42no—
53C2×#41×C53—21no—
54#135×C53—21no—
55#135×C53—21no—
56#135×C53C2×C412yes—
57#135×C53D412yes—
58C2×Q16×C53D411yescharacteristic
59C2×SD16×C53—21no—
60C2×D8×C53D411yescharacteristic
61C22×C8×C53D411yescharacteristic
62C22×C8×C53C2311yescharacteristic
63C4×Q8×C53—22no—
64C4×D4×C53—22no—
65C4×D4×C53—42no—
66#117×C53—21no—
67#117×C53—41no—
68C2×C42×C53—41no—
69C2×C42×C53D411yescharacteristic
70#107×C53—21no—
71#106×C53D412yes—
72#103×C53—22no—
73#102×C53—22no—
74C4×C8×C53—21no—
75C4×C8×C53—21no—
76#41×C53—22no—
77#41×C53—22no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—41no—
83C22×C4×C53—21no—
84C22×C4×C53—42no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C2×C22⋊D53—41no—
88#11719945—41no—
89#11719945—41no—
90#11719945—41no—
91#11719945—41no—
92C2×D212—21no—
93C2×C4×D53—41no—
94C2×Dic106—21no—
95Q16×C53—21no—
96Q16×C53C22⋊C412yes—
97SD16×C53—22no—
98SD16×C53—22no—
99D8×C53—21no—
100D8×C53C22⋊C412yes—
101C2×C8×C53—21no—
102C2×C8×C53—21no—
103C2×C8×C53—21no—
104C2×C8×C53C22×C411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53#4111yescharacteristic
109C2×C8×C53#4111yescharacteristic
110C4⋊C4×C53—22no—
111C4⋊C4×C53—42no—
112C22⋊C4×C53—42no—
113C22⋊C4×C53—41no—
114C42×C53—21no—
115C42×C53—42no—
116C42×C53—41no—
117C42×C53D811yescharacteristic
118C42×C53D811yescharacteristic
119#11719917—41no—
120#11719917—41no—
121C2×#88322—41no—
122Dic212—21no—
123Dic212—21no—
124D424—21no—
125D424—21no—
126#11719913—41no—
127#11719913—41no—
128#11719908—22no—
129#11719908—41no—
130C23×C53—41no—
131C23×C53AGL(1,ℤ/8)11yescharacteristic
132C22×D53—41no—
133Q8×C53—22no—
134Q8×C53—41no—
135D4×C53—22no—
136D4×C53—42no—
137D4×C53—42no—
138D4×C53—41no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—42no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—84no—
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53C2×D811yescharacteristic
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C22⋊D53—81no—
154C22⋊D53—81no—
155C2×Dic53—41no—
156D212—41no—
157D212—41no—
158C4×D53—81no—
159C4×D53—81no—
160Dic106—41no—
161Dic106—41no—
162C8×C53—41no—
163C8×C53C2×C22⋊C411yescharacteristic
164C8×C53C4×D411yescharacteristic
165C8×C53#12111yescharacteristic
166C8×C53#12311yescharacteristic
167#88322—81no—
168#13027—531nomaximal, Sylow
169C22×C53—41no—
170C22×C53—42no—
171C22×C53—82no—
172C22×C53#44211yescharacteristic
173C22×C53#44711yescharacteristic
174C22×C53#46311yescharacteristic
175C2×D53—82no—
176C2×D53—81no—
177C4×C53—22no—
178C4×C53—41no—
179C4×C53—42no—
180C4×C53—82no—
181C4×C53#38611yescharacteristic
182C4×C53C4×D811yescharacteristic
183C4×C53#44911yescharacteristic
184C4×C53#45911yescharacteristic
185Dic53—81no—
186#3414—531no—
187C4×#135D5311yescharacteristic, complemented
188#2145—531no—
189#2143—531no—
190#1974—531no—
191#1389—531no—
192#1376—531no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—21no—
196C2×C53—82no—
197C2×C53#187911yescharacteristic
198D53—161no—
199C2×#135—531no—
200C2×#135C2×D5311yescharacteristic
201C4×#41—21no—
202#510—532no—
203#510—1061no—
204#509—532no—
205#509—1061no—
206#508—1062no—
207C2×M32—531no—
208C2×M32—531no—
209C4×Q16—21no—
210C4×SD16—22no—
211C4×D8—21no—
212C2×#108—531no—
213#427C2×D5311yescharacteristic
214C2×#104—1061no—
215#416—21no—
216#408—1061no—
217C2×C4×C8C2×D5311yescharacteristic
218#360—1062no—
219#359—1062no—
220C53#1302711yescharacteristic, complemented, Sylow
221C2×#41—21no—
222C2×#41—1061no—
223#135—21no—
224#135—21no—
225#135—1061no—
226#135—1061no—
227#135—1061no—
228#135—1061no—
229#135C4×D5312yes—
230#135D21212yes—
231C2×Q16—531no—
232C2×Q16C22⋊D5311yescharacteristic
233C2×SD16—21no—
234C2×SD16—1061no—
235C2×D8—531no—
236C2×D8C22⋊D5311yescharacteristic
237C2×M16—1061no—
238C22×C8C22⋊D5311yescharacteristic
239C22×C8C22×D5311yescharacteristic
240C4×Q8—22no—
241C4×D4—22no—
242C4×D4—42no—
243#117—21no—
244#117—41no—
245C2×C42—41no—
246C2×C42C22⋊D5311yescharacteristic
247Q32—1062no—
248SD32—1062no—
249SD32—1062no—
250D16—1062no—
251M32—532no—
252M32—1061no—
253M32—1061no—
254M32—1061no—
255C2×C16—1061no—
256C2×C16—1061no—
257#108—1061no—
258#108—1061no—
259#107—21no—
260#106C22⋊D5312yes—
261#104—2121no—
262#104—2121no—
263#103—22no—
264#102—22no—
265C22⋊C8—2122no—
266C4×C8—21no—
267C4×C8—21no—
268#41—22no—
269#41—22no—
270#41—41no—
271#41—41no—
272#41—2121no—
273#41—2121no—
274#41—2121no—
275#41—2121no—
276C2×Q8—21no—
277C2×Q8—1061no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—1061no—
281C2×D4—2121no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—2121no—
286C22×C4C2×C22⋊D5311yescharacteristic
287Q16—21no—
288Q16—1061no—
289Q16—1061no—
290Q16#1171992112yes—
291SD16—22no—
292SD16—22no—
293SD16—2121no—
294SD16—2121no—
295D8—21no—
296D8—1061no—
297D8—1061no—
298D8#1171992112yes—
299M16—2121no—
300M16—2121no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—2121no—
305C2×C8C2×C4×D5311yescharacteristic
306C2×C8C2×D21211yescharacteristic
307C2×C8#1171994511yescharacteristic
308C2×C8#1171994511yescharacteristic
309C2×C8C2×C22⋊D5311yescharacteristic
310C2×C8C2×C22⋊D5311yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C42—21no—
316C42—42no—
317C42—41no—
318C42#1171992211yescharacteristic
319C42#1171992211yescharacteristic
320C16—1062no—
321C16—2121no—
322C23—41no—
323C23—2121no—
324C23#1313324311yescharacteristic
325Q8—22no—
326Q8—41no—
327Q8—2121no—
328Q8—2121no—
329D4—22no—
330D4—42no—
331D4—42no—
332D4—41no—
333D4—4241no—
334D4—4241no—
335D4—2121no—
336D4—2121no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—84no—
345C2×C4—4241no—
346C2×C4—4241no—
347C2×C4—2121no—
348C2×C4C4×C22⋊D5311yescharacteristic
349C2×C4#1313318611yescharacteristic
350C2×C4#1313319111yescharacteristic
351C2×C4C2×#1171992211yescharacteristic
352C2×C4#1313323711yescharacteristic
353C2×C4#1313324211yescharacteristic
354C8—41no—
355C8—4241no—
356C8C2×#1171992111yescharacteristic
357C8C4×C22⋊D5311yescharacteristic
358C8#1313319111yescharacteristic
359C8#1313320011yescharacteristic
360C22—41no—
361C22—42no—
362C22—82no—
363C22—4242no—
364C22—4241no—
365C22#1681073311yescharacteristic
366C22#1681075711yescharacteristic
367C22#1681083811yescharacteristic
368C4—22no—
369C4—41no—
370C4—42no—
371C4—82no—
372C4—4241no—
373C4#1681050211yescharacteristic
374C4C4×#1171992211yescharacteristic
375C4#1681076811yescharacteristic
376C4#1681081811yescharacteristic
377C2—21no—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—8481no—
382C2#2027506311yescharacteristic
subgroupG/H
centerC4#16810818
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC4×#135×C53C2
socleC2×C53#1879
prime radical#20287849C2
cosocleC22×C8×C53C23
hypercenterC4×#135D53
hypocenterC53#13027
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
2. #20276450C2semidirect
3. #20276443C2semidirect
4. #20275603C2semidirect
5. #20272831C2semidirect
6. #20272797C2semidirect
7. C4×#135×C53C2semidirect
10. #16811063C4semidirect
12. #16811058C4semidirect
34. C2×Dic212C2×C4semidirect
36. C2×D424C2×C4semidirect
45. #13133157C2×C4semidirect
187. C4×#135D53semidirect
220. C53#13027semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202764502
#202764432
#202756032
#202728312
#202727972
C4×#135×C532
#1302753