Finite Group Databasebeta

← #27301177

Subgroups — #27301177

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811053C211yesmaximal, characteristic
2#20276451C211yesmaximal, characteristic, complemented
3#20276442C211yesmaximal, characteristic, complemented
4#20275613C211yesmaximal, characteristic, complemented
5#20272812C211yesmaximal, characteristic, complemented
6#20272804C211yesmaximal, characteristic, complemented
7C53×#2957C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic
9#16811053—22no—
10#16811053—22no—
11C2×#13133163C411yescharacteristic, complemented
12C2×#13133161C411yescharacteristic, complemented
13C2×#13133160C411yescharacteristic, complemented
14C2×#13133158C411yescharacteristic, complemented
15C2×#13133157C2211yescharacteristic
16C22×#11719908C2211yescharacteristic
17#16810661C2211yescharacteristic
18#16810639—21no—
19#16810596—21no—
20#16810433—22no—
21#16810429—22no—
22C2×#135×C53C2211yescharacteristic
23C53×#568—21no—
24C53×#444—22no—
25C53×#443—22no—
26C2×#108×C53C2211yescharacteristic
27C53×#421—21no—
28C53×#405C2211yescharacteristic
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239—21no—
33#13133239—21no—
34C2×Dic212C2×C411yescharacteristic
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic
37#13133176—21no—
38#13133163—22no—
39#13133161—22no—
40#13133160—22no—
41#13133158—22no—
42#13133157—21no—
43#13133157—21no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47C2×#11719908C2×C411yescharacteristic
48C2×#11719908C2×C411yescharacteristic
49#13133151—21no—
50#13133149—21no—
51#13133143—42no—
52#13133131—42no—
53C2×#41×C53—21no—
54#135×C53—21no—
55#135×C53—21no—
56#135×C53D412yes—
57#135×C53D412yes—
58C2×Q16×C53C2×C411yescharacteristic
59C2×SD16×C53—21no—
60C2×D8×C53C2×C411yescharacteristic
61#131×C53—24no—
62#131×C53—42no—
63C2×M16×C53—21no—
64C2×M16×C53—42no—
65C2×M16×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67#117×C53D411yescharacteristic
68#108×C53—21no—
69#108×C53D412yes—
70#104×C53—24no—
71#97×C53—21no—
72C4×C8×C53—21no—
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—41no—
81C22×C4×C53C2×D411yescharacteristic
82C2×C22⋊D53—41no—
83#11719945—41no—
84#11719945—41no—
85#11719945—41no—
86#11719945—41no—
87C2×D212—21no—
88C2×C4×D53—41no—
89C2×Dic106—21no—
90Q16×C53—21no—
91Q16×C53C22⋊C412yes—
92SD16×C53—22no—
93SD16×C53—22no—
94D8×C53—21no—
95D8×C53C22⋊C412yes—
96M16×C53—21no—
97M16×C53—22no—
98M16×C53—44no—
99M16×C53—41no—
100M16×C53D812yes—
101C2×C8×C53—21no—
102C2×C8×C53—44no—
103C2×C8×C53—41no—
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—21no—
111C22⋊C4×C53—21no—
112C42×C53—21no—
113C22⋊Dic53—41no—
114C4⋊Dic53—41no—
115C4×Dic53—41no—
116Dic212—21no—
117Dic212—21no—
118D424—21no—
119D424—21no—
120#11719913—41no—
121#11719913—41no—
122#11719908—21no—
123#11719908—21no—
124#11719908—41no—
125C23×C53—41no—
126C23×C53AGL(1,ℤ/8)11yescharacteristic
127C22×D53—41no—
128Q8×C53—22no—
129Q8×C53—41no—
130D4×C53—22no—
131D4×C53—42no—
132D4×C53—42no—
133D4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—41no—
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53C2×D811yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C22⋊D53—81no—
145C22⋊D53—81no—
146C2×Dic53—41no—
147C2×Dic53—81no—
148D212—41no—
149D212—41no—
150C4×D53—81no—
151C4×D53—81no—
152Dic106—41no—
153Dic106—41no—
154C8×C53—22no—
155C8×C53—42no—
156C8×C53C2×C22⋊C411yescharacteristic
157C8×C53C4×D411yescharacteristic
158C8×C53#12111yescharacteristic
159C8×C53#12311yescharacteristic
160#13016—531nomaximal, Sylow
161C22×C53—41no—
162C22×C53—42no—
163C22×C53—161no—
164C22×C53#44211yescharacteristic
165C22×C53#44911yescharacteristic
166C22×C53#45911yescharacteristic
167C2×D53—81no—
168C2×D53—161no—
169C4×C53—42no—
170C4×C53—81no—
171C4×C53#38611yescharacteristic
172C4×C53C4×D811yescharacteristic
173C4×C53#44711yescharacteristic
174C4×C53#46311yescharacteristic
175Dic53—81no—
176Dic53—161no—
177C2×#508—531no—
178#2957D5311yescharacteristic, complemented
179#2145—531no—
180#2143—531no—
181#1976—531no—
182#1382—531no—
183#1379—531no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—82no—
188C2×C53#187911yescharacteristic
189D53—161no—
190C2×#135—531no—
191C2×#135C2×D5311yescharacteristic
192#568—21no—
193#508—1062no—
194#508—1062no—
195C2×Q32—531no—
196C2×SD32—531no—
197C2×SD32—531no—
198C2×D16—531no—
199C2×M32—531no—
200C22×C16—531no—
201#444—22no—
202#443—22no—
203C2×#108C2×D5311yescharacteristic
204#427—531no—
205#421—21no—
206#421—1061no—
207#410—1061no—
208#405C2×D5311yescharacteristic
209#360—1062no—
210#359—1062no—
211C53#1301611yescharacteristic, complemented, Sylow
212C2×#41—21no—
213C2×#41—1061no—
214#135—21no—
215#135—21no—
216#135—1061no—
217#135—1061no—
218#135—1061no—
219#135—1061no—
220#135D21212yes—
221#135C22⋊D5312yes—
222C2×Q16—531no—
223C2×Q16C4×D5311yescharacteristic
224C2×SD16—21no—
225C2×SD16—1061no—
226C2×D8—531no—
227C2×D8C4×D5311yescharacteristic
228#131—24no—
229#131—42no—
230C2×M16—21no—
231C2×M16—42no—
232C2×M16C22⋊D5311yescharacteristic
233C22×C8C22×D5311yescharacteristic
234#117—1061no—
235#117C22⋊D5311yescharacteristic
236Q32—1062no—
237SD32—1062no—
238SD32—1062no—
239D16—1062no—
240M32—1061no—
241M32—1061no—
242C2×C16—531no—
243C2×C16—531no—
244C2×C16—1061no—
245C2×C16—1061no—
246C2×C16—1061no—
247#108—21no—
248#108C22⋊D5312yes—
249#107—1061no—
250#106—1061no—
251#104—24no—
252#104—2122no—
253#99—2122no—
254#97—21no—
255C4×C8—21no—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260#41—2121no—
261#41—2121no—
262#41—2121no—
263#41—2121no—
264C2×Q8—21no—
265C2×Q8—1061no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—1061no—
269C2×D4—2121no—
270C22×C4—41no—
271C22×C4—2121no—
272C22×C4C2×C22⋊D5311yescharacteristic
273Q16—21no—
274Q16—1061no—
275Q16—1061no—
276Q16#1171992112yes—
277SD16—22no—
278SD16—22no—
279SD16—2121no—
280SD16—2121no—
281D8—21no—
282D8—1061no—
283D8—1061no—
284D8#1171992112yes—
285M16—21no—
286M16—22no—
287M16—44no—
288M16—41no—
289M16#1171992212yes—
290C2×C8—21no—
291C2×C8—44no—
292C2×C8—41no—
293C2×C8C2×C4×D5311yescharacteristic
294C2×C8C2×D21211yescharacteristic
295C2×C8#1171994511yescharacteristic
296C2×C8#1171994511yescharacteristic
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—2121no—
301C22⋊C4—21no—
302C22⋊C4—2121no—
303C42—21no—
304C42—2121no—
305C16—1061no—
306C16—1061no—
307C16—2121no—
308C23—41no—
309C23—2121no—
310C23#1313324211yescharacteristic
311Q8—22no—
312Q8—41no—
313Q8—2121no—
314Q8—2121no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—4241no—
320D4—4241no—
321D4—2121no—
322D4—2121no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—4241no—
328C2×C4—4241no—
329C2×C4—4241no—
330C2×C4—2121no—
331C2×C4C4×C22⋊D5311yescharacteristic
332C2×C4#1313318611yescharacteristic
333C2×C4#1313319111yescharacteristic
334C2×C4C2×#1171992211yescharacteristic
335C2×C4#1313323711yescharacteristic
336C2×C4#1313324311yescharacteristic
337C8—22no—
338C8—42no—
339C8C2×#1171992111yescharacteristic
340C8C4×C22⋊D5311yescharacteristic
341C8#1313319111yescharacteristic
342C8#1313320011yescharacteristic
343C22—41no—
344C22—42no—
345C22—161no—
346C22—8481no—
347C22—4241no—
348C22#1681073311yescharacteristic
349C22#1681076811yescharacteristic
350C22#1681081811yescharacteristic
351C4—42no—
352C4—81no—
353C4—8481no—
354C4—4241no—
355C4#1681050211yescharacteristic
356C4C4×#1171992211yescharacteristic
357C4#1681075711yescharacteristic
358C4#1681083811yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—82no—
363C2—8481no—
364C2#2027506311yescharacteristic
subgroupG/H
centerC4#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2957C2
socleC2×C53#1879
prime radicalC2×#16811053C2
cosocleC22×C8×C53C23
hypercenter#2957D53
hypocenterC53#13016
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. #20276451C2semidirect
3. #20276442C2semidirect
4. #20275613C2semidirect
5. #20272812C2semidirect
6. #20272804C2semidirect
7. C53×#2957C2semidirect
11. C2×#13133163C4semidirect
12. C2×#13133161C4semidirect
13. C2×#13133160C4semidirect
14. C2×#13133158C4semidirect
178. #2957D53semidirect
211. C53#13016semidirect

Maximal subgroups

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

subgroupindex
C2×#168110532
#202764512
#202764422
#202756132
#202728122
#202728042
C53×#29572
#1301653