Finite Group Databasebeta

← #27301345

Subgroups — #27301345

Order 13568.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811058C211yesmaximal, characteristic
2C2×#16810435C211yesmaximal, characteristic, complemented
3C2×#16810433C211yesmaximal, characteristic, complemented
4#20275591C211yesmaximal, characteristic, complemented
5#20272815C211yesmaximal, characteristic, complemented
6#20272795C211yesmaximal, characteristic, complemented
7C2×C53×#444C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9#16811058—22no—
10#16811058—22no—
11C2×#13133160C412yescomplemented
12C2×#13133158C412yescomplemented
13C2×#13133157C2211yescharacteristic
14C2×#13133157C2211yescharacteristic
15C2×#13133149C2211yescharacteristic
16C2×#13133139—21no—
17#16810525—21no—
18#16810435—22no—
19#16810433—22no—
20C2×#135×C53C2211yescharacteristic
21C2×#131×C53—21no—
22C53×#444—22no—
23C53×#444—22no—
24C2×#108×C53C2211yescharacteristic
25C2×#104×C53—21no—
26C2×#97×C53C2211yescharacteristic
27C22×D212—21no—
28C2×D424—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424C2×C412yes—
34C2×C4⋊Dic53—21no—
35C2×#11719921—42no—
36#13133160—24no—
37#13133158—24no—
38#13133157—21no—
39#13133157—21no—
40#13133157—21no—
41#13133157—21no—
42C2×#11719908—21no—
43C2×#11719908C2×C412yes—
44#13133149—21no—
45#13133149—21no—
46#13133139—42no—
47C2×#41×C53—21no—
48#135×C53—21no—
49#135×C53—21no—
50#135×C53D412yes—
51#135×C53D412yes—
52C2×Q16×C53C2×C411yescharacteristic
53C2×SD16×C53—21no—
54C2×D8×C53C2×C411yescharacteristic
55#131×C53—24no—
56#131×C53—42no—
57C2×M16×C53—21no—
58C2×M16×C53—41no—
59C22×C8×C53—41no—
60C22×C8×C53D411yescharacteristic
61C22×C8×C53C2311yescharacteristic
62C2×C42×C53D411yescharacteristic
63#108×C53—21no—
64#108×C53D412yes—
65#104×C53—24no—
66#97×C53—21no—
67#97×C53—21no—
68C23×D53—41no—
69#41×C53—22no—
70#41×C53—22no—
71#41×C53—41no—
72#41×C53—41no—
73C2×Q8×C53—21no—
74C2×D4×C53—21no—
75C2×D4×C53—41no—
76C22×C4×C53—21no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×Dic53—41no—
80C2×D212—22no—
81C2×D212—41no—
82C2×D212—41no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86Q16×C53—21no—
87Q16×C53C22⋊C412yescomplemented
88SD16×C53—22no—
89SD16×C53—22no—
90D8×C53—21no—
91D8×C53C22⋊C412yescomplemented
92M16×C53—22no—
93M16×C53—44no—
94M16×C53—41no—
95C2×C8×C53—21no—
96C2×C8×C53—21no—
97C2×C8×C53—44no—
98C2×C8×C53—41no—
99C2×C8×C53D812yes—
100C2×C8×C53C22×C411yescharacteristic
101C2×C8×C53C2×D411yescharacteristic
102C2×C8×C53C2×D411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53#4111yescharacteristic
105C2×C8×C53#4111yescharacteristic
106C42×C53—21no—
107C42×C53—21no—
108#11719921—84no—
109C4⋊Dic53—41no—
110C4⋊Dic53—41no—
111D424—22no—
112D424—22no—
113D424—41no—
114D424—41no—
115#11719908—22no—
116#11719908—41no—
117C23×C53—41no—
118C23×C53AGL(1,ℤ/8)11yescharacteristic
119C22×D53—42no—
120C22×D53—82no—
121C22×D53—82no—
122C22×D53—81no—
123C22×D53—81no—
124Q8×C53—22no—
125Q8×C53—41no—
126D4×C53—22no—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×C4×C53C2×D811yescharacteristic
139C2×C4×C53#13511yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C2×Dic53—82no—
142C2×Dic53—81no—
143D212—42no—
144D212—42no—
145D212—81no—
146D212—81no—
147C8×C53—22no—
148C8×C53—42no—
149C8×C53C2×C22⋊C411yescharacteristic
150C8×C53C4×D411yescharacteristic
151C8×C53#12111yescharacteristic
152C8×C53#12311yescharacteristic
153#13040—531nomaximal, Sylow
154C22×C53—22no—
155C22×C53—42no—
156C22×C53—82no—
157C22×C53#44211yescharacteristic
158C22×C53#44911yescharacteristic
159C22×C53#45911yescharacteristic
160C2×D53—82no—
161C2×D53—84no—
162C2×D53—161no—
163C2×D53—161no—
164C2×D53—162no—
165C4×C53—42no—
166C4×C53—81no—
167C4×C53#38611yescharacteristic
168C4×C53C4×D811yescharacteristic
169C4×C53#44711yescharacteristic
170C4×C53#46311yescharacteristic
171Dic53—161no—
172C2×#509—531no—
173C2×#444D5311yescharacteristic, complemented
174C2×#361—531no—
175C2×#360—531no—
176#1972—531no—
177#1383—531no—
178#1375—531no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—82no—
182C2×C53#187911yescharacteristic
183C2×C53AGL(1,ℤ/16)12yes—
184D53—162no—
185C2×#135C2×D5311yescharacteristic
186C22×D8—531no—
187C2×#131—21no—
188#509—1062no—
189#509—1062no—
190C2×SD32—532no—
191C2×D16—532no—
192C2×M32—531no—
193C2×M32—531no—
194#444—22no—
195#444—22no—
196C2×#108C2×D5311yescharacteristic
197C2×#106—531no—
198C2×#104—21no—
199C2×#102—1061no—
200C2×#97C2×D5311yescharacteristic
201#390—1061no—
202#361—1062no—
203#360—1062no—
204C53#1304011yescharacteristic, complemented, Sylow
205C2×#41—21no—
206C22×D4—1061no—
207#135—21no—
208#135—21no—
209#135D21212yes—
210#135C22⋊D5312yes—
211C2×Q16C4×D5311yescharacteristic
212C2×SD16—21no—
213C2×D8—532no—
214C2×D8—1061no—
215C2×D8—1061no—
216C2×D8—1061no—
217C2×D8—1061no—
218C2×D8—1061no—
219C2×D8C4×D5311yescharacteristic
220#131—24no—
221#131—42no—
222C2×M16—21no—
223C2×M16—41no—
224C22×C8—41no—
225C22×C8C22⋊D5311yescharacteristic
226C22×C8C22×D5311yescharacteristic
227C2×C4⋊C4—1061no—
228C2×C22⋊C4—2122no—
229C2×C42C22⋊D5311yescharacteristic
230SD32—1064no—
231D16—1064no—
232M32—1061no—
233M32—1061no—
234M32—1061no—
235M32—1061no—
236C2×C16—532no—
237C2×C16—1061no—
238#108—21no—
239#108C22⋊D5312yes—
240#106—1061no—
241#106—1061no—
242#104—24no—
243#102—2122no—
244#97—21no—
245#97—21no—
246C24—2121no—
247#41—22no—
248#41—22no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—1062no—
255C2×D4—2121no—
256C2×D4—2121no—
257C2×D4—2121no—
258C2×D4—2121no—
259C2×D4—2121no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—2121no—
263C22×C4C2×C22⋊D5311yescharacteristic
264Q16—21no—
265Q16#1171992112yescomplemented
266SD16—22no—
267SD16—22no—
268D8—21no—
269D8—1062no—
270D8—1062no—
271D8—2121no—
272D8—2121no—
273D8#1171992112yescomplemented
274M16—22no—
275M16—44no—
276M16—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—44no—
280C2×C8—41no—
281C2×C8#1171992212yes—
282C2×C8C2×C4×D5311yescharacteristic
283C2×C8C2×D21211yescharacteristic
284C2×C8#1171994511yescharacteristic
285C2×C8#1171994511yescharacteristic
286C2×C8C2×C22⋊D5311yescharacteristic
287C2×C8C2×C22⋊D5311yescharacteristic
288C4⋊C4—2121no—
289C4⋊C4—2121no—
290C22⋊C4—4244no—
291C42—21no—
292C42—21no—
293C16—1062no—
294C16—2121no—
295C23—41no—
296C23—4242no—
297C23—4242no—
298C23—4241no—
299C23—4241no—
300C23—2122no—
301C23#1313324211yescharacteristic
302Q8—22no—
303Q8—41no—
304D4—22no—
305D4—42no—
306D4—42no—
307D4—41no—
308D4—4241no—
309D4—4241no—
310D4—2122no—
311D4—2122no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—4242no—
318C2×C4—4241no—
319C2×C4C4×C22⋊D5311yescharacteristic
320C2×C4#1313318611yescharacteristic
321C2×C4#1313319111yescharacteristic
322C2×C4C2×#1171992211yescharacteristic
323C2×C4#1313323711yescharacteristic
324C2×C4#1313324311yescharacteristic
325C8—22no—
326C8—42no—
327C8C2×#1171992111yescharacteristic
328C8C4×C22⋊D5311yescharacteristic
329C8#1313319111yescharacteristic
330C8#1313320011yescharacteristic
331C22—22no—
332C22—42no—
333C22—82no—
334C22—8482no—
335C22—8481no—
336C22—8481no—
337C22—4242no—
338C22—4244no—
339C22#1681073311yescharacteristic
340C22#1681076811yescharacteristic
341C22#1681081811yescharacteristic
342C4—42no—
343C4—81no—
344C4—8481no—
345C4#1681050211yescharacteristic
346C4C4×#1171992211yescharacteristic
347C4#1681075711yescharacteristic
348C4#1681083811yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2—8482no—
353C2#2027506311yescharacteristic
354C2#2027661312yes—
subgroupG/H
centerC22#16810733
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#444C2
socleC22×C53#442
prime radicalC2×#16811058C2
cosocleC22×C8×C53C23
hypercenterC2×#444D53
hypocenterC53#13040
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. C2×#16810435C2semidirect
3. C2×#16810433C2semidirect
4. #20275591C2semidirect
5. #20272815C2semidirect
6. #20272795C2semidirect
7. C2×C53×#444C2semidirect
11. C2×#13133160C4semidirect
12. C2×#13133158C4semidirect
87. Q16×C53C22⋊C4semidirect
91. D8×C53C22⋊C4semidirect
173. C2×#444D53semidirect
204. C53#13040semidirect
265. Q16#11719921semidirect
273. D8#11719921semidirect

Maximal subgroups

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

subgroupindex
C2×#168110582
C2×#168104352
C2×#168104332
#202755912
#202728152
#202727952
C2×C53×#4442
#1304053