Finite Group Databasebeta

← #27301233

Subgroups — #27301233

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287849C211yesmaximal, characteristic
2#20276464C211yesmaximal, characteristic, complemented
3#20276442C211yesmaximal, characteristic, complemented
4#20275608C211yesmaximal, characteristic, complemented
5#20272826C211yesmaximal, characteristic, complemented
6#20272799C211yesmaximal, characteristic, complemented
7C2×C53×#444C211yesmaximal, 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#16810439—21no—
20#16810436—21no—
21#16810429—22no—
22C2×#135×C53C2211yescharacteristic
23C2×#131×C53—21no—
24C53×#444—22no—
25C53×#444—22no—
26C2×#108×C53C2211yescharacteristic
27C2×#104×C53—21no—
28C2×#97×C53C2211yescharacteristic
29C2×#11719945—21no—
30#13133239—21no—
31#13133239—21no—
32#13133239—21no—
33#13133239—21no—
34C2×Dic212C2×C411yescharacteristic
35C2×#11719913—21no—
36C2×D424C2×C411yescharacteristic
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×C412yes—
46C2×#11719908—21no—
47C2×#11719908—21no—
48#13133153—21no—
49#13133153—21no—
50#13133145—42no—
51#13133129—42no—
52C2×#41×C53—21no—
53#135×C53—21no—
54#135×C53—21no—
55#135×C53C2×C412yes—
56#135×C53D412yes—
57C2×Q16×C53D411yescharacteristic
58C2×SD16×C53—21no—
59C2×D8×C53D411yescharacteristic
60#131×C53—24no—
61#131×C53—42no—
62C2×M16×C53—21no—
63C2×M16×C53—41no—
64C22×C8×C53—41no—
65C22×C8×C53D411yescharacteristic
66C22×C8×C53C2311yescharacteristic
67C2×C42×C53D411yescharacteristic
68#108×C53—21no—
69#108×C53D412yes—
70#104×C53—24no—
71#97×C53—21no—
72#97×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—21no—
81C22×C4×C53—41no—
82C22×C4×C53C2×D411yescharacteristic
83C2×C22⋊D53—41no—
84#11719945—41no—
85#11719945—41no—
86#11719945—41no—
87#11719945—41no—
88C2×D212—21no—
89C2×C4×D53—41no—
90C2×Dic106—21no—
91Q16×C53—21no—
92Q16×C53C22⋊C412yes—
93SD16×C53—22no—
94SD16×C53—22no—
95D8×C53—21no—
96D8×C53C22⋊C412yes—
97M16×C53—22no—
98M16×C53—44no—
99M16×C53—41no—
100C2×C8×C53—21no—
101C2×C8×C53—21no—
102C2×C8×C53—44no—
103C2×C8×C53—41no—
104C2×C8×C53D812yes—
105C2×C8×C53C22×C411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53C2×D411yescharacteristic
108C2×C8×C53C2×D411yescharacteristic
109C2×C8×C53#4111yescharacteristic
110C2×C8×C53#4111yescharacteristic
111C42×C53—21no—
112C42×C53—21no—
113#11719917—41no—
114#11719917—41no—
115C2×#88322—41no—
116Dic212—21no—
117Dic212—21no—
118D424—21no—
119D424—21no—
120#11719913—41no—
121#11719913—41no—
122#11719908—22no—
123#11719908—41no—
124C23×C53—41no—
125C23×C53#13511yescharacteristic
126C22×D53—41no—
127Q8×C53—22no—
128Q8×C53—41no—
129D4×C53—22no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—81no—
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53C2×D811yescharacteristic
142C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
144C22⋊D53—81no—
145C22⋊D53—81no—
146C2×Dic53—41no—
147D212—41no—
148D212—41no—
149C4×D53—81no—
150C4×D53—81no—
151Dic106—41no—
152Dic106—41no—
153C8×C53—22no—
154C8×C53—42no—
155C8×C53C2×C22⋊C411yescharacteristic
156C8×C53C4×D411yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12311yescharacteristic
159#88322—81no—
160#13024—531nomaximal, Sylow
161C22×C53—41no—
162C22×C53—42no—
163C22×C53—82no—
164C22×C53C4×D811yescharacteristic
165C22×C53#44911yescharacteristic
166C22×C53#46311yescharacteristic
167C2×D53—82no—
168C2×D53—81no—
169C4×C53—42no—
170C4×C53—81no—
171C4×C53#38611yescharacteristic
172C4×C53#44211yescharacteristic
173C4×C53#44711yescharacteristic
174C4×C53#45911yescharacteristic
175Dic53—81no—
176#3414—531no—
177C2×#444D5311yescharacteristic, complemented
178#2148—531no—
179#2143—531no—
180#1975—531no—
181#1388—531no—
182#1377—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—82no—
187C2×C53#187911yescharacteristic
188D53—161no—
189C2×#135—531no—
190C2×#135C2×D5311yescharacteristic
191C2×#131—21no—
192#510—532no—
193#510—1061no—
194#509—532no—
195#509—1061no—
196#508—1062no—
197C2×M32—531no—
198C2×M32—531no—
199#444—22no—
200#444—22no—
201C2×#108—531no—
202C2×#108C2×D5311yescharacteristic
203C2×#104—21no—
204C2×#104—1061no—
205#408—1061no—
206C2×#97C2×D5311yescharacteristic
207#362—1061no—
208#361—1061no—
209#359—1062no—
210C53#1302411yescharacteristic, complemented, Sylow
211C2×#41—21no—
212C2×#41—1061no—
213#135—21no—
214#135—21no—
215#135—1061no—
216#135—1061no—
217#135—1061no—
218#135—1061no—
219#135C4×D5312yes—
220#135C22⋊D5312yes—
221C2×Q16—531no—
222C2×Q16D21211yescharacteristic
223C2×SD16—21no—
224C2×SD16—1061no—
225C2×D8—531no—
226C2×D8D21211yescharacteristic
227#131—24no—
228#131—42no—
229C2×M16—21no—
230C2×M16—41no—
231C2×M16—1061no—
232C22×C8—41no—
233C22×C8C22⋊D5311yescharacteristic
234C22×C8C22×D5311yescharacteristic
235C2×C42C22⋊D5311yescharacteristic
236Q32—1062no—
237SD32—1062no—
238SD32—1062no—
239D16—1062no—
240M32—532no—
241M32—1061no—
242M32—1061no—
243M32—1061no—
244C2×C16—1061no—
245C2×C16—1061no—
246#108—21no—
247#108—1061no—
248#108—1061no—
249#108C22⋊D5312yes—
250#104—24no—
251#104—2122no—
252C22⋊C8—2122no—
253#97—21no—
254#97—21no—
255#41—22no—
256#41—22no—
257#41—41no—
258#41—41no—
259#41—2121no—
260#41—2121no—
261#41—2121no—
262#41—2121no—
263C2×Q8—21no—
264C2×Q8—1061no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—1061no—
268C2×D4—2121no—
269C22×C4—21no—
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—22no—
286M16—44no—
287M16—41no—
288M16—2121no—
289M16—2121no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—44no—
293C2×C8—41no—
294C2×C8—2121no—
295C2×C8#1171992212yes—
296C2×C8C2×C4×D5311yescharacteristic
297C2×C8C2×D21211yescharacteristic
298C2×C8#1171994511yescharacteristic
299C2×C8#1171994511yescharacteristic
300C2×C8C2×C22⋊D5311yescharacteristic
301C2×C8C2×C22⋊D5311yescharacteristic
302C42—21no—
303C42—21no—
304C16—1062no—
305C16—2121no—
306C23—41no—
307C23—2121no—
308C23#1313323711yescharacteristic
309Q8—22no—
310Q8—41no—
311Q8—2121no—
312Q8—2121no—
313D4—22no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—4241no—
318D4—4241no—
319D4—2121no—
320D4—2121no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—4241no—
327C2×C4—4241no—
328C2×C4—2121no—
329C2×C4C4×C22⋊D5311yescharacteristic
330C2×C4#1313318611yescharacteristic
331C2×C4#1313319111yescharacteristic
332C2×C4C2×#1171992211yescharacteristic
333C2×C4#1313324211yescharacteristic
334C2×C4#1313324311yescharacteristic
335C8—22no—
336C8—42no—
337C8—4241no—
338C8C2×#1171992111yescharacteristic
339C8C4×C22⋊D5311yescharacteristic
340C8#1313319111yescharacteristic
341C8#1313320011yescharacteristic
342C22—41no—
343C22—42no—
344C22—82no—
345C22—4242no—
346C22—4241no—
347C22C4×#1171992211yescharacteristic
348C22#1681076811yescharacteristic
349C22#1681083811yescharacteristic
350C4—42no—
351C4—81no—
352C4—4241no—
353C4#1681050211yescharacteristic
354C4#1681073311yescharacteristic
355C4#1681075711yescharacteristic
356C4#1681081811yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—82no—
361C2—8481no—
362C2#2027506311yescharacteristic
subgroupG/H
centerC4#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#444C2
socleC2×C53#1879
prime radical#20287849C2
cosocleC22×C8×C53C23
hypercenterC2×#444D53
hypocenterC53#13024
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. #20276464C2semidirect
3. #20276442C2semidirect
4. #20275608C2semidirect
5. #20272826C2semidirect
6. #20272799C2semidirect
7. C2×C53×#444C2semidirect
10. #16811063C4semidirect
12. #16811058C4semidirect
177. C2×#444D53semidirect
210. C53#13024semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202764642
#202764422
#202756082
#202728262
#202727992
C2×C53×#4442
#1302453