Finite Group Databasebeta

← #27302257

Subgroups — #27302257

Order 13568.

442 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#20283425C211yesmaximal, characteristic
2#20276470C211yesmaximal, characteristic, complemented
3C2×#16810408C211yesmaximal, characteristic, complemented
4C2×#16810407C211yesmaximal, characteristic, complemented
5C2×#16810406C211yesmaximal, characteristic, complemented
6C2×#16810398C211yesmaximal, characteristic, complemented
7C2×C53×#350C211yesmaximal, characteristic, complemented
8C2×#13133223C2211yescharacteristic
9#16811169C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C22×#11719908C2211yescharacteristic
12#16810699—22no—
13C8×C22⋊D53—22no—
14#16810646C411yescharacteristic
15#16810591C411yescharacteristic
16#16810590C411yescharacteristic
17C2×C8×Dic53C2211yescharacteristic
18#16810441—22no—
19#16810408—22no—
20#16810407—22no—
21#16810406—22no—
22#16810398—22no—
23C2×#131×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C22×C16×C53C2211yescharacteristic
26C53×#350—22no—
27C2×#11719945C2×C411yescharacteristic
28#13133223—21no—
29#13133223—21no—
30#13133223—21no—
31#13133223—21no—
32C2×#11719917C2×C411yescharacteristic
33C2×#11719912—21no—
34C22×#88322C2×C411yescharacteristic
35C2×C8×D53—21no—
36#13133215C811yescharacteristic
37#13133213C811yescharacteristic
38#13133206C811yescharacteristic
39#13133204C811yescharacteristic
40#13133189—22no—
41C4×C22⋊D53—22no—
42C2×C4×Dic53C2×C411yescharacteristic
43#13133157—21no—
44#13133157—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48C2×#11719908D412yescomplemented
49#13133146—22no—
50C22⋊#88322—22no—
51#13133129—22no—
52C8×Dic53D412yes—
53C8×Dic53D412yes—
54C2×#41×C53C2×C411yescharacteristic
55#131×C53—21no—
56#131×C53—21no—
57#131×C53—21no—
58#131×C53—21no—
59C2×M16×C53—21no—
60C2×M16×C53C2×C411yescharacteristic
61C22×C8×C53—21no—
62C22×C8×C53C2311yescharacteristic
63M32×C53—21no—
64M32×C53—21no—
65C2×C16×C53—21no—
66C2×C16×C53—21no—
67C2×C16×C53—21no—
68C2×C16×C53D412yescomplemented
69#41×C53—21no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73C2×Q8×C53C2×C811yescharacteristic
74C2×D4×C53—21no—
75C2×D4×C53C2×C811yescharacteristic
76C22×C4×C53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C2×C22⋊D53—21no—
79C2×C22⋊D53—22no—
80C22×Dic53M1612yes—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945—21no—
85C2×D212C2×C811yescharacteristic
86C2×C4×D53—21no—
87C2×Dic106C2×C811yescharacteristic
88M16×C53—21no—
89M16×C53—21no—
90M16×C53—41no—
91M16×C53—41no—
92C2×C8×C53—21no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53#4111yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C22⋊Dic53—22no—
102#11719921—22no—
103#11719919—22no—
104C4×Dic53C22⋊C412yes—
105C4×Dic53C2×C811yescharacteristic
106C4×Dic53C2×C811yescharacteristic
107#11719917—21no—
108#11719917—21no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322C22⋊C412yes—
113#11719912—41no—
114#11719912—41no—
115C8×D53—41no—
116C8×D53—41no—
117C16×C53—22no—
118C16×C53—41no—
119#11719908—22no—
120#11719908—41no—
121C23×C53—21no—
122C23×C53C2×M1611yescharacteristic
123C22×D53—21no—
124Q8×C53—21no—
125Q8×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—41no—
129D4×C53—41no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53#13111yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C22⋊D53—44no—
140C22⋊D53—41no—
141C22⋊D53—41no—
142C2×Dic53—21no—
143C2×Dic53—42no—
144C2×Dic53C22⋊C812yes—
145C2×Dic53C22⋊C812yes—
146D212—21no—
147D212—21no—
148C4×D53—41no—
149C4×D53—41no—
150Dic106—21no—
151Dic106—21no—
152C8×C53—41no—
153C8×C53#12111yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12411yescharacteristic
156C8×C53#12711yescharacteristic
157#88322—22no—
158#88322—41no—
159#13192—531nomaximal, Sylow
160C22×C53—22no—
161C22×C53—42no—
162C22×C53—41no—
163C22×C53C2×C22⋊C811yescharacteristic
164C22×C53#43611yescharacteristic
165C22×C53#43611yescharacteristic
166C2×D53—42no—
167C2×D53—41no—
168C4×C53—41no—
169C4×C53#39011yescharacteristic
170C4×C53#40811yescharacteristic
171C4×C53C8×D411yescharacteristic
172C4×C53C8×D411yescharacteristic
173Dic53—22no—
174Dic53—22no—
175Dic53—41no—
176#2931—531no—
177#2150—531no—
178C2×#350—531no—
179C2×#350—531no—
180C2×#350—531no—
181C2×#350D5311yescharacteristic, complemented
182C2×#346—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—81no—
186C2×C53#195411yescharacteristic
187C2×C53#211712yes—
188D53—81no—
189C2×#131—531no—
190C2×#131C2×D5311yescharacteristic
191#532—531no—
192C2×M32—531no—
193C2×M32C2×D5311yescharacteristic
194C22×C16—531no—
195C22×C16C2×D5311yescharacteristic
196#436—1062no—
197C8×D4—1062no—
198#423—531no—
199#408—531no—
200#408—531no—
201C2×C4×C8—531no—
202C4⋊C16—1062no—
203#350—22no—
204#350—1062no—
205#350—1062no—
206#350—1062no—
207#346—1062no—
208C53#1319211yescharacteristic, complemented, Sylow
209C2×#41—531no—
210C2×#41C4×D5311yescharacteristic
211#131—21no—
212#131—21no—
213#131—21no—
214#131—21no—
215#131—1061no—
216#131—1061no—
217#131—1061no—
218#131—1061no—
219C2×M16—21no—
220C2×M16—531no—
221C2×M16—1061no—
222C2×M16C4×D5311yescharacteristic
223C22×C8—21no—
224C22×C8—531no—
225C22×C8—1061no—
226C22×C8C22×D5311yescharacteristic
227C4⋊Q8—531no—
228#127—531no—
229#124—531no—
230#124—531no—
231#121—1062no—
232C4×D4—1062no—
233C2×C42—531no—
234M32—21no—
235M32—21no—
236M32—1061no—
237M32—1061no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—532no—
242C2×C16—1061no—
243C2×C16—1061no—
244C2×C16—1061no—
245C2×C16C22⋊D5312yescomplemented
246C4⋊C8—1062no—
247C22⋊C8—1062no—
248C22⋊C8—1062no—
249C4×C8—532no—
250C4×C8—532no—
251#41—21no—
252#41—21no—
253#41—21no—
254#41—21no—
255#41—1061no—
256#41—1061no—
257#41—1061no—
258#41—1061no—
259C2×Q8—531no—
260C2×Q8C8×D5311yescharacteristic
261C2×D4—21no—
262C2×D4—531no—
263C2×D4—1062no—
264C2×D4—1061no—
265C2×D4C8×D5311yescharacteristic
266C22×C4—21no—
267C22×C4—532no—
268C22×C4—1061no—
269C22×C4C2×C4×D5311yescharacteristic
270M16—21no—
271M16—21no—
272M16—41no—
273M16—41no—
274M16—1061no—
275M16—1061no—
276M16—2121no—
277M16—2121no—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—532no—
282C2×C8—1061no—
283C2×C8—1061no—
284C2×C8—1061no—
285C2×C8—2121no—
286C2×C8—2121no—
287C2×C8#1171994511yescharacteristic
288C2×C8D4×D5311yescharacteristic
289C2×C8D4×D5311yescharacteristic
290C2×C8D4×D5311yescharacteristic
291C2×C8#1171994711yescharacteristic
292C2×C8C2×C22⋊D5311yescharacteristic
293C4⋊C4—1062no—
294C22⋊C4—1062no—
295C22⋊C4—1062no—
296C42—532no—
297C42—531no—
298C42—531no—
299C16—22no—
300C16—41no—
301C16—1062no—
302C16—2121no—
303C23—21no—
304C23—1061no—
305C23M16×D5311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—1061no—
309Q8—1061no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—41no—
314D4—1061no—
315D4—1061no—
316D4—2124no—
317D4—2121no—
318D4—2121no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—532no—
323C2×C4—532no—
324C2×C4—1061no—
325C2×C4—2122no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4C22⋊C4×D5311yescharacteristic
329C2×C4#1313317711yescharacteristic
330C2×C4C4×C22⋊D5311yescharacteristic
331C2×C4C2×C8×D5311yescharacteristic
332C2×C4#1313322311yescharacteristic
333C2×C4#1313322511yescharacteristic
334C8—41no—
335C8—1062no—
336C8—2121no—
337C8#1313318911yescharacteristic
338C8#1313319011yescharacteristic
339C8#1313320311yescharacteristic
340C8#1313321311yescharacteristic
341C22—22no—
342C22—41no—
343C22—42no—
344C22—2121no—
345C22—2122no—
346C22C22⋊C8×D5311yescharacteristic
347C22#1681069811yescharacteristic
348C22#1681069911yescharacteristic
349C4—41no—
350C4—1062no—
351C4—1062no—
352C4—2121no—
353C4#1681052311yescharacteristic
354C4#1681058811yescharacteristic
355C4#1681068611yescharacteristic
356C4C8×C22⋊D5311yescharacteristic
357C2—21no—
358C2—21no—
359C2—81no—
360C2—4241no—
361C2#2027550711yescharacteristic
362C2#2027634112yes—
subgroupG/H
centerC2×C8D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#350C2
socleC22×C53C2×C22⋊C8
prime radical#16811169C4
cosocleC22×C8×C53C23
hypercenterC2×#350D53
hypocenterC53#13192
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. #20276470C2semidirect
3. C2×#16810408C2semidirect
4. C2×#16810407C2semidirect
5. C2×#16810406C2semidirect
6. C2×#16810398C2semidirect
7. C2×C53×#350C2semidirect
48. C2×#11719908D4semidirect
68. C2×C16×C53D4semidirect
181. C2×#350D53semidirect
208. C53#13192semidirect
245. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202834252
#202764702
C2×#168104082
C2×#168104072
C2×#168104062
C2×#168103982
C2×C53×#3502
#1319253