Finite Group Databasebeta

← #27301012

Subgroups — #27301012

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287817C211yesmaximal, characteristic, complemented
2#20283431C211yesmaximal, characteristic
3#20276343C211yesmaximal, characteristic, complemented
4#20276335C211yesmaximal, characteristic, complemented
5#20272794C211yesmaximal, characteristic, complemented
6#20272794C211yesmaximal, characteristic, complemented
7C53×#2117C211yesmaximal, characteristic, complemented
8C2×#13133223C2211yescharacteristic
9#16811169C411yescharacteristic
10#16811041—22no—
11#16811039—22no—
12C2×#13133157C2211yescharacteristic
13C2×#13133157C2211yescharacteristic
14M32×D53—22no—
15#16810694—22no—
16#16810692—22no—
17#16810646C411yescharacteristic
18#16810591C411yescharacteristic
19#16810590C411yescharacteristic
20C2×#13133127C2211yescharacteristic
21#16810407—22no—
22#16810404—22no—
23C2×#131×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C53×#350—22no—
27C2×#11719945C2×C411yescharacteristic
28#13133223—21no—
29#13133223—21no—
30#13133223—21no—
31#13133223—21no—
32C2×#11719917C2×C411yescharacteristic
33C2×#11719912D412yes—
34C22×#88322C2×C411yescharacteristic
35C2×C8×D53D412yes—
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—
45#13133157—21no—
46#13133157—21no—
47#13133156—24no—
48C2×#11719908—21no—
49C2×#11719908D412yescomplemented
50C16×D53—24no—
51#13133146—22no—
52C22⋊#88322—22no—
53#13133129—22no—
54#13133127—21no—
55#13133127—21no—
56C2×#41×C53C2×C411yescharacteristic
57#131×C53—21no—
58#131×C53—21no—
59#131×C53—21no—
60#131×C53—21no—
61C2×M16×C53—21no—
62C2×M16×C53C2×C411yescharacteristic
63C22×C8×C53—21no—
64C22×C8×C53C2311yescharacteristic
65M32×C53—21no—
66M32×C53—21no—
67M32×C53—21no—
68M32×C53—21no—
69C2×C16×C53—21no—
70C2×C16×C53D412yescomplemented
71#41×C53—21no—
72#41×C53—21no—
73#41×C53—21no—
74#41×C53—21no—
75C2×Q8×C53C2×C811yescharacteristic
76C2×D4×C53—21no—
77C2×D4×C53C2×C811yescharacteristic
78C22×C4×C53—21no—
79C22×C4×C53C22×C411yescharacteristic
80C2×C22⋊D53—22no—
81C2×C22⋊D53M1612yes—
82C22×Dic53—21no—
83#11719945—21no—
84#11719945—21no—
85#11719945—21no—
86#11719945—21no—
87C2×D212C2×C811yescharacteristic
88C2×C4×D53C22⋊C412yes—
89C2×Dic106C2×C811yescharacteristic
90M16×C53—21no—
91M16×C53—21no—
92M16×C53—41no—
93M16×C53—41no—
94C2×C8×C53—21no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53C2×D411yescharacteristic
101C2×C8×C53#4111yescharacteristic
102C2×C8×C53#4111yescharacteristic
103C22⋊Dic53—22no—
104#11719921—22no—
105#11719919—22no—
106C4×Dic53—21no—
107C4×Dic53C2×C811yescharacteristic
108C4×Dic53C2×C811yescharacteristic
109#11719917—21no—
110#11719917—21no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322—21no—
114C2×#88322C22⋊C412yes—
115#11719912—22no—
116#11719912—22no—
117C8×D53—22no—
118C8×D53—22no—
119C16×C53—22no—
120C16×C53—41no—
121#11719908—22no—
122#11719908—41no—
123C23×C53—21no—
124C23×C53C2×M1611yescharacteristic
125C22×D53C22⋊C812yes—
126Q8×C53—21no—
127Q8×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—41no—
131D4×C53—41no—
132C2×C4×C53—21no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C22×C811yescharacteristic
139C2×C4×C53#13111yescharacteristic
140C2×C4×C53#13111yescharacteristic
141C22⋊D53—22no—
142C22⋊D53—22no—
143C22⋊D53—82no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—81no—
147C2×Dic53C22⋊C812yes—
148D212—21no—
149D212—21no—
150C4×D53—22no—
151C4×D53—22no—
152Dic106—21no—
153Dic106—21no—
154C8×C53—41no—
155C8×C53#12011yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12311yescharacteristic
159#88322—22no—
160#88322—41no—
161#12988—531nomaximal, Sylow
162C22×C53—41no—
163C22×C53—42no—
164C22×C53—41no—
165C22×C53C2×C22⋊C811yescharacteristic
166C22×C53#43511yescharacteristic
167C22×C53#43511yescharacteristic
168C2×D53—22no—
169C2×D53—42no—
170C4×C53—41no—
171C4×C53#38611yescharacteristic
172C4×C53#40811yescharacteristic
173C4×C53C8×D411yescharacteristic
174C4×C53C8×D411yescharacteristic
175Dic53—22no—
176Dic53—41no—
177Dic53—41no—
178#3407—531no—
179#2932—531no—
180#2117—531no—
181#2117D5311yescharacteristic, complemented
182#2115—531no—
183#1375—531no—
184#1375—531no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—81no—
189C2×C53#186911yescharacteristic
190D53—42no—
191C2×#131—531no—
192C2×#131C2×D5311yescharacteristic
193#532—531no—
194#504—1062no—
195#504—1062no—
196C2×M32—531no—
197C2×M32—531no—
198C2×M32—1062no—
199C2×M32C2×D5311yescharacteristic
200C2×M32C2×D5311yescharacteristic
201#435—1062no—
202#435—1062no—
203#423—531no—
204#408—531no—
205#408—531no—
206C2×#97—531no—
207#350—22no—
208#350—1062no—
209#349—1062no—
210C53#1298811yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41C4×D5311yescharacteristic
213#131—21no—
214#131—21no—
215#131—21no—
216#131—21no—
217#131—1061no—
218#131—1061no—
219#131—1061no—
220#131—1061no—
221C2×M16—21no—
222C2×M16—531no—
223C2×M16—532no—
224C2×M16C4×D5311yescharacteristic
225C22×C8—21no—
226C22×C8—532no—
227C22×C8—531no—
228C22×C8C22×D5311yescharacteristic
229C4⋊Q8—531no—
230#127—531no—
231#124—531no—
232#124—531no—
233#121—1062no—
234C4×D4—1062no—
235C2×C42—531no—
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32—1064no—
241M32—1061no—
242M32—1061no—
243M32—1061no—
244M32—1061no—
245C2×C16—21no—
246C2×C16—532no—
247C2×C16—1064no—
248C2×C16—1061no—
249C2×C16C22⋊D5312yescomplemented
250C4⋊C8—1062no—
251C22⋊C8—1062no—
252C22⋊C8—1062no—
253#97—1061no—
254#97—1061no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—1061no—
260#41—1061no—
261#41—1061no—
262#41—1061no—
263C2×Q8—531no—
264C2×Q8C8×D5311yescharacteristic
265C2×D4—21no—
266C2×D4—532no—
267C2×D4—531no—
268C2×D4—1062no—
269C2×D4C8×D5311yescharacteristic
270C22×C4—21no—
271C22×C4—532no—
272C22×C4—1061no—
273C22×C4C2×C4×D5311yescharacteristic
274M16—21no—
275M16—21no—
276M16—41no—
277M16—41no—
278M16—1061no—
279M16—1061no—
280M16—1062no—
281M16—1062no—
282C2×C8—21no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—532no—
286C2×C8—1062no—
287C2×C8—1062no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8#1171994511yescharacteristic
292C2×C8D4×D5311yescharacteristic
293C2×C8D4×D5311yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8#1171994711yescharacteristic
296C2×C8C2×C22⋊D5311yescharacteristic
297C4⋊C4—1062no—
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C42—531no—
301C42—531no—
302C42—1061no—
303C16—22no—
304C16—41no—
305C16—1062no—
306C16—2121no—
307C23—21no—
308C23—532no—
309C23M16×D5311yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—1061no—
313Q8—1061no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—41no—
318D4—1062no—
319D4—1062no—
320D4—1061no—
321D4—1061no—
322D4—4242no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—532no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—4241no—
332C2×C4C22⋊C4×D5311yescharacteristic
333C2×C4#1313317711yescharacteristic
334C2×C4C4×C22⋊D5311yescharacteristic
335C2×C4C2×C8×D5311yescharacteristic
336C2×C4#1313322311yescharacteristic
337C2×C4#1313322511yescharacteristic
338C8—41no—
339C8—1062no—
340C8—2121no—
341C8#1313318511yescharacteristic
342C8#1313318811yescharacteristic
343C8#1313319011yescharacteristic
344C8#1313319911yescharacteristic
345C22—41no—
346C22—42no—
347C22—41no—
348C22—1062no—
349C22—2122no—
350C22C22⋊C8×D5311yescharacteristic
351C22#1681069211yescharacteristic
352C22#1681069711yescharacteristic
353C4—41no—
354C4—1062no—
355C4—2121no—
356C4—2121no—
357C4#1681049911yescharacteristic
358C4#1681058811yescharacteristic
359C4#1681068611yescharacteristic
360C4C8×C22⋊D5311yescharacteristic
361C2—21no—
362C2—21no—
363C2—21no—
364C2—81no—
365C2—2122no—
366C2#2027500411yescharacteristic
subgroupG/H
centerC4#16810499
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#2117C2
socleC2×C53#1869
prime radical#16811169C4
cosocleC22×C8×C53C23
hypercenter#2117D53
hypocenterC53#12988
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
1. #20287817C2semidirect
3. #20276343C2semidirect
4. #20276335C2semidirect
5. #20272794C2semidirect
6. #20272794C2semidirect
7. C53×#2117C2semidirect
49. C2×#11719908D4semidirect
70. C2×C16×C53D4semidirect
181. #2117D53semidirect
210. C53#12988semidirect
249. C2×C16C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202878172
#202834312
#202763432
#202763352
#202727942
C53×#21172
#1298853