Finite Group Databasebeta

← #27357136

Subgroups — #27357136

Order 13568.

1338 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 1338 conjugacy classes of subgroups — too many to draw. Only its 203 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284711C212yesmaximal, complemented
2#20279612C211yesmaximal, characteristic, complemented
3#20273701C214yesmaximal, complemented
4#20273633C214yesmaximal, complemented
5#20273275C212yesmaximal, complemented
6#20273036C211yesmaximal, characteristic, complemented
7C53×#3496C211yesmaximal, characteristic, complemented
8C2×#13133188—21no—
9C2×C4×D212C2211yescharacteristic, complemented
10#16810934C2214yescomplemented
11#16810900—22no—
12#16810815—22no—
13#16810756C2214yescomplemented
14C4×#11719922—24no—
15#16810676C2212yescomplemented
16#16810548C2212yescomplemented
17#16810532C2212yescomplemented
18#16810502—22no—
19#16810489C2211yescharacteristic, complemented
20#16810480—21no—
21#16810357C2218yescomplemented
22#16810347C2214yescomplemented
23C2×H(5,𝔽2)×C53—21no—
24C53×#547—22no—
25C53×#546—24no—
26D42×C53—22no—
27C53×#537—21no—
28C53×#534C2214yes—
29C53×#532C2212yes—
30C2×#121×C53—22no—
31C2×C4×D4×C53C2211yescharacteristic
32C22×D212C2311yescharacteristic
33C22×C22⋊D53—22no—
34C22×C4×D53—21no—
35C2×#11719922—24no—
36#13133188—44no—
37C4×D212—22no—
38C4×D212—22no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2311yescharacteristic
41C2×C22⋊Dic53—22no—
42C2×#11719921—21no—
43C2×#11719921—22no—
44#13133150—24no—
45C4⋊#88322—22no—
46#13133138—24no—
47#13133137—24no—
48C22⋊#88322C2314yes—
49C4×#88322—22no—
50#13133124—21no—
51#13133124—22no—
52#13133124C2312yes—
53H(5,𝔽2)×C53—44no—
54H(5,𝔽2)×C53—44no—
55C2×#41×C53—24no—
56C2×#41×C53—22no—
57C22×D4×C53—24no—
58C22×D4×C53—24no—
59C22×D4×C53—22no—
60C22×D4×C53D411yescharacteristic
61C23×C4×C53D412yes—
62C4⋊Q8×C53D412yescomplemented
63#127×C53D412yescomplemented
64#124×C53—22no—
65#123×C53—28no—
66C22⋊Q8×C53—24no—
67#121×C53—24no—
68#121×C53—24no—
69#121×C53—24no—
70#121×C53—22no—
71#121×C53—24no—
72#121×C53C2314yes—
73#120×C53—28no—
74C4×D4×C53—22no—
75C4×D4×C53—22no—
76C4×D4×C53—22no—
77C4×D4×C53D414yescomplemented
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53D411yescharacteristic
80C2×C22⋊C4×C53—24no—
81C2×C22⋊C4×C53C2312yes—
82C2×C42×C53C2311yescharacteristic
83C24×C53—24no—
84C24×C53C2×D412yes—
85C23×D53—21no—
86#41×C53—416no—
87#41×C53—44no—
88#41×C53—44no—
89C2×Q8×C53—22no—
90C2×D4×C53—28no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—22no—
94C2×D4×C53—22no—
95C2×D4×C53—22no—
96C2×D4×C53—22no—
97C2×D4×C53—416no—
98C2×D4×C53—48no—
99C2×D4×C53—48no—
100C2×D4×C53—84no—
101C2×D4×C53D814yescomplemented
102C2×D4×C53C2×D414yes—
103C22×C4×C53—24no—
104C22×C4×C53—22no—
105C22×C4×C53—24no—
106C22×C4×C53—22no—
107C22×C4×C53—82no—
108C22×C4×C53C2×D412yes—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2411yescharacteristic
112C2×C22⋊D53—44no—
113C2×C22⋊D53—44no—
114C2×C22⋊D53—82no—
115C22×Dic53—22no—
116C22×Dic53—21no—
117C2×D212—22no—
118C2×D212—42no—
119C2×C4×D53—42no—
120C2×C4×D53—42no—
121C2×C4×D53—42no—
122C4⋊C4×C53—24no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53C2×D414yes—
126C22⋊C4×C53—28no—
127C22⋊C4×C53—24no—
128C22⋊C4×C53—24no—
129C22⋊C4×C53—24no—
130C42×C53C2×D412yes—
131C42×C53C2×D412yes—
132C22⋊Dic53—44no—
133#11719922—48no—
134#11719921—41no—
135#11719921—41no—
136#11719921—82no—
137C4⋊Dic53—22no—
138C4⋊Dic53—42no—
139C2×#88322—24no—
140C23×C53—22no—
141C23×C53—22no—
142C23×C53—24no—
143C23×C53—42no—
144C23×C53—42no—
145C23×C53—48no—
146C23×C53—84no—
147C23×C53—84no—
148C23×C53C2×D814yes—
149C23×C53C22×D411yescharacteristic
150C22×D53—22no—
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—41no—
155C22×D53—41no—
156C22×D53—41no—
157Q8×C53—44no—
158D4×C53—28no—
159D4×C53—44no—
160D4×C53—44no—
161D4×C53—416no—
162D4×C53—44no—
163D4×C53—44no—
164D4×C53—44no—
165D4×C53—84no—
166C2×C4×C53—24no—
167C2×C4×C53—22no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—44no—
171C2×C4×C53—48no—
172C2×C4×C53—48no—
173C2×C4×C53—42no—
174C2×C4×C53—82no—
175C2×C4×C53#12011yescharacteristic
176C2×C4×C53#12011yescharacteristic
177C2×C4×C53#12012yes—
178C2×C4×C53C2×D812yes—
179C2×C4×C53AGL(1,ℤ/8)12yes—
180C2×C4×C53C22×D412yes—
181C22⋊D53—88no—
182C2×Dic53—42no—
183C2×Dic53—42no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—42no—
187C2×Dic53—41no—
188D212—24no—
189D212—42no—
190C4×D53—84no—
191#88322—44no—
192#18210—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—24no—
196C22×C53—44no—
197C22×C53—42no—
198C22×C53—42no—
199C22×C53—44no—
200C22×C53—44no—
201C22×C53—44no—
202C22×C53—82no—
203C22×C53—168no—
204C22×C53C2×#12011yescharacteristic
205C22×C53C22×D811yescharacteristic
206C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
207C2×D53—42no—
208C2×D53—42no—
209C2×D53—42no—
210C2×D53—81no—
211C2×D53—81no—
212C2×D53—81no—
213C2×D53—81no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—22no—
217C4×C53—44no—
218C4×C53—42no—
219C4×C53#44714yes—
220Dic53—82no—
221Dic53—81no—
222#3496D5311yescharacteristic, complemented
223#3100—532no—
224#2535—531no—
225#1607—534no—
226#1595—534no—
227#1517—532no—
228#1460—531no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—44no—
232C2×C53—84no—
233C2×C53C2×#44711yescharacteristic
234C2×C53#301911yescharacteristic
235C2×C53#302511yescharacteristic
236D53—82no—
237C2×H(5,𝔽2)—21no—
238#547—22no—
239#546—24no—
240D42—22no—
241#537—21no—
242#534C2×D5314yes—
243#532C2×D5312yes—
244C2×#121—22no—
245C2×#121—1061no—
246C2×C4×D4—531no—
247C2×C4×D4C2×D5311yescharacteristic
248#480—534no—
249#474—1062no—
250#459—1062no—
251#447—534no—
252C4×D8—1064no—
253#431—532no—
254#397—532no—
255#392—532no—
256#386—1062no—
257#382—531no—
258#380—1061no—
259#327—538no—
260#323—534no—
261C53#1821011yescharacteristic, complemented, Sylow
262H(5,𝔽2)—44no—
263H(5,𝔽2)—44no—
264C2×#41—24no—
265C2×#41—22no—
266C22×D4—24no—
267C22×D4—24no—
268C22×D4—22no—
269C22×D4—531no—
270C22×D4—1062no—
271C22×D4C22⋊D5311yescharacteristic
272C23×C4—1061no—
273C23×C4C22⋊D5312yes—
274C2×D8—1064no—
275C4⋊Q8C22⋊D5312yescomplemented
276#127C22⋊D5312yescomplemented
277#124—22no—
278#123—28no—
279C22⋊Q8—24no—
280#121—24no—
281#121—24no—
282#121—24no—
283#121—22no—
284#121—24no—
285#121—2124no—
286#121C22×D5314yes—
287#120—28no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—1062no—
292C4×D4—1062no—
293C4×D4C22⋊D5314yescomplemented
294C2×C4⋊C4—22no—
295C2×C4⋊C4—531no—
296C2×C4⋊C4—1061no—
297C2×C4⋊C4C22⋊D5311yescharacteristic
298C2×C22⋊C4—24no—
299C2×C22⋊C4—1062no—
300C2×C22⋊C4—1062no—
301C2×C22⋊C4—1061no—
302C2×C22⋊C4C22×D5312yes—
303C2×C42C22×D5311yescharacteristic
304#107—1064no—
305C4⋊C8—1062no—
306#102—1064no—
307#102—1064no—
308C22⋊C8—534no—
309C4×C8—1062no—
310#95—532no—
311#95—1062no—
312#95—1061no—
313C24—24no—
314C24—1061no—
315C24C2×C22⋊D5312yes—
316#41—416no—
317#41—44no—
318#41—44no—
319C2×Q8—22no—
320C2×D4—28no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—416no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—84no—
331C2×D4—1062no—
332C2×D4—4242no—
333C2×D4—2122no—
334C2×D4—2124no—
335C2×D4—2124no—
336C2×D4#1171992214yescomplemented
337C2×D4C2×C22⋊D5314yes—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—82no—
343C22×C4—1062no—
344C22×C4—1061no—
345C22×C4—2122no—
346C22×C4—2122no—
347C22×C4—2122no—
348C22×C4C2×C22⋊D5312yes—
349C22×C4C2×C22⋊D5311yescharacteristic
350C22×C4C2×C22⋊D5311yescharacteristic
351C22×C4C23×D5311yescharacteristic
352D8—2128no—
353C2×C8—1064no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—1062no—
358C4⋊C4—2122no—
359C4⋊C4C2×C22⋊D5314yes—
360C22⋊C4—28no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—4242no—
365C22⋊C4—2124no—
366C22⋊C4—2121no—
367C22⋊C4—2121no—
368C42C2×C22⋊D5312yes—
369C42C2×C22⋊D5312yes—
370C23—22no—
371C23—22no—
372C23—24no—
373C23—42no—
374C23—42no—
375C23—48no—
376C23—84no—
377C23—84no—
378C23—1062no—
379C23—2121no—
380C23—2121no—
381C23—2121no—
382C23—2121no—
383C23—2121no—
384C23—2121no—
385C23C2×#1171992214yes—
386C23C22×C22⋊D5311yescharacteristic
387Q8—44no—
388D4—28no—
389D4—44no—
390D4—44no—
391D4—416no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—84no—
396D4—1064no—
397D4—4248no—
398D4—2122no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—4244no—
409C2×C4—2122no—
410C2×C4—2122no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2122no—
414C2×C4—2121no—
415C2×C4#1313318611yescharacteristic
416C2×C4#1313318611yescharacteristic
417C2×C4#1313318612yes—
418C2×C4C2×#1171992212yes—
419C2×C4#1313324312yes—
420C2×C4C22×C22⋊D5312yes—
421C8—2124no—
422C22—21no—
423C22—21no—
424C22—24no—
425C22—44no—
426C22—42no—
427C22—42no—
428C22—44no—
429C22—44no—
430C22—44no—
431C22—82no—
432C22—168no—
433C22—4241no—
434C22—4241no—
435C22—4241no—
436C22—4241no—
437C22—2122no—
438C22—2122no—
439C22—2122no—
440C22C2×#1313318611yescharacteristic
441C22C22×#1171992211yescharacteristic
442C22C2×#1313324311yescharacteristic
443C4—21no—
444C4—21no—
445C4—22no—
446C4—42no—
447C4—44no—
448C4—4242no—
449C4—4241no—
450C4#1681075714yes—
451C2—21no—
452C2—21no—
453C2—44no—
454C2—84no—
455C2—4242no—
456C2C2×#1681075711yescharacteristic
457C2#2028407211yescharacteristic
458C2#2028410711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#3496C2
socleC22×C53C2×#120
hypercenter#3496D53
hypocenterC53#18210
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #20284711C2semidirect
2. #20279612C2semidirect
3. #20273701C2semidirect
4. #20273633C2semidirect
5. #20273275C2semidirect
6. #20273036C2semidirect
7. C53×#3496C2semidirect
9. C2×C4×D212C22semidirect
10. #16810934C22semidirect
13. #16810756C22semidirect
15. #16810676C22semidirect
16. #16810548C22semidirect
17. #16810532C22semidirect
19. #16810489C22semidirect
21. #16810357C22semidirect
22. #16810347C22semidirect
62. C4⋊Q8×C53D4semidirect
63. #127×C53D4semidirect
77. C4×D4×C53D4semidirect
101. C2×D4×C53D8semidirect
222. #3496D53semidirect
261. C53#18210semidirect
275. C4⋊Q8C22⋊D53semidirect
276. #127C22⋊D53semidirect
293. C4×D4C22⋊D53semidirect
336. C2×D4#11719922semidirect

Maximal subgroups

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

subgroupindex
#202847112
#202796122
#202737012
#202736332
#202732752
#202730362
C53×#34962
#1821053