Finite Group Databasebeta

← #27299267

Subgroups — #27299267

Order 13568.

628 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 123 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#20277437C211yesmaximal, characteristic
2C2×#16810415C212yesmaximal
3C4×#13133132C211yesmaximal, characteristic, complemented
4#20272839C212yesmaximal, complemented
5C4×#99×C53C211yesmaximal, characteristic, complemented
6C2×#13133213C2211yescharacteristic
7C2×C4×D212C411yescharacteristic, complemented
8C2×#13133132C2212yes—
9#16810526C2211yescharacteristic
10#16810524C411yescharacteristic, complemented
11#16810502—21no—
12#16810502—21no—
13#16810499—21no—
14#16810499—21no—
15#16810488C411yescharacteristic, complemented
16C4×C22⋊Dic53—21no—
17#16810469C411yescharacteristic, complemented
18#16810415C418yescomplemented
19#16810390—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#99×C53C2212yes—
22C4×C22⋊C4×C53—21no—
23C53×#342—21no—
24C22×D212C2×C411yescharacteristic
25C22×C22⋊D53—21no—
26C22×C22⋊D53—21no—
27C22×C4×D53—21no—
28#13133213—22no—
29#13133213C2×C414yes—
30C4×D212—41no—
31C4×D212—41no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×C4×Dic53—41no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42#13133132—22no—
43#13133132C2×C414yescomplemented
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—41no—
49C22×D4×C53C2311yescharacteristic
50C23×C4×C53D411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C4×D4×C53—41no—
53C4×D4×C53—41no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53D411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53—41no—
59C2×C42×C53C2×C411yescharacteristic
60#99×C53—22no—
61#99×C53C2×C414yescomplemented
62#95×C53—41no—
63C24×C53C2×D411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C23×D53—21no—
66C2×D4×C53—41no—
67C2×D4×C53—41no—
68C2×D4×C53C22×C412yes—
69C2×D4×C53#4112yes—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—81no—
76C22×C4×C53—81no—
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22×C411yescharacteristic
82C2×C22⋊D53—24no—
83C2×C22⋊D53—24no—
84C2×C22⋊D53—81no—
85C2×C22⋊D53—81no—
86C2×C22⋊D53—81no—
87C2×C22⋊D53—81no—
88C22×Dic53—21no—
89C22×Dic53—21no—
90C22×Dic53—21no—
91C22×Dic53—42no—
92C2×D212—21no—
93C2×D212—41no—
94C2×D212—41no—
95C2×D212C4212yescomplemented
96C2×C4×D53—42no—
97C2×C4×D53—81no—
98C2×C4×D53—81no—
99C4⋊C4×C53—41no—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—42no—
104C42×C53—21no—
105C42×C53—21no—
106C42×C53—82no—
107C22⋊Dic53—24no—
108C22⋊Dic53—42no—
109C22⋊Dic53—81no—
110C22⋊Dic53—81no—
111#11719921—41no—
112#11719921—41no—
113#11719921—81no—
114#11719921—81no—
115C4⋊Dic53—41no—
116C4×Dic53—21no—
117C4×Dic53—41no—
118C4×Dic53—82no—
119C4×Dic53C4212yescomplemented
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—21no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53C4×D412yes—
130C23×C53C4×D412yes—
131C22×D53—22no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138D4×C53—42no—
139D4×C53—42no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—44no—
152C2×C4×C53—81no—
153C2×C4×C53—81no—
154C2×C4×C53—82no—
155C2×C4×C53—82no—
156C2×C4×C53#9911yescharacteristic
157C2×C4×C53#9911yescharacteristic
158C2×C4×C53C2×C4211yescharacteristic
159C2×C4×C53#11711yescharacteristic
160C22⋊D53—84no—
161C22⋊D53—84no—
162C2×Dic53—22no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—44no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—82no—
173C2×Dic53—82no—
174D212—42no—
175D212—42no—
176C4×D53—84no—
177#12628—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—22no—
181C22×C53—22no—
182C22×C53—42no—
183C22×C53—42no—
184C22×C53—41no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—81no—
189C22×C53—81no—
190C22×C53C4×C22⋊C411yescharacteristic
191C22×C53C2×#9911yescharacteristic
192C22×C53#41011yescharacteristic
193C2×D53—42no—
194C2×D53—42no—
195C2×D53—42no—
196C2×D53—81no—
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—84no—
208Dic53—42no—
209Dic53—81no—
210Dic53—84no—
211Dic53—81no—
212#2328—531no—
213C2×#353—532no—
214C4×#99—531no—
215C4×#99D5311yescharacteristic, complemented
216#1393—532no—
217C2×C53—21no—
218C2×C53—21no—
219C2×C53—42no—
220C2×C53—42no—
221C2×C53C4×#9911yescharacteristic
222C2×C53#212711yescharacteristic
223C2×C53#212811yescharacteristic
224D53—82no—
225C2×#127—531no—
226C2×C4×D4—531no—
227C2×C4×D4C2×D5311yescharacteristic
228C2×#99—532no—
229C2×#99C2×D5312yes—
230#390—531no—
231#390—531no—
232#386—1061no—
233#386—1061no—
234#386—1061no—
235#386—1061no—
236#382—531no—
237C4×C22⋊C4—21no—
238C4×C22⋊C4—1061no—
239#376—531no—
240#353—538no—
241#342—21no—
242#342—1061no—
243C53#1262811yescharacteristic, complemented, Sylow
244C22×D4—531no—
245C22×D4—1061no—
246C22×D4—1061no—
247C22×D4C22×D5311yescharacteristic
248C23×C4—1061no—
249C23×C4D21211yescharacteristic
250C23×C4C22⋊D5311yescharacteristic
251#127—534no—
252#127—1062no—
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—2121no—
256C4×D4—2121no—
257C2×C4⋊C4—531no—
258C2×C4⋊C4C4×D5311yescharacteristic
259C2×C22⋊C4—22no—
260C2×C22⋊C4—1062no—
261C2×C22⋊C4—1061no—
262C2×C22⋊C4—1061no—
263C2×C22⋊C4—1061no—
264C2×C22⋊C4—1061no—
265C2×C22⋊C4—1061no—
266C2×C22⋊C4D21211yescharacteristic
267C2×C22⋊C4C22⋊D5311yescharacteristic
268C2×C42—41no—
269C2×C42—531no—
270C2×C42—531no—
271C2×C42—2121no—
272C2×C42C4×D5311yescharacteristic
273#99—22no—
274#99—534no—
275#99—1062no—
276#99C4×D5314yescomplemented
277#95—41no—
278#95—1061no—
279#95—1061no—
280#95—1061no—
281#95—1061no—
282#95—2121no—
283C24—1061no—
284C24C2×D21211yescharacteristic
285C24C2×C22⋊D5311yescharacteristic
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—532no—
289C2×D4—1064no—
290C2×D4—1064no—
291C2×D4—1061no—
292C2×D4—4241no—
293C2×D4—4241no—
294C2×D4—4241no—
295C2×D4—4241no—
296C2×D4—2121no—
297C2×D4—2121no—
298C2×D4C2×C4×D5312yes—
299C2×D4#1171994512yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—42no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4—1061no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—4241no—
311C22×C4—4241no—
312C22×C4—2122no—
313C22×C4—2122no—
314C22×C4#1171992111yescharacteristic
315C22×C4#1171992111yescharacteristic
316C22×C4#1171992111yescharacteristic
317C22×C4#1171992111yescharacteristic
318C22×C4C2×C4×D5311yescharacteristic
319C4⋊C4—41no—
320C4⋊C4—2121no—
321C22⋊C4—24no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—42no—
325C22⋊C4—1064no—
326C22⋊C4—4241no—
327C22⋊C4—4241no—
328C22⋊C4—4241no—
329C22⋊C4—4241no—
330C22⋊C4—2122no—
331C22⋊C4—2121no—
332C22⋊C4—2121no—
333C42—21no—
334C42—21no—
335C42—82no—
336C42—532no—
337C42—1061no—
338C42—4242no—
339C42—2121no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—1062no—
349C23—2121no—
350C23—2121no—
351C23—2121no—
352C23—2121no—
353C23—2121no—
354C23—2121no—
355C23C2×#1171992111yescharacteristic
356C23C4×C22⋊D5312yes—
357C23C4×D21212yes—
358D4—42no—
359D4—42no—
360D4—4244no—
361D4—4244no—
362D4—2122no—
363D4—2122no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—1062no—
381C2×C4—4244no—
382C2×C4—4242no—
383C2×C4—4242no—
384C2×C4—2124no—
385C2×C4—2121no—
386C2×C4—2121no—
387C2×C4—2121no—
388C2×C4—2121no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4#1313313111yescharacteristic
394C2×C4#1313313111yescharacteristic
395C2×C4C42×D5311yescharacteristic
396C2×C4#1313317411yescharacteristic
397C22—21no—
398C22—21no—
399C22—22no—
400C22—22no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—81no—
408C22—81no—
409C22—4241no—
410C22—4241no—
411C22—4241no—
412C22—4241no—
413C22—2122no—
414C22—2122no—
415C22—2122no—
416C22C4×#1171992111yescharacteristic
417C22C2×#1313313111yescharacteristic
418C22#1681059611yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—84no—
427C4—4241no—
428C4—4244no—
429C4—4241no—
430C4—2122no—
431C2—21no—
432C2—21no—
433C2—42no—
434C2—42no—
435C2—4242no—
436C2C4×#1313313111yescharacteristic
437C2#2027638211yescharacteristic
438C2#2027638611yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radicalC2×#13133213C22
cosocleC22×D4×C53C23
hypercenterC4×#99D53
hypocenterC53#12628
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. C4×#13133132C2semidirect
4. #20272839C2semidirect
5. C4×#99×C53C2semidirect
7. C2×C4×D212C4semidirect
10. #16810524C4semidirect
15. #16810488C4semidirect
17. #16810469C4semidirect
18. #16810415C4semidirect
43. #13133132C2×C4semidirect
61. #99×C53C2×C4semidirect
95. C2×D212C42semidirect
119. C4×Dic53C42semidirect
215. C4×#99D53semidirect
243. C53#12628semidirect
276. #99C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202774372
C2×#168104152
C4×#131331322
#202728392
C4×#99×C532
#1262853