Finite Group Databasebeta

← #28480210

Subgroups — #28480210

Order 15104.

726 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 135 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#21423862C211yesmaximal, characteristic, complemented
2#21422716C212yesmaximal, complemented
3#21422556C212yesmaximal, complemented
4#21422430C211yesmaximal, characteristic, complemented
5C59×#3491C211yesmaximal, characteristic
6#16837459C2212yescomplemented
7C2×C22⋊(C59⋊C8)—21no—
8C2×#14273638C2211yescharacteristic, complemented
9C2×C4×C59⋊C8C2211yescharacteristic, complemented
10#16837157—21no—
11#16837144—22no—
12#16837142—22no—
13Q8⋊(C59⋊C8)—22no—
14D4⋊(C59⋊C8)—22no—
15#16837128—21no—
16#16837126—21no—
17C59×#552—22no—
18C59×#552—22no—
19C59×#551—22no—
20C59×#551—22no—
21C59×#543C411yescharacteristic
22C59×#539—21no—
23C59×#539C412yes—
24C59×#534C411yescharacteristic
25C59×#529—21no—
26C59×#529C2212yes—
27C2×#123×C59—21no—
28C59×#518—21no—
29C2×#117×C59C2211yescharacteristic
30C22×C59⋊C8—21no—
31C22×C59⋊C8—21no—
32C4⋊(C59⋊C8)—22no—
33C22⋊(C59⋊C8)—24no—
34C22⋊(C59⋊C8)—22no—
35#14273638—22no—
36#14273638—22no—
37C4×C59⋊C8—22no—
38C22×D4×C59—21no—
39C23×C4×C59D411yescharacteristic
40#126×C59—22no—
41#126×C59—22no—
42#126×C59—22no—
43#126×C59—22no—
44#125×C59—21no—
45#125×C59—22no—
46#125×C59—22no—
47#125×C59D412yes—
48#124×C59—21no—
49#124×C59D412yes—
50#123×C59—22no—
51#123×C59—22no—
52#123×C59—22no—
53#123×C59—22no—
54#123×C59—22no—
55#123×C59—22no—
56#123×C59—22no—
57C22⋊Q8×C59—22no—
58C22⋊Q8×C59—22no—
59C22⋊Q8×C59—22no—
60C22⋊Q8×C59—21no—
61C22⋊Q8×C59—22no—
62C22⋊Q8×C59C2×C412yes—
63#121×C59—21no—
64#121×C59C2×C412yes—
65#120×C59—22no—
66C4×Q8×C59—21no—
67C4×Q8×C59D412yes—
68C4×D4×C59—22no—
69C4×D4×C59—22no—
70C4×D4×C59—21no—
71C4×D4×C59—22no—
72C4×D4×C59—22no—
73C4×D4×C59D412yes—
74#117×C59—22no—
75#117×C59—22no—
76#117×C59—22no—
77#117×C59—22no—
78#117×C59—22no—
79C2×C4⋊C4×C59—21no—
80C2×C4⋊C4×C59—21no—
81C2×C4⋊C4×C59C2×C411yescharacteristic
82C2×C4⋊C4×C59D411yescharacteristic
83C2×C22⋊C4×C59—22no—
84C2×C22⋊C4×C59—21no—
85C2×C22⋊C4×C59—21no—
86C2×C22⋊C4×C59—21no—
87C2×C22⋊C4×C59C2×C411yescharacteristic
88C2×C22⋊C4×C59D411yescharacteristic
89C2×C42×C59D411yescharacteristic
90C2×C42×C59C2311yescharacteristic
91C24×C59—21no—
92C24×C59C22⋊C411yescharacteristic
93C2×Q8×C59—21no—
94C2×Q8×C59C22⋊C412yes—
95C2×D4×C59—22no—
96C2×D4×C59—22no—
97C2×D4×C59—21no—
98C2×D4×C59—42no—
99C2×D4×C59—42no—
100C2×D4×C59C22⋊C412yes—
101C22×C4×C59—21no—
102C22×C4×C59—22no—
103C22×C4×C59—21no—
104C22×C4×C59—21no—
105C22×C4×C59—22no—
106C22×C4×C59—21no—
107C22×C4×C59—21no—
108C22×C4×C59—22no—
109C22×C4×C59—41no—
110C22×C4×C59C22⋊C411yescharacteristic
111C22×C4×C59C22⋊C411yescharacteristic
112C22×C4×C59C22⋊C411yescharacteristic
113C22×C4×C59C22×C411yescharacteristic
114C22×C4×C59C2×D411yescharacteristic
115C22×C4×C59C2×D411yescharacteristic
116C4⋊C4×C59—22no—
117C4⋊C4×C59—22no—
118C4⋊C4×C59—22no—
119C4⋊C4×C59—22no—
120C4⋊C4×C59—21no—
121C4⋊C4×C59—21no—
122C4⋊C4×C59—22no—
123C4⋊C4×C59—22no—
124C4⋊C4×C59—22no—
125C4⋊C4×C59—22no—
126C4⋊C4×C59—22no—
127C4⋊C4×C59—22no—
128C4⋊C4×C59C22⋊C412yes—
129C4⋊C4×C59C22⋊C412yes—
130C22⋊C4×C59—22no—
131C22⋊C4×C59—22no—
132C22⋊C4×C59—22no—
133C22⋊C4×C59—22no—
134C22⋊C4×C59—22no—
135C22⋊C4×C59—22no—
136C22⋊C4×C59—22no—
137C22⋊C4×C59—22no—
138C22⋊C4×C59—22no—
139C22⋊C4×C59—22no—
140C22⋊C4×C59—22no—
141C22⋊C4×C59—22no—
142C42×C59—22no—
143C42×C59—21no—
144C42×C59—21no—
145C42×C59—22no—
146C42×C59C2×D412yes—
147C42×C59C2×D412yes—
148C2×C59⋊C8—22no—
149C2×C59⋊C8—22no—
150C2×C59⋊C8—41no—
151C2×C59⋊C8—41no—
152C2×C59⋊C8—41no—
153C2×C59⋊C8—41no—
154C2×C59⋊C8—42no—
155C23×C59—21no—
156C23×C59—21no—
157C23×C59—22no—
158C23×C59—41no—
159C23×C59—41no—
160C23×C59—42no—
161C23×C59—81no—
162C23×C59—81no—
163C23×C59#10412yes—
164C23×C59C2×C22⋊C411yescharacteristic
165Q8×C59—42no—
166Q8×C59—81no—
167D4×C59—42no—
168D4×C59—44no—
169D4×C59—42no—
170D4×C59—42no—
171D4×C59—81no—
172C2×C4×C59—21no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—22no—
177C2×C4×C59—22no—
178C2×C4×C59—22no—
179C2×C4×C59—21no—
180C2×C4×C59—21no—
181C2×C4×C59—21no—
182C2×C4×C59—41no—
183C2×C4×C59—41no—
184C2×C4×C59—42no—
185C2×C4×C59—42no—
186C2×C4×C59—42no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—82no—
190C2×C4×C59—81no—
191C2×C4×C59—82no—
192C2×C4×C59—81no—
193C2×C4×C59C2×C22⋊C412yes—
194C2×C4×C59#12012yes—
195C2×C4×C59#12011yescharacteristic
196C2×C4×C59#12011yescharacteristic
197C59⋊C8—42no—
198C59⋊C8—42no—
199C59⋊C8—82no—
200#8348—591nomaximal, Sylow
201C22×C59—21no—
202C22×C59—21no—
203C22×C59—22no—
204C22×C59—42no—
205C22×C59—41no—
206C22×C59—41no—
207C22×C59—42no—
208C22×C59—42no—
209C22×C59—42no—
210C22×C59—81no—
211C22×C59—162no—
212C22×C59#37911yescharacteristic
213C22×C59C2×#10411yescharacteristic
214C22×C59#42111yescharacteristic
215C4×C59—22no—
216C4×C59—21no—
217C4×C59—21no—
218C4×C59—22no—
219C4×C59—41no—
220C4×C59—42no—
221C4×C59—41no—
222C4×C59—42no—
223C4×C59—41no—
224C4×C59—42no—
225#3491D5911yescharacteristic
226#1750—591no—
227#1532—592no—
228#1494—592no—
229#1464—591no—
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—42no—
233C2×C59—82no—
234C2×C59#178911yescharacteristic
235C2×C59#179111yescharacteristic
236C2×C59#179411yescharacteristic
237#552—22no—
238#552—22no—
239#551—22no—
240#551—22no—
241#543Dic5911yescharacteristic
242#539—21no—
243#539Dic5912yes—
244#534Dic5911yescharacteristic
245#529—21no—
246#529C2×D5912yes—
247C2×#123—21no—
248#518—21no—
249C2×#117C2×D5911yescharacteristic
250#432—592no—
251C2×C22⋊C8—1181no—
252C2×#97—591no—
253C2×C4×C8—591no—
254#336—1181no—
255#330—1182no—
256#329—1182no—
257Q8⋊C8—1182no—
258D4⋊C8—1182no—
259#324—1181no—
260#323—1181no—
261C59#834811yescharacteristic, complemented, Sylow
262C22×D4—21no—
263C23×C4C22⋊D5911yescharacteristic
264C22×C8—1181no—
265C22×C8—1181no—
266#126—22no—
267#126—22no—
268#126—22no—
269#126—22no—
270#125—21no—
271#125—22no—
272#125—22no—
273#125C22⋊D5912yes—
274#124—21no—
275#124C22⋊D5912yes—
276#123—22no—
277#123—22no—
278#123—22no—
279#123—22no—
280#123—22no—
281#123—22no—
282#123—22no—
283C22⋊Q8—22no—
284C22⋊Q8—22no—
285C22⋊Q8—22no—
286C22⋊Q8—21no—
287C22⋊Q8—22no—
288C22⋊Q8C2×Dic5912yes—
289#121—21no—
290#121C2×Dic5912yes—
291#120—22no—
292C4×Q8—21no—
293C4×Q8C22⋊D5912yes—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—21no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4C22⋊D5912yes—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C2×Dic5911yescharacteristic
308C2×C4⋊C4C22⋊D5911yescharacteristic
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4C2×Dic5911yescharacteristic
314C2×C22⋊C4C22⋊D5911yescharacteristic
315C2×C42C22⋊D5911yescharacteristic
316C2×C42C22×D5911yescharacteristic
317C4⋊C8—1182no—
318C22⋊C8—1184no—
319C22⋊C8—1182no—
320#97—1182no—
321#97—1182no—
322C4×C8—1182no—
323C24—21no—
324C24C22⋊Dic5911yescharacteristic
325C2×Q8—21no—
326C2×Q8C22⋊Dic5912yes—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4C22⋊Dic5912yes—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4C22⋊Dic5911yescharacteristic
343C22×C4C22⋊Dic5911yescharacteristic
344C22×C4C22⋊Dic5911yescharacteristic
345C22×C4C22×Dic5911yescharacteristic
346C22×C4C2×C22⋊D5911yescharacteristic
347C22×C4C2×C22⋊D5911yescharacteristic
348C2×C8—1182no—
349C2×C8—1182no—
350C2×C8—2362no—
351C2×C8—2361no—
352C2×C8—2361no—
353C2×C8—2361no—
354C2×C8—2361no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4C22⋊Dic5912yes—
368C4⋊C4C22⋊Dic5912yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C42—22no—
382C42—21no—
383C42—21no—
384C42—22no—
385C42C2×C22⋊D5912yes—
386C42C2×C22⋊D5912yes—
387C23—21no—
388C23—21no—
389C23—22no—
390C23—41no—
391C23—41no—
392C23—42no—
393C23—81no—
394C23—81no—
395C23#1427365412yes—
396C23C2×C22⋊Dic5911yescharacteristic
397Q8—42no—
398Q8—81no—
399D4—42no—
400D4—44no—
401D4—42no—
402D4—42no—
403D4—81no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4C2×C22⋊Dic5912yes—
426C2×C4#1427369611yescharacteristic
427C2×C4#1427369611yescharacteristic
428C2×C4#1427369612yes—
429C8—4722no—
430C8—2362no—
431C8—2362no—
432C22—21no—
433C22—21no—
434C22—22no—
435C22—42no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—81no—
442C22—162no—
443C22#1683725811yescharacteristic
444C22C2×#1427365411yescharacteristic
445C22#1683741911yescharacteristic
446C4—21no—
447C4—21no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—42no—
452C4—41no—
453C4—42no—
454C4—41no—
455C4—42no—
456C2—21no—
457C2—21no—
458C2—42no—
459C2—82no—
460C2#2142404011yescharacteristic
461C2#2142404711yescharacteristic
462C2#2142406011yescharacteristic
subgroupG/H
centerC22#16837258
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3491C2
socleC22×C59#379
prime radicalC59×#534C4
cosocleC2×C42×C59C23
hypercenter#3491D59
hypocenterC59#8348
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3491

Complements

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

NH ≅ G/Ntype
1. #21423862C2semidirect
2. #21422716C2semidirect
3. #21422556C2semidirect
4. #21422430C2semidirect
6. #16837459C22semidirect
8. C2×#14273638C22semidirect
9. C2×C4×C59⋊C8C22semidirect
261. C59#8348semidirect

Maximal subgroups

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

subgroupindex
#214238622
#214227162
#214225562
#214224302
C59×#34912
#834859