Finite Group Databasebeta

← #28485006

Subgroups — #28485006

Order 15104.

452 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#16837740C211yesmaximal, characteristic
2#21432750C211yesmaximal, characteristic
3#21424193C211yesmaximal, characteristic
4#21422616C211yesmaximal, characteristic
5#21421972C211yesmaximal, characteristic
6#21421942C211yesmaximal, characteristic
7C59×#1472C211yesmaximal, characteristic
8C2×#14273686C2211yescharacteristic
9#16837740—21no—
10#16837740—21no—
11#16837740—21no—
12#16837740—21no—
13#16837460—22no—
14#16837456—21no—
15#16837452—21no—
16C2×#14273660C411yescharacteristic, complemented
17C2×#14273658C411yescharacteristic, complemented
18#16837428—21no—
19C2×#14273651C411yescharacteristic, complemented
20C2×#14273651C411yescharacteristic, complemented
21#16837366C2211yescharacteristic
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23#16837357—21no—
24C4×C22⋊Dic59C2211yescharacteristic
25#16837163—21no—
26#16837163—21no—
27#16837139—22no—
28C2×C22⋊Q8×C59C2211yescharacteristic
29C59×#407C2211yescharacteristic
30C2×C22⋊C8×C59C2211yescharacteristic
31C59×#324—21no—
32C59×#324—21no—
33C2×#11745664—21no—
34C22×C59⋊C8C2×C411yescharacteristic
35C22×C59⋊C8C2×C411yescharacteristic
36#14273686—21no—
37#14273686—21no—
38#14273686D411yescharacteristic
39#14273686D411yescharacteristic
40#14273686D411yescharacteristic
41#14273686D411yescharacteristic
42C2×C4⋊Dic59C2×C411yescharacteristic
43C2×C4⋊Dic59C2×C411yescharacteristic
44C2×C22⋊Dic59—21no—
45C2×C22⋊Dic59—21no—
46C2×C4×Dic59—21no—
47C2×C4×Dic59—21no—
48#14273660—21no—
49#14273660—21no—
50#14273658—21no—
51#14273658—21no—
52Dic59⋊C8—22no—
53Dic59⋊C8—22no—
54#14273651—21no—
55#14273651—21no—
56#14273651—21no—
57#14273651—21no—
58C22⋊(C59⋊C8)—21no—
59C22⋊(C59⋊C8)—21no—
60C22⋊(C59⋊C8)—21no—
61C22⋊(C59⋊C8)D412yes—
62#14273637—22no—
63C8×Dic59—22no—
64C59⋊#95—21no—
65C22×Q8×C59C2×C411yescharacteristic
66C23×C4×C59C2311yescharacteristic
67C2×M16×C59—21no—
68C22×C8×C59—21no—
69C22⋊Q8×C59—21no—
70C22⋊Q8×C59—21no—
71C22⋊Q8×C59—21no—
72C22⋊Q8×C59—21no—
73C2×C4⋊C4×C59—21no—
74C2×C4⋊C4×C59C2×C411yescharacteristic
75C2×C22⋊C4×C59—21no—
76C22⋊C8×C59—21no—
77C22⋊C8×C59—21no—
78C22⋊C8×C59—21no—
79C22⋊C8×C59D411yescharacteristic
80C22⋊C8×C59D411yescharacteristic
81C24×C59#4111yescharacteristic
82C2×Q8×C59—21no—
83C2×Q8×C59—21no—
84C2×Q8×C59—41no—
85C2×Q8×C59—41no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59C22×C411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59#4111yescharacteristic
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—41no—
99M16×C59—41no—
100M16×C59—41no—
101C2×C8×C59—21no—
102C2×C8×C59—21no—
103C2×C8×C59—41no—
104C2×C8×C59—41no—
105C2×C8×C59—41no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C22⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112C22⋊Dic59—21no—
113C22⋊Dic59—21no—
114C22⋊Dic59—21no—
115C22⋊Dic59—21no—
116C22⋊Dic59—22no—
117C22⋊Dic59—41no—
118C4⋊Dic59—21no—
119C4⋊Dic59—21no—
120C4⋊Dic59C22⋊C411yescharacteristic
121C4⋊Dic59C22⋊C411yescharacteristic
122C4⋊Dic59C22⋊C411yescharacteristic
123C4⋊Dic59C22⋊C411yescharacteristic
124C4×Dic59—21no—
125C4×Dic59—21no—
126C4×Dic59—21no—
127C4×Dic59—21no—
128C4×Dic59—41no—
129C4×Dic59—41no—
130#11745664—41no—
131#11745664—41no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8—22no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—41no—
137C23×C59—21no—
138C23×C59—21no—
139C23×C59—41no—
140C23×C59—41no—
141C23×C59C4×D411yescharacteristic
142C23×C59#12111yescharacteristic
143C23×C59#12311yescharacteristic
144Q8×C59—41no—
145Q8×C59—41no—
146Q8×C59—81no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—81no—
161C2×C4×C59C2×C22⋊C411yescharacteristic
162C2×C4×C59C4×D411yescharacteristic
163C2×C4×C59#12011yescharacteristic
164C2×C4×C59#12111yescharacteristic
165C2×Dic59—21no—
166C2×Dic59—21no—
167C2×Dic59—21no—
168C2×Dic59—21no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—42no—
172C2×Dic59—81no—
173C2×Dic59—81no—
174C2×Dic59—81no—
175C8×C59—41no—
176C8×C59—41no—
177C8×C59—81no—
178C59⋊C8—42no—
179C59⋊C8—41no—
180C59⋊C8—41no—
181#9240—591nomaximal, Sylow
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—81no—
191C22×C59—81no—
192C22×C59#38611yescharacteristic
193C22×C59#44311yescharacteristic
194C22×C59#44411yescharacteristic
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—21no—
198C4×C59—81no—
199C4×C59—81no—
200C4×C59#41011yescharacteristic
201C4×C59#41011yescharacteristic
202Dic59—41no—
203Dic59—41no—
204Dic59—41no—
205Dic59—41no—
206Dic59—82no—
207C2×#485—591no—
208#2924—591no—
209#1826—591no—
210#1508—591no—
211#1472D5911yescharacteristic
212#1293—591no—
213#1282—591no—
214C2×C59—21no—
215C2×C59—21no—
216C2×C59—41no—
217C2×C59—41no—
218C2×C59#187611yescharacteristic
219C2×C59#189911yescharacteristic
220C2×C59#190211yescharacteristic
221C2×C22⋊Q8C2×D5911yescharacteristic
222C2×#117—591no—
223#485—1181no—
224#485—1181no—
225#485—1181no—
226#485—1181no—
227#433—1182no—
228#432—1181no—
229#431—1181no—
230C2×#107—591no—
231C2×#106—591no—
232#424—1181no—
233C2×#103—591no—
234C2×#103—591no—
235#407—591no—
236#407C2×D5911yescharacteristic
237C2×C22⋊C8—591no—
238C2×C22⋊C8C2×D5911yescharacteristic
239#405—1181no—
240C4×C22⋊C4—591no—
241#339—1181no—
242#339—1181no—
243#328—1182no—
244#324—21no—
245#324—21no—
246C59#924011yescharacteristic, complemented, Sylow
247C22×Q8C4×D5911yescharacteristic
248C23×C4C22×D5911yescharacteristic
249C2×M16—21no—
250C2×M16—1181no—
251C22×C8—21no—
252C22×C8—591no—
253C22×C8—591no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258#117—591no—
259#117—591no—
260#117—591no—
261#117—591no—
262#117—1181no—
263#117—1181no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—591no—
266C2×C4⋊C4—591no—
267C2×C4⋊C4C4×D5911yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—1181no—
270C2×C22⋊C4—1181no—
271C2×C42—1181no—
272C2×C42—1181no—
273#107—1181no—
274#107—1181no—
275#106—1181no—
276#106—1181no—
277C4⋊C8—1182no—
278C4⋊C8—1182no—
279#103—1181no—
280#103—1181no—
281#103—1181no—
282#103—1181no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—592no—
287C22⋊C8—1181no—
288C22⋊C8—1181no—
289C22⋊C8—1181no—
290C22⋊C8C22⋊D5911yescharacteristic
291C22⋊C8C22⋊D5911yescharacteristic
292#97—1182no—
293C4×C8—1182no—
294#95—1181no—
295C24#1174568511yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—1181no—
305C22×C4—1181no—
306C22×C4—2361no—
307C22×C4C2×C4×D5911yescharacteristic
308C22×C4D4×D5911yescharacteristic
309C22×C4D4×D5911yescharacteristic
310C22×C4D4×D5911yescharacteristic
311C22×C4#1174568711yescharacteristic
312C22×C4C2×C22⋊D5911yescharacteristic
313M16—41no—
314M16—41no—
315M16—2361no—
316M16—2361no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—1181no—
323C2×C8—1181no—
324C2×C8—1182no—
325C2×C8—2361no—
326C2×C8—2361no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—591no—
332C4⋊C4—591no—
333C4⋊C4—591no—
334C4⋊C4—591no—
335C4⋊C4—1181no—
336C4⋊C4—1181no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—1181no—
340C22⋊C4—1181no—
341C22⋊C4—1181no—
342C22⋊C4—1181no—
343C22⋊C4—1182no—
344C22⋊C4—2361no—
345C42—1181no—
346C42—1181no—
347C42—1181no—
348C42—1181no—
349C42—2361no—
350C42—2361no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23C4×C22⋊D5911yescharacteristic
356C23#1427370011yescharacteristic
357C23#1427370911yescharacteristic
358Q8—41no—
359Q8—41no—
360Q8—81no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—1181no—
376C2×C4—1181no—
377C2×C4—1181no—
378C2×C4—1181no—
379C2×C4—4721no—
380C2×C4—4721no—
381C2×C4—4721no—
382C2×C4—2362no—
383C2×C4—2361no—
384C2×C4—2361no—
385C2×C4C22⋊C4×D5911yescharacteristic
386C2×C4(C22⋊D59)⋊C411yescharacteristic
387C2×C4#1427369511yescharacteristic
388C2×C4#1427369811yescharacteristic
389C8—41no—
390C8—41no—
391C8—81no—
392C8—2362no—
393C8—2361no—
394C8—2361no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22#1683727811yescharacteristic
406C22#1683751711yescharacteristic
407C22#1683752211yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—81no—
412C4—81no—
413C4—4722no—
414C4—2361no—
415C4—2361no—
416C4—2361no—
417C4—2361no—
418C4#1683737411yescharacteristic
419C4#1683737411yescharacteristic
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2#2142442611yescharacteristic
425C2#2142455111yescharacteristic
426C2#2142457211yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1472C2
socleC22×C59#386
prime radicalC24×C59#41
cosocleC23×C4×C59C23
hypercenter#1472D59
hypocenterC59#9240
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
16. C2×#14273660C4semidirect
17. C2×#14273658C4semidirect
19. C2×#14273651C4semidirect
20. C2×#14273651C4semidirect
246. C59#9240semidirect

Maximal subgroups

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

subgroupindex
C2×#168377402
#214327502
#214241932
#214226162
#214219722
#214219422
C59×#14722
#924059