Finite Group Databasebeta

← #28493086

Subgroups — #28493086

Order 15104.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#21434868C211yesmaximal, characteristic
2#21425081C211yesmaximal, characteristic
3#21424520C211yesmaximal, characteristic
4#21424517C211yesmaximal, characteristic
5#21424276C211yesmaximal, characteristic
6#21421963C211yesmaximal, characteristic
7C59×#1996C211yesmaximal, characteristic
8C23×C59⋊C8C2211yescharacteristic
9C2×#14273706C2211yescharacteristic
10#16837737—21no—
11#16837735—21no—
12#16837641—21no—
13#16837641—21no—
14#16837558—21no—
15#16837557—21no—
16C2×C4⋊(C59⋊C8)—21no—
17C2×#14273651—21no—
18C2×#14273650—21no—
19C2×C22⋊(C59⋊C8)C2211yescharacteristic
20#16837262C2211yescharacteristic
21#16837168—21no—
22#16837168—21no—
23#16837164—21no—
24#16837164—21no—
25#16837157—21no—
26C2×C22⋊Q8×C59C2211yescharacteristic
27C59×#380C2211yescharacteristic
28C4×C22⋊C4×C59C2211yescharacteristic
29C59×#342—21no—
30C59×#342—21no—
31C22×Dic118D411yescharacteristic
32C2×#11745672—21no—
33C2×#11745672—21no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—42no—
38C22×C59⋊C8D411yescharacteristic
39C22×C59⋊C8D411yescharacteristic
40#14273706—21no—
41#14273706—21no—
42#14273706—21no—
43#14273706—21no—
44C2×C4⋊Dic59—21no—
45C2×C4⋊Dic59D411yescharacteristic
46C2×Dic59⋊C4—21no—
47C2×C22⋊Dic59—21no—
48C2×C22⋊Dic59—21no—
49#14273660—21no—
50#14273660—21no—
51C4⋊(C59⋊C8)—42no—
52#14273651—21no—
53#14273651—21no—
54#14273651—42no—
55#14273650—21no—
56#14273650—21no—
57#14273650—42no—
58C22⋊(C59⋊C8)—21no—
59C22⋊(C59⋊C8)—21no—
60C59⋊#95—21no—
61C22×Q8×C59D411yescharacteristic
62C23×C4×C59C2311yescharacteristic
63C22⋊Q8×C59—21no—
64C22⋊Q8×C59—21no—
65C22⋊Q8×C59—21no—
66C22⋊Q8×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C22⋊C4×C59—21no—
71C2×C22⋊C4×C59—21no—
72C2×C22⋊C4×C59D411yescharacteristic
73C2×C22⋊C4×C59D411yescharacteristic
74C2×C42×C59—21no—
75#95×C59—21no—
76#95×C59—21no—
77C24×C59C2×D411yescharacteristic
78C2×Q8×C59—21no—
79C2×Q8×C59—21no—
80C2×Q8×C59—41no—
81C2×Q8×C59—41no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59—41no—
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C22×C4×C59#4111yescharacteristic
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—41no—
98C2×Dic118—21no—
99C2×Dic118—21no—
100C2×Dic118—41no—
101C2×Dic118—41no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—41no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C22⋊C4×C59—21no—
109C22⋊C4×C59—21no—
110C22⋊C4×C59—21no—
111C22⋊C4×C59—41no—
112C22⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C22⋊C4×C59—41no—
115C22⋊C4×C59Q1612yescomplemented
116C42×C59—41no—
117C42×C59—41no—
118C22⋊Dic59—41no—
119C22⋊Dic59—41no—
120C22⋊Dic59—41no—
121C22⋊Dic59—41no—
122#11745672—81no—
123#11745672—81no—
124C4⋊Dic59—21no—
125C4⋊Dic59—21no—
126C4⋊Dic59—41no—
127C4⋊Dic59—41no—
128Dic59⋊C4—41no—
129Dic59⋊C4—41no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8—21no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—41no—
137C2×C59⋊C8—41no—
138C2×C59⋊C8—41no—
139C2×C59⋊C8—41no—
140C2×C59⋊C8—41no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59#12111yescharacteristic
144C23×C59#12111yescharacteristic
145C23×C59#12111yescharacteristic
146C23×C59C2×Q1611yescharacteristic
147C23×C59#13511yescharacteristic
148C23×C59#13711yescharacteristic
149C23×C59#13711yescharacteristic
150Q8×C59—42no—
151Q8×C59—41no—
152Q8×C59—41no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—22no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59—81no—
173C2×C4×C59—81no—
174C2×C4×C59—81no—
175C2×C4×C59#12011yescharacteristic
176C2×C4×C59#12311yescharacteristic
177C2×C4×C59#12311yescharacteristic
178C2×C4×C59#12411yescharacteristic
179C2×Dic59—41no—
180C2×Dic59—41no—
181C2×Dic59—81no—
182C2×Dic59—81no—
183C2×Dic59—81no—
184C2×Dic59—81no—
185C2×Dic59—81no—
186Dic118—42no—
187Dic118—41no—
188Dic118—41no—
189C59⋊C8—42no—
190C59⋊C8—41no—
191C59⋊C8—41no—
192#10566—591nomaximal, Sylow
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—21no—
196C22×C59—21no—
197C22×C59—21no—
198C22×C59—21no—
199C22×C59—41no—
200C22×C59—41no—
201C22×C59—41no—
202C22×C59—41no—
203C22×C59#39411yescharacteristic
204C22×C59#46211yescharacteristic
205C22×C59#46211yescharacteristic
206C22×C59#46411yescharacteristic
207C22×C59#46411yescharacteristic
208C22×C59#46711yescharacteristic
209C22×C59#47011yescharacteristic
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—21no—
213C4×C59—42no—
214C4×C59—81no—
215C4×C59—81no—
216C4×C59—81no—
217C4×C59—161no—
218C4×C59B(4,𝔽2)11yescharacteristic
219C4×C59#45811yescharacteristic
220Dic59—81no—
221Dic59—81no—
222Dic59—161no—
223#3189—591no—
224#1996—591no—
225#1996D5911yescharacteristic
226#1892—591no—
227#1892—591no—
228#1850—591no—
229#1289—591no—
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—21no—
233C2×C59—21no—
234C2×C59—21no—
235C2×C59—21no—
236C2×C59#204411yescharacteristic
237C2×C59#205311yescharacteristic
238C2×C59#205611yescharacteristic
239C23×C8—591no—
240C2×C22⋊Q8—591no—
241C2×C22⋊Q8C2×D5911yescharacteristic
242#484—1181no—
243#484—1181no—
244#467—1181no—
245#467—1181no—
246#451—1181no—
247#451—1181no—
248C2×C4⋊C8—1181no—
249C2×#103—1181no—
250C2×#103—1181no—
251C2×C22⋊C8—591no—
252#380—591no—
253#380C2×D5911yescharacteristic
254C4×C22⋊C4C2×D5911yescharacteristic
255#342—21no—
256#342—21no—
257#342—1181no—
258#342—1181no—
259#340—1181no—
260#340—1181no—
261#336—1181no—
262C59#1056611yescharacteristic, complemented, Sylow
263C22×Q8—591no—
264C22×Q8D23611yescharacteristic
265C23×C4C22×D5911yescharacteristic
266C2×Q16—1181no—
267C2×Q16—1181no—
268C22×C8—591no—
269C22×C8—591no—
270C22×C8—1181no—
271C22×C8—1181no—
272C22×C8—1181no—
273C22×C8—2362no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—1181no—
279C22⋊Q8—1181no—
280C22⋊Q8—1181no—
281C22⋊Q8—1181no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—591no—
285C2×C4⋊C4—1181no—
286C2×C4⋊C4—1181no—
287C2×C4⋊C4D23611yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—1181no—
291C2×C22⋊C4—1181no—
292C2×C22⋊C4C22⋊D5911yescharacteristic
293C2×C22⋊C4C22⋊D5911yescharacteristic
294C2×C42—21no—
295#107—1181no—
296#107—1181no—
297C4⋊C8—2362no—
298#103—1181no—
299#103—1181no—
300#103—1181no—
301#103—1181no—
302#103—2362no—
303#103—2362no—
304C22⋊C8—1181no—
305C22⋊C8—1181no—
306#95—21no—
307#95—21no—
308#95—1181no—
309C24C2×C22⋊D5911yescharacteristic
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—1181no—
315C2×Q8—1181no—
316C2×Q8—2361no—
317C2×Q8—2361no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—1181no—
326C22×C4—1181no—
327C22×C4—2361no—
328C22×C4C2×D23611yescharacteristic
329C22×C4#1174568511yescharacteristic
330C22×C4D4×D5911yescharacteristic
331C22×C4D4×D5911yescharacteristic
332C22×C4#1174568711yescharacteristic
333C22×C4#1174568711yescharacteristic
334Q16—4721no—
335Q16—4721no—
336C2×C8—1182no—
337C2×C8—1182no—
338C2×C8—1182no—
339C2×C8—1181no—
340C2×C8—1181no—
341C2×C8—2361no—
342C2×C8—2361no—
343C2×C8—2361no—
344C2×C8—2361no—
345C2×C8—2361no—
346C2×C8—2361no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—1181no—
354C4⋊C4—1181no—
355C4⋊C4—2361no—
356C4⋊C4—2361no—
357C4⋊C4—2361no—
358C4⋊C4—2361no—
359C22⋊C4—21no—
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—2361no—
367C22⋊C4—2361no—
368C22⋊C4—2361no—
369C22⋊C4—2361no—
370C22⋊C4#1174567212yescomplemented
371C42—41no—
372C42—41no—
373C23—41no—
374C23—41no—
375C23#1427369811yescharacteristic
376C23#1427369911yescharacteristic
377C23#1427370111yescharacteristic
378C23C2×#1174567211yescharacteristic
379C23#1427374711yescharacteristic
380C23#1427375711yescharacteristic
381C23#1427375811yescharacteristic
382Q8—42no—
383Q8—41no—
384Q8—41no—
385Q8—2362no—
386Q8—2361no—
387Q8—2361no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—4721no—
411C2×C4—4721no—
412C2×C4—4721no—
413C2×C4—4721no—
414C2×C4—4721no—
415C2×C4—2361no—
416C2×C4—2361no—
417C2×C4#1427369711yescharacteristic
418C2×C4#1427370911yescharacteristic
419C2×C4#1427371111yescharacteristic
420C2×C4#1427371311yescharacteristic
421C8—2362no—
422C8—2361no—
423C8—2361no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22#1683731811yescharacteristic
435C22#1683760911yescharacteristic
436C22#1683761211yescharacteristic
437C22#1683762111yescharacteristic
438C22#1683762411yescharacteristic
439C22#1683764111yescharacteristic
440C22#1683765811yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—161no—
449C4—9441no—
450C4—4721no—
451C4—4721no—
452C4#1683759011yescharacteristic
453C4#1683759311yescharacteristic
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2#2142534611yescharacteristic
461C2#2142540111yescharacteristic
462C2#2142541811yescharacteristic
subgroupG/H
centerC22#16837318
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1996C2
socleC22×C59#394
prime radicalC24×C59C2×D4
hypercenter#1996D59
hypocenterC59#10566
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
115. C22⋊C4×C59Q16semidirect
262. C59#10566semidirect
370. C22⋊C4#11745672semidirect

Maximal subgroups

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

subgroupindex
#214348682
#214250812
#214245202
#214245172
#214242762
#214219632
C59×#19962
#1056659