Finite Group Databasebeta

← #28504658

Subgroups — #28504658

Order 15104.

540 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C4×#14273708C211yesmaximal, characteristic
2C2×#16837193C212yesmaximal
3#21424084C211yesmaximal, characteristic
4#21422215C212yesmaximal
5C59×#1800C211yesmaximal, characteristic
6C2×#14273708C2211yescharacteristic
7C22×C4×Dic59C411yescharacteristic, complemented
8C2×#14273642C2212yes—
9#16837290—21no—
10#16837283—21no—
11#16837282C2211yescharacteristic
12#16837274C411yescharacteristic, complemented
13#16837263C411yescharacteristic, complemented
14#16837262—21no—
15C4×Dic59⋊C4—21no—
16C4×C22⋊Dic59—21no—
17C4×C22⋊Dic59C411yescharacteristic, complemented
18#16837193—24no—
19#16837169—21no—
20C2×C4×D4×C59C2211yescharacteristic
21C59×#423—21no—
22C2×#100×C59C2212yes—
23C59×#343—21no—
24C23×Dic59C2×C411yescharacteristic
25C23×Dic59C2×C411yescharacteristic
26#14273708—21no—
27#14273708—21no—
28#14273708D412yes—
29#14273708Q812yes—
30C2×C4⋊Dic59—41no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—22no—
36C2×C22⋊Dic59C2×C411yescharacteristic
37C2×C22⋊Dic59C2×C411yescharacteristic
38C2×C4×Dic59—21no—
39C2×C4×Dic59—21no—
40C2×C4×Dic59—21no—
41C2×C4×Dic59—41no—
42C2×C4×Dic59—41no—
43#14273642—22no—
44#14273642—22no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—41no—
51C22×D4×C59C2311yescharacteristic
52C23×C4×C59C2×C411yescharacteristic
53C23×C4×C59D411yescharacteristic
54C2×M16×C59—22no—
55C4×D4×C59—41no—
56C4×D4×C59—41no—
57C2×C4⋊C4×C59D411yescharacteristic
58C2×C22⋊C4×C59C2×C411yescharacteristic
59C2×C22⋊C4×C59D411yescharacteristic
60C2×C42×C59D411yescharacteristic
61C4⋊C8×C59—41no—
62C4⋊C8×C59—41no—
63#100×C59—22no—
64#100×C59—22no—
65C24×C59C22×C411yescharacteristic
66C24×C59C2×D411yescharacteristic
67C2×D4×C59—41no—
68C2×D4×C59—41no—
69C2×D4×C59C2×D411yescharacteristic
70C2×D4×C59C2×Q811yescharacteristic
71C2×D4×C59#4111yescharacteristic
72C2×D4×C59#4111yescharacteristic
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—42no—
78C22×C4×C59—42no—
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C22×Dic59—42no—
90C22×Dic59—41no—
91C22×Dic59—41no—
92C22×Dic59—41no—
93C22×Dic59—42no—
94C22×Dic59C4⋊C412yescomplemented
95M16×C59—42no—
96M16×C59—42no—
97C2×C8×C59—42no—
98C4⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C42×C59—21no—
102C42×C59—21no—
103C22⋊Dic59—21no—
104C22⋊Dic59—22no—
105C22⋊Dic59—22no—
106C22⋊Dic59—42no—
107C22⋊Dic59—41no—
108C22⋊Dic59—42no—
109C22⋊Dic59—42no—
110C22⋊Dic59C4⋊C412yescomplemented
111C4⋊Dic59—82no—
112Dic59⋊C4—22no—
113Dic59⋊C4—22no—
114Dic59⋊C4—42no—
115C4×Dic59—42no—
116C4×Dic59—42no—
117C4×Dic59—41no—
118C4×Dic59—41no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59C2×C22⋊C411yescharacteristic
128C23×C59C2×C4⋊C411yescharacteristic
129C23×C59#11711yescharacteristic
130C23×C59#12111yescharacteristic
131C23×C59C22⋊Q811yescharacteristic
132D4×C59—41no—
133D4×C59—41no—
134D4×C59—41no—
135D4×C59—41no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—22no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—42no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—81no—
149C2×C4×C59#9911yescharacteristic
150C2×C4×C59#9911yescharacteristic
151C2×C4×C59#12111yescharacteristic
152C2×C4×C59C22⋊Q811yescharacteristic
153C2×Dic59—22no—
154C2×Dic59—22no—
155C2×Dic59—22no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—42no—
159C2×Dic59—41no—
160C2×Dic59—42no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—42no—
165C2×Dic59—42no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—41no—
169C2×Dic59—82no—
170C2×Dic59—82no—
171C2×Dic59—82no—
172C8×C59—82no—
173#12794—591nomaximal, Sylow
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—21no—
180C22×C59—41no—
181C22×C59—41no—
182C22×C59—41no—
183C22×C59—41no—
184C22×C59—42no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—41no—
188C22×C59—41no—
189C22×C59—81no—
190C22×C59#38011yescharacteristic
191C22×C59C2×#9911yescharacteristic
192C22×C59#41011yescharacteristic
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—41no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59#35212yescomplemented
200Dic59—42no—
201Dic59—42no—
202Dic59—42no—
203Dic59—81no—
204Dic59—162no—
205C4×#123—591no—
206C2×#352—592no—
207#1800D5911yescharacteristic
208#1799—591no—
209#1390—592no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59—41no—
213C2×C59—41no—
214C2×C59—41no—
215C2×C59—41no—
216C2×C59#179911yescharacteristic
217C2×C59C2×#35211yescharacteristic
218C2×C59#212011yescharacteristic
219C2×#123—591no—
220C2×C4×D4C2×D5911yescharacteristic
221C22×C42—591no—
222#423—21no—
223C2×#100C2×D5912yes—
224C2×#99—592no—
225#388—1181no—
226#387—1181no—
227#386—591no—
228#385—591no—
229#381—591no—
230#380—1181no—
231C4×C4⋊C4—1181no—
232C4×C22⋊C4—591no—
233C4×C22⋊C4—1181no—
234#352—1184no—
235#343—21no—
236#342—1181no—
237C59#1279411yescharacteristic, complemented, Sylow
238C22×D4C22×D5911yescharacteristic
239C23×C4—591no—
240C23×C4—591no—
241C23×C4C4×D5911yescharacteristic
242C23×C4D23611yescharacteristic
243C2×M16—22no—
244#123—592no—
245#123—592no—
246#123—1181no—
247#123—1181no—
248C4×D4—41no—
249C4×D4—41no—
250C2×C4⋊C4—1181no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—2361no—
253C2×C4⋊C4C22⋊D5911yescharacteristic
254C2×C22⋊C4—591no—
255C2×C22⋊C4—591no—
256C2×C22⋊C4—1181no—
257C2×C22⋊C4—1181no—
258C2×C22⋊C4—1182no—
259C2×C22⋊C4C4×D5911yescharacteristic
260C2×C22⋊C4D23611yescharacteristic
261C2×C42—1181no—
262C2×C42—1181no—
263C2×C42—1181no—
264C2×C42—2361no—
265C2×C42—2361no—
266C2×C42C22⋊D5911yescharacteristic
267C4⋊C8—41no—
268C4⋊C8—41no—
269#100—22no—
270#100—22no—
271#99—1182no—
272#99—1182no—
273#95—1181no—
274#95—1181no—
275#95—1181no—
276#95—1181no—
277#95—1181no—
278#95—2361no—
279C24C2×C4×D5911yescharacteristic
280C24C2×D23611yescharacteristic
281C2×D4—41no—
282C2×D4—41no—
283C2×D4D4×D5911yescharacteristic
284C2×D4#1174568711yescharacteristic
285C2×D4Q8×D5911yescharacteristic
286C2×D4#1174568911yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—592no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4—1181no—
297C22×C4—1181no—
298C22×C4—1181no—
299C22×C4—2362no—
300C22×C4—2362no—
301C22×C4—2361no—
302C22×C4—2361no—
303C22×C4—2361no—
304C22×C4(C2×D59)⋊C411yescharacteristic
305C22×C4(C2×D59)⋊C411yescharacteristic
306C22×C4(C2×D59)⋊C411yescharacteristic
307C22×C4(C2×D59)⋊C411yescharacteristic
308C22×C4C2×C22⋊D5911yescharacteristic
309M16—42no—
310M16—42no—
311C2×C8—42no—
312C4⋊C4—41no—
313C4⋊C4—1182no—
314C4⋊C4—1182no—
315C4⋊C4—4722no—
316C4⋊C4—2362no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—592no—
320C22⋊C4—1181no—
321C22⋊C4—1182no—
322C22⋊C4—1182no—
323C22⋊C4—2362no—
324C22⋊C4—2362no—
325C22⋊C4—2362no—
326C22⋊C4—2361no—
327C42—21no—
328C42—21no—
329C42—2362no—
330C42—2362no—
331C42—2361no—
332C42—2361no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C2×(C2×D59)⋊C411yescharacteristic
342C23C4⋊C4×D5911yescharacteristic
343C23#1427368511yescharacteristic
344C23#1427370211yescharacteristic
345C23#1427370711yescharacteristic
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—1182no—
364C2×C4—1182no—
365C2×C4—1182no—
366C2×C4—4722no—
367C2×C4—4722no—
368C2×C4—4722no—
369C2×C4—2361no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2362no—
376C2×C4—2362no—
377C2×C4—2361no—
378C2×C4—2361no—
379C2×C4—2361no—
380C2×C4—2362no—
381C2×C4—2362no—
382C2×C4#1427364111yescharacteristic
383C2×C4#1427364111yescharacteristic
384C2×C4#1427369811yescharacteristic
385C2×C4#1427370311yescharacteristic
386C8—82no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22#1683725911yescharacteristic
404C22C2×#1427364111yescharacteristic
405C22#1683737511yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—9442no—
413C4—4721no—
414C4—2362no—
415C4—2362no—
416C4—2362no—
417C4#1683719312yescomplemented
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—41no—
424C2#2142408111yescharacteristic
425C2C2×#1683719311yescharacteristic
426C2#2142573811yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1800C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenter#1800D59
hypocenterC59#12794
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
7. C22×C4×Dic59C4semidirect
12. #16837274C4semidirect
13. #16837263C4semidirect
17. C4×C22⋊Dic59C4semidirect
94. C22×Dic59C4⋊C4semidirect
110. C22⋊Dic59C4⋊C4semidirect
199. C4×C59#352semidirect
237. C59#12794semidirect
417. C4#16837193semidirect

Maximal subgroups

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

subgroupindex
C4×#142737082
C2×#168371932
#214240842
#214222152
C59×#18002
#1279459