Finite Group Databasebeta

← #28499508

Subgroups — #28499508

Order 15104.

450 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 79 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#21429980C211yesmaximal, characteristic
2#21422434C211yesmaximal, characteristic, complemented
3#21422122C212yesmaximal, complemented
4#21422098C212yesmaximal, complemented
5C59×#1800C211yesmaximal, characteristic, complemented
6C2×#14273701C411yescharacteristic
7#16837366—21no—
8#16837341C2211yescharacteristic
9#16837338C411yescharacteristic
10#16837319C2211yescharacteristic
11#16837317—21no—
12#16837311C411yescharacteristic
13#16837310—21no—
14#16837293—21no—
15#16837281—21no—
16#16837265C411yescharacteristic
17#16837128C2212yes—
18#16837126—21no—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#423—21no—
21C2×#100×C59—21no—
22C59×#343C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C2×#11745664—41no—
27#14273701—41no—
28#14273701—41no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)—41no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C22×D4×C59D411yescharacteristic
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×M16×C59—22no—
48C4×D4×C59—41no—
49C4×D4×C59—41no—
50C2×C4⋊C4×C59C2311yescharacteristic
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C4⋊C8×C59—41no—
55C4⋊C8×C59—41no—
56#100×C59—42no—
57C24×C59C22⋊C411yescharacteristic
58C24×C59C22⋊C411yescharacteristic
59C23×D59—21no—
60C2×D4×C59—21no—
61C2×D4×C59—21no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—81no—
69C22×C4×C59—81no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—42no—
76C2×C22⋊D59—42no—
77C2×C22⋊D59—81no—
78C2×C22⋊D59—81no—
79C2×C22⋊D59—81no—
80C2×C22⋊D59—81no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84C2×D236—41no—
85C2×D236—81no—
86M16×C59—42no—
87M16×C59—42no—
88C2×C8×C59—42no—
89C4⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C42×C59—21no—
93C42×C59—21no—
94C22⋊Dic59—81no—
95C22⋊Dic59—81no—
96(C2×D59)⋊C4—41no—
97(C2×D59)⋊C4—41no—
98(C2×D59)⋊C4—41no—
99(C2×D59)⋊C4—41no—
100(C2×D59)⋊C4—81no—
101(C2×D59)⋊C4—81no—
102C4⋊Dic59—41no—
103Dic59⋊C4—81no—
104#11745664—82no—
105C2×C59⋊C8—42no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59C2×C22⋊C411yescharacteristic
117C22×D59—41no—
118C22×D59—41no—
119C22×D59—41no—
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—41no—
124D4×C59—42no—
125D4×C59—42no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—41no—
132C2×C4×C59—42no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—81no—
137C2×C4×C59—81no—
138C2×C4×C59—81no—
139C2×C4×C59#9911yescharacteristic
140C2×C4×C59#9911yescharacteristic
141C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C59#13711yescharacteristic
143C22⋊D59—84no—
144C22⋊D59—84no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154D236—82no—
155C8×C59—82no—
156C59⋊C8—82no—
157#11832—591nomaximal, Sylow
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—42no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59C2×#9911yescharacteristic
171C22×C59#41711yescharacteristic
172C22×C59#42111yescharacteristic
173C2×D59—81no—
174C2×D59—81no—
175C2×D59—81no—
176C2×D59—81no—
177C2×D59—81no—
178C2×D59—81no—
179C2×D59—81no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—41no—
185C4×C59—41no—
186C4×C59—41no—
187Dic59—81no—
188Dic59—81no—
189Dic59—81no—
190#2635—591no—
191#1800D5911yescharacteristic, complemented
192#1465—591no—
193#1353—592no—
194#1345—592no—
195C2×C59—21no—
196C2×C59—21no—
197C2×C59—81no—
198C2×C59—81no—
199C2×C59#151311yescharacteristic
200C2×C59#212111yescharacteristic
201C2×C59#213411yescharacteristic
202D59—161no—
203C2×#121—591no—
204C2×C4×D4C2×D5911yescharacteristic
205#423—21no—
206C2×#100—21no—
207#407—1181no—
208#400—591no—
209#399—591no—
210#394—591no—
211#394—1181no—
212#392—591no—
213#392—1181no—
214#388—1181no—
215#386—1181no—
216#381—591no—
217#343C2×D5912yes—
218#324—592no—
219#323—1181no—
220C59#1183211yescharacteristic, complemented, Sylow
221C22×D4—591no—
222C22×D4—1181no—
223C22×D4—1181no—
224C22×D4C22⋊D5911yescharacteristic
225C23×C4C4×D5911yescharacteristic
226C23×C4D23611yescharacteristic
227C2×M16—22no—
228C2×M16—2361no—
229#121—2361no—
230#121—2361no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—591no—
234C2×C4⋊C4—1181no—
235C2×C4⋊C4C22×D5911yescharacteristic
236C2×C22⋊C4—1181no—
237C2×C22⋊C4—1181no—
238C2×C22⋊C4—1181no—
239C2×C22⋊C4—1181no—
240C2×C22⋊C4—1181no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4C4×D5911yescharacteristic
243C2×C22⋊C4D23611yescharacteristic
244C2×C42C22⋊D5911yescharacteristic
245C4⋊C8—41no—
246C4⋊C8—41no—
247#100—42no—
248C22⋊C8—1182no—
249C22⋊C8—2361no—
250#95—591no—
251#95—591no—
252#95—1181no—
253#95—1181no—
254#95—1181no—
255C24—1181no—
256C24(C2×D59)⋊C411yescharacteristic
257C24(C2×D59)⋊C411yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—4721no—
263C2×D4—4721no—
264C2×D4—4721no—
265C2×D4—4721no—
266C2×D4—4721no—
267C2×D4—2361no—
268C2×D4—2362no—
269C2×D4—2362no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—1181no—
277C22×C4—1181no—
278C22×C4—1181no—
279C22×C4(C2×D59)⋊C411yescharacteristic
280C22×C4(C2×D59)⋊C411yescharacteristic
281C22×C4C2×C4×D5911yescharacteristic
282C22×C4C2×D23611yescharacteristic
283C22×C4C2×C22⋊D5911yescharacteristic
284M16—42no—
285M16—42no—
286M16—4722no—
287C2×C8—42no—
288C2×C8—2362no—
289C4⋊C4—41no—
290C4⋊C4—4721no—
291C4⋊C4—2361no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—4721no—
295C22⋊C4—4721no—
296C22⋊C4—4721no—
297C22⋊C4—4721no—
298C22⋊C4—2361no—
299C22⋊C4—2361no—
300C22⋊C4—2361no—
301C22⋊C4—2361no—
302C42—21no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—2361no—
315C23—2361no—
316C23—2361no—
317C23—2361no—
318C23—2361no—
319C23—2361no—
320C23—2361no—
321C23C2×(C2×D59)⋊C411yescharacteristic
322D4—42no—
323D4—42no—
324D4—4722no—
325D4—4724no—
326D4—4724no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4—2361no—
349C2×C4#1427364111yescharacteristic
350C2×C4#1427364111yescharacteristic
351C2×C4#1427375311yescharacteristic
352C2×C4#1427375811yescharacteristic
353C8—82no—
354C8—4722no—
355C22—21no—
356C22—21no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—4721no—
368C22—4721no—
369C22—4721no—
370C22—4721no—
371C22—4721no—
372C22—4721no—
373C22—4721no—
374C22C2×#1427364111yescharacteristic
375C22#1683740111yescharacteristic
376C22#1683742011yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—4721no—
385C4—4721no—
386C4—4721no—
387C2—21no—
388C2—21no—
389C2—81no—
390C2—81no—
391C2—9441no—
392C2#2142263711yescharacteristic
393C2#2142574311yescharacteristic
394C2#2142578711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1800C2
socleC22×C59C2×#99
prime radical#21429980C2
cosocleC2×C4⋊C4×C59C23
hypercenter#1800D59
hypocenterC59#11832
LCSG ▷ C22×C4×C59 ▷ C23×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
2. #21422434C2semidirect
3. #21422122C2semidirect
4. #21422098C2semidirect
5. C59×#1800C2semidirect
191. #1800D59semidirect
220. C59#11832semidirect

Maximal subgroups

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

subgroupindex
#214299802
#214224342
#214221222
#214220982
C59×#18002
#1183259