Finite Group Databasebeta

← #28498606

Subgroups — #28498606

Order 15104.

422 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C8×D4×D59C211yesmaximal, characteristic
2#21425898C211yesmaximal, characteristic, complemented
3#21425893C211yesmaximal, characteristic, complemented
4#21422052C211yesmaximal, characteristic, complemented
5D4⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422050C211yesmaximal, characteristic, complemented
7C59×D4⋊C16C211yesmaximal, characteristic, complemented
8C22×C8×D59—21no—
9C4×D4×D59C411yescharacteristic
10C8×D236C2211yescharacteristic
11D4×C59⋊C8C411yescharacteristic
12#16837467C411yescharacteristic
13C8×C22⋊D59—21no—
14(C22⋊D59)⋊C8—21no—
15C4⋊C8×D59C411yescharacteristic
16C22⋊C8×D59—21no—
17C4×C8×D59C2211yescharacteristic
18C4⋊(C59⋊C16)C2211yescharacteristic, complemented
19Dic59⋊C16—21no—
20(C2×D59)⋊C16—21no—
21(C2×D59)⋊C16—21no—
22#16837177—21no—
23C4×C59⋊C16C2211yescharacteristic, complemented
24C8×D4×C59C2211yescharacteristic
25C59×C4⋊C16C2211yescharacteristic, complemented
26C4×C16×C59C2211yescharacteristic, complemented
27C2×D4×D59C811yescharacteristic
28C22×C4×D59—21no—
29C22×C59⋊C8—21no—
30C2×C8×D59—21no—
31C2×C8×D59—21no—
32C2×C8×D59—21no—
33C2×C8×D59—42no—
34C2×C8×D59—42no—
35C2×C8×D59D411yescharacteristic
36C2×C8×D59D411yescharacteristic
37C4×D236C2×C411yescharacteristic
38D4×Dic59C811yescharacteristic
39#14273689C811yescharacteristic
40C4×C22⋊D59—21no—
41(C22⋊D59)⋊C4—21no—
42C4⋊C4×D59C811yescharacteristic
43C22⋊C4×D59—21no—
44C42×D59C2×C411yescharacteristic
45C2×C59⋊C16—21no—
46C2×C59⋊C16—21no—
47C4⋊(C59⋊C8)C2×C411yescharacteristic
48Dic59⋊C8—21no—
49C22⋊(C59⋊C8)—21no—
50(C2×D59)⋊C8—21no—
51(C2×D59)⋊C8—21no—
52C4×C59⋊C8C2×C411yescharacteristic
53C8×Dic59D411yescharacteristic
54C8×Dic59D411yescharacteristic
55C22×C8×C59—21no—
56C4×D4×C59C2×C411yescharacteristic
57C2×C16×C59—21no—
58C2×C16×C59—21no—
59C4⋊C8×C59C2×C411yescharacteristic
60C22⋊C8×C59—21no—
61C4×C8×C59C2311yescharacteristic
62C23×D59—21no—
63C2×D4×C59C2×C811yescharacteristic
64C22×C4×C59—21no—
65C2×C22⋊D59—21no—
66C22×Dic59—21no—
67D4×D59—22no—
68D4×D59—22no—
69C2×D236C2×C811yescharacteristic
70C2×C4×D59—21no—
71C2×C4×D59—21no—
72C2×C4×D59—21no—
73C2×C4×D59—42no—
74C2×C4×D59—42no—
75C2×C4×D59C22⋊C411yescharacteristic
76C2×C4×D59C2×C811yescharacteristic
77C2×C4×D59M1611yescharacteristic
78C2×C8×C59—21no—
79C2×C8×C59—41no—
80C2×C8×C59—41no—
81C2×C8×C59C2×D411yescharacteristic
82C2×C8×C59C2×D411yescharacteristic
83C4⋊C4×C59C2×C811yescharacteristic
84C22⋊C4×C59—21no—
85C42×C59C22×C411yescharacteristic
86C22⋊Dic59—21no—
87(C2×D59)⋊C4—21no—
88(C2×D59)⋊C4—21no—
89C4⋊Dic59C2×C811yescharacteristic
90Dic59⋊C4—21no—
91C4×Dic59C22⋊C411yescharacteristic
92C4×Dic59C2×C811yescharacteristic
93C4×Dic59M1611yescharacteristic
94C2×C59⋊C8—21no—
95C2×C59⋊C8—42no—
96C2×C59⋊C8C22⋊C411yescharacteristic
97C2×C59⋊C8C22⋊C411yescharacteristic
98C8×D59—21no—
99C8×D59—21no—
100C8×D59—41no—
101C8×D59—42no—
102C8×D59—42no—
103C8×D59—41no—
104C8×D59—41no—
105C16×C59—41no—
106C16×C59—41no—
107C59⋊C16—41no—
108C59⋊C16—41no—
109C23×C59—21no—
110C22×D59—24no—
111C22×D59—21no—
112C22×D59—42no—
113C22×D59—42no—
114C22×D59C22⋊C811yescharacteristic
115D4×C59—21no—
116D4×C59M3212yes—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59C2×C22⋊C411yescharacteristic
121C2×C4×C59C22×C811yescharacteristic
122C2×C4×C59C2×M1611yescharacteristic
123C22⋊D59—24no—
124C2×Dic59—21no—
125C2×Dic59—42no—
126C2×Dic59C22⋊C811yescharacteristic
127C2×Dic59C22⋊C811yescharacteristic
128C2×Dic59C22⋊C811yescharacteristic
129D236—21no—
130D236M3212yes—
131C4×D59—22no—
132C4×D59—21no—
133C4×D59—21no—
134C4×D59—21no—
135C4×D59—21no—
136C4×D59—42no—
137C4×D59—42no—
138C4×D59—41no—
139C4×D59—41no—
140C8×C59—21no—
141C8×C59—41no—
142C8×C59AGL(1,ℤ/8)11yescharacteristic
143C8×C59AGL(1,ℤ/8)11yescharacteristic
144C59⋊C8—21no—
145C59⋊C8—21no—
146C59⋊C8—41no—
147#11631—591nomaximal, Sylow
148C22×C59—22no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59C2×C22⋊C811yescharacteristic
152C2×D59—21no—
153C2×D59—21no—
154C2×D59—22no—
155C2×D59—48no—
156C2×D59—41no—
157C2×D59—41no—
158C2×D59#35012yes—
159C4×C59—21no—
160C4×C59—41no—
161C4×C59#41811yescharacteristic
162C4×C59#42111yescharacteristic
163C4×C59C2×M3211yescharacteristic
164C4×C59C2×M3211yescharacteristic
165Dic59—21no—
166Dic59—21no—
167Dic59—21no—
168Dic59—41no—
169C2×C8×D4—591no—
170#2163—591no—
171#2162—591no—
172D4⋊C16—591no—
173D4⋊C16—591no—
174D4⋊C16—591no—
175D4⋊C16D5911yescharacteristic, complemented
176C2×C59—42no—
177C2×C59#147811yescharacteristic
178C2×C59#211211yescharacteristic
179C2×C59C2×#35011yescharacteristic
180D59—22no—
181D59—42no—
182C23×C8—1181no—
183C2×C4×D4—591no—
184C8×D4—591no—
185C8×D4—591no—
186C8×D4—591no—
187C8×D4—1181no—
188C8×D4—1181no—
189C8×D4C2×D5911yescharacteristic
190C2×C4⋊C8—591no—
191C2×C22⋊C8—1181no—
192C2×C4×C8—591no—
193C4⋊C16—591no—
194C4⋊C16—1181no—
195C4⋊C16C2×D5911yescharacteristic, complemented
196C22⋊C16—1181no—
197C22⋊C16—1181no—
198#346—1181no—
199C4×C16—591no—
200C4×C16C2×D5911yescharacteristic, complemented
201C59#1163111yescharacteristic, complemented, Sylow
202C22×D4—591no—
203C23×C4—1181no—
204C22×C8—21no—
205C22×C8—591no—
206C22×C8—591no—
207C22×C8—1181no—
208C22×C8—1181no—
209C22×C8—1181no—
210C22×C8—1181no—
211C22×C8—2362no—
212C22×C8—2362no—
213C4×D4—591no—
214C4×D4—591no—
215C4×D4—591no—
216C4×D4—1181no—
217C4×D4—1181no—
218C4×D4C4×D5911yescharacteristic
219C2×C4⋊C4—591no—
220C2×C22⋊C4—1181no—
221C2×C42—591no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—1181no—
225C2×C16—1181no—
226C4⋊C8—591no—
227C4⋊C8—1181no—
228C4⋊C8C4×D5911yescharacteristic
229C22⋊C8—21no—
230C22⋊C8—1181no—
231C22⋊C8—1181no—
232C22⋊C8—1181no—
233C4×C8—591no—
234C4×C8—591no—
235C4×C8—591no—
236C4×C8C22×D5911yescharacteristic
237C24—1181no—
238C2×D4—591no—
239C2×D4—1181no—
240C2×D4—1182no—
241C2×D4—1182no—
242C2×D4C8×D5911yescharacteristic
243C22×C4—21no—
244C22×C4—591no—
245C22×C4—591no—
246C22×C4—591no—
247C22×C4—1181no—
248C22×C4—1181no—
249C22×C4—1181no—
250C22×C4—1181no—
251C22×C4—2362no—
252C22×C4—2362no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—591no—
257C2×C8—591no—
258C2×C8—1181no—
259C2×C8—1181no—
260C2×C8—1181no—
261C2×C8—2361no—
262C2×C8—2362no—
263C2×C8—2362no—
264C2×C8—2362no—
265C2×C8—2361no—
266C2×C8—2361no—
267C2×C8D4×D5911yescharacteristic
268C2×C8D4×D5911yescharacteristic
269C4⋊C4—591no—
270C4⋊C4—1181no—
271C4⋊C4C8×D5911yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—1181no—
274C22⋊C4—1181no—
275C22⋊C4—1181no—
276C42—591no—
277C42—591no—
278C42—591no—
279C42C2×C4×D5911yescharacteristic
280C16—41no—
281C16—41no—
282C16—2361no—
283C16—2361no—
284C23—21no—
285C23—591no—
286C23—1184no—
287C23—1181no—
288C23—2362no—
289C23—2362no—
290D4—21no—
291D4—592no—
292D4—1184no—
293D4—1181no—
294D4#1427366612yes—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—591no—
299C2×C4—591no—
300C2×C4—591no—
301C2×C4—1182no—
302C2×C4—1181no—
303C2×C4—1181no—
304C2×C4—1181no—
305C2×C4—1181no—
306C2×C4—1181no—
307C2×C4—2362no—
308C2×C4—2362no—
309C2×C4—2362no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4C22⋊C4×D5911yescharacteristic
313C2×C4C2×C8×D5911yescharacteristic
314C2×C4M16×D5911yescharacteristic
315C8—21no—
316C8—41no—
317C8—1181no—
318C8—1181no—
319C8—2361no—
320C8#1427375111yescharacteristic
321C8#1427375411yescharacteristic
322C22—22no—
323C22—41no—
324C22—41no—
325C22—592no—
326C22—1181no—
327C22—1181no—
328C22—1182no—
329C22—2368no—
330C22—2361no—
331C22—2361no—
332C22C22⋊C8×D5911yescharacteristic
333C4—21no—
334C4—41no—
335C4—1181no—
336C4—1181no—
337C4—1181no—
338C4—2361no—
339C4#1683740511yescharacteristic
340C4#1683741711yescharacteristic
341C4M32×D5911yescharacteristic
342C4C2×#1427366611yescharacteristic
343C2—42no—
344C2—1182no—
345C2—2362no—
346C2#2142248511yescharacteristic
347C2#2142570511yescharacteristic
348C2#350×D5911yescharacteristic
subgroupG/H
centerC2×C8D4×D59
commutatorC2×C4×C59C22×C8
FrattiniC4×C8C22×D59
FittingC59×D4⋊C16C2
socleC22×C59C2×C22⋊C8
prime radicalC2×D4×D59C8
cosocleC4×C8×C59C23
hypercenterD4⋊C16D59
hypocenterC59#11631
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ D4⋊C16

Complements

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

NH ≅ G/Ntype
2. #21425898C2semidirect
3. #21425893C2semidirect
4. #21422052C2semidirect
5. D4⋊(C59⋊C16)C2semidirect
6. #21422050C2semidirect
7. C59×D4⋊C16C2semidirect
18. C4⋊(C59⋊C16)C22semidirect
23. C4×C59⋊C16C22semidirect
25. C59×C4⋊C16C22semidirect
26. C4×C16×C59C22semidirect
175. D4⋊C16D59semidirect
195. C4⋊C16C2×D59semidirect
200. C4×C16C2×D59semidirect
201. C59#11631semidirect

Maximal subgroups

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

subgroupindex
C8×D4×D592
#214258982
#214258932
#214220522
D4⋊(C59⋊C16)2
#214220502
C59×D4⋊C162
#1163159