Finite Group Databasebeta

← #28504672

Subgroups — #28504672

Order 15104.

680 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16837195C212yesmaximal
2#21424084C212yesmaximal, complemented
3#21422221C212yesmaximal, complemented
4C4×#127×C59C211yesmaximal, characteristic
5C2×#14273642C2214yes—
6#16837262—22no—
7#16837195—24no—
8#16837169—22no—
9C2×#127×C59C2211yescharacteristic
10C2×C4×D4×C59—22no—
11C2×C4×D4×C59C411yescharacteristic, complemented
12C2×C4×D4×C59C2211yescharacteristic
13C59×#390—22no—
14C59×#390C411yescharacteristic, complemented
15C59×#390C2211yescharacteristic
16C59×#384C411yescharacteristic, complemented
17C43×C59C411yescharacteristic, complemented
18C2×C4⋊Dic59—42no—
19C2×C22⋊Dic59—24no—
20#14273642—24no—
21#14273642—24no—
22C59⋊#95—42no—
23C22×D4×C59—22no—
24C22×D4×C59C2×C411yescharacteristic
25C22×D4×C59C2311yescharacteristic
26C23×C4×C59—21no—
27C23×C4×C59D412yes—
28#127×C59—22no—
29#127×C59D412yes—
30#127×C59Q812yes—
31C4×D4×C59—41no—
32C4×D4×C59—41no—
33C4×D4×C59—48no—
34C4×D4×C59—41no—
35C4×D4×C59—41no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59C2×C411yescharacteristic
38C2×C4⋊C4×C59C2×C411yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—21no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59D412yes—
43C2×C42×C59—21no—
44C2×C42×C59—21no—
45C2×C42×C59C2×C411yescharacteristic
46C2×C42×C59C2×C411yescharacteristic
47C2×C42×C59C2×C411yescharacteristic
48C24×C59—21no—
49C24×C59C2×D412yes—
50C2×D4×C59—24no—
51C2×D4×C59—24no—
52C2×D4×C59—21no—
53C2×D4×C59—41no—
54C2×D4×C59—41no—
55C2×D4×C59—44no—
56C2×D4×C59—44no—
57C2×D4×C59—41no—
58C2×D4×C59—41no—
59C2×D4×C59C4⋊C412yescomplemented
60C2×D4×C59C2×D411yescharacteristic
61C2×D4×C59C2×Q811yescharacteristic
62C2×D4×C59#4112yes—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—21no—
68C22×C4×C59—44no—
69C22×C4×C59—44no—
70C22×C4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C412yes—
75C22×C4×C59C22×C411yescharacteristic
76C22×Dic59—44no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—44no—
79C4⋊C4×C59—41no—
80C22⋊C4×C59—42no—
81C22⋊C4×C59—44no—
82C22⋊C4×C59—44no—
83C22⋊C4×C59—42no—
84C42×C59—24no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—42no—
89C42×C59—41no—
90C42×C59C4⋊C412yescomplemented
91C22⋊Dic59—44no—
92C22⋊Dic59—44no—
93C4⋊Dic59—84no—
94C23×C59—22no—
95C23×C59—22no—
96C23×C59—22no—
97C23×C59—42no—
98C23×C59—42no—
99C23×C59—42no—
100C23×C59—42no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104C23×C59#12112yes—
105C23×C59C22⋊Q812yes—
106D4×C59—41no—
107D4×C59—41no—
108D4×C59—42no—
109D4×C59—42no—
110D4×C59—48no—
111D4×C59—48no—
112D4×C59—42no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—21no—
121C2×C4×C59—22no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—44no—
125C2×C4×C59—41no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—41no—
129C2×C4×C59—82no—
130C2×C4×C59—82no—
131C2×C4×C59—82no—
132C2×C4×C59#9911yescharacteristic
133C2×C4×C59#9911yescharacteristic
134C2×C4×C59C2×C4⋊C411yescharacteristic
135C2×C4×C59#11711yescharacteristic
136C2×Dic59—84no—
137C2×Dic59—84no—
138C2×Dic59—84no—
139#12797—591nomaximal, Sylow
140C22×C59—21no—
141C22×C59—21no—
142C22×C59—22no—
143C22×C59—22no—
144C22×C59—42no—
145C22×C59—42no—
146C22×C59—44no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59—82no—
151C22×C59—82no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59#38011yescharacteristic
155C22×C59C2×#9911yescharacteristic
156C22×C59#41011yescharacteristic
157C4×C59—24no—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—42no—
162C4×C59—42no—
163C4×C59—41no—
164C4×C59—81no—
165C4×C59#35312yescomplemented
166Dic59—164no—
167C4×#127D5911yescharacteristic
168C2×#353—592no—
169#1799—592no—
170#1393—592no—
171C2×C59—21no—
172C2×C59—21no—
173C2×C59—42no—
174C2×C59—42no—
175C2×C59—82no—
176C2×C59#179911yescharacteristic
177C2×C59C2×#35311yescharacteristic
178C2×C59#212611yescharacteristic
179C2×#127C2×D5911yescharacteristic
180C2×C4×D4—22no—
181C2×C4×D4Dic5911yescharacteristic, complemented
182C2×C4×D4C2×D5911yescharacteristic
183C2×#99—594no—
184#390—22no—
185#390Dic5911yescharacteristic, complemented
186#390C2×D5911yescharacteristic
187#384Dic5911yescharacteristic, complemented
188#380—1182no—
189C43Dic5911yescharacteristic, complemented
190#353—1184no—
191#342—1182no—
192C59#1279711yescharacteristic, complemented, Sylow
193C22×D4—22no—
194C22×D4C2×Dic5911yescharacteristic
195C22×D4C22×D5911yescharacteristic
196C23×C4—21no—
197C23×C4C22⋊D5912yes—
198#127—22no—
199#127Dic11812yes—
200#127D23612yes—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—48no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—2362no—
208C2×C4⋊C4C2×Dic5911yescharacteristic
209C2×C4⋊C4C2×Dic5911yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—1184no—
214C2×C22⋊C4C22⋊D5912yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C2×Dic5911yescharacteristic
218C2×C42C2×Dic5911yescharacteristic
219C2×C42C2×Dic5911yescharacteristic
220#99—1184no—
221#99—1184no—
222#95—2362no—
223C24—21no—
224C24C2×C22⋊D5912yes—
225C2×D4—24no—
226C2×D4—24no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4C4⋊Dic5912yescomplemented
235C2×D4C2×Dic11811yescharacteristic
236C2×D4C2×D23611yescharacteristic
237C2×D4#1174568512yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—2364no—
248C22×C4C22⋊Dic5911yescharacteristic
249C22×C4C22⋊Dic5912yes—
250C22×C4C22⋊Dic5911yescharacteristic
251C22×C4C22×Dic5911yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—4724no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—2364no—
261C22⋊C4—2364no—
262C42—24no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—41no—
267C42—42no—
268C42C4⋊Dic5912yescomplemented
269C23—22no—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—41no—
278C23C2×C22⋊Dic5911yescharacteristic
279C23#1427370112yes—
280C23#1427370612yes—
281D4—41no—
282D4—41no—
283D4—42no—
284D4—42no—
285D4—48no—
286D4—48no—
287D4—42no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—4724no—
308C2×C4—4724no—
309C2×C4—4724no—
310C2×C4#1427364211yescharacteristic
311C2×C4#1427364211yescharacteristic
312C2×C4C2×C4⋊Dic5911yescharacteristic
313C2×C4#1427368611yescharacteristic
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1683726211yescharacteristic
329C22C2×#1427364211yescharacteristic
330C22#1683737611yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—42no—
336C4—42no—
337C4—41no—
338C4—81no—
339C4—9444no—
340C4#1683719512yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—42no—
345C2—82no—
346C2#2142408411yescharacteristic
347C2C2×#1683719511yescharacteristic
348C2#2142576111yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#127×C59C2
socleC22×C59C2×#99
prime radicalC2×#127×C59C22
cosocleC22×D4×C59C23
hypercenterC4×#127D59
hypocenterC59#12797
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. #21424084C2semidirect
3. #21422221C2semidirect
11. C2×C4×D4×C59C4semidirect
14. C59×#390C4semidirect
16. C59×#384C4semidirect
17. C43×C59C4semidirect
59. C2×D4×C59C4⋊C4semidirect
90. C42×C59C4⋊C4semidirect
165. C4×C59#353semidirect
181. C2×C4×D4Dic59semidirect
185. #390Dic59semidirect
187. #384Dic59semidirect
189. C43Dic59semidirect
192. C59#12797semidirect
234. C2×D4C4⋊Dic59semidirect
268. C42C4⋊Dic59semidirect
340. C4#16837195semidirect

Maximal subgroups

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

subgroupindex
C2×#168371952
#214240842
#214222212
C4×#127×C592
#1279759