Finite Group Databasebeta

← #28486356

Subgroups — #28486356

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#21433125C211yesmaximal, characteristic
2#21429292C211yesmaximal, characteristic
3#21422722C212yesmaximal, complemented
4#21421953C212yesmaximal, complemented
5C59×#1466C211yesmaximal, characteristic
6C2×#14273719C2211yescharacteristic
7#16837474—24no—
8#16837459C2212yes—
9#16837366—21no—
10C2×C22⋊(C59⋊C8)—21no—
11#16837325—21no—
12#16837325—21no—
13#16837305C411yescharacteristic
14#16837305C411yescharacteristic
15#16837275—21no—
16#16837275—21no—
17#16837271C411yescharacteristic
18#16837267C411yescharacteristic
19C4×C4⋊Dic59C2211yescharacteristic
20#16837143—24no—
21C2×C4×D4×C59C2211yescharacteristic
22C59×#432C2212yes—
23C59×#323—21no—
24C59×#323—21no—
25C2×#11745664—22no—
26C22×C59⋊C8—22no—
27#14273719—22no—
28#14273719—22no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—21no—
35C2×Dic59⋊C4—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C4×Dic59C2×C411yescharacteristic
41C2×C4×Dic59C2×C411yescharacteristic
42C4⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)—22no—
44C22⋊(C59⋊C8)—22no—
45C22⋊(C59⋊C8)—22no—
46#14273638D414yes—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C22×D4×C59C2×C411yescharacteristic
51C23×C4×C59D411yescharacteristic
52C23×C4×C59D411yescharacteristic
53C4×D4×C59—22no—
54C4×D4×C59—22no—
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C22⋊C4×C59D411yescharacteristic
57C2×C22⋊C4×C59D411yescharacteristic
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—22no—
60C22⋊C8×C59—22no—
61#97×C59—22no—
62C24×C59C22⋊C411yescharacteristic
63C24×C59C22⋊C411yescharacteristic
64C2×D4×C59—22no—
65C2×D4×C59—22no—
66C2×D4×C59—41no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—81no—
72C22×C4×C59—81no—
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×Dic59—21no—
79C22×Dic59—21no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C2×C8×C59—42no—
83C4⋊C4×C59—22no—
84C22⋊C4×C59—22no—
85C22⋊C4×C59—22no—
86C42×C59C2×D412yes—
87C42×C59#4112yes—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C22⋊Dic59—81no—
91C22⋊Dic59—81no—
92C4⋊Dic59—22no—
93C4⋊Dic59—22no—
94C4⋊Dic59—42no—
95Dic59⋊C4—41no—
96Dic59⋊C4—42no—
97Dic59⋊C4—41no—
98Dic59⋊C4—81no—
99Dic59⋊C4—81no—
100C4×Dic59—41no—
101C4×Dic59—41no—
102#11745664—82no—
103C2×C59⋊C8—24no—
104C2×C59⋊C8—82no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59#10112yes—
115C23×C59C2×C22⋊C411yescharacteristic
116D4×C59—44no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—81no—
128C2×C4×C59—81no—
129C2×C4×C59C4×D412yes—
130C2×C4×C59#12011yescharacteristic
131C2×C4×C59#12112yes—
132C2×C4×C59#12311yescharacteristic
133C2×Dic59—22no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C8×C59—82no—
146C59⋊C8—44no—
147#9449—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59—22no—
151C22×C59—42no—
152C22×C59—42no—
153C22×C59—41no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59#38611yescharacteristic
161C22×C59C2×#10111yescharacteristic
162C22×C59#41311yescharacteristic
163C4×C59—22no—
164C4×C59—22no—
165C4×C59—21no—
166C4×C59—21no—
167C4×C59—41no—
168C4×C59—41no—
169Dic59—42no—
170Dic59—81no—
171Dic59—81no—
172Dic59—81no—
173#2973—591no—
174#2566—591no—
175#1534—592no—
176#1466D5911yescharacteristic
177#1285—592no—
178C2×C59—21no—
179C2×C59—21no—
180C2×C59—42no—
181C2×C59—81no—
182C2×C59#188411yescharacteristic
183C2×C59#190111yescharacteristic
184C2×C59#191111yescharacteristic
185C2×#125—591no—
186C2×C4×D4C2×D5911yescharacteristic
187#435—1184no—
188#432—592no—
189#432C2×D5912yes—
190#407—1181no—
191C2×C22⋊C8—1181no—
192#396—1181no—
193#396—1181no—
194#390—591no—
195#390—591no—
196#385—1181no—
197#385—1181no—
198#384—591no—
199#382—591no—
200C4×C4⋊C4—591no—
201#330—1184no—
202#323—21no—
203#323—21no—
204C59#944911yescharacteristic, complemented, Sylow
205C22×D4C4×D5911yescharacteristic
206C23×C4D23611yescharacteristic
207C23×C4D23611yescharacteristic
208C2×M16—1182no—
209C22×C8—1182no—
210#125—1182no—
211#125—1182no—
212C4×D4—22no—
213C4×D4—22no—
214C2×C4⋊C4—591no—
215C2×C4⋊C4—591no—
216C2×C4⋊C4—1181no—
217C2×C4⋊C4—1181no—
218C2×C4⋊C4—1181no—
219C2×C4⋊C4—1181no—
220C2×C4⋊C4—1181no—
221C2×C4⋊C4C4×D5911yescharacteristic
222C2×C22⋊C4—1181no—
223C2×C22⋊C4—1181no—
224C2×C22⋊C4—1181no—
225C2×C22⋊C4—1181no—
226C2×C22⋊C4C22⋊D5911yescharacteristic
227C2×C22⋊C4C22⋊D5911yescharacteristic
228C2×C42—591no—
229C2×C42—591no—
230C2×C42C22×D5911yescharacteristic
231C4⋊C8—22no—
232C4⋊C8—1182no—
233C22⋊C8—22no—
234C22⋊C8—1182no—
235C22⋊C8—1182no—
236C22⋊C8—1182no—
237#97—22no—
238#97—594no—
239#95—1181no—
240#95—1181no—
241#95—1181no—
242C24(C2×D59)⋊C411yescharacteristic
243C24(C2×D59)⋊C411yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—1181no—
254C22×C4—1181no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4(C2×D59)⋊C411yescharacteristic
258C22×C4(C2×D59)⋊C411yescharacteristic
259C22×C4C2×C4×D5911yescharacteristic
260C22×C4C2×D23611yescharacteristic
261C22×C4C2×C22⋊D5911yescharacteristic
262M16—4722no—
263C2×C8—42no—
264C2×C8—1184no—
265C2×C8—4722no—
266C4⋊C4—22no—
267C4⋊C4—1182no—
268C4⋊C4—1182no—
269C4⋊C4—4721no—
270C4⋊C4—4721no—
271C4⋊C4—2362no—
272C4⋊C4—2361no—
273C4⋊C4—2362no—
274C4⋊C4—2361no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—4721no—
278C22⋊C4—4721no—
279C22⋊C4—2362no—
280C22⋊C4—2362no—
281C42—2361no—
282C42—2361no—
283C42D4×D5912yes—
284C42#1174568712yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1427364512yes—
295C23C2×(C2×D59)⋊C411yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—1182no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4—2361no—
319C2×C4—2361no—
320C2×C4—2361no—
321C2×C4(C22⋊D59)⋊C412yes—
322C2×C4#1427369711yescharacteristic
323C2×C4#1427369912yes—
324C2×C4#1427371111yescharacteristic
325C8—82no—
326C8—2364no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1683727911yescharacteristic
340C22C2×#1427364511yescharacteristic
341C22#1683738411yescharacteristic
342C4—22no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—4721no—
349C4—4721no—
350C4—4721no—
351C4—2362no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#2142447711yescharacteristic
357C2#2142456211yescharacteristic
358C2#2142462611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1466C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#1466D59
hypocenterC59#9449
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #21422722C2semidirect
4. #21421953C2semidirect
204. C59#9449semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214292922
#214227222
#214219532
C59×#14662
#944959