Finite Group Databasebeta

← #28501669

Subgroups — #28501669

Order 15104.

398 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425449C211yesmaximal, characteristic
2#21423086C211yesmaximal, characteristic
3#21423086C211yesmaximal, characteristic
4#21422096C211yesmaximal, characteristic
5#21422096C211yesmaximal, characteristic
6#21422044C211yesmaximal, characteristic
7C59×#2722C211yesmaximal, characteristic
8#16838028C2211yescharacteristic
9#16837732—21no—
10#16837732—21no—
11#16837275—21no—
12#16837169C2211yescharacteristic
13#16837168—21no—
14#16837138—21no—
15#16837138—21no—
16#16837126C2211yescharacteristic
17#16837126C2211yescharacteristic
18C2×#126×C59—21no—
19C59×#398—21no—
20C59×#398—21no—
21C59×#397C2211yescharacteristic
22C59×#397C2211yescharacteristic
23C59×#389—21no—
24C59×#387—21no—
25C59×#385C2211yescharacteristic
26C59×#380—21no—
27#14273706—21no—
28#14273706—21no—
29#14273704—41no—
30#14273704—41no—
31C2×Dic59⋊C4—41no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34#14273658—41no—
35#14273658—41no—
36#14273650—41no—
37#14273650—41no—
38C22⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C59⋊#95—41no—
41C23×C4×C59D411yescharacteristic
42#126×C59—42no—
43#126×C59—42no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59C2311yescharacteristic
56C2×C42×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62#95×C59D411yescharacteristic
63#95×C59D411yescharacteristic
64C24×C59C2×D411yescharacteristic
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—42no—
72C22×C4×C59—41no—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—41no—
78C2×Dic118—21no—
79C2×Dic118—21no—
80C4⋊C4×C59—42no—
81C4⋊C4×C59—42no—
82C4⋊C4×C59—42no—
83C4⋊C4×C59—42no—
84C4⋊C4×C59—42no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—81no—
88C22⋊C4×C59—21no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59SD1612yes—
95C42×C59—42no—
96C22⋊Dic59—22no—
97C22⋊Dic59—41no—
98C22⋊Dic59—41no—
99C22⋊Dic59—41no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102Dic59⋊C4—41no—
103Dic59⋊C4—41no—
104Dic59⋊C4—82no—
105C2×C59⋊C8—41no—
106C2×C59⋊C8—41no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59#12011yescharacteristic
112C23×C59C2×SD1611yescharacteristic
113C23×C59AGL(1,ℤ/8)11yescharacteristic
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—81no—
138C2×Dic59—42no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—81no—
142C2×Dic59—81no—
143C2×Dic59—81no—
144C2×Dic59—81no—
145Dic118—81no—
146Dic118—81no—
147C59⋊C8—81no—
148C59⋊C8—81no—
149#12244—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—41no—
155C22×C59—41no—
156C22×C59—41no—
157C22×C59—41no—
158C22×C59—81no—
159C22×C59#45011yescharacteristic
160C22×C59#45611yescharacteristic
161C22×C59B(4,𝔽2)11yescharacteristic
162C4×C59—42no—
163C4×C59—41no—
164C4×C59—42no—
165C4×C59—41no—
166C4×C59—81no—
167C4×C59—81no—
168C4×C59—81no—
169C4×C59—81no—
170Dic59—82no—
171Dic59—81no—
172Dic59—81no—
173#2722D5911yescharacteristic
174#2062—591no—
175#1608—591no—
176#1608—591no—
177#1344—591no—
178#1344—591no—
179#1326—591no—
180C2×C59—21no—
181C2×C59—21no—
182C2×C59—41no—
183C2×C59—41no—
184C2×C59#160411yescharacteristic
185C2×C59#219711yescharacteristic
186C2×C59#220111yescharacteristic
187#543—591no—
188C2×#126—21no—
189#483—1181no—
190#483—1181no—
191#398—21no—
192#398—21no—
193#397C2×D5911yescharacteristic
194#397C2×D5911yescharacteristic
195#389—21no—
196#387—21no—
197#385—1181no—
198#385C2×D5911yescharacteristic
199#380—21no—
200#342—591no—
201#342—1181no—
202#328—1181no—
203#328—1181no—
204#323—591no—
205#323—591no—
206C59#1224411yescharacteristic, complemented, Sylow
207C23×C4C22⋊D5911yescharacteristic
208#126—42no—
209#126—42no—
210C22⋊Q8—1181no—
211C22⋊Q8—1181no—
212C22⋊Q8—2361no—
213C22⋊Q8—2361no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—2361no—
221C2×C4⋊C4C22⋊D5911yescharacteristic
222C2×C4⋊C4C22⋊D5911yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—1181no—
226C2×C22⋊C4—1181no—
227C2×C22⋊C4C22⋊D5911yescharacteristic
228C2×C22⋊C4C22×D5911yescharacteristic
229C2×C42—21no—
230#106—2361no—
231#106—2361no—
232#103—2361no—
233#103—2361no—
234C22⋊C8—1181no—
235C22⋊C8—1181no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—2361no—
242#95C22⋊D5911yescharacteristic
243#95C22⋊D5911yescharacteristic
244C24C2×C22⋊D5911yescharacteristic
245C2×Q8—1181no—
246C2×Q8—1181no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4—2361no—
258C22×C4C2×C22⋊D5911yescharacteristic
259C22×C4C2×C22⋊D5911yescharacteristic
260C2×C8—2361no—
261C2×C8—2361no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—81no—
270C4⋊C4—1181no—
271C4⋊C4—1181no—
272C4⋊C4—4722no—
273C4⋊C4—2361no—
274C4⋊C4—2361no—
275C22⋊C4—21no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—1182no—
282C22⋊C4—2361no—
283C22⋊C4—2361no—
284C22⋊C4—2361no—
285C22⋊C4#1174567012yes—
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#1427369611yescharacteristic
292C23C2×#1174567011yescharacteristic
293C23#1427375311yescharacteristic
294Q8—4721no—
295Q8—4721no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—4721no—
321C2×C4—4721no—
322C2×C4—4721no—
323C2×C4—4721no—
324C2×C4—2362no—
325C2×C4—2361no—
326C2×C4—2361no—
327C8—4721no—
328C8—4721no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22#1683755311yescharacteristic
339C22#1683758511yescharacteristic
340C22#1683758911yescharacteristic
341C4—42no—
342C4—41no—
343C4—42no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—4722no—
350C4—4721no—
351C4—4721no—
352C2—21no—
353C2—21no—
354C2—41no—
355C2—41no—
356C2#2142306511yescharacteristic
357C2#2142608611yescharacteristic
358C2#2142610911yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2722C2
socleC22×C59B(4,𝔽2)
prime radicalC24×C59C2×D4
hypercenter#2722D59
hypocenterC59#12244
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2722

Complements

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

NH ≅ G/Ntype
206. C59#12244semidirect

Maximal subgroups

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

subgroupindex
#214254492
#214230862
#214220962
#214220442
C59×#27222
#1224459