Finite Group Databasebeta

← #28498319

Subgroups — #28498319

Order 15104.

412 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 91 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#21422440C211yesmaximal, characteristic, complemented
2#21422420C211yesmaximal, characteristic, complemented
3#21422030C212yesmaximal, complemented
4#21422028C212yesmaximal
5C59×#2545C211yesmaximal, characteristic
6#16837459—21no—
7#16837455—21no—
8#16837128—21no—
9#16837128—21no—
10#16837128C2212yes—
11#16837128C2212yes—
12C2×C4⋊Q8×C59C411yescharacteristic
13C2×#124×C59C411yescharacteristic
14C2×#124×C59C411yescharacteristic
15C2×C4×Q8×C59C2211yescharacteristic
16C59×#396—21no—
17C59×#396—21no—
18C59×#391C2211yescharacteristic
19C59×#390C2211yescharacteristic
20C59×#388—21no—
21C59×#388—21no—
22C4×C4⋊C4×C59C411yescharacteristic
23C4⋊(C59⋊C8)—41no—
24C4⋊(C59⋊C8)—41no—
25C22⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—22no—
27#14273638—41no—
28C4×C59⋊C8—41no—
29C22×Q8×C59C2×C411yescharacteristic
30C22×Q8×C59C2×C411yescharacteristic
31C22×D4×C59C2×C411yescharacteristic
32C4⋊Q8×C59—42no—
33#124×C59—41no—
34#124×C59—41no—
35#124×C59—42no—
36C4×Q8×C59—41no—
37C4×Q8×C59—41no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C4⋊C4×C59C2311yescharacteristic
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C42×C59C2×C411yescharacteristic
51C2×C42×C59C2×C411yescharacteristic
52C2×C42×C59C2×C411yescharacteristic
53C2×C42×C59D411yescharacteristic
54C2×C42×C59D411yescharacteristic
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58C24×C59—21no—
59C2×Q8×C59—22no—
60C2×Q8×C59—22no—
61C2×Q8×C59—22no—
62C2×Q8×C59—41no—
63C2×Q8×C59—41no—
64C2×Q8×C59—41no—
65C2×D4×C59—22no—
66C2×D4×C59—41no—
67C2×D4×C59—41no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—42no—
82C4⋊C4×C59—42no—
83C4⋊C4×C59—42no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—42no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—42no—
91C42×C59—21no—
92C42×C59—21no—
93C42×C59—21no—
94C42×C59—21no—
95C42×C59—21no—
96C42×C59—21no—
97C42×C59—41no—
98C42×C59—41no—
99C2×C59⋊C8—42no—
100C2×C59⋊C8—42no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59C2×C22⋊C411yescharacteristic
109Q8×C59—44no—
110Q8×C59—82no—
111D4×C59—82no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—22no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
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#9911yescharacteristic
136C2×C4×C59#9911yescharacteristic
137C2×C4×C59#10012yes—
138C2×C4×C59#13112yes—
139C59⋊C8—82no—
140C59⋊C8—82no—
141#11565—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59—81no—
145C22×C59—81no—
146C22×C59—81no—
147C22×C59—81no—
148C22×C59—81no—
149C22×C59—81no—
150C22×C59—81no—
151C22×C59#40811yescharacteristic
152C22×C59C2×#9911yescharacteristic
153C22×C59C2×#10011yescharacteristic
154C4×C59—21no—
155C4×C59—21no—
156C4×C59—22no—
157C4×C59—22no—
158C4×C59—42no—
159C4×C59—41no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—81no—
164C4×C59—81no—
165#2545D5911yescharacteristic
166#1467—591no—
167#1461—591no—
168#1319—592no—
169#1318—592no—
170C2×C59—21no—
171C2×C59—21no—
172C2×C59—161no—
173C2×C59#147111yescharacteristic
174C2×C59#212611yescharacteristic
175C2×C59#213311yescharacteristic
176C2×C4⋊Q8Dic5911yescharacteristic
177C2×#124Dic5911yescharacteristic
178C2×#124Dic5911yescharacteristic
179C2×C4×Q8C2×D5911yescharacteristic
180#432—1181no—
181#431—1181no—
182#396—21no—
183#396—21no—
184#391C2×D5911yescharacteristic
185#390C2×D5911yescharacteristic
186#388—21no—
187#388—21no—
188C4×C4⋊C4Dic5911yescharacteristic
189#324—592no—
190#324—592no—
191#324—1181no—
192#324—1181no—
193C59#1156511yescharacteristic, complemented, Sylow
194C22×Q8C2×Dic5911yescharacteristic
195C22×Q8C2×Dic5911yescharacteristic
196C22×D4C2×Dic5911yescharacteristic
197C4⋊Q8—42no—
198#124—41no—
199#124—41no—
200#124—42no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C22⋊D5911yescharacteristic
207C2×C4⋊C4C22⋊D5911yescharacteristic
208C2×C4⋊C4C22×D5911yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42C2×Dic5911yescharacteristic
216C2×C42C2×Dic5911yescharacteristic
217C2×C42C2×Dic5911yescharacteristic
218C2×C42C22⋊D5911yescharacteristic
219C2×C42C22⋊D5911yescharacteristic
220C4⋊C8—2361no—
221C4⋊C8—2361no—
222C22⋊C8—1182no—
223C22⋊C8—1182no—
224#97—2361no—
225C4×C8—2361no—
226#95—21no—
227#95—21no—
228#95—21no—
229C24—21no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C22⋊Dic5911yescharacteristic
243C22×C4C22⋊Dic5911yescharacteristic
244C22×C4C22⋊Dic5911yescharacteristic
245C22×C4C22⋊Dic5911yescharacteristic
246C22×C4C22×Dic5911yescharacteristic
247C22×C4C2×C22⋊D5911yescharacteristic
248C22×C4C2×C22⋊D5911yescharacteristic
249C2×C8—2362no—
250C2×C8—2362no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—41no—
271C42—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×C22⋊Dic5911yescharacteristic
280Q8—44no—
281Q8—82no—
282D4—82no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4#1427364211yescharacteristic
307C2×C4#1427364211yescharacteristic
308C2×C4#1427364412yes—
309C2×C4#1427373412yes—
310C8—4722no—
311C8—4722no—
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#1683736911yescharacteristic
322C22C2×#1427364211yescharacteristic
323C22C2×#1427364411yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—81no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—161no—
338C2#2142245611yescharacteristic
339C2#2142576111yescharacteristic
340C2#2142578511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2545C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2545D59
hypocenterC59#11565
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2545

Complements

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

NH ≅ G/Ntype
1. #21422440C2semidirect
2. #21422420C2semidirect
3. #21422030C2semidirect
193. C59#11565semidirect

Maximal subgroups

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

subgroupindex
#214224402
#214224202
#214220302
#214220282
C59×#25452
#1156559