Finite Group Databasebeta

← #28499729

Subgroups — #28499729

Order 15104.

440 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 95 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#21429001C211yesmaximal, characteristic
2#21422420C211yesmaximal, characteristic, complemented
3#21422101C212yesmaximal, complemented
4#21422098C212yesmaximal
5C59×#1801C211yesmaximal, characteristic, complemented
6C2×#14273703—21no—
7C2×C4×D236C411yescharacteristic
8#16837455—21no—
9#16837332C411yescharacteristic
10#16837332C2211yescharacteristic
11#16837319C411yescharacteristic
12#16837319C2211yescharacteristic
13#16837293—21no—
14#16837293—21no—
15#16837268C411yescharacteristic
16#16837259—21no—
17#16837128—21no—
18#16837128C2212yes—
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#423—21no—
21C2×#101×C59—21no—
22C59×#343C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C4×D59—21no—
25#14273703—42no—
26#14273703—42no—
27C4×D236—41no—
28C4×D236—41no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C4⋊(C59⋊C8)—41no—
39C22⋊(C59⋊C8)—22no—
40C4×C59⋊C8—41no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C22×Q8×C59D411yescharacteristic
47C2×M16×C59—22no—
48C4×Q8×C59—41no—
49C4×Q8×C59—41no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C4⋊C4×C59C2311yescharacteristic
53C2×C42×C59C2×C411yescharacteristic
54C2×C42×C59D411yescharacteristic
55C2×C42×C59D411yescharacteristic
56C4⋊C8×C59—41no—
57C4⋊C8×C59—41no—
58#101×C59—42no—
59C23×D59—21no—
60C2×Q8×C59—41no—
61C2×Q8×C59—41no—
62C2×Q8×C59D812yes—
63C2×Q8×C59SD1612yes—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C2×D236—41no—
75C2×D236—42no—
76C2×C4×D59—42no—
77C2×C4×D59—42no—
78C2×C4×D59—42no—
79M16×C59—42no—
80M16×C59—42no—
81C2×C8×C59—42no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C42×C59—21no—
86C42×C59—21no—
87C42×C59—21no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—21no—
91(C2×D59)⋊C4—42no—
92(C2×D59)⋊C4—42no—
93(C2×D59)⋊C4—41no—
94(C2×D59)⋊C4—41no—
95(C2×D59)⋊C4—42no—
96(C2×D59)⋊C4—41no—
97(C2×D59)⋊C4—41no—
98C4⋊Dic59—42no—
99C4⋊Dic59—41no—
100Dic59⋊C4—42no—
101Dic59⋊C4—41no—
102Dic59⋊C4—41no—
103C2×C59⋊C8—42no—
104C23×C59C2×C22⋊C411yescharacteristic
105C22×D59—22no—
106C22×D59—41no—
107C22×D59—41no—
108C22×D59—41no—
109C22×D59—41no—
110C22×D59—41no—
111C22×D59—41no—
112Q8×C59—42no—
113Q8×C59—42no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
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#9911yescharacteristic
128C2×C4×C59#9911yescharacteristic
129C2×C4×C59#10212yes—
130C2×C4×C59#10212yes—
131C2×C4×C59C2×D811yescharacteristic
132C2×C4×C59C2×SD1611yescharacteristic
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142D236—42no—
143D236—42no—
144C4×D59—84no—
145C8×C59—82no—
146C59⋊C8—82no—
147#11879—591nomaximal, Sylow
148C22×C59—21no—
149C22×C59—21no—
150C22×C59C2×#9911yescharacteristic
151C22×C59C2×#10211yescharacteristic
152C22×C59#42111yescharacteristic
153C2×D59—42no—
154C2×D59—42no—
155C2×D59—42no—
156C2×D59—81no—
157C2×D59—81no—
158C2×D59—81no—
159C2×D59—81no—
160C4×C59—21no—
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—22no—
164C4×C59—22no—
165C4×C59—21no—
166C4×C59—41no—
167C4×C59—41no—
168C4×C59—41no—
169Dic59—81no—
170Dic59—81no—
171Dic59—81no—
172#2536—591no—
173#1801D5911yescharacteristic, complemented
174#1461—591no—
175#1346—592no—
176#1345—592no—
177C2×C59—21no—
178C2×C59—21no—
179C2×C59#151911yescharacteristic
180C2×C59#212011yescharacteristic
181C2×C59#213311yescharacteristic
182D59—82no—
183C2×C22⋊Q8—1181no—
184C2×C4×Q8C2×D5911yescharacteristic
185C2×C4×D4—591no—
186#431—1181no—
187#423—21no—
188C2×#101—21no—
189#398—591no—
190#398—591no—
191#394—591no—
192#394—591no—
193#388—1181no—
194#388—1181no—
195#382—591no—
196#380—1181no—
197#343C2×D5912yes—
198#324—592no—
199#324—1181no—
200C59#1187911yescharacteristic, complemented, Sylow
201C22×Q8C22⋊D5911yescharacteristic
202C22×D4—591no—
203C23×C4—1181no—
204C2×M16—22no—
205C22⋊Q8—2362no—
206C22⋊Q8—2362no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—2361no—
210C4×D4—2361no—
211C2×C4⋊C4—591no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4—1181no—
215C2×C4⋊C4C4×D5911yescharacteristic
216C2×C4⋊C4D23611yescharacteristic
217C2×C4⋊C4C22×D5911yescharacteristic
218C2×C22⋊C4—1181no—
219C2×C22⋊C4—1181no—
220C2×C22⋊C4—1181no—
221C2×C22⋊C4—1181no—
222C2×C22⋊C4—1181no—
223C2×C42C4×D5911yescharacteristic
224C2×C42D23611yescharacteristic
225C2×C42C22⋊D5911yescharacteristic
226C4⋊C8—41no—
227C4⋊C8—41no—
228C4⋊C8—2361no—
229#101—42no—
230C22⋊C8—1182no—
231C4×C8—2361no—
232#95—591no—
233#95—591no—
234#95—1181no—
235#95—1181no—
236#95—1181no—
237C24—1181no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8#1174566912yes—
241C2×Q8#1174567112yes—
242C2×D4—2361no—
243C2×D4—2362no—
244C22×C4—1181no—
245C22×C4—1181no—
246C22×C4—1181no—
247C22×C4—2362no—
248C22×C4—2362no—
249C22×C4—2362no—
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4(C2×D59)⋊C411yescharacteristic
252C22×C4(C2×D59)⋊C411yescharacteristic
253C22×C4(C2×D59)⋊C411yescharacteristic
254C22×C4C2×C4×D5911yescharacteristic
255C22×C4C2×D23611yescharacteristic
256C22×C4C2×C22⋊D5911yescharacteristic
257M16—42no—
258M16—42no—
259C2×C8—42no—
260C2×C8—2362no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—2361no—
265C4⋊C4—2362no—
266C4⋊C4—2362no—
267C4⋊C4—2361no—
268C4⋊C4—2361no—
269C22⋊C4—2362no—
270C22⋊C4—2362no—
271C22⋊C4—2361no—
272C22⋊C4—2361no—
273C22⋊C4—2362no—
274C22⋊C4—2361no—
275C22⋊C4—2361no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—1182no—
283C23—2361no—
284C23—2361no—
285C23—2361no—
286C23—2361no—
287C23—2361no—
288C23—2361no—
289C23C2×(C2×D59)⋊C411yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—2362no—
293D4—2362no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—4724no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4#1427364111yescharacteristic
318C2×C4#1427364111yescharacteristic
319C2×C4#1427364712yes—
320C2×C4#1427364712yes—
321C2×C4C2×#1174566911yescharacteristic
322C2×C4C2×#1174567111yescharacteristic
323C8—82no—
324C8—4722no—
325C22—21no—
326C22—21no—
327C22—4721no—
328C22—4721no—
329C22—4721no—
330C22—4721no—
331C22—2362no—
332C22—2362no—
333C22—2362no—
334C22C2×#1427364111yescharacteristic
335C22C2×#1427364711yescharacteristic
336C22#1683742011yescharacteristic
337C4—21no—
338C4—22no—
339C4—22no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—4721no—
347C4—4721no—
348C4—4721no—
349C2—21no—
350C2—21no—
351C2—4722no—
352C2#2142266311yescharacteristic
353C2#2142573811yescharacteristic
354C2#2142578311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1801C2
socleC22×C59C2×#99
prime radicalC22×D236C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1801D59
hypocenterC59#11879
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #21422420C2semidirect
3. #21422101C2semidirect
5. C59×#1801C2semidirect
173. #1801D59semidirect
200. C59#11879semidirect

Maximal subgroups

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

subgroupindex
#214290012
#214224202
#214221012
#214220982
C59×#18012
#1187959