Finite Group Databasebeta

← #28485478

Subgroups — #28485478

Order 15104.

416 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 105 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#21433903C211yesmaximal, characteristic
2#21433179C211yesmaximal, characteristic
3#21424108C211yesmaximal, characteristic
4#21422659C211yesmaximal, characteristic
5#21422500C211yesmaximal, characteristic
6#21421991C211yesmaximal, characteristic
7C59×#1286C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9C2×C4×Dic118C2211yescharacteristic
10#16837710—22no—
11#16837698C412yescomplemented
12#16837688C412yescomplemented
13#16837532—22no—
14#16837529—22no—
15#16837456—22no—
16#16837455C2211yescharacteristic
17#16837444—21no—
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19#16837412—21no—
20C2×#14273650—21no—
21#16837164—21no—
22#16837138—22no—
23#16837134—22no—
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C59×#382C2211yescharacteristic
27C22×Dic118D411yescharacteristic
28C22×C59⋊C8—21no—
29#14273727C2×C411yescharacteristic
30#14273727C2×C411yescharacteristic
31#14273719—21no—
32#14273706—21no—
33#14273706—21no—
34C4×Dic118—21no—
35C4×Dic118—21no—
36C4×Dic118—21no—
37C4×Dic118—21no—
38C4×Dic118C2×C412yes—
39C4×Dic118D412yes—
40#14273686—21no—
41C2×C4⋊Dic59D411yescharacteristic
42C2×Dic59⋊C4—21no—
43C2×C4×Dic59—21no—
44#14273661—22no—
45#14273659—22no—
46C4⋊(C59⋊C8)—21no—
47C4⋊(C59⋊C8)—21no—
48C4⋊(C59⋊C8)—21no—
49Dic59⋊C8—24no—
50#14273652—22no—
51#14273652—22no—
52#14273650—42no—
53C22⋊(C59⋊C8)D412yes—
54#14273637—24no—
55C4×C59⋊C8—21no—
56C2×M16×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C42×C59C2311yescharacteristic
59C4⋊C8×C59—21no—
60C4⋊C8×C59—21no—
61C4⋊C8×C59C2×C412yes—
62C22⋊C8×C59—21no—
63C4×C8×C59D412yes—
64#95×C59—21no—
65#95×C59—21no—
66C22×C4×C59—41no—
67C22×C4×C59C2×D411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59#4111yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C2×Dic118—21no—
73C2×Dic118—21no—
74C2×Dic118—21no—
75C2×Dic118—21no—
76C2×Dic118—21no—
77C2×Dic118—21no—
78C2×Dic118—21no—
79C2×Dic118C22⋊C412yes—
80M16×C59—41no—
81M16×C59—41no—
82C2×C8×C59—22no—
83C2×C8×C59—22no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C42×C59C22×C411yescharacteristic
87C42×C59C2×D411yescharacteristic
88C42×C59C2×D411yescharacteristic
89C42×C59#4111yescharacteristic
90C22⋊Dic59—41no—
91C4⋊Dic59—21no—
92C4⋊Dic59—21no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95C4⋊Dic59—21no—
96C4⋊Dic59C22⋊C412yes—
97Dic59⋊C4—22no—
98Dic59⋊C4—22no—
99Dic59⋊C4—41no—
100Dic59⋊C4—41no—
101C4×Dic59—22no—
102C4×Dic59—22no—
103C4×Dic59—41no—
104C2×C59⋊C8—22no—
105C2×C59⋊C8—41no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C23×C59#12111yescharacteristic
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—81no—
112C2×C4×C59—81no—
113C2×C4×C59—81no—
114C2×C4×C59C2×C22⋊C411yescharacteristic
115C2×C4×C59C4×D411yescharacteristic
116C2×C4×C59C4×D411yescharacteristic
117C2×C4×C59#12011yescharacteristic
118C2×C4×C59#12111yescharacteristic
119C2×C4×C59#12311yescharacteristic
120C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C59#13711yescharacteristic
122C2×C4×C59#13711yescharacteristic
123C2×C4×C59#13711yescharacteristic
124C2×Dic59—22no—
125C2×Dic59—22no—
126C2×Dic59—41no—
127C2×Dic59—41no—
128C2×Dic59—41no—
129C2×Dic59—41no—
130C2×Dic59—81no—
131Dic118—22no—
132Dic118—22no—
133Dic118—41no—
134Dic118—41no—
135Dic118—42no—
136Dic118—81no—
137C8×C59—42no—
138C8×C59—42no—
139C59⋊C8—42no—
140C59⋊C8—81no—
141#9308—591nomaximal, Sylow
142C22×C59—21no—
143C22×C59—21no—
144C22×C59#38611yescharacteristic
145C22×C59#46811yescharacteristic
146C22×C59#47011yescharacteristic
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—21no—
150C4×C59—21no—
151C4×C59—161no—
152C4×C59#41711yescharacteristic
153C4×C59#41911yescharacteristic
154C4×C59#46811yescharacteristic
155C4×C59#47011yescharacteristic
156Dic59—42no—
157Dic59—42no—
158Dic59—81no—
159Dic59—81no—
160#3078—591no—
161#2983—591no—
162#1805—591no—
163#1518—591no—
164#1481—591no—
165#1301—591no—
166#1286D5911yescharacteristic
167C2×C59—21no—
168C2×C59—21no—
169C2×C59#188111yescharacteristic
170C2×C59#189411yescharacteristic
171C2×C59#190511yescharacteristic
172#533—591no—
173C2×C4×Q8—591no—
174#479—1182no—
175#477—592no—
176#475—592no—
177#446—1182no—
178#446—1182no—
179#432—1182no—
180#431—591no—
181#431C2×D5911yescharacteristic
182#428—1181no—
183#423C2×D5911yescharacteristic
184C2×C4⋊C8—591no—
185#419—1181no—
186C2×#103—1181no—
187#382C2×D5911yescharacteristic
188#340—1181no—
189#328—1182no—
190Q8⋊C8—1182no—
191C59#930811yescharacteristic, complemented, Sylow
192C22×Q8—591no—
193C2×M16—21no—
194C22×C8—1181no—
195C4⋊Q8—591no—
196C4⋊Q8—591no—
197#125—1181no—
198C22⋊Q8—1181no—
199C22⋊Q8—1181no—
200C4×Q8—592no—
201C4×Q8—592no—
202C4×Q8—1181no—
203C4×Q8—1181no—
204C4×Q8—1181no—
205C4×Q8—1181no—
206#117—1181no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—591no—
209C2×C4⋊C4—1181no—
210C2×C42—1181no—
211C2×C42C22×D5911yescharacteristic
212#107—1182no—
213#106—1182no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—1184no—
217C4⋊C8—1181no—
218C4⋊C8—1181no—
219C4⋊C8—1181no—
220C4⋊C8C4×D5912yes—
221#103—1182no—
222#103—1182no—
223#103—2362no—
224C22⋊C8—21no—
225C22⋊C8—592no—
226#97—1184no—
227C4×C8—1181no—
228C4×C8D23612yes—
229#95—21no—
230#95—21no—
231C2×Q8—592no—
232C2×Q8—1181no—
233C2×Q8—1181no—
234C2×Q8—1181no—
235C2×Q8—1181no—
236C2×Q8—1181no—
237C2×Q8—1181no—
238C2×Q8—1181no—
239C22×C4—41no—
240C22×C4—1181no—
241C22×C4—1181no—
242C22×C4#1174568511yescharacteristic
243C22×C4D4×D5911yescharacteristic
244C22×C4D4×D5911yescharacteristic
245M16—41no—
246M16—41no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8—1182no—
250C2×C8—2361no—
251C2×C8—2361no—
252C2×C8—2361no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—592no—
256C4⋊C4—1182no—
257C4⋊C4—1181no—
258C4⋊C4—1182no—
259C4⋊C4—1181no—
260C4⋊C4—1181no—
261C4⋊C4—1181no—
262C4⋊C4—1181no—
263C4⋊C4—2361no—
264C4⋊C4—2361no—
265C22⋊C4—2361no—
266C42—1182no—
267C42—1182no—
268C42—2361no—
269C42C2×C4×D5911yescharacteristic
270C42C2×D23611yescharacteristic
271C42D4×D5911yescharacteristic
272C42#1174568911yescharacteristic
273C23#1427370011yescharacteristic
274Q8—1182no—
275Q8—1182no—
276Q8—4721no—
277Q8—2362no—
278Q8—2361no—
279Q8—2361no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—1182no—
286C2×C4—1182no—
287C2×C4—4721no—
288C2×C4—2361no—
289C2×C4—2361no—
290C2×C4—2361no—
291C2×C4—2361no—
292C2×C4C22⋊C4×D5911yescharacteristic
293C2×C4#1427368911yescharacteristic
294C2×C4C4×D23611yescharacteristic
295C2×C4#1427369711yescharacteristic
296C2×C4#1427370211yescharacteristic
297C2×C4#1427371211yescharacteristic
298C2×C4#1427375111yescharacteristic
299C2×C4#1427375611yescharacteristic
300C2×C4#1427375611yescharacteristic
301C2×C4#1427375911yescharacteristic
302C8—42no—
303C8—42no—
304C8—4721no—
305C8—2362no—
306C22—21no—
307C22—21no—
308C22#1683728011yescharacteristic
309C22#1683764711yescharacteristic
310C22#1683765911yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—161no—
316C4—4721no—
317C4—4721no—
318C4—2362no—
319C4—2362no—
320C4#1683740011yescharacteristic
321C4#1683740911yescharacteristic
322C4#1683764911yescharacteristic
323C4#1683766111yescharacteristic
324C2—21no—
325C2—21no—
326C2#2142445711yescharacteristic
327C2#2142452811yescharacteristic
328C2#2142459111yescharacteristic
subgroupG/H
centerC22#16837280
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1286C2
socleC22×C59#386
prime radicalC23×C59#121
cosocleC2×C42×C59C23
hypercenter#1286D59
hypocenterC59#9308
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
11. #16837698C4semidirect
12. #16837688C4semidirect
191. C59#9308semidirect

Maximal subgroups

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

subgroupindex
#214339032
#214331792
#214241082
#214226592
#214225002
#214219912
C59×#12862
#930859