Finite Group Databasebeta

← #28564951

Subgroups — #28564951

Order 15104.

1004 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 195 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#21434553C211yesmaximal, characteristic
2#21434218C212yesmaximal
3#21423236C214yesmaximal, complemented
4#21423174C214yesmaximal, complemented
5#21422822C211yesmaximal, characteristic, complemented
6#21422543C212yesmaximal, complemented
7C2×Q82×C59C211yesmaximal, characteristic, complemented
8C2×#14273758—22no—
9#16837952C2211yescharacteristic
10#16837794—22no—
11#16837781C2212yes—
12#16837776C2212yes—
13#16837715—22no—
14#16837697C2214yes—
15#16837602C2214yes—
16#16837541—22no—
17#16837500—24no—
18#16837459C2212yes—
19C4×#11745664C2211yescharacteristic
20#16837151C2214yes—
21Q8⋊(C59⋊C8)C2218yes—
22Q82×C59—22no—
23Q82×C59—24no—
24Q82×C59C2214yes—
25C2×C4⋊Q8×C59—22no—
26C2×C4⋊Q8×C59—22no—
27C2×C4⋊Q8×C59C2211yescharacteristic
28C2×C4×Q8×C59—22no—
29C2×C4×Q8×C59C2212yes—
30C2×#11745685—21no—
31#14273758—48no—
32C2×#11745672—24no—
33C2×#11745671—24no—
34C2×#11745671—24no—
35C2×#11745664—21no—
36#14273727C2311yescharacteristic
37#14273724C2311yescharacteristic
38#14273715—21no—
39#14273701—22no—
40C4×D236—22no—
41#14273660—24no—
42C4⋊(C59⋊C8)C2314yes—
43#14273651—24no—
44#14273647—24no—
45C22⋊(C59⋊C8)—22no—
46#14273638—22no—
47#14273638—21no—
48C4×C59⋊C8C2312yes—
49C22×Q8×C59—22no—
50C22×Q8×C59D412yes—
51C22×Q8×C59D412yes—
52C4⋊Q8×C59—28no—
53C4⋊Q8×C59—28no—
54C4⋊Q8×C59—28no—
55C4⋊Q8×C59—28no—
56C4⋊Q8×C59—22no—
57C4⋊Q8×C59D412yes—
58C4⋊Q8×C59C2312yes—
59C4×Q8×C59—28no—
60C4×Q8×C59—28no—
61C4×Q8×C59—22no—
62C4×Q8×C59—22no—
63C4×Q8×C59D414yes—
64C4×Q8×C59C2314yes—
65C2×C4⋊C4×C59—24no—
66C2×C4⋊C4×C59—22no—
67C2×C4⋊C4×C59—22no—
68C2×C4⋊C4×C59D412yes—
69C2×C42×C59—22no—
70C2×C42×C59—22no—
71C2×C42×C59C2311yescharacteristic
72C2×Q8×C59—28no—
73C2×Q8×C59—28no—
74C2×Q8×C59—22no—
75C2×Q8×C59—22no—
76C2×Q8×C59—44no—
77C2×Q8×C59—84no—
78C2×Q8×C59C2×D414yes—
79C2×Q8×C59C2×D414yes—
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—22no—
83C22×C4×C59C2×D412yes—
84C22×C4×C59C2×D411yescharacteristic
85C2×C22⋊D59—41no—
86#11745685—44no—
87C2×D236—22no—
88C2×D236—21no—
89C2×C4×D59—41no—
90C2×Dic118—21no—
91C4⋊C4×C59—216no—
92C4⋊C4×C59—24no—
93C4⋊C4×C59—24no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—24no—
96C4⋊C4×C59—24no—
97C4⋊C4×C59C2×D414yes—
98C42×C59—28no—
99C42×C59—24no—
100C42×C59—24no—
101C42×C59C2×D411yescharacteristic
102C42×C59C2×D412yes—
103C42×C59C2411yescharacteristic
104#11745672—48no—
105#11745671—48no—
106#11745671—48no—
107(C2×D59)⋊C4—42no—
108C4⋊Dic59—22no—
109#11745664—24no—
110C2×C59⋊C8—22no—
111C2×C59⋊C8—24no—
112C23×C59#12011yescharacteristic
113C22×D59—41no—
114Q8×C59—28no—
115Q8×C59—24no—
116Q8×C59—44no—
117Q8×C59—82no—
118Q8×C59—88no—
119Q8×C59#13718yescomplemented
120C2×C4×C59—28no—
121C2×C4×C59—24no—
122C2×C4×C59—22no—
123C2×C4×C59—24no—
124C2×C4×C59—24no—
125C2×C4×C59—24no—
126C2×C4×C59—44no—
127C2×C4×C59—44no—
128C2×C4×C59#12012yes—
129C2×C4×C59#12011yescharacteristic
130C2×C4×C59C22×D411yescharacteristic
131C2×C4×C59C22×D412yes—
132C22⋊D59—82no—
133C2×Dic59—41no—
134D236—42no—
135D236—44no—
136C4×D59—82no—
137Dic118—42no—
138C59⋊C8—24no—
139C59⋊C8—44no—
140#18546—591nomaximal, Sylow
141C22×C59—22no—
142C22×C59#45312yes—
143C22×C59C2×#12011yescharacteristic
144C2×D59—82no—
145C2×D59—81no—
146C4×C59—28no—
147C4×C59—22no—
148C4×C59—21no—
149C4×C59—21no—
150C4×C59—44no—
151C4×C59—44no—
152C4×C59C2×#13714yes—
153Dic59—81no—
154C2×Q82D5911yescharacteristic, complemented
155#3152—591no—
156#3112—592no—
157#1633—594no—
158#1623—594no—
159#1558—591no—
160#1491—592no—
161C2×C59—22no—
162C2×C59#300412yes—
163C2×C59C2×#45311yescharacteristic
164D59—161no—
165C2×#137—1182no—
166Q82—22no—
167Q82—24no—
168Q82C2×D5914yes—
169#532—591no—
170C2×C4⋊Q8—22no—
171C2×C4⋊Q8—22no—
172C2×C4⋊Q8C2×D5911yescharacteristic
173C2×C4×Q8—22no—
174C2×C4×Q8C2×D5912yes—
175#497—1182no—
176#494—592no—
177#492—592no—
178#480—1182no—
179#477—594no—
180#460—594no—
181#448—1182no—
182#440—1184no—
183#432—592no—
184C4×M16—591no—
185#333—594no—
186Q8⋊C8—598no—
187C59#1854611yescharacteristic, complemented, Sylow
188C2×#41—1181no—
189C22×Q8—22no—
190C22×Q8C22⋊D5912yes—
191C22×Q8C22⋊D5912yes—
192#137—2368no—
193C2×Q16—1184no—
194C2×SD16—1184no—
195C2×SD16—1184no—
196C2×M16—1181no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—591no—
203C4⋊Q8C22⋊D5912yes—
204C4⋊Q8C22×D5912yes—
205#127—591no—
206#124—1181no—
207#121—1182no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8C22⋊D5914yes—
213C4×Q8C22×D5914yes—
214C4×D4—1182no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C22⋊D5912yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C22×D5911yescharacteristic
222#107—1184no—
223C4⋊C8—594no—
224#103—1184no—
225#102—1184no—
226C22⋊C8—1182no—
227#97—1181no—
228#97—1182no—
229C4×C8—592no—
230#41—2364no—
231C2×Q8—28no—
232C2×Q8—28no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—84no—
237C2×Q8—1181no—
238C2×Q8C2×C22⋊D5914yes—
239C2×Q8C2×C22⋊D5914yes—
240C2×D4—1182no—
241C2×D4—1181no—
242C2×D4—2361no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—2361no—
247C22×C4C2×C22⋊D5912yes—
248C22×C4C2×C22⋊D5911yescharacteristic
249Q16—2368no—
250SD16—2368no—
251SD16—2368no—
252M16—1184no—
253C2×C8—1182no—
254C2×C8—1184no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—1182no—
262C4⋊C4C2×C22⋊D5914yes—
263C22⋊C4—2362no—
264C42—28no—
265C42—24no—
266C42—24no—
267C42C2×C22⋊D5912yes—
268C42C2×C22⋊D5911yescharacteristic
269C42C23×D5911yescharacteristic
270C23—2361no—
271C23#1427369611yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275Q8—82no—
276Q8—88no—
277Q8—2362no—
278Q8#1427375818yescomplemented
279D4—4722no—
280D4—2362no—
281D4—2364no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—4722no—
291C2×C4—2361no—
292C2×C4#1427369612yes—
293C2×C4#1427369611yescharacteristic
294C2×C4C22×C22⋊D5911yescharacteristic
295C2×C4C22×C22⋊D5912yes—
296C8—1184no—
297C8—2364no—
298C22—22no—
299C22—4721no—
300C22—4722no—
301C22#1683757012yes—
302C22C2×#1427369611yescharacteristic
303C4—21no—
304C4—21no—
305C4—28no—
306C4—22no—
307C4—44no—
308C4—44no—
309C4—4721no—
310C4C2×#1427375814yes—
311C2—22no—
312C2—9441no—
313C2#2143333012yes—
314C2C2×#1683757011yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC2×Q82×C59C2
socleC22×C59C2×#120
prime radical#16837952C22
hypercenterC2×Q82D59
hypocenterC59#18546
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #21423236C2semidirect
4. #21423174C2semidirect
5. #21422822C2semidirect
6. #21422543C2semidirect
7. C2×Q82×C59C2semidirect
119. Q8×C59#137semidirect
154. C2×Q82D59semidirect
187. C59#18546semidirect
278. Q8#14273758semidirect

Maximal subgroups

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

subgroupindex
#214345532
#214342182
#214232362
#214231742
#214228222
#214225432
C2×Q82×C592
#1854659