Finite Group Databasebeta

← #28486025

Subgroups — #28486025

Order 15104.

582 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 111 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#21432515C211yesmaximal, characteristic, complemented
2#21422694C212yesmaximal, complemented
3#21422430C211yesmaximal, characteristic, complemented
4#21421989C212yesmaximal, complemented
5C59×#2610C211yesmaximal, characteristic
6C22×#11745664—21no—
7#16837459C2212yes—
8#16837358—24no—
9C4×#11745664—21no—
10C4×#11745664C2212yes—
11#16837142—24no—
12#16837128—21no—
13#16837126—21no—
14C2×#124×C59C2211yescharacteristic
15C2×#117×C59C2211yescharacteristic
16C59×#395C411yescharacteristic
17C59×#394C411yescharacteristic
18C59×#393C411yescharacteristic
19C59×#392C411yescharacteristic
20C59×#390—21no—
21C59×#388—21no—
22C59×#388—21no—
23C59×#386—21no—
24C59×#376C2211yescharacteristic
25C2×#11745664—24no—
26C2×#11745664—24no—
27C2×#11745664—22no—
28C2×#11745664—22no—
29C22×C59⋊C8—22no—
30C4⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32#14273638—22no—
33#14273638—22no—
34#14273638D414yescomplemented
35C4×C59⋊C8—22no—
36C4×C59⋊C8—22no—
37C22×Q8×C59C2×C411yescharacteristic
38C22×D4×C59—21no—
39C22×D4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41#124×C59—22no—
42#124×C59—22no—
43#117×C59—22no—
44#117×C59—22no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59C2×C411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C42×C59—21no—
57C2×C42×C59D411yescharacteristic
58C2×C42×C59C2311yescharacteristic
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62#95×C59C2×C411yescharacteristic
63#95×C59C2×C411yescharacteristic
64C24×C59—21no—
65C24×C59C22⋊C411yescharacteristic
66C2×Q8×C59—22no—
67C2×Q8×C59—81no—
68C2×D4×C59—22no—
69C2×D4×C59—42no—
70C2×D4×C59—42no—
71C2×D4×C59—81no—
72C2×D4×C59—81no—
73C22×C4×C59—22no—
74C22×C4×C59—22no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C4⋊C4×C59—22no—
87C4⋊C4×C59—22no—
88C4⋊C4×C59—81no—
89C22⋊C4×C59—22no—
90C22⋊C4×C59—22no—
91C22⋊C4×C59—42no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—42no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—81no—
97C22⋊C4×C59—81no—
98C42×C59—21no—
99C42×C59—21no—
100C42×C59—42no—
101C42×C59C2×D412yes—
102C42×C59#4112yes—
103#11745664—28no—
104#11745664—24no—
105#11745664—24no—
106C2×C59⋊C8—28no—
107C2×C59⋊C8—24no—
108C2×C59⋊C8—42no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59#10412yes—
121C23×C59C2×C22⋊C411yescharacteristic
122Q8×C59—82no—
123D4×C59—82no—
124D4×C59—84no—
125C2×C4×C59—22no—
126C2×C4×C59—22no—
127C2×C4×C59—24no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—22no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59#10412yes—
148C2×C4×C59C4×D412yes—
149C2×C4×C59#12011yescharacteristic
150C2×C4×C59#12112yes—
151C2×C4×C59#12311yescharacteristic
152C59⋊C8—28no—
153C59⋊C8—82no—
154#9387—591nomaximal, Sylow
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—22no—
158C22×C59—42no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—81no—
162C22×C59—81no—
163C22×C59—81no—
164C22×C59—81no—
165C22×C59—81no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59#38611yescharacteristic
170C22×C59C2×#10411yescharacteristic
171C22×C59C2×#10411yescharacteristic
172C4×C59—22no—
173C4×C59—21no—
174C4×C59—22no—
175C4×C59—21no—
176C4×C59—22no—
177C4×C59—42no—
178C4×C59—41no—
179C4×C59—41no—
180C4×C59—81no—
181C4×C59—81no—
182#2873—591no—
183#2610D5911yescharacteristic
184#1527—592no—
185#1464—591no—
186#1300—592no—
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—42no—
190C2×C59—161no—
191C2×C59#188411yescharacteristic
192C2×C59#189611yescharacteristic
193C2×C59#191411yescharacteristic
194C22×M16—1181no—
195C2×#124C2×D5911yescharacteristic
196C2×#117C2×D5911yescharacteristic
197#432—592no—
198#405—1184no—
199C4×M16—592no—
200C4×M16—1181no—
201#395Dic5911yescharacteristic
202#394Dic5911yescharacteristic
203#393Dic5911yescharacteristic
204#392Dic5911yescharacteristic
205#390—21no—
206#388—21no—
207#388—21no—
208#386—21no—
209#376C2×D5911yescharacteristic
210#329—1184no—
211#324—1181no—
212#323—1181no—
213C59#938711yescharacteristic, complemented, Sylow
214C22×Q8C2×Dic5911yescharacteristic
215C22×D4—21no—
216C22×D4C2×Dic5911yescharacteristic
217C23×C4C22⋊D5911yescharacteristic
218C2×M16—1182no—
219C2×M16—1184no—
220C2×M16—1184no—
221C2×M16—1182no—
222C22×C8—1182no—
223#124—22no—
224#124—22no—
225#117—22no—
226#117—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic5911yescharacteristic
229C2×C4⋊C4C22⋊D5911yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4C2×Dic5911yescharacteristic
237C2×C22⋊C4C22⋊D5911yescharacteristic
238C2×C42—21no—
239C2×C42C22⋊D5911yescharacteristic
240C2×C42C22×D5911yescharacteristic
241C4⋊C8—1182no—
242C22⋊C8—1182no—
243#97—594no—
244#97—1182no—
245#97—1182no—
246C4×C8—1182no—
247C4×C8—1182no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95C2×Dic5911yescharacteristic
252#95C2×Dic5911yescharacteristic
253C24—21no—
254C24C22⋊Dic5911yescharacteristic
255C2×Q8—22no—
256C2×Q8—81no—
257C2×D4—22no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—81no—
261C2×D4—81no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4C22⋊Dic5911yescharacteristic
270C22×C4C22⋊Dic5911yescharacteristic
271C22×C4C22⋊Dic5911yescharacteristic
272C22×C4C22×Dic5911yescharacteristic
273C22×C4C2×C22⋊D5911yescharacteristic
274C22×C4C2×C22⋊D5911yescharacteristic
275M16—1188no—
276M16—1184no—
277M16—1184no—
278C2×C8—1188no—
279C2×C8—1184no—
280C2×C8—2362no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—81no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C42—21no—
294C42—21no—
295C42—42no—
296C42D4×D5912yes—
297C42#1174568712yes—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1427365412yes—
310C23C2×C22⋊Dic5911yescharacteristic
311Q8—82no—
312D4—82no—
313D4—84no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4#1427365412yes—
337C2×C4D4×Dic5912yes—
338C2×C4#1427369511yescharacteristic
339C2×C4#1427369912yes—
340C2×C4#1427370811yescharacteristic
341C8—1188no—
342C8—4722no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#1683728211yescharacteristic
358C22C2×#1427365411yescharacteristic
359C22C2×#1427365411yescharacteristic
360C4—22no—
361C4—22no—
362C4—21no—
363C4—22no—
364C4—21no—
365C4—42no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—161no—
374C2#2142448011yescharacteristic
375C2#2142454011yescharacteristic
376C2#2142464411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2610C2
socleC22×C59#386
prime radicalC59×#392C4
cosocleC2×C42×C59C23
hypercenter#2610D59
hypocenterC59#9387
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #21432515C2semidirect
2. #21422694C2semidirect
3. #21422430C2semidirect
4. #21421989C2semidirect
34. #14273638D4semidirect
213. C59#9387semidirect

Maximal subgroups

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

subgroupindex
#214325152
#214226942
#214224302
#214219892
C59×#26102
#938759