Finite Group Databasebeta

← #28483888

Subgroups — #28483888

Order 15104.

476 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 109 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#21434105C211yesmaximal, characteristic
2#21433036C211yesmaximal, characteristic, complemented
3#21424894C211yesmaximal, characteristic
4#21422751C211yesmaximal, characteristic, complemented
5#21422522C211yesmaximal, characteristic
6#21421992C211yesmaximal, characteristic, complemented
7C59×#1281C211yesmaximal, characteristic, complemented
8#16837959C2211yescharacteristic
9C2×C4×D236C2211yescharacteristic
10#16837733C412yes—
11#16837693—21no—
12#16837608—21no—
13#16837555C412yes—
14#16837530—22no—
15#16837530—22no—
16C4×#11745660—22no—
17#16837428—21no—
18#16837426C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×(C2×D59)⋊C8—21no—
21C2×(C2×D59)⋊C8—21no—
22#16837357—21no—
23#16837150—21no—
24#16837149—21no—
25#16837131—22no—
26C59×#431C2211yescharacteristic
27C59×#431C2211yescharacteristic
28C2×#97×C59C2211yescharacteristic
29C22×D236C2×C411yescharacteristic
30C22×C4×D59—21no—
31C2×#11745660—22no—
32C2×#11745664—21no—
33C22×C59⋊C8—21no—
34#14273727D411yescharacteristic
35#14273727D411yescharacteristic
36#14273719—21no—
37#14273706—21no—
38#14273706C2×C412yes—
39C4×Dic118—21no—
40C4×Dic118D412yes—
41C4×D236—21no—
42C4×D236—21no—
43C4×D236—21no—
44C4×D236—21no—
45#14273686—21no—
46C2×C4⋊Dic59C2×C411yescharacteristic
47C2×(C2×D59)⋊C4—21no—
48#14273659—22no—
49C4⋊(C59⋊C8)—21no—
50C4⋊(C59⋊C8)—21no—
51C4⋊(C59⋊C8)—21no—
52C4⋊(C59⋊C8)D412yes—
53Dic59⋊C8—24no—
54#14273652—22no—
55#14273649—22no—
56(C2×D59)⋊C8—42no—
57(C2×D59)⋊C8—42no—
58#14273637—22no—
59C8×Dic59—22no—
60C22×C8×C59D411yescharacteristic
61C22×C8×C59D411yescharacteristic
62C2×C42×C59C2311yescharacteristic
63C4⋊C8×C59—21no—
64C4⋊C8×C59—21no—
65C22⋊C8×C59—21no—
66C22⋊C8×C59C2×C412yes—
67#97×C59—21no—
68#97×C59—21no—
69C4×C8×C59—21no—
70C4×C8×C59—21no—
71C23×D59—21no—
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59#4111yescharacteristic
75C22×Dic59—21no—
76C2×D236—21no—
77C2×D236—21no—
78C2×D236—42no—
79C2×C4×D59—41no—
80C2×C4×D59—41no—
81C2×C4×D59—41no—
82C2×C4×D59—41no—
83C2×C4×D59—41no—
84C2×C4×D59—41no—
85C2×Dic118—21no—
86C2×Dic118C22⋊C412yes—
87C2×C8×C59—21no—
88C2×C8×C59—21no—
89C2×C8×C59—22no—
90C2×C8×C59—21no—
91C2×C8×C59—21no—
92C2×C8×C59—21no—
93C2×C8×C59—41no—
94C2×C8×C59SD1612yescomplemented
95C42×C59C2×D411yescharacteristic
96C42×C59C2×D411yescharacteristic
97C42×C59C2×D411yescharacteristic
98C42×C59#4111yescharacteristic
99C22⋊Dic59—22no—
100(C2×D59)⋊C4—41no—
101(C2×D59)⋊C4—41no—
102C4⋊Dic59—21no—
103C4⋊Dic59—21no—
104C4⋊Dic59—21no—
105C4⋊Dic59—21no—
106C4⋊Dic59—21no—
107C4⋊Dic59C22⋊C412yes—
108Dic59⋊C4—22no—
109Dic59⋊C4—22no—
110C4×Dic59—22no—
111#11745664—41no—
112#11745664—41no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117#11745660—44no—
118C23×C59C4×D411yescharacteristic
119C22×D59—22no—
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—41no—
124C22×D59—41no—
125C22×D59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59C2×C22⋊C411yescharacteristic
129C2×C4×C59C4×D411yescharacteristic
130C2×C4×C59#12011yescharacteristic
131C2×C4×C59#12111yescharacteristic
132C2×C4×C59#12111yescharacteristic
133C2×C4×C59#12311yescharacteristic
134C2×C4×C59C2×SD1611yescharacteristic
135C2×C4×C59#13511yescharacteristic
136C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C59#13711yescharacteristic
138C2×Dic59—22no—
139C2×Dic59—22no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143D236—22no—
144D236—22no—
145D236—41no—
146D236—41no—
147C4×D59—81no—
148C4×D59—81no—
149C4×D59—81no—
150C4×D59—81no—
151Dic118—42no—
152Dic118—81no—
153C8×C59—22no—
154C8×C59—42no—
155C8×C59—41no—
156C8×C59—81no—
157C59⋊C8—42no—
158C59⋊C8—81no—
159#9050—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—21no—
162C22×C59#38611yescharacteristic
163C22×C59C4×SD1611yescharacteristic
164C22×C59#44011yescharacteristic
165C2×D59—42no—
166C2×D59—42no—
167C2×D59—42no—
168C2×D59—81no—
169C2×D59—81no—
170C2×D59—81no—
171C2×D59—81no—
172C4×C59—21no—
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—21no—
176C4×C59#44811yescharacteristic
177C4×C59#44911yescharacteristic
178C4×C59#46011yescharacteristic
179C4×C59#46411yescharacteristic
180Dic59—42no—
181Dic59—42no—
182Dic59—81no—
183#3101—591no—
184#2961—591no—
185#1955—591no—
186#1542—591no—
187#1486—591no—
188#1301—591no—
189#1281D5911yescharacteristic, complemented
190C2×C59—21no—
191C2×C59—21no—
192C2×C59#188011yescharacteristic
193C2×C59#188111yescharacteristic
194C2×C59#189111yescharacteristic
195D59—82no—
196#533—591no—
197C2×C4×D4—591no—
198#483—592no—
199#476—1181no—
200#461—1181no—
201#450—592no—
202#446—1182no—
203#446—1182no—
204C4×SD16—1182no—
205#431C2×D5911yescharacteristic
206#431C2×D5911yescharacteristic
207#424—1181no—
208#423—591no—
209C2×C4⋊C8—591no—
210C2×C22⋊C8—1181no—
211C2×C22⋊C8—1181no—
212#405—1181no—
213C2×#97C2×D5911yescharacteristic
214#333—1181no—
215#332—1181no—
216D4⋊C8—1182no—
217C59#905011yescharacteristic, complemented, Sylow
218C22×D4—591no—
219C23×C4—1181no—
220C2×SD16—1182no—
221C2×M16—1181no—
222C22×C8—1181no—
223C22×C8D23611yescharacteristic
224C22×C8D23611yescharacteristic
225C4⋊Q8—591no—
226C4⋊Q8—591no—
227#125—1181no—
228C22⋊Q8—592no—
229C22⋊Q8—1181no—
230C4×Q8—592no—
231C4×Q8—1181no—
232C4×D4—1181no—
233C4×D4—1181no—
234C4×D4—1181no—
235C4×D4—1181no—
236#117—1181no—
237C2×C4⋊C4—591no—
238C2×C22⋊C4—1181no—
239C2×C42C22×D5911yescharacteristic
240#106—1182no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—592no—
244C4⋊C8—1184no—
245C4⋊C8—1181no—
246C4⋊C8—1181no—
247C4⋊C8—1181no—
248#103—1182no—
249#102—1182no—
250C22⋊C8—21no—
251C22⋊C8—2362no—
252C22⋊C8—2362no—
253C22⋊C8C4×D5912yes—
254#97—21no—
255#97—21no—
256#97—1182no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—1182no—
260C24—1181no—
261C2×Q8—592no—
262C2×Q8—1181no—
263C2×D4—1181no—
264C2×D4—1181no—
265C2×D4—2362no—
266C22×C4—1181no—
267C22×C4—2361no—
268C22×C4—2361no—
269C22×C4—2361no—
270C22×C4—2361no—
271C22×C4—2361no—
272C22×C4—2361no—
273C22×C4C2×C4×D5911yescharacteristic
274C22×C4C2×D23611yescharacteristic
275C22×C4#1174568511yescharacteristic
276SD16—2364no—
277M16—2361no—
278M16—2361no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—41no—
286C2×C8—1182no—
287C2×C8—2361no—
288C2×C8—2361no—
289C2×C8—2361no—
290C2×C8#1174566012yescomplemented
291C4⋊C4—592no—
292C4⋊C4—1182no—
293C4⋊C4—1181no—
294C4⋊C4—1181no—
295C4⋊C4—1181no—
296C4⋊C4—1182no—
297C4⋊C4—1181no—
298C4⋊C4—1181no—
299C22⋊C4—1182no—
300C22⋊C4—2361no—
301C22⋊C4—2361no—
302C42—1182no—
303C42D4×D5911yescharacteristic
304C42D4×D5911yescharacteristic
305C42D4×D5911yescharacteristic
306C42#1174568911yescharacteristic
307C23—1182no—
308C23—2361no—
309C23—2361no—
310C23—2361no—
311C23—2361no—
312C23—2361no—
313C23—2361no—
314C23C4×D23611yescharacteristic
315Q8—4721no—
316Q8—2362no—
317D4—1182no—
318D4—1182no—
319D4—2361no—
320D4—2361no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—1182no—
324C2×C4—1182no—
325C2×C4—4721no—
326C2×C4—4721no—
327C2×C4—4721no—
328C2×C4—4721no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4C22⋊C4×D5911yescharacteristic
333C2×C4#1427368911yescharacteristic
334C2×C4#1427369711yescharacteristic
335C2×C4#1427370011yescharacteristic
336C2×C4#1427370211yescharacteristic
337C2×C4#1427371211yescharacteristic
338C2×C4C2×#1174566011yescharacteristic
339C2×C4#1427374911yescharacteristic
340C2×C4#1427375511yescharacteristic
341C2×C4#1427376011yescharacteristic
342C8—22no—
343C8—42no—
344C8—41no—
345C8—81no—
346C8—4721no—
347C8—2362no—
348C22—21no—
349C22—21no—
350C22—4721no—
351C22—4721no—
352C22—4721no—
353C22—4721no—
354C22—2362no—
355C22—2362no—
356C22—2362no—
357C22#1683728011yescharacteristic
358C22C4×#1174566011yescharacteristic
359C22#1683750411yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—4721no—
365C4—2362no—
366C4—2362no—
367C4#1683754311yescharacteristic
368C4#1683754911yescharacteristic
369C4#1683760311yescharacteristic
370C4#1683762511yescharacteristic
371C2—21no—
372C2—21no—
373C2—4722no—
374C2#2142445011yescharacteristic
375C2#2142445711yescharacteristic
376C2#2142451311yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1281C2
socleC22×C59#386
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#1281D59
hypocenterC59#9050
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
2. #21433036C2semidirect
4. #21422751C2semidirect
6. #21421992C2semidirect
7. C59×#1281C2semidirect
94. C2×C8×C59SD16semidirect
189. #1281D59semidirect
217. C59#9050semidirect
290. C2×C8#11745660semidirect

Maximal subgroups

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

subgroupindex
#214341052
#214330362
#214248942
#214227512
#214225222
#214219922
C59×#12812
#905059