Finite Group Databasebeta

← #28477491

Subgroups — #28477491

Order 15104.

504 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 85 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#21431113C211yesmaximal, characteristic
2#21425426C211yesmaximal, characteristic, complemented
3#21425234C211yesmaximal, characteristic, complemented
4#21421921C211yesmaximal, characteristic, complemented
5#21421918C211yesmaximal, characteristic, complemented
6#21421917C211yesmaximal, characteristic
7C59×#2013C211yesmaximal, characteristic
8C2×#14273704—21no—
9C2×#14273658—21no—
10C2×#14273650—21no—
11#16837338—21no—
12#16837335—21no—
13#16837325—21no—
14#16837308C2211yescharacteristic
15#16837300C2211yescharacteristic
16#16837282—21no—
17#16837277—21no—
18#16837164—21no—
19#16837164—21no—
20#16837157C2211yescharacteristic
21#16837157C2211yescharacteristic
22C22×SD16×C59—21no—
23C2×C4⋊Q8×C59C2211yescharacteristic
24C2×#103×C59—21no—
25C59×#390C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×Dic118D411yescharacteristic
28C23×Dic59—21no—
29C22×C59⋊C8—21no—
30C22×C59⋊C8—21no—
31#14273704—42no—
32#14273704—42no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59D411yescharacteristic
35C2×C4⋊Dic59D411yescharacteristic
36C2×C4⋊Dic59D411yescharacteristic
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—21no—
39C2×Dic59⋊C4—21no—
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43#14273658—42no—
44#14273650—42no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C22×Q8×C59—21no—
50C22×D4×C59D411yescharacteristic
51C2×SD16×C59—42no—
52C2×SD16×C59—42no—
53C2×SD16×C59—42no—
54C22×C8×C59—21no—
55C4⋊Q8×C59—42no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59D411yescharacteristic
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59—21no—
61C2×C42×C59C2311yescharacteristic
62#103×C59—42no—
63C24×C59—21no—
64C2×Q8×C59—22no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C2×D4×C59—21no—
71C2×D4×C59—41no—
72C2×D4×C59—41no—
73C2×D4×C59SD1612yes—
74C22×C4×C59—21no—
75C22×C4×C59—41no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×Dic59—21no—
80C22×Dic59—21no—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—42no—
84C22×Dic59—81no—
85C22×Dic59—81no—
86C2×Dic118—21no—
87C2×Dic118—41no—
88C2×Dic118—41no—
89C2×Dic118SD1612yescomplemented
90SD16×C59—48no—
91C2×C8×C59—22no—
92C2×C8×C59—41no—
93C2×C8×C59—41no—
94C4⋊C4×C59—22no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59—41no—
98C4⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C42×C59—41no—
104C22⋊Dic59—42no—
105C22⋊Dic59—81no—
106C22⋊Dic59—81no—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59—42no—
110C4⋊Dic59—41no—
111C4⋊Dic59—41no—
112Dic59⋊C4—42no—
113Dic59⋊C4—41no—
114Dic59⋊C4—41no—
115Dic59⋊C4—81no—
116C2×C59⋊C8—22no—
117C2×C59⋊C8—41no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—41no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8—41no—
122C23×C59—22no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59#12011yescharacteristic
130Q8×C59—42no—
131Q8×C59—42no—
132Q8×C59—81no—
133Q8×C59—81no—
134D4×C59—42no—
135D4×C59—81no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—42no—
146C2×C4×C59—81no—
147C2×C4×C59—81no—
148C2×C4×C59C2×SD1611yescharacteristic
149C2×C4×C59C2×SD1611yescharacteristic
150C2×C4×C59#13511yescharacteristic
151C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C59#13711yescharacteristic
153C2×C4×C59#13711yescharacteristic
154C2×Dic59—22no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—84no—
167Dic118—42no—
168Dic118—81no—
169C8×C59—42no—
170C59⋊C8—42no—
171C59⋊C8—81no—
172#7753—591nomaximal, Sylow
173C22×C59—42no—
174C22×C59—42no—
175C22×C59—42no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59—81no—
180C22×C59#44811yescharacteristic
181C22×C59#45011yescharacteristic
182C22×C59#45211yescharacteristic
183C22×C59#45311yescharacteristic
184C22×C59#45511yescharacteristic
185C22×C59#45611yescharacteristic
186C22×C59B(4,𝔽2)11yescharacteristic
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—82no—
194C4×C59—81no—
195Dic59—42no—
196Dic59—81no—
197Dic59—81no—
198Dic59—81no—
199#2732—591no—
200#2058—591no—
201#2024—591no—
202#2013D5911yescharacteristic
203#1272—591no—
204#1271—591no—
205#1270—591no—
206C2×C59—82no—
207C2×C59#159611yescharacteristic
208C2×C59#160111yescharacteristic
209C2×C59#160311yescharacteristic
210C2×C59#163111yescharacteristic
211C2×C59#164311yescharacteristic
212C2×C59#165111yescharacteristic
213C2×C59#166111yescharacteristic
214C22×SD16—21no—
215C2×C4⋊Q8C2×D5911yescharacteristic
216C2×C22⋊Q8—1181no—
217C2×#106—1181no—
218C2×#103—21no—
219C2×#103—1181no—
220#399—1181no—
221#398—1181no—
222#396—1181no—
223#391—591no—
224#390C2×D5911yescharacteristic
225#389—591no—
226#386—1181no—
227#385—1181no—
228#340—1181no—
229#340—1181no—
230#336—591no—
231#336—591no—
232#336C2×D5911yescharacteristic
233C59#775311yescharacteristic, complemented, Sylow
234C22×Q8—21no—
235C22×Q8—591no—
236C22×D4C22⋊D5911yescharacteristic
237C23×C4—1181no—
238C2×SD16—42no—
239C2×SD16—42no—
240C2×SD16—42no—
241C22×C8—21no—
242C22×C8—1181no—
243C22×C8—1181no—
244C4⋊Q8—42no—
245C22⋊Q8—2362no—
246C22⋊Q8—2362no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—591no—
250C2×C4⋊C4—591no—
251C2×C4⋊C4—591no—
252C2×C4⋊C4—1181no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4—1181no—
255C2×C4⋊C4—1181no—
256C2×C4⋊C4C22⋊D5911yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—1181no—
260C2×C22⋊C4—1181no—
261C2×C22⋊C4—1181no—
262C2×C42C22×D5911yescharacteristic
263#106—2362no—
264#103—42no—
265#103—2362no—
266#95—1181no—
267#95—1181no—
268#95—1181no—
269#95—1181no—
270C24—21no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—592no—
278C2×Q8—1181no—
279C2×Q8—2361no—
280C2×Q8—2361no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4#1174567012yes—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—1181no—
288C22×C4—1181no—
289C22×C4—1181no—
290C22×C4—1181no—
291C22×C4—4721no—
292C22×C4—4721no—
293C22×C4—2362no—
294C22×C4D4×D5911yescharacteristic
295C22×C4D4×D5911yescharacteristic
296C22×C4C2×C22⋊D5911yescharacteristic
297SD16—48no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—1182no—
302C2×C8—2361no—
303C2×C8—2361no—
304C2×C8—2361no—
305C2×C8—2361no—
306C2×C8—2361no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—1181no—
313C4⋊C4—1181no—
314C4⋊C4—4721no—
315C4⋊C4—2361no—
316C4⋊C4—2361no—
317C4⋊C4—2362no—
318C4⋊C4—2362no—
319C4⋊C4—2361no—
320C4⋊C4—2361no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—4721no—
326C22⋊C4—4721no—
327C22⋊C4—2362no—
328C42—41no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1427369511yescharacteristic
337Q8—42no—
338Q8—42no—
339Q8—81no—
340Q8—81no—
341Q8—4721no—
342Q8—2362no—
343D4—42no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—1182no—
358C2×C4—4724no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2361no—
362C2×C4—2361no—
363C2×C4—2361no—
364C2×C4—2361no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4—2361no—
368C2×C4—2361no—
369C2×C4—2361no—
370C2×C4SD16×D5911yescharacteristic
371C2×C4C2×#1174567011yescharacteristic
372C2×C4#1427374811yescharacteristic
373C2×C4#1427375411yescharacteristic
374C2×C4#1427375611yescharacteristic
375C2×C4#1427375711yescharacteristic
376C8—42no—
377C8—4721no—
378C8—2362no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1683754011yescharacteristic
387C22#1683755111yescharacteristic
388C22#1683756111yescharacteristic
389C22#1683756911yescharacteristic
390C22#1683757811yescharacteristic
391C22#1683758311yescharacteristic
392C22#1683758811yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—81no—
400C4—82no—
401C4—4721no—
402C4—4721no—
403C4—4721no—
404C4—2362no—
405C2—82no—
406C2#2142301811yescharacteristic
407C2#2142304611yescharacteristic
408C2#2142305611yescharacteristic
409C2#2142322011yescharacteristic
410C2#2142330111yescharacteristic
411C2#2142335711yescharacteristic
412C2#2142341011yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2013C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2013D59
hypocenterC59#7753
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #21425426C2semidirect
3. #21425234C2semidirect
4. #21421921C2semidirect
5. #21421918C2semidirect
89. C2×Dic118SD16semidirect
233. C59#7753semidirect

Maximal subgroups

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

subgroupindex
#214311132
#214254262
#214252342
#214219212
#214219182
#214219172
C59×#20132
#775359