Finite Group Databasebeta

← #28477329

Subgroups — #28477329

Order 15104.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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#21432434C211yesmaximal, characteristic
2#21424163C211yesmaximal, characteristic
3#21424163C211yesmaximal, characteristic
4#21421921C211yesmaximal, characteristic, complemented
5#21421921C211yesmaximal, characteristic, complemented
6#21421919C211yesmaximal, characteristic, complemented
7C59×#1271C211yesmaximal, characteristic, complemented
8C2×#14273707C411yescharacteristic
9C2×C42×D59C2211yescharacteristic
10C2×(C2×D59)⋊C8—21no—
11C2×(C2×D59)⋊C8—21no—
12#16837308C2211yescharacteristic
13#16837307C411yescharacteristic
14#16837298C411yescharacteristic
15#16837289—21no—
16#16837288—21no—
17#16837273—21no—
18#16837264—21no—
19#16837259C411yescharacteristic
20#16837157C2211yescharacteristic
21#16837157C2211yescharacteristic
22#16837156—21no—
23#16837156—21no—
24C59×#389C2211yescharacteristic
25C59×#336C2211yescharacteristic
26C59×#336C2211yescharacteristic
27C22×Dic118C2×C411yescharacteristic
28C22×C4×D59C2×C411yescharacteristic
29C22×C4×D59D411yescharacteristic
30C22×C4×D59D411yescharacteristic
31C22×C59⋊C8—21no—
32C22×C59⋊C8—21no—
33#14273707—42no—
34C2×C4⋊Dic59—21no—
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×Dic59⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×C4×Dic59C2×C411yescharacteristic
40C2×C4×Dic59D411yescharacteristic
41C2×C4×Dic59D411yescharacteristic
42C42×D59—42no—
43(C2×D59)⋊C8—42no—
44(C2×D59)⋊C8—42no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C22×C8×C59—21no—
52C22×C8×C59—21no—
53C2×C4⋊C4×C59—21no—
54C2×C4⋊C4×C59C2×C411yescharacteristic
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C42×C59C2311yescharacteristic
57#95×C59—21no—
58C23×D59C22⋊C411yescharacteristic
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59C22×C411yescharacteristic
62C22×C4×C59C2×D411yescharacteristic
63C22×C4×C59C2×D411yescharacteristic
64C22×Dic59—21no—
65C22×Dic59—21no—
66C22×Dic59C22⋊C411yescharacteristic
67C22×Dic59C22⋊C411yescharacteristic
68C22×Dic59C22⋊C411yescharacteristic
69C2×C4×D59—22no—
70C2×C4×D59—22no—
71C2×C4×D59—22no—
72C2×C4×D59—21no—
73C2×C4×D59—21no—
74C2×C4×D59—21no—
75C2×C4×D59—21no—
76C2×C4×D59—21no—
77C2×C4×D59—21no—
78C2×C4×D59—41no—
79C2×C4×D59—41no—
80C2×C4×D59—41no—
81C2×Dic118—21no—
82C2×Dic118—21no—
83C2×Dic118—41no—
84C2×Dic118—41no—
85C2×C8×C59—41no—
86C2×C8×C59—41no—
87C2×C8×C59—41no—
88C2×C8×C59—41no—
89C2×C8×C59—41no—
90C2×C8×C59—41no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—21no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C4⋊C4×C59—41no—
96C42×C59—41no—
97(C2×D59)⋊C4—42no—
98(C2×D59)⋊C4—42no—
99C4⋊Dic59—42no—
100C4⋊Dic59—41no—
101Dic59⋊C4—42no—
102C4×Dic59—41no—
103C4×Dic59—41no—
104C4×Dic59—41no—
105C2×C59⋊C8—41no—
106C2×C59⋊C8—41no—
107C2×C59⋊C8—41no—
108C2×C59⋊C8—41no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C23×C59C2×C22⋊C411yescharacteristic
112C22×D59—21no—
113C22×D59—21no—
114C22×D59—21no—
115C22×D59—21no—
116C22×D59—21no—
117C22×D59—21no—
118C22×D59#9912yes—
119C2×C4×C59—21no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59#13111yescharacteristic
132C2×C4×C59#13111yescharacteristic
133C2×C4×C59#13511yescharacteristic
134C2×C4×C59#13511yescharacteristic
135C2×C4×C59#13711yescharacteristic
136C2×C4×C59#13711yescharacteristic
137C2×Dic59—21no—
138C2×Dic59—21no—
139C2×Dic59—21no—
140C2×Dic59—21no—
141C2×Dic59—21no—
142C2×Dic59—21no—
143C2×Dic59—21no—
144C2×Dic59—21no—
145C2×Dic59—21no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C4×D59—42no—
153C4×D59—42no—
154C4×D59—42no—
155C4×D59—42no—
156C4×D59—42no—
157C4×D59—42no—
158Dic118—81no—
159Dic118—81no—
160C8×C59—81no—
161C8×C59—81no—
162C59⋊C8—81no—
163C59⋊C8—81no—
164#7721—591nomaximal, Sylow
165C22×C59#40811yescharacteristic
166C22×C59C2×#9911yescharacteristic
167C22×C59#41311yescharacteristic
168C22×C59#41611yescharacteristic
169C22×C59#41911yescharacteristic
170C22×C59#42111yescharacteristic
171C22×C59#42111yescharacteristic
172C2×D59—22no—
173C2×D59—22no—
174C2×D59—22no—
175C2×D59—41no—
176C2×D59—41no—
177C2×D59—41no—
178C2×D59—41no—
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—81no—
186C4×C59—81no—
187Dic59—41no—
188Dic59—41no—
189Dic59—41no—
190Dic59—81no—
191Dic59—81no—
192#2855—591no—
193#1817—591no—
194#1817—591no—
195#1272—591no—
196#1272—591no—
197#1271—591no—
198#1271D5911yescharacteristic, complemented
199C2×C59#146811yescharacteristic
200C2×C59#148211yescharacteristic
201C2×C59#149711yescharacteristic
202C2×C59#151111yescharacteristic
203C2×C59#151611yescharacteristic
204C2×C59#153511yescharacteristic
205C2×C59#155411yescharacteristic
206D59—42no—
207C2×C22⋊Q8—591no—
208C22×C42—591no—
209C2×C22⋊C8—1181no—
210C2×C22⋊C8—1181no—
211#391—591no—
212#391—591no—
213#389—591no—
214#389C2×D5911yescharacteristic
215#387—1181no—
216#387—1181no—
217#385—1181no—
218#381—1181no—
219#380—591no—
220#336—591no—
221#336—591no—
222#336—1181no—
223#336—1181no—
224#336C2×D5911yescharacteristic
225#336C2×D5911yescharacteristic
226C59#772111yescharacteristic, complemented, Sylow
227C22×Q8—591no—
228C23×C4—591no—
229C23×C4—591no—
230C23×C4—591no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—1181no—
234C22×C8—1181no—
235C22⋊Q8—2362no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—591no—
238C2×C4⋊C4—1181no—
239C2×C4⋊C4—1181no—
240C2×C4⋊C4C4×D5911yescharacteristic
241C2×C4⋊C4C4×D5911yescharacteristic
242C2×C22⋊C4—1181no—
243C2×C22⋊C4—1181no—
244C2×C42—591no—
245C2×C42—591no—
246C2×C42—591no—
247C2×C42—2362no—
248C2×C42C22×D5911yescharacteristic
249C22⋊C8—2362no—
250C22⋊C8—2362no—
251#95—21no—
252#95—1181no—
253#95—1181no—
254#95—1181no—
255#95—1181no—
256#95—1181no—
257#95—1181no—
258C24—591no—
259C2×Q8—1181no—
260C2×Q8—1181no—
261C2×Q8—2361no—
262C2×Q8—2361no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—591no—
266C22×C4—591no—
267C22×C4—591no—
268C22×C4—1182no—
269C22×C4—1182no—
270C22×C4—1182no—
271C22×C4—1181no—
272C22×C4—1181no—
273C22×C4—1181no—
274C22×C4—1181no—
275C22×C4—1181no—
276C22×C4—1181no—
277C22×C4—1181no—
278C22×C4—1181no—
279C22×C4—2361no—
280C22×C4—2361no—
281C22×C4—2361no—
282C22×C4C2×C4×D5911yescharacteristic
283C22×C4D4×D5911yescharacteristic
284C22×C4D4×D5911yescharacteristic
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—2361no—
292C2×C8—2361no—
293C2×C8—2361no—
294C2×C8—2361no—
295C2×C8—2361no—
296C2×C8—2361no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—2361no—
303C4⋊C4—2362no—
304C4⋊C4—2362no—
305C22⋊C4—2362no—
306C22⋊C4—2362no—
307C42—41no—
308C42—2361no—
309C42—2361no—
310C42—2361no—
311C23—592no—
312C23—1181no—
313C23—1181no—
314C23—1181no—
315C23—1181no—
316C23—1181no—
317C23—1181no—
318C23C22⋊C4×D5911yescharacteristic
319Q8—4721no—
320Q8—4721no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—1181no—
334C2×C4—1181no—
335C2×C4—1181no—
336C2×C4—1181no—
337C2×C4—1181no—
338C2×C4—1181no—
339C2×C4—1181no—
340C2×C4—1181no—
341C2×C4—1181no—
342C2×C4—2362no—
343C2×C4—2362no—
344C2×C4—2362no—
345C2×C4—2362no—
346C2×C4—2362no—
347C2×C4—2362no—
348C2×C4—2361no—
349C2×C4—2361no—
350C2×C4—2361no—
351C2×C4—2361no—
352C2×C4—2361no—
353C2×C4—2361no—
354C2×C4#1427373311yescharacteristic
355C2×C4#1427373511yescharacteristic
356C2×C4#1427374611yescharacteristic
357C2×C4#1427374811yescharacteristic
358C2×C4#1427375611yescharacteristic
359C2×C4#1427375911yescharacteristic
360C8—81no—
361C8—81no—
362C8—4721no—
363C8—4721no—
364C22—1182no—
365C22—1182no—
366C22—1182no—
367C22—2361no—
368C22—2361no—
369C22—2361no—
370C22—2361no—
371C22#1683736711yescharacteristic
372C22#99×D5911yescharacteristic
373C22#1683738511yescharacteristic
374C22#1683739511yescharacteristic
375C22#1683740911yescharacteristic
376C22#1683741711yescharacteristic
377C22#1683741711yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—81no—
385C4—81no—
386C4—4721no—
387C4—4721no—
388C4—2361no—
389C4—2361no—
390C4—2361no—
391C2—2362no—
392C2#2142244311yescharacteristic
393C2#2142250111yescharacteristic
394C2#2142256711yescharacteristic
395C2#2142262711yescharacteristic
396C2#2142264911yescharacteristic
397C2#2142272611yescharacteristic
398C2#2142280311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1271C2
socleC23×C59C2×C22⋊C4
prime radicalC23×D59C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1271D59
hypocenterC59#7721
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #21421921C2semidirect
5. #21421921C2semidirect
6. #21421919C2semidirect
7. C59×#1271C2semidirect
198. #1271D59semidirect
226. C59#7721semidirect

Maximal subgroups

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

subgroupindex
#214324342
#214241632
#214219212
#214219192
C59×#12712
#772159