Finite Group Databasebeta

← #28477370

Subgroups — #28477370

Order 15104.

520 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 107 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#21432397C211yesmaximal, characteristic
2#21424164C211yesmaximal, characteristic, complemented
3#21421921C212yesmaximal, complemented
4#21421916C212yesmaximal, complemented
5C59×#1797C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic
7C2×#14273706C411yescharacteristic
8C2×#14273701C411yescharacteristic
9C2×C22⋊(C59⋊C8)—21no—
10#16837308C2211yescharacteristic
11#16837304C2211yescharacteristic
12#16837293—21no—
13#16837293—21no—
14#16837281—21no—
15#16837281—21no—
16#16837268C411yescharacteristic
17#16837157—21no—
18#16837157C2212yes—
19C22×C42×C59C2211yescharacteristic
20C2×C4⋊C8×C59—21no—
21C2×C22⋊C8×C59—21no—
22C59×#336C2212yes—
23C22×Dic118C2×C411yescharacteristic
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C22×C59⋊C8—22no—
27#14273715—42no—
28#14273706—42no—
29#14273701—42no—
30C2×C4⋊Dic59C2×C411yescharacteristic
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×C22⋊Dic59—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C22⋊(C59⋊C8)—42no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C23×C4×C59C2×C411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C22×C8×C59—22no—
48C2×C42×C59—42no—
49C2×C42×C59C2×C411yescharacteristic
50C2×C42×C59D411yescharacteristic
51C2×C42×C59D411yescharacteristic
52C2×C42×C59C2311yescharacteristic
53C4⋊C8×C59—42no—
54C22⋊C8×C59—42no—
55C24×C59C22⋊C411yescharacteristic
56C23×D59—21no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59—41no—
66C22×C4×C59—41no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C2×C22⋊D59—42no—
74C2×C22⋊D59—42no—
75C2×C22⋊D59—42no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C2×D236—22no—
80C2×D236—41no—
81C2×D236—41no—
82C2×Dic118—22no—
83C2×Dic118—41no—
84C2×Dic118—41no—
85C2×C8×C59—42no—
86C2×C8×C59—42no—
87C2×C8×C59—42no—
88C42×C59—22no—
89C42×C59—22no—
90C42×C59—41no—
91C42×C59—41no—
92C22⋊Dic59—42no—
93(C2×D59)⋊C4—42no—
94(C2×D59)⋊C4—42no—
95(C2×D59)⋊C4—42no—
96(C2×D59)⋊C4—42no—
97(C2×D59)⋊C4—41no—
98(C2×D59)⋊C4—41no—
99C4⋊Dic59—41no—
100C4⋊Dic59—41no—
101Dic59⋊C4—42no—
102C2×C59⋊C8—42no—
103C2×C59⋊C8—42no—
104C2×C59⋊C8—42no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59#10012yes—
112C23×C59C2×C22⋊C411yescharacteristic
113C22×D59—41no—
114C22×D59—41no—
115C22×D59—41no—
116C22×D59—41no—
117C22×D59—41no—
118C22×D59—41no—
119C22×D59—41no—
120C2×C4×C59—21no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—44no—
135C2×C4×C59—44no—
136C2×C4×C59—42no—
137C2×C4×C59—42no—
138C2×C4×C59#9912yes—
139C2×C4×C59#13112yes—
140C2×C4×C59#13512yes—
141C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C59#13711yescharacteristic
143C22⋊D59—84no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153D236—82no—
154Dic118—82no—
155C8×C59—82no—
156C59⋊C8—82no—
157#7730—591nomaximal, Sylow
158C22×C59—22no—
159C22×C59—22no—
160C22×C59—22no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59#40811yescharacteristic
166C22×C59C2×#9911yescharacteristic
167C22×C59C2×#10011yescharacteristic
168C22×C59#41611yescharacteristic
169C22×C59#41711yescharacteristic
170C22×C59#42111yescharacteristic
171C22×C59#42111yescharacteristic
172C2×D59—81no—
173C2×D59—81no—
174C2×D59—81no—
175C2×D59—81no—
176C2×D59—81no—
177C2×D59—81no—
178C2×D59—81no—
179C4×C59—22no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—22no—
183C4×C59—22no—
184C4×C59—41no—
185C4×C59—41no—
186Dic59—81no—
187Dic59—81no—
188Dic59—81no—
189#2850—591no—
190#1817—591no—
191#1797D5911yescharacteristic, complemented
192#1272—592no—
193#1270—592no—
194C2×C59—42no—
195C2×C59#147111yescharacteristic
196C2×C59#148211yescharacteristic
197C2×C59#149511yescharacteristic
198C2×C59#151311yescharacteristic
199C2×C59#151611yescharacteristic
200C2×C59#153311yescharacteristic
201C2×C59#155111yescharacteristic
202D59—161no—
203C2×#124—591no—
204C2×C22⋊Q8—591no—
205C2×#121—591no—
206C22×C42C2×D5911yescharacteristic
207C2×C4⋊C8—21no—
208C2×C22⋊C8—21no—
209C2×C22⋊C8—1181no—
210#391—591no—
211#390—591no—
212#388—1181no—
213#388—1181no—
214#386—1181no—
215#386—1181no—
216#382—591no—
217#336—592no—
218#336—1181no—
219#336C2×D5912yes—
220C59#773011yescharacteristic, complemented, Sylow
221C22×Q8—591no—
222C22×D4—591no—
223C22×D4—1181no—
224C23×C4C4×D5911yescharacteristic
225C23×C4D23611yescharacteristic
226C23×C4C22⋊D5911yescharacteristic
227C22×C8—22no—
228C22×C8—1182no—
229#124—2362no—
230C22⋊Q8—2362no—
231#121—2362no—
232C2×C4⋊C4—591no—
233C2×C4⋊C4—591no—
234C2×C4⋊C4—1181no—
235C2×C22⋊C4—1181no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4—1181no—
238C2×C22⋊C4—1181no—
239C2×C22⋊C4—1181no—
240C2×C22⋊C4—1181no—
241C2×C42—42no—
242C2×C42C4×D5911yescharacteristic
243C2×C42D23611yescharacteristic
244C2×C42C22⋊D5911yescharacteristic
245C2×C42C22×D5911yescharacteristic
246C4⋊C8—42no—
247C22⋊C8—42no—
248C22⋊C8—2362no—
249#95—1181no—
250#95—1181no—
251#95—1181no—
252#95—1181no—
253C24—1181no—
254C24(C2×D59)⋊C411yescharacteristic
255C2×Q8—1182no—
256C2×Q8—2361no—
257C2×Q8—2361no—
258C2×D4—1182no—
259C2×D4—2361no—
260C2×D4—2361no—
261C2×D4—2362no—
262C2×D4—2362no—
263C2×D4—2362no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—1181no—
275C22×C4—1181no—
276C22×C4—1181no—
277C22×C4(C2×D59)⋊C411yescharacteristic
278C22×C4(C2×D59)⋊C411yescharacteristic
279C22×C4(C2×D59)⋊C411yescharacteristic
280C22×C4C2×C4×D5911yescharacteristic
281C22×C4C2×D23611yescharacteristic
282C22×C4C2×C22⋊D5911yescharacteristic
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—2362no—
287C2×C8—2362no—
288C2×C8—2362no—
289C4⋊C4—2361no—
290C4⋊C4—2361no—
291C4⋊C4—2362no—
292C22⋊C4—2362no—
293C22⋊C4—2362no—
294C22⋊C4—2362no—
295C22⋊C4—2362no—
296C22⋊C4—2362no—
297C22⋊C4—2361no—
298C22⋊C4—2361no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—41no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—2361no—
310C23—2361no—
311C23—2361no—
312C23—2361no—
313C23—2361no—
314C23—2361no—
315C23—2361no—
316C23#1427364312yes—
317C23C2×(C2×D59)⋊C411yescharacteristic
318Q8—4722no—
319D4—4722no—
320D4—4724no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4#1427364112yes—
349C2×C4#1427373512yes—
350C2×C4#1427374912yes—
351C2×C4#1427375311yescharacteristic
352C2×C4#1427375811yescharacteristic
353C8—82no—
354C8—4722no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—4721no—
363C22—4721no—
364C22—4721no—
365C22—4721no—
366C22—4721no—
367C22—4721no—
368C22—4721no—
369C22#1683737011yescharacteristic
370C22C2×#1427364111yescharacteristic
371C22C2×#1427364311yescharacteristic
372C22#1683739911yescharacteristic
373C22#1683740111yescharacteristic
374C22#1683741811yescharacteristic
375C22#1683742011yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—4721no—
384C4—4721no—
385C4—4721no—
386C2—42no—
387C2—9441no—
388C2#2142245711yescharacteristic
389C2#2142250511yescharacteristic
390C2#2142255911yescharacteristic
391C2#2142263711yescharacteristic
392C2#2142265311yescharacteristic
393C2#2142271911yescharacteristic
394C2#2142278911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1797C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C4
cosocleC2×C42×C59C23
hypercenter#1797D59
hypocenterC59#7730
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #21424164C2semidirect
3. #21421921C2semidirect
4. #21421916C2semidirect
5. C59×#1797C2semidirect
191. #1797D59semidirect
220. C59#7730semidirect

Maximal subgroups

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

subgroupindex
#214323972
#214241642
#214219212
#214219162
C59×#17972
#773059