Finite Group Databasebeta

← #28491821

Subgroups — #28491821

Order 15104.

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

normal subgroups only

This group has 416 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#21427390C211yesmaximal, characteristic
2#21425076C211yesmaximal, characteristic
3#21422596C211yesmaximal, characteristic
4#21422594C211yesmaximal, characteristic
5#21421997C211yesmaximal, characteristic
6#21421924C211yesmaximal, characteristic
7C59×#1303C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9C2×C4×Dic118—21no—
10#16837455C2211yescharacteristic
11#16837336—21no—
12#16837335—21no—
13#16837323—21no—
14#16837322—21no—
15C4×Dic59⋊C4—21no—
16C4×C4⋊Dic59C2211yescharacteristic
17#16837248C2211yescharacteristic
18#16837163—21no—
19#16837163—21no—
20#16837140—21no—
21#16837140—21no—
22#16837138—21no—
23#16837138—21no—
24C59×#431C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C4×C4⋊C4×C59C2211yescharacteristic
27C22×Dic118—21no—
28#14273727Q811yescharacteristic, complemented
29#14273727Q811yescharacteristic, complemented
30#14273719—21no—
31#14273706—21no—
32#14273706—21no—
33C4×Dic118—21no—
34C4×Dic118—21no—
35C4×Dic118—42no—
36C4×Dic118—42no—
37#14273686—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59—21no—
40C2×Dic59⋊C4—21no—
41C2×Dic59⋊C4—21no—
42C2×Dic59⋊C4—21no—
43C2×Dic59⋊C4—21no—
44C2×C4×Dic59—21no—
45C2×C4×Dic59D411yescharacteristic
46C2×C4×Dic59D411yescharacteristic
47C4⋊(C59⋊C8)—21no—
48C22⋊(C59⋊C8)—21no—
49C4×C59⋊C8—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C59⋊#95D411yescharacteristic
55C59⋊#95D411yescharacteristic
56C2×M16×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C42×C59—21no—
59C2×C42×C59C2311yescharacteristic
60C4⋊C8×C59—21no—
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C22⋊C8×C59—21no—
64C4×C8×C59—21no—
65#95×C59D411yescharacteristic
66#95×C59D411yescharacteristic
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C2×Dic118—21no—
77C2×Dic118—21no—
78C2×Dic118—42no—
79C2×Dic118—42no—
80C2×Dic118—42no—
81M16×C59—41no—
82M16×C59—41no—
83C2×C8×C59—41no—
84C2×C8×C59—41no—
85C4⋊C4×C59—42no—
86C42×C59—41no—
87C42×C59—41no—
88C42×C59C2×Q811yescharacteristic
89C42×C59#4111yescharacteristic
90C42×C59#4111yescharacteristic
91C42×C59#4111yescharacteristic
92C22⋊Dic59—41no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95C4⋊Dic59—21no—
96C4⋊Dic59—21no—
97C4⋊Dic59—42no—
98C4⋊Dic59—42no—
99Dic59⋊C4—42no—
100Dic59⋊C4—42no—
101Dic59⋊C4—42no—
102Dic59⋊C4—42no—
103Dic59⋊C4—41no—
104Dic59⋊C4—41no—
105C4×Dic59—21no—
106C4×Dic59—21no—
107C4×Dic59—41no—
108C4×Dic59—41no—
109C4×Dic59—41no—
110C4×Dic59SD1612yes—
111C4×Dic59Q1612yes—
112C2×C59⋊C8—41no—
113C23×C59#12011yescharacteristic
114C2×C4×C59—22no—
115C2×C4×C59—41no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59C22⋊Q811yescharacteristic
123C2×C4×C59C22⋊Q811yescharacteristic
124C2×C4×C59C22⋊Q811yescharacteristic
125C2×C4×C59#12411yescharacteristic
126C2×C4×C59#12411yescharacteristic
127C2×C4×C59#12411yescharacteristic
128C2×C4×C59C2×SD1611yescharacteristic
129C2×C4×C59C2×Q1611yescharacteristic
130C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C59#13711yescharacteristic
132C2×Dic59—22no—
133C2×Dic59—22no—
134C2×Dic59—22no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—41no—
140C2×Dic59—41no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146Dic118—48no—
147Dic118—81no—
148Dic118—81no—
149C8×C59—81no—
150C8×C59—81no—
151C59⋊C8—81no—
152#10371—591nomaximal, Sylow
153C22×C59—21no—
154C22×C59—21no—
155C22×C59#39311yescharacteristic
156C22×C59#45011yescharacteristic
157C22×C59#45111yescharacteristic
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—21no—
162C4×C59—42no—
163C4×C59—81no—
164C4×C59#47511yescharacteristic
165C4×C59#47711yescharacteristic
166C4×C59#48711yescharacteristic
167C4×C59#48911yescharacteristic
168Dic59—42no—
169Dic59—42no—
170Dic59—42no—
171Dic59—81no—
172Dic59—81no—
173Dic59—81no—
174#2399—591no—
175#1995—591no—
176#1503—591no—
177#1503—591no—
178#1303—591no—
179#1303D5911yescharacteristic
180#1274—591no—
181C2×C59—21no—
182C2×C59—21no—
183C2×C59#202411yescharacteristic
184C2×C59#203111yescharacteristic
185C2×C59#203111yescharacteristic
186#533—591no—
187C2×C4×Q8—1181no—
188#431—591no—
189#431C2×D5911yescharacteristic
190#423C2×D5911yescharacteristic
191#398—1181no—
192#398—1181no—
193#395—1181no—
194#395—1181no—
195C4×C4⋊C4—591no—
196C4×C4⋊C4—1181no—
197C4×C4⋊C4C2×D5911yescharacteristic
198#376—591no—
199#339—1181no—
200#339—1181no—
201#328—1181no—
202#328—1181no—
203#328—1181no—
204#328—1181no—
205C59#1037111yescharacteristic, complemented, Sylow
206C22×Q8—1181no—
207C2×M16—21no—
208C4⋊Q8—591no—
209C4⋊Q8—591no—
210#125—1181no—
211C22⋊Q8—1181no—
212C22⋊Q8—1181no—
213C4×Q8—1181no—
214C4×Q8—1181no—
215C4×Q8—2362no—
216C4×Q8—2362no—
217#117—1181no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—1181no—
220C2×C4⋊C4—1181no—
221C2×C4⋊C4—1181no—
222C2×C4⋊C4—1181no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4—1181no—
225C2×C42—21no—
226C2×C42—591no—
227C2×C42—591no—
228C2×C42—1181no—
229C2×C42C22×D5911yescharacteristic
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—1181no—
234C22⋊C8—21no—
235C22⋊C8—1181no—
236C4×C8—21no—
237C4×C8—1181no—
238#95—591no—
239#95—591no—
240#95—1181no—
241#95—1181no—
242#95—1181no—
243#95—1181no—
244#95D23611yescharacteristic
245#95D23611yescharacteristic
246C2×Q8—1181no—
247C2×Q8—1181no—
248C2×Q8—2362no—
249C2×Q8—2362no—
250C2×Q8—2362no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—1181no—
254C22×C4—1181no—
255C22×C4—1181no—
256C22×C4—1181no—
257C22×C4C2×D23611yescharacteristic
258C22×C4D4×D5911yescharacteristic
259C22×C4D4×D5911yescharacteristic
260M16—41no—
261M16—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—2361no—
265C4⋊C4—42no—
266C4⋊C4—1181no—
267C4⋊C4—1181no—
268C4⋊C4—1181no—
269C4⋊C4—1181no—
270C4⋊C4—2362no—
271C4⋊C4—2362no—
272C4⋊C4—2362no—
273C4⋊C4—2362no—
274C4⋊C4—2362no—
275C4⋊C4—2362no—
276C4⋊C4—2361no—
277C4⋊C4—2361no—
278C22⋊C4—2361no—
279C42—41no—
280C42—41no—
281C42—592no—
282C42—592no—
283C42—1181no—
284C42—1181no—
285C42—2361no—
286C42—2361no—
287C42—2361no—
288C42#1174568511yescharacteristic
289C42#1174568511yescharacteristic
290C42#1174568711yescharacteristic
291C42Q8×D5911yescharacteristic
292C23#1427369711yescharacteristic
293Q8—4721no—
294Q8—4721no—
295Q8—2368no—
296C2×C4—22no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—1182no—
305C2×C4—1182no—
306C2×C4—1182no—
307C2×C4—2361no—
308C2×C4—2361no—
309C2×C4—2361no—
310C2×C4—2361no—
311C2×C4—2361no—
312C2×C4—2361no—
313C2×C4—2361no—
314C2×C4—2361no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4—2361no—
318C2×C4#1427370511yescharacteristic
319C2×C4#1427370511yescharacteristic
320C2×C4#1427370711yescharacteristic
321C2×C4#1427371311yescharacteristic
322C2×C4#1427371311yescharacteristic
323C2×C4#1427371511yescharacteristic
324C2×C4SD16×D5911yescharacteristic
325C2×C4Q16×D5911yescharacteristic
326C2×C4#1427375111yescharacteristic
327C2×C4#1427375611yescharacteristic
328C8—81no—
329C8—81no—
330C8—4721no—
331C22—21no—
332C22—21no—
333C22#1683731511yescharacteristic
334C22#1683755411yescharacteristic
335C22#1683755911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—81no—
342C4—4721no—
343C4—4721no—
344C4—4721no—
345C4—2362no—
346C4—2362no—
347C4—2362no—
348C4#1683768611yescharacteristic
349C4#1683769611yescharacteristic
350C4#1683775111yescharacteristic
351C4#1683776011yescharacteristic
352C2—21no—
353C2—21no—
354C2#2142523611yescharacteristic
355C2#2142526911yescharacteristic
356C2#2142527111yescharacteristic
subgroupG/H
centerC2×C4#14273713
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1303C2
socleC22×C59#393
prime radicalC23×C59#120
hypercenter#1303D59
hypocenterC59#10371
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
28. #14273727Q8semidirect
29. #14273727Q8semidirect
205. C59#10371semidirect

Maximal subgroups

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

subgroupindex
#214273902
#214250762
#214225962
#214225942
#214219972
#214219242
C59×#13032
#1037159