Finite Group Databasebeta

← #28499561

Subgroups — #28499561

Order 15104.

390 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 69 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#21429666C211yesmaximal, characteristic
2#21424172C211yesmaximal, characteristic
3#21422407C211yesmaximal, characteristic
4#21422117C211yesmaximal, characteristic
5#21422097C211yesmaximal, characteristic
6#21422095C211yesmaximal, characteristic
7C59×#1351C211yesmaximal, characteristic
8C2×Q8×Dic59C411yescharacteristic, complemented
9C2×#14273683C2211yescharacteristic
10#16837452—21no—
11#16837334C411yescharacteristic, complemented
12#16837327C2211yescharacteristic
13#16837321C411yescharacteristic, complemented
14#16837296—21no—
15#16837289—21no—
16#16837285—21no—
17#16837277—21no—
18#16837269C411yescharacteristic, complemented
19#16837263—21no—
20#16837169C2211yescharacteristic
21#16837168—21no—
22#16837127—21no—
23#16837126C2211yescharacteristic
24C59×#393C2211yescharacteristic
25C59×#342C2211yescharacteristic
26C59×#323C2211yescharacteristic
27C23×Dic59D411yescharacteristic
28Q8×Dic59—41no—
29Q8×Dic59—41no—
30#14273683—41no—
31#14273683—41no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4C2×C411yescharacteristic
37C2×Dic59⋊C4D411yescharacteristic
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59D411yescharacteristic
41C2×C4×Dic59—21no—
42C2×C4×Dic59C2×C411yescharacteristic
43C2×C4×Dic59D411yescharacteristic
44Dic59⋊C8—41no—
45C22⋊(C59⋊C8)—21no—
46C8×Dic59—41no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C59⋊#95—41no—
53C59⋊#95—41no—
54C59⋊#95C2×C411yescharacteristic
55C22×Q8×C59C2×C411yescharacteristic
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59C2311yescharacteristic
59C22⋊C8×C59—21no—
60#95×C59—21no—
61#95×C59C2×C411yescharacteristic
62C24×C59C2×D411yescharacteristic
63C2×Q8×C59—42no—
64C2×Q8×C59—41no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—41no—
68C22×C4×C59C22×C411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×Dic59—21no—
71C22×Dic59—21no—
72C22×Dic59—21no—
73C22×Dic59—21no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—41no—
77C22×Dic59—41no—
78C22×Dic59C22⋊C411yescharacteristic
79C22×Dic59C22⋊C411yescharacteristic
80C22×Dic59C22⋊C411yescharacteristic
81C22×Dic59C22⋊C411yescharacteristic
82C2×C8×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊Dic59—41no—
88C22⋊Dic59—41no—
89C22⋊Dic59—41no—
90C22⋊Dic59—41no—
91C22⋊Dic59—41no—
92C4⋊Dic59—42no—
93C4⋊Dic59—41no—
94C4⋊Dic59—41no—
95C4⋊Dic59—41no—
96Dic59⋊C4—41no—
97Dic59⋊C4—41no—
98Dic59⋊C4—42no—
99C4×Dic59—21no—
100C4×Dic59—21no—
101C4×Dic59—21no—
102C4×Dic59—21no—
103C4×Dic59—42no—
104C2×C59⋊C8—41no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59#13511yescharacteristic
111C23×C59#13511yescharacteristic
112Q8×C59—42no—
113Q8×C59—42no—
114C2×C4×C59—22no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—81no—
125C2×C4×C59—81no—
126C2×C4×C59—81no—
127C2×Dic59—21no—
128C2×Dic59—21no—
129C2×Dic59—21no—
130C2×Dic59—21no—
131C2×Dic59—21no—
132C2×Dic59—21no—
133C2×Dic59—21no—
134C2×Dic59—41no—
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—
146C2×Dic59—81no—
147C2×Dic59—81no—
148C2×Dic59—81no—
149C2×Dic59#9911yescharacteristic
150C2×Dic59#9911yescharacteristic
151C8×C59—81no—
152C59⋊C8—81no—
153#11842—591nomaximal, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—41no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—81no—
163C22×C59C2×#9911yescharacteristic
164C22×C59#41611yescharacteristic
165C22×C59#42111yescharacteristic
166C4×C59—42no—
167C4×C59—41no—
168C4×C59—41no—
169C4×C59—81no—
170C4×C59—161no—
171Dic59—21no—
172Dic59—21no—
173Dic59—41no—
174Dic59—41no—
175Dic59—41no—
176Dic59—81no—
177Dic59—81no—
178Dic59—161no—
179#2599—591no—
180#1820—591no—
181#1458—591no—
182#1351—591no—
183#1351D5911yescharacteristic
184#1344—591no—
185#1344—591no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—41no—
189C2×C59—41no—
190C2×C59#151611yescharacteristic
191C2×C59#212011yescharacteristic
192C2×C59#212911yescharacteristic
193C2×C4×Q8—591no—
194C2×#117—591no—
195#431—1181no—
196#398—591no—
197#397—591no—
198#395—591no—
199#393C2×D5911yescharacteristic
200#388—1181no—
201#387—1181no—
202#387—1181no—
203#385—1181no—
204#383—591no—
205#381—1181no—
206#342—591no—
207#342—1181no—
208#342C2×D5911yescharacteristic
209#324—1181no—
210#323—591no—
211#323C2×D5911yescharacteristic
212C59#1184211yescharacteristic, complemented, Sylow
213C22×Q8C4×D5911yescharacteristic
214C23×C4—591no—
215C4×Q8—2361no—
216C4×Q8—2361no—
217#117—2361no—
218#117—2361no—
219C2×C4⋊C4—591no—
220C2×C4⋊C4—591no—
221C2×C4⋊C4—591no—
222C2×C4⋊C4—1181no—
223C2×C4⋊C4—1181no—
224C2×C4⋊C4—1181no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—591no—
228C2×C22⋊C4—1181no—
229C2×C22⋊C4—1181no—
230C2×C22⋊C4C22×D5911yescharacteristic
231C2×C42—591no—
232C2×C42—591no—
233C2×C42—1181no—
234C4⋊C8—2361no—
235C22⋊C8—21no—
236C22⋊C8—1181no—
237C4×C8—2361no—
238#95—21no—
239#95—591no—
240#95—1181no—
241#95—1181no—
242#95—1181no—
243#95—1181no—
244#95—1181no—
245#95—2361no—
246#95—2361no—
247#95C4×D5911yescharacteristic
248C24D4×D5911yescharacteristic
249C2×Q8—42no—
250C2×Q8—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—591no—
255C22×C4—591no—
256C22×C4—591no—
257C22×C4—591no—
258C22×C4—1181no—
259C22×C4—1181no—
260C22×C4—1181no—
261C22×C4—1181no—
262C22×C4—1181no—
263C22×C4—1181no—
264C22×C4—2361no—
265C22×C4—2361no—
266C22×C4C2×C4×D5911yescharacteristic
267C22×C4D4×D5911yescharacteristic
268C2×C8—41no—
269C2×C8—2361no—
270C4⋊C4—2361no—
271C4⋊C4—2361no—
272C4⋊C4—2361no—
273C4⋊C4—2362no—
274C4⋊C4—2361no—
275C4⋊C4—2361no—
276C4⋊C4—2362no—
277C22⋊C4—41no—
278C22⋊C4—42no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—2361no—
282C22⋊C4—2361no—
283C22⋊C4—2361no—
284C22⋊C4—2361no—
285C22⋊C4—2361no—
286C42—1181no—
287C42—1181no—
288C42—1181no—
289C42—1181no—
290C42—2362no—
291C23—21no—
292C23—21no—
293C23—41no—
294C23—41no—
295C23C22⋊C4×D5911yescharacteristic
296C23#1427374811yescharacteristic
297C23#1427375011yescharacteristic
298Q8—42no—
299Q8—42no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—591no—
314C2×C4—591no—
315C2×C4—1181no—
316C2×C4—1181no—
317C2×C4—1181no—
318C2×C4—1181no—
319C2×C4—1181no—
320C2×C4—1181no—
321C2×C4—1181no—
322C2×C4—4721no—
323C2×C4—4721no—
324C2×C4—4721no—
325C2×C4—2361no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4—2361no—
329C2×C4—2361no—
330C2×C4—2361no—
331C2×C4—2361no—
332C2×C4—2361no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4—2361no—
336C2×C4—2361no—
337C8—81no—
338C8—4721no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22#99×D5911yescharacteristic
349C22#1683739711yescharacteristic
350C22#1683741711yescharacteristic
351C4—42no—
352C4—41no—
353C4—41no—
354C4—81no—
355C4—161no—
356C4—1181no—
357C4—1181no—
358C4—9441no—
359C4—4721no—
360C4—4721no—
361C4—2361no—
362C4—2361no—
363C4—2361no—
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2#2142265111yescharacteristic
369C2#2142573411yescharacteristic
370C2#2142576911yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1351C2
socleC22×C59C2×#99
prime radicalC24×C59C2×D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1351D59
hypocenterC59#11842
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1351

Complements

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

NH ≅ G/Ntype
8. C2×Q8×Dic59C4semidirect
11. #16837334C4semidirect
13. #16837321C4semidirect
18. #16837269C4semidirect
212. C59#11842semidirect

Maximal subgroups

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

subgroupindex
#214296662
#214241722
#214224072
#214221172
#214220972
#214220952
C59×#13512
#1184259