Finite Group Databasebeta

← C59×#9728 #28421837

Subgroups — C59×#9728 #28421837

Order 15104.

414 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3167C211yesmaximal, characteristic
2C4×#108×C59C211yesmaximal, characteristic, complemented
3C59×#1526C211yesmaximal, characteristic, complemented
4C59×#1525C211yesmaximal, characteristic
5C59×#1485C211yesmaximal, characteristic, complemented
6C59×#1484C211yesmaximal, characteristic
7C59×#1451C211yesmaximal, characteristic, complemented
8C59×#533C2211yescharacteristic
9C59×#532C2211yescharacteristic
10C59×#498C411yescharacteristic
11C59×#492C411yescharacteristic
12C59×#487C411yescharacteristic
13C59×#486C411yescharacteristic
14C59×#465—21no—
15C59×#465—21no—
16C4×SD16×C59—21no—
17C4×SD16×C59—21no—
18C59×#431C2211yescharacteristic
19C2×#108×C59—21no—
20C4×M16×C59C2211yescharacteristic
21C4×M16×C59C2211yescharacteristic
22C4×M16×C59C2211yescharacteristic
23C2×C4×C8×C59C2211yescharacteristic
24C59×#341—21no—
25C59×#330—22no—
26C59×#329—22no—
27C59×Q8⋊C8—22no—
28C59×D4⋊C8—22no—
29C59×#322—21no—
30C59×#322—21no—
31C2×#41×C59—21no—
32C2×SD16×C59—21no—
33C2×SD16×C59—21no—
34C2×M16×C59—21no—
35C2×M16×C59—21no—
36C2×M16×C59—21no—
37C22×C8×C59D411yescharacteristic
38C22×C8×C59D411yescharacteristic
39C4⋊Q8×C59C2×C411yescharacteristic
40C4⋊Q8×C59C2×C411yescharacteristic
41C4⋊Q8×C59C2×C411yescharacteristic
42#127×C59C2×C411yescharacteristic
43#125×C59—21no—
44#124×C59—21no—
45C22⋊Q8×C59—21no—
46C22⋊Q8×C59—21no—
47#121×C59—21no—
48#121×C59—21no—
49C4×Q8×C59—21no—
50C4×Q8×C59—21no—
51C4×D4×C59—21no—
52C4×D4×C59—21no—
53#117×C59—21no—
54C2×C42×C59C2311yescharacteristic
55#108×C59—24no—
56#106×C59—21no—
57#106×C59—21no—
58C4⋊C8×C59D412yes—
59#103×C59—21no—
60#103×C59—21no—
61#102×C59—21no—
62#102×C59—21no—
63C22⋊C8×C59—21no—
64#97×C59—21no—
65#97×C59—21no—
66#97×C59D412yes—
67C4×C8×C59—21no—
68C4×C8×C59—21no—
69C4×C8×C59—21no—
70C4×C8×C59—21no—
71C4×C8×C59C2×C411yescharacteristic
72C4×C8×C59C2×C411yescharacteristic
73C4×C8×C59D411yescharacteristic
74C4×C8×C59D411yescharacteristic
75#41×C59—82no—
76C2×Q8×C59—21no—
77C2×Q8×C59—21no—
78C2×Q8×C59—21no—
79C2×D4×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59#4111yescharacteristic
86C22×C4×C59#4111yescharacteristic
87SD16×C59—42no—
88SD16×C59—42no—
89M16×C59—22no—
90M16×C59—22no—
91M16×C59—42no—
92M16×C59—41no—
93M16×C59—41no—
94C2×C8×C59—22no—
95C2×C8×C59—22no—
96C2×C8×C59—22no—
97C2×C8×C59—41no—
98C2×C8×C59—41no—
99C2×C8×C59—41no—
100C2×C8×C59C22⋊C411yescharacteristic
101C2×C8×C59C22⋊C411yescharacteristic
102C2×C8×C59C22⋊C411yescharacteristic
103C2×C8×C59C22⋊C411yescharacteristic
104C4⋊C4×C59—21no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59—41no—
112C22⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C22⋊C4×C59—41no—
115C42×C59—41no—
116C42×C59C22×C411yescharacteristic
117C42×C59C2×D411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59C2×D411yescharacteristic
120C23×C59—41no—
121C23×C59#12411yescharacteristic
122Q8×C59—41no—
123Q8×C59—41no—
124Q8×C59—42no—
125Q8×C59—81no—
126D4×C59—41no—
127D4×C59—41no—
128D4×C59—42no—
129D4×C59—81no—
130D4×C59—82no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—82no—
139C2×C4×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59C4×D411yescharacteristic
141C2×C4×C59C4×D411yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12111yescharacteristic
144C2×C4×C59#12711yescharacteristic
145C8×C59—21no—
146C8×C59—21no—
147C8×C59—21no—
148C8×C59—42no—
149C8×C59—42no—
150C8×C59—42no—
151C8×C59—42no—
152C8×C59#10412yes—
153#9728C5911yesmaximal, characteristic, direct product factor, Sylow
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—81no—
158C22×C59—81no—
159C22×C59—81no—
160C22×C59#39011yescharacteristic
161C4×C59—21no—
162C4×C59—21no—
163C4×C59—21no—
164C4×C59—81no—
165C4×C59—81no—
166C4×C59—81no—
167C4×C59C2×#10411yescharacteristic
168C4×C59#42111yescharacteristic
169C4×C59#44412yes—
170C4×C59#47112yes—
171#3167C2×C5911yescharacteristic
172C4×#108C2×C5911yescharacteristic, complemented
173#1526C2×C5911yescharacteristic, complemented
174#1525C2×C5911yescharacteristic
175#1485C2×C5911yescharacteristic, complemented
176#1484C2×C5911yescharacteristic
177#1451C2×C5911yescharacteristic, complemented
178C2×C59—41no—
179C2×C59—161no—
180C2×C59#197511yescharacteristic
181C2×C59#197711yescharacteristic
182C2×C59#198011yescharacteristic
183#533C22×C5911yescharacteristic
184#532C22×C5911yescharacteristic
185#498C4×C5911yescharacteristic
186#492C4×C5911yescharacteristic
187#487C4×C5911yescharacteristic
188#486C4×C5911yescharacteristic
189#465—21no—
190#465—21no—
191C4×SD16—21no—
192C4×SD16—21no—
193#431C22×C5911yescharacteristic
194C2×#108—21no—
195C4×M16C22×C5911yescharacteristic
196C4×M16C22×C5911yescharacteristic
197C4×M16C22×C5911yescharacteristic
198C2×C4×C8C22×C5911yescharacteristic
199#341—21no—
200#330—22no—
201#329—22no—
202Q8⋊C8—22no—
203D4⋊C8—22no—
204#322—21no—
205#322—21no—
206C59#972811yescharacteristic, direct product factor, Sylow
207C2×#41—21no—
208C2×SD16—21no—
209C2×SD16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C22×C8D4×C5911yescharacteristic
214C22×C8D4×C5911yescharacteristic
215C4⋊Q8C2×C4×C5911yescharacteristic
216C4⋊Q8C2×C4×C5911yescharacteristic
217C4⋊Q8C2×C4×C5911yescharacteristic
218#127C2×C4×C5911yescharacteristic
219#125—21no—
220#124—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223#121—21no—
224#121—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—21no—
228C4×D4—21no—
229#117—21no—
230C2×C42C23×C5911yescharacteristic
231#108—24no—
232#106—21no—
233#106—21no—
234C4⋊C8D4×C5912yes—
235#103—21no—
236#103—21no—
237#102—21no—
238#102—21no—
239C22⋊C8—21no—
240#97—21no—
241#97—21no—
242#97D4×C5912yes—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8C2×C4×C5911yescharacteristic
248C4×C8C2×C4×C5911yescharacteristic
249C4×C8D4×C5911yescharacteristic
250C4×C8D4×C5911yescharacteristic
251#41—82no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C22×C4—41no—
260C22×C4C2×D4×C5911yescharacteristic
261C22×C4#41×C5911yescharacteristic
262C22×C4#41×C5911yescharacteristic
263SD16—42no—
264SD16—42no—
265M16—22no—
266M16—22no—
267M16—42no—
268M16—41no—
269M16—41no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8C22⋊C4×C5911yescharacteristic
277C2×C8C22⋊C4×C5911yescharacteristic
278C2×C8C22⋊C4×C5911yescharacteristic
279C2×C8C22⋊C4×C5911yescharacteristic
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42—41no—
292C42C22×C4×C5911yescharacteristic
293C42C2×D4×C5911yescharacteristic
294C42C2×D4×C5911yescharacteristic
295C42C2×D4×C5911yescharacteristic
296C23—41no—
297C23#124×C5911yescharacteristic
298Q8—41no—
299Q8—41no—
300Q8—42no—
301Q8—81no—
302D4—41no—
303D4—41no—
304D4—42no—
305D4—81no—
306D4—82no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—82no—
315C2×C4C2×C22⋊C4×C5911yescharacteristic
316C2×C4C4×D4×C5911yescharacteristic
317C2×C4C4×D4×C5911yescharacteristic
318C2×C4#121×C5911yescharacteristic
319C2×C4#121×C5911yescharacteristic
320C2×C4#127×C5911yescharacteristic
321C8—21no—
322C8—21no—
323C8—21no—
324C8—42no—
325C8—42no—
326C8—42no—
327C8—42no—
328C8#104×C5912yes—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C59×#39011yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4C2×#104×C5911yescharacteristic
343C4C59×#42111yescharacteristic
344C4C59×#44412yes—
345C4C59×#47112yes—
346C2—41no—
347C2—161no—
348C2C59×#197511yescharacteristic
349C2C59×#197711yescharacteristic
350C2C59×#198011yescharacteristic
subgroupG/H
centerC2×C4×C59#121
commutatorC42C22×C4×C59
Frattini, cosocleC2×C42C23×C59
socleC22×C59#390
prime radicalC59×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#108×C59C2semidirect
3. C59×#1526C2semidirect
5. C59×#1485C2semidirect
7. C59×#1451C2semidirect
153. #9728C59direct (N × H)
172. C4×#108C2×C59semidirect
173. #1526C2×C59semidirect
175. #1485C2×C59semidirect
177. #1451C2×C59semidirect
206. C59#9728direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#31672
C4×#108×C592
C59×#15262
C59×#15252
C59×#14852
C59×#14842
C59×#14512
#972859