Finite Group Databasebeta

← C59×#10161 #28422270

Subgroups — C59×#10161 #28422270

Order 15104.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3208C211yesmaximal, characteristic
2C59×#3165C211yesmaximal, characteristic, complemented
3C2×C59×#486C211yesmaximal, characteristic
4C59×#1965C211yesmaximal, characteristic
5C59×#1548C211yesmaximal, characteristic, complemented
6C59×#1493C211yesmaximal, characteristic, complemented
7C59×#1278C211yesmaximal, characteristic, complemented
8C59×#533C2211yescharacteristic
9C2×C4⋊Q8×C59C2211yescharacteristic
10C2×#127×C59C2211yescharacteristic
11C59×#501—22no—
12C59×#500—21no—
13C59×#498—21no—
14C59×#492—21no—
15C59×#486—22no—
16C59×#486—21no—
17C59×#486—22no—
18C59×#460—22no—
19C59×#450—22no—
20C59×#431C2211yescharacteristic
21C59×#428—22no—
22C59×#423C2211yescharacteristic
23C2×#102×C59—22no—
24C2×#100×C59—22no—
25C4×M16×C59C2211yescharacteristic
26C2×C4×C8×C59C2211yescharacteristic
27C59×#333—22no—
28C59×Q8⋊C8—22no—
29C22×Q8×C59—21no—
30C22×D4×C59—21no—
31C22×D4×C59—21no—
32C22×D4×C59D412yes—
33C2×SD16×C59—24no—
34C2×M16×C59—21no—
35C2×M16×C59D412yes—
36C22×C8×C59—21no—
37C4⋊Q8×C59—21no—
38C4⋊Q8×C59—21no—
39C4⋊Q8×C59—21no—
40C4⋊Q8×C59D411yescharacteristic
41C4⋊Q8×C59D412yes—
42C4⋊Q8×C59D411yescharacteristic
43#127×C59—21no—
44#127×C59—21no—
45#127×C59—21no—
46#127×C59—21no—
47#125×C59—21no—
48C22⋊Q8×C59—21no—
49C22⋊Q8×C59—21no—
50C4×Q8×C59—21no—
51C4×Q8×C59—21no—
52#117×C59—21no—
53C2×C4⋊C4×C59—21no—
54C2×C4⋊C4×C59—21no—
55C2×C42×C59C2311yescharacteristic
56#107×C59—24no—
57#106×C59—24no—
58C4⋊C8×C59—21no—
59C4⋊C8×C59—21no—
60C4⋊C8×C59D412yes—
61#102×C59—28no—
62#102×C59—24no—
63#100×C59—44no—
64C22⋊C8×C59—21no—
65#97×C59—21no—
66C4×C8×C59—21no—
67C4×C8×C59—21no—
68C4×C8×C59—21no—
69C4×C8×C59D412yes—
70C24×C59—22no—
71C2×Q8×C59—21no—
72C2×Q8×C59—21no—
73C2×Q8×C59—22no—
74C2×Q8×C59—41no—
75C2×Q8×C59—82no—
76C2×D4×C59—22no—
77C2×D4×C59—22no—
78C2×D4×C59—44no—
79C2×D4×C59—41no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—42no—
84C2×D4×C59—42no—
85C2×D4×C59—82no—
86C22×C4×C59—41no—
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59#4111yescharacteristic
90SD16×C59—84no—
91M16×C59—22no—
92M16×C59—42no—
93M16×C59SD1614yescomplemented
94C2×C8×C59—22no—
95C2×C8×C59—22no—
96C2×C8×C59—22no—
97C2×C8×C59—42no—
98C2×C8×C59—41no—
99C4⋊C4×C59—22no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—21no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—41no—
106C4⋊C4×C59—41no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C42×C59—41no—
111C42×C59C2×D411yescharacteristic
112C42×C59C2×D411yescharacteristic
113C42×C59C2×D411yescharacteristic
114C42×C59C2×D411yescharacteristic
115C23×C59—42no—
116C23×C59—44no—
117C23×C59—82no—
118C23×C59—82no—
119C23×C59#12111yescharacteristic
120Q8×C59—42no—
121Q8×C59—42no—
122Q8×C59—82no—
123Q8×C59—82no—
124D4×C59—44no—
125D4×C59—44no—
126D4×C59—82no—
127D4×C59—82no—
128D4×C59—82no—
129D4×C59—81no—
130D4×C59—81no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—42no—
136C2×C4×C59—82no—
137C2×C4×C59#12011yescharacteristic
138C2×C4×C59#12011yescharacteristic
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#12711yescharacteristic
143C2×C4×C59C2×SD1612yes—
144C2×C4×C59#13712yes—
145C8×C59—24no—
146C8×C59—42no—
147C8×C59—42no—
148C8×C59—81no—
149#10161C5911yesmaximal, characteristic, direct product factor, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—82no—
153C22×C59—82no—
154C22×C59—82no—
155C22×C59—164no—
156C22×C59#39211yescharacteristic
157C22×C59#46112yes—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—81no—
161C4×C59—82no—
162C4×C59—81no—
163C4×C59#44812yes—
164C4×C59#45512yes—
165C4×C59#47212yes—
166C4×C59#49211yescharacteristic
167C4×C59#49711yescharacteristic
168#3208C2×C5911yescharacteristic
169#3165C2×C5911yescharacteristic, complemented
170C2×#486C2×C5911yescharacteristic
171#1965C2×C5911yescharacteristic
172#1548C2×C5911yescharacteristic, complemented
173#1493C2×C5911yescharacteristic, complemented
174#1278C2×C5911yescharacteristic, complemented
175C2×C59—22no—
176C2×C59—162no—
177C2×C59#200411yescharacteristic
178C2×C59#201311yescharacteristic
179C2×C59#201811yescharacteristic
180#533C22×C5911yescharacteristic
181C2×C4⋊Q8C22×C5911yescharacteristic
182C2×#127C22×C5911yescharacteristic
183#501—22no—
184#500—21no—
185#498—21no—
186#492—21no—
187#486—22no—
188#486—21no—
189#486—22no—
190#460—22no—
191#450—22no—
192#431C22×C5911yescharacteristic
193#428—22no—
194#423C22×C5911yescharacteristic
195C2×#102—22no—
196C2×#100—22no—
197C4×M16C22×C5911yescharacteristic
198C2×C4×C8C22×C5911yescharacteristic
199#333—22no—
200Q8⋊C8—22no—
201C59#1016111yescharacteristic, direct product factor, Sylow
202C22×Q8—21no—
203C22×D4—21no—
204C22×D4—21no—
205C22×D4D4×C5912yes—
206C2×SD16—24no—
207C2×M16—21no—
208C2×M16D4×C5912yes—
209C22×C8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8D4×C5912yes—
214C4⋊Q8D4×C5911yescharacteristic
215C4⋊Q8D4×C5911yescharacteristic
216#127—21no—
217#127—21no—
218#127—21no—
219#127—21no—
220#125—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225#117—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C42C23×C5911yescharacteristic
229#107—24no—
230#106—24no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8D4×C5912yes—
234#102—28no—
235#102—24no—
236#100—44no—
237C22⋊C8—21no—
238#97—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8D4×C5912yes—
243C24—22no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—82no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—82no—
259C22×C4—41no—
260C22×C4C2×D4×C5911yescharacteristic
261C22×C4C2×D4×C5911yescharacteristic
262C22×C4#41×C5911yescharacteristic
263SD16—84no—
264M16—22no—
265M16—42no—
266M16SD16×C5914yescomplemented
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—42no—
271C2×C8—41no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C22⋊C4—41no—
283C42—41no—
284C42C2×D4×C5911yescharacteristic
285C42C2×D4×C5911yescharacteristic
286C42C2×D4×C5911yescharacteristic
287C42C2×D4×C5911yescharacteristic
288C23—42no—
289C23—44no—
290C23—82no—
291C23—82no—
292C23#121×C5911yescharacteristic
293Q8—42no—
294Q8—42no—
295Q8—82no—
296Q8—82no—
297D4—44no—
298D4—44no—
299D4—82no—
300D4—82no—
301D4—82no—
302D4—81no—
303D4—81no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4#120×C5911yescharacteristic
311C2×C4#120×C5911yescharacteristic
312C2×C4#120×C5911yescharacteristic
313C2×C4#121×C5911yescharacteristic
314C2×C4#121×C5911yescharacteristic
315C2×C4#127×C5911yescharacteristic
316C2×C4C2×SD16×C5912yes—
317C2×C4#137×C5912yes—
318C8—24no—
319C8—42no—
320C8—42no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—164no—
328C22C59×#39211yescharacteristic
329C22C59×#46112yes—
330C4—21no—
331C4—21no—
332C4—81no—
333C4—82no—
334C4—81no—
335C4C59×#44812yes—
336C4C59×#45512yes—
337C4C59×#47212yes—
338C4C59×#49211yescharacteristic
339C4C59×#49711yescharacteristic
340C2—22no—
341C2—162no—
342C2C59×#200411yescharacteristic
343C2C59×#201311yescharacteristic
344C2C59×#201811yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC2×#127×C59C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3165C2semidirect
5. C59×#1548C2semidirect
6. C59×#1493C2semidirect
7. C59×#1278C2semidirect
93. M16×C59SD16semidirect
149. #10161C59direct (N × H)
169. #3165C2×C59semidirect
172. #1548C2×C59semidirect
173. #1493C2×C59semidirect
174. #1278C2×C59semidirect
201. C59#10161direct (N × H)
266. M16SD16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#32082
C59×#31652
C2×C59×#4862
C59×#19652
C59×#15482
C59×#14932
C59×#12782
#1016159