Finite Group Databasebeta

← C59×#10172 #28422281

Subgroups — C59×#10172 #28422281

Order 15104.

562 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×#3227C211yesmaximal, characteristic
2C2×C59×#498C211yesmaximal, characteristic, complemented
3C59×#3152C211yesmaximal, characteristic
4C59×#1997C211yesmaximal, characteristic, complemented
5C59×#1546C211yesmaximal, characteristic, complemented
6C59×#1492C211yesmaximal, characteristic
7C59×#1278C211yesmaximal, characteristic, complemented
8C2×#137×C59—22no—
9C59×#532C2211yescharacteristic
10C2×C4⋊Q8×C59C2211yescharacteristic
11C2×C4⋊Q8×C59C2211yescharacteristic
12C59×#498—22no—
13C59×#498—21no—
14C59×#498—22no—
15C59×#497—22no—
16C59×#494—21no—
17C59×#492—21no—
18C59×#486—21no—
19C59×#481—22no—
20C59×#476—22no—
21C59×#431C2211yescharacteristic
22C2×#106×C59—22no—
23C59×#423C2211yescharacteristic
24C4×M16×C59C2211yescharacteristic
25C2×C4×C8×C59C2211yescharacteristic
26C59×#343—22no—
27C59×#332—22no—
28C59×D4⋊C8—22no—
29C2×#41×C59—21no—
30C22×Q8×C59—21no—
31C22×Q8×C59—21no—
32#137×C59—44no—
33#137×C59—44no—
34C2×Q16×C59—24no—
35C2×SD16×C59—24no—
36C2×M16×C59—21no—
37C2×M16×C59D412yes—
38C22×C8×C59—21no—
39C4⋊Q8×C59—21no—
40C4⋊Q8×C59—21no—
41C4⋊Q8×C59—21no—
42C4⋊Q8×C59—21no—
43C4⋊Q8×C59—21no—
44C4⋊Q8×C59—21no—
45C4⋊Q8×C59—21no—
46C4⋊Q8×C59D411yescharacteristic
47C4⋊Q8×C59D412yes—
48#127×C59D411yescharacteristic
49#124×C59—21no—
50#121×C59—21no—
51#121×C59—21no—
52C4×D4×C59—21no—
53C4×D4×C59—21no—
54C2×C4⋊C4×C59—21no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59D412yes—
58C2×C42×C59C2311yescharacteristic
59#106×C59—28no—
60#106×C59—24no—
61C4⋊C8×C59—21no—
62C4⋊C8×C59—21no—
63C4⋊C8×C59D412yes—
64#102×C59—24no—
65C22⋊C8×C59—21no—
66#97×C59—21no—
67C4×C8×C59—21no—
68C4×C8×C59—21no—
69C4×C8×C59—21no—
70C4×C8×C59D412yes—
71#41×C59—42no—
72#41×C59—42no—
73C2×Q8×C59—21no—
74C2×Q8×C59—22no—
75C2×Q8×C59—41no—
76C2×Q8×C59—41no—
77C2×Q8×C59—41no—
78C2×Q8×C59—42no—
79C2×Q8×C59—42no—
80C2×Q8×C59—82no—
81C2×D4×C59—21no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—41no—
85C22×C4×C59—22no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59#4111yescharacteristic
91Q16×C59—48no—
92SD16×C59—48no—
93M16×C59—22no—
94M16×C59—42no—
95M16×C59SD1614yes—
96C2×C8×C59—24no—
97C2×C8×C59—22no—
98C2×C8×C59—22no—
99C2×C8×C59—22no—
100C2×C8×C59—41no—
101C4⋊C4×C59—22no—
102C4⋊C4×C59—22no—
103C4⋊C4×C59—22no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—22no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112C22⋊C4×C59—41no—
113C42×C59C2×D411yescharacteristic
114C42×C59C2×D411yescharacteristic
115C42×C59C2×D411yescharacteristic
116C42×C59C2×D411yescharacteristic
117C23×C59—41no—
118C23×C59#12111yescharacteristic
119Q8×C59—44no—
120Q8×C59—42no—
121Q8×C59—82no—
122Q8×C59—82no—
123Q8×C59—81no—
124Q8×C59—81no—
125D4×C59—42no—
126D4×C59—42no—
127D4×C59—42no—
128D4×C59—82no—
129C2×C4×C59—22no—
130C2×C4×C59—22no—
131C2×C4×C59—41no—
132C2×C4×C59—42no—
133C2×C4×C59—42no—
134C2×C4×C59—82no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59—82no—
138C2×C4×C59#12011yescharacteristic
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59#12011yescharacteristic
141C2×C4×C59#12111yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59#12711yescharacteristic
144C2×C4×C59C2×SD1612yes—
145C2×C4×C59AGL(1,ℤ/8)12yes—
146C8×C59—24no—
147C8×C59—24no—
148C8×C59—42no—
149C8×C59—81no—
150#10172C5911yesmaximal, characteristic, direct product factor, Sylow
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59#39211yescharacteristic
157C22×C59#46012yes—
158C4×C59—21no—
159C4×C59—21no—
160C4×C59—81no—
161C4×C59—82no—
162C4×C59—82no—
163C4×C59#45012yes—
164C4×C59#45612yes—
165C4×C59#47112yes—
166C4×C59#49211yescharacteristic
167C4×C59#49611yescharacteristic
168#3227C2×C5911yescharacteristic
169C2×#498C2×C5911yescharacteristic, complemented
170#3152C2×C5911yescharacteristic
171#1997C2×C5911yescharacteristic, complemented
172#1546C2×C5911yescharacteristic, complemented
173#1492C2×C5911yescharacteristic
174#1278C2×C5911yescharacteristic, complemented
175C2×C59—22no—
176C2×C59—161no—
177C2×C59#200511yescharacteristic
178C2×C59#201411yescharacteristic
179C2×C59#201911yescharacteristic
180C2×#137—22no—
181#532C22×C5911yescharacteristic
182C2×C4⋊Q8C22×C5911yescharacteristic
183C2×C4⋊Q8C22×C5911yescharacteristic
184#498—22no—
185#498—21no—
186#498—22no—
187#497—22no—
188#494—21no—
189#492—21no—
190#486—21no—
191#481—22no—
192#476—22no—
193#431C22×C5911yescharacteristic
194C2×#106—22no—
195#423C22×C5911yescharacteristic
196C4×M16C22×C5911yescharacteristic
197C2×C4×C8C22×C5911yescharacteristic
198#343—22no—
199#332—22no—
200D4⋊C8—22no—
201C59#1017211yescharacteristic, direct product factor, Sylow
202C2×#41—21no—
203C22×Q8—21no—
204C22×Q8—21no—
205#137—44no—
206#137—44no—
207C2×Q16—24no—
208C2×SD16—24no—
209C2×M16—21no—
210C2×M16D4×C5912yes—
211C22×C8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8D4×C5912yes—
220C4⋊Q8D4×C5911yescharacteristic
221#127D4×C5911yescharacteristic
222#124—21no—
223#121—21no—
224#121—21no—
225C4×D4—21no—
226C4×D4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4D4×C5912yes—
231C2×C42C23×C5911yescharacteristic
232#106—28no—
233#106—24no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8D4×C5912yes—
237#102—24no—
238C22⋊C8—21no—
239#97—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8D4×C5912yes—
244#41—42no—
245#41—42no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—82no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4C2×D4×C5911yescharacteristic
262C22×C4C2×D4×C5911yescharacteristic
263C22×C4#41×C5911yescharacteristic
264Q16—48no—
265SD16—48no—
266M16—22no—
267M16—42no—
268M16SD16×C5914yes—
269C2×C8—24no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—41no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C42C2×D4×C5911yescharacteristic
287C42C2×D4×C5911yescharacteristic
288C42C2×D4×C5911yescharacteristic
289C42C2×D4×C5911yescharacteristic
290C23—41no—
291C23#121×C5911yescharacteristic
292Q8—44no—
293Q8—42no—
294Q8—82no—
295Q8—82no—
296Q8—81no—
297Q8—81no—
298D4—42no—
299D4—42no—
300D4—42no—
301D4—82no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—82no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—82no—
311C2×C4#120×C5911yescharacteristic
312C2×C4#120×C5911yescharacteristic
313C2×C4#120×C5911yescharacteristic
314C2×C4#121×C5911yescharacteristic
315C2×C4#121×C5911yescharacteristic
316C2×C4#127×C5911yescharacteristic
317C2×C4C2×SD16×C5912yes—
318C2×C4AGL(1,ℤ/8)×C5912yes—
319C8—24no—
320C8—24no—
321C8—42no—
322C8—81no—
323C22—21no—
324C22—21no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C59×#39211yescharacteristic
329C22C59×#46012yes—
330C4—21no—
331C4—21no—
332C4—81no—
333C4—82no—
334C4—82no—
335C4C59×#45012yes—
336C4C59×#45612yes—
337C4C59×#47112yes—
338C4C59×#49211yescharacteristic
339C4C59×#49611yescharacteristic
340C2—22no—
341C2—161no—
342C2C59×#200511yescharacteristic
343C2C59×#201411yescharacteristic
344C2C59×#201911yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC59×#532C22
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. C2×C59×#498C2semidirect
4. C59×#1997C2semidirect
5. C59×#1546C2semidirect
7. C59×#1278C2semidirect
150. #10172C59direct (N × H)
169. C2×#498C2×C59semidirect
171. #1997C2×C59semidirect
172. #1546C2×C59semidirect
174. #1278C2×C59semidirect
201. C59#10172direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#32272
C2×C59×#4982
C59×#31522
C59×#19972
C59×#15462
C59×#14922
C59×#12782
#1017259