Finite Group Databasebeta

← C59×#12730 #28424839

Subgroups — C59×#12730 #28424839

Order 15104.

666 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C59×#3509C211yesmaximal, characteristic
2C59×#2133C212yesmaximal
3C2×C59×#356C212yesmaximal, complemented
4C59×#1830C212yesmaximal, complemented
5C59×#557—22no—
6C59×#556—22no—
7C59×#552—24no—
8C59×#541—22no—
9C59×#536C411yescharacteristic
10C59×#536C2211yescharacteristic
11C59×#533—22no—
12C59×#533C411yescharacteristic
13C2×C4⋊Q8×C59C2211yescharacteristic
14C59×#519C411yescharacteristic
15C59×#519C2211yescharacteristic
16C4×#41×C59C411yescharacteristic
17C2×#101×C59C2214yes—
18C59×#356—22no—
19C59×#356—22no—
20C59×#356—22no—
21C59×#355—22no—
22C59×#344—24no—
23C2×#41×C59C2×C411yescharacteristic
24C22×Q8×C59C2×C411yescharacteristic
25C22×Q8×C59C2311yescharacteristic
26C2×M16×C59—24no—
27C4⋊Q8×C59—21no—
28C4⋊Q8×C59—22no—
29C4⋊Q8×C59—21no—
30C4⋊Q8×C59D412yes—
31C4⋊Q8×C59D412yes—
32C4⋊Q8×C59D412yes—
33#126×C59—24no—
34#126×C59—24no—
35#125×C59—24no—
36#125×C59—24no—
37#125×C59—24no—
38#125×C59—21no—
39#124×C59—21no—
40#124×C59D412yes—
41#123×C59—22no—
42#123×C59—22no—
43C22⋊Q8×C59—24no—
44C22⋊Q8×C59—22no—
45C22⋊Q8×C59—22no—
46C22⋊Q8×C59—22no—
47C4×Q8×C59—24no—
48C4×Q8×C59—21no—
49C4×Q8×C59—22no—
50C4×Q8×C59—22no—
51C4×Q8×C59—22no—
52C4×Q8×C59—22no—
53C4×Q8×C59—22no—
54C4×D4×C59—21no—
55C4×D4×C59—22no—
56#117×C59—24no—
57#117×C59—22no—
58#117×C59—21no—
59#117×C59C2×C411yescharacteristic
60#117×C59C2×C411yescharacteristic
61#117×C59C2×C411yescharacteristic
62#117×C59D414yes—
63C2×C4⋊C4×C59—22no—
64C2×C42×C59—21no—
65C2×C42×C59C2×C411yescharacteristic
66#101×C59—24no—
67#101×C59—24no—
68#41×C59—42no—
69C2×Q8×C59—22no—
70C2×Q8×C59—21no—
71C2×Q8×C59—42no—
72C2×Q8×C59—42no—
73C2×Q8×C59C22⋊C412yes—
74C2×Q8×C59C22⋊C411yescharacteristic
75C2×Q8×C59C2×D412yes—
76C2×Q8×C59C2×D412yes—
77C2×D4×C59—21no—
78C2×D4×C59C22⋊C411yescharacteristic
79C22×C4×C59—22no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C2×D412yes—
85M16×C59—44no—
86M16×C59—44no—
87C2×C8×C59—44no—
88C4⋊C4×C59—22no—
89C4⋊C4×C59—22no—
90C4⋊C4×C59—24no—
91C4⋊C4×C59—24no—
92C4⋊C4×C59—22no—
93C4⋊C4×C59—22no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—21no—
96C4⋊C4×C59—21no—
97C4⋊C4×C59—24no—
98C4⋊C4×C59—24no—
99C4⋊C4×C59—24no—
100C4⋊C4×C59—24no—
101C22⋊C4×C59—22no—
102C22⋊C4×C59—24no—
103C22⋊C4×C59—21no—
104C22⋊C4×C59—24no—
105C22⋊C4×C59C22⋊C412yes—
106C22⋊C4×C59C22⋊C412yes—
107C42×C59—22no—
108C42×C59—24no—
109C42×C59—21no—
110C42×C59—21no—
111C42×C59—21no—
112C42×C59—24no—
113C42×C59C22⋊C411yescharacteristic
114C42×C59C22⋊C411yescharacteristic
115C42×C59C22⋊C412yes—
116C23×C59—21no—
117C23×C59C2×C22⋊C411yescharacteristic
118Q8×C59—42no—
119Q8×C59—42no—
120Q8×C59—42no—
121Q8×C59—41no—
122Q8×C59—44no—
123Q8×C59—42no—
124D4×C59—41no—
125D4×C59—42no—
126C2×C4×C59—21no—
127C2×C4×C59—24no—
128C2×C4×C59—22no—
129C2×C4×C59—22no—
130C2×C4×C59—22no—
131C2×C4×C59—22no—
132C2×C4×C59—22no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—42no—
136C2×C4×C59—44no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59C2×C22⋊C411yescharacteristic
141C2×C4×C59#12014yes—
142C8×C59—84no—
143#12730C5911yesmaximal, characteristic, direct product factor, Sylow
144C22×C59—41no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59#37911yescharacteristic
148C22×C59#41012yes—
149C4×C59—22no—
150C4×C59—24no—
151C4×C59—21no—
152C4×C59—21no—
153C4×C59—42no—
154C4×C59—41no—
155C4×C59—42no—
156C4×C59—42no—
157C4×C59—41no—
158C4×C59—42no—
159#3509C2×C5911yescharacteristic
160#2133C2×C5912yes—
161C2×#356C2×C5912yescomplemented
162#1830C2×C5912yescomplemented
163C2×C59—21no—
164C2×C59—22no—
165C2×C59—81no—
166C2×C59#178211yescharacteristic
167#557—22no—
168#556—22no—
169#552—24no—
170#541—22no—
171#536C4×C5911yescharacteristic
172#536C22×C5911yescharacteristic
173#533—22no—
174#533C4×C5911yescharacteristic
175C2×C4⋊Q8C22×C5911yescharacteristic
176#519C4×C5911yescharacteristic
177#519C22×C5911yescharacteristic
178C4×#41C4×C5911yescharacteristic
179C2×#101C22×C5914yes—
180#356—22no—
181#356—22no—
182#356—22no—
183#355—22no—
184#344—24no—
185C59#1273011yescharacteristic, direct product factor, Sylow
186C2×#41C2×C4×C5911yescharacteristic
187C22×Q8C2×C4×C5911yescharacteristic
188C22×Q8C23×C5911yescharacteristic
189C2×M16—24no—
190C4⋊Q8—21no—
191C4⋊Q8—22no—
192C4⋊Q8—21no—
193C4⋊Q8D4×C5912yes—
194C4⋊Q8D4×C5912yes—
195C4⋊Q8D4×C5912yes—
196#126—24no—
197#126—24no—
198#125—24no—
199#125—24no—
200#125—24no—
201#125—21no—
202#124—21no—
203#124D4×C5912yes—
204#123—22no—
205#123—22no—
206C22⋊Q8—24no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C4×Q8—24no—
211C4×Q8—21no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×D4—21no—
218C4×D4—22no—
219#117—24no—
220#117—22no—
221#117—21no—
222#117C2×C4×C5911yescharacteristic
223#117C2×C4×C5911yescharacteristic
224#117C2×C4×C5911yescharacteristic
225#117D4×C5914yes—
226C2×C4⋊C4—22no—
227C2×C42—21no—
228C2×C42C2×C4×C5911yescharacteristic
229#101—24no—
230#101—24no—
231#41—42no—
232C2×Q8—22no—
233C2×Q8—21no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8C22⋊C4×C5912yes—
237C2×Q8C22⋊C4×C5911yescharacteristic
238C2×Q8C2×D4×C5912yes—
239C2×Q8C2×D4×C5912yes—
240C2×D4—21no—
241C2×D4C22⋊C4×C5911yescharacteristic
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4C22×C4×C5911yescharacteristic
247C22×C4C2×D4×C5912yes—
248M16—44no—
249M16—44no—
250C2×C8—44no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C22⋊C4—22no—
265C22⋊C4—24no—
266C22⋊C4—21no—
267C22⋊C4—24no—
268C22⋊C4C22⋊C4×C5912yes—
269C22⋊C4C22⋊C4×C5912yes—
270C42—22no—
271C42—24no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—24no—
276C42C22⋊C4×C5911yescharacteristic
277C42C22⋊C4×C5911yescharacteristic
278C42C22⋊C4×C5912yes—
279C23—21no—
280C23C2×C22⋊C4×C5911yescharacteristic
281Q8—42no—
282Q8—42no—
283Q8—42no—
284Q8—41no—
285Q8—44no—
286Q8—42no—
287D4—41no—
288D4—42no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4C2×C22⋊C4×C5911yescharacteristic
303C2×C4C2×C22⋊C4×C5911yescharacteristic
304C2×C4#120×C5914yes—
305C8—84no—
306C22—41no—
307C22—42no—
308C22—41no—
309C22C59×#37911yescharacteristic
310C22C59×#41012yes—
311C4—22no—
312C4—24no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—41no—
317C4—42no—
318C4—42no—
319C4—41no—
320C4—42no—
321C2—21no—
322C2—22no—
323C2—81no—
324C2C59×#178211yescharacteristic
subgroupG/H
center, socleC2×C59#1782
commutatorC22×C4C22×C4×C59
Frattini, cosocleC22×Q8C23×C59
prime radicalC2×#41×C59C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C59 ◁ C23×C59 ◁ C59×#519 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C59×#356C2semidirect
4. C59×#1830C2semidirect
143. #12730C59direct (N × H)
161. C2×#356C2×C59semidirect
162. #1830C2×C59semidirect
185. C59#12730direct (N × H)

Maximal subgroups

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

subgroupindex
C59×#35092
C59×#21332
C2×C59×#3562
C59×#18302
#1273059