Finite Group Databasebeta

← C61×#18529 #29521962

Subgroups — C61×#18529 #29521962

Order 15616.

1630 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 246 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
1C2×D42×C61C211yesmaximal, characteristic
2C61×#3138C211yesmaximal, characteristic, complemented
3C61×#3110C212yesmaximal, complemented
4C61×#1639C218yesmaximal, complemented
5C61×#1555C211yesmaximal, characteristic, complemented
6C61×#1490C212yesmaximal, complemented
7C23×D4×C61—22no—
8D42×C61—22no—
9D42×C61—24no—
10D42×C61C2214yes—
11C61×#533C2211yescharacteristic, complemented
12C2×#127×C61C2211yescharacteristic
13C2×#121×C61—22no—
14C2×#120×C61—22no—
15C2×C4×D4×C61C2212yes—
16C61×#490—22no—
17C61×#486C2214yescomplemented
18C61×#484—22no—
19C61×#474C2214yescomplemented
20C61×#461C2214yescomplemented
21C61×#450—22no—
22C61×#440—24no—
23C61×#432C2212yescomplemented
24C61×#418—22no—
25C4×M16×C61C2211yescharacteristic, complemented
26C61×#331C2214yescomplemented
27C61×D4⋊C8C2218yescomplemented
28C25×C61—22no—
29C22×D4×C61—28no—
30C22×D4×C61—24no—
31C22×D4×C61—28no—
32C22×D4×C61—22no—
33C22×D4×C61—416no—
34C22×D4×C61D412yes—
35C22×D4×C61D412yes—
36C23×C4×C61—22no—
37C2×SD16×C61—24no—
38C2×M16×C61—21no—
39C4⋊Q8×C61C2312yescomplemented
40#127×C61—22no—
41#127×C61D412yes—
42#127×C61C2312yes—
43#125×C61—21no—
44C22⋊Q8×C61—22no—
45#121×C61—28no—
46#121×C61—28no—
47#120×C61—28no—
48#120×C61—28no—
49C4×Q8×C61—22no—
50C4×D4×C61—22no—
51C4×D4×C61—22no—
52C4×D4×C61D414yes—
53C4×D4×C61C2314yes—
54#117×C61—21no—
55C2×C4⋊C4×C61D412yes—
56C2×C22⋊C4×C61—22no—
57C2×C22⋊C4×C61—22no—
58C2×C42×C61C2311yescharacteristic
59#107×C61—24no—
60C4⋊C8×C61C2314yescomplemented
61#103×C61—24no—
62#102×C61—28no—
63#102×C61—24no—
64C22⋊C8×C61—22no—
65#97×C61—21no—
66#97×C61—22no—
67C4×C8×C61C2312yescomplemented
68C24×C61—22no—
69C24×C61—28no—
70C24×C61—22no—
71C24×C61—416no—
72C24×C61—84no—
73C2×Q8×C61—22no—
74C2×D4×C61—28no—
75C2×D4×C61—216no—
76C2×D4×C61—22no—
77C2×D4×C61—24no—
78C2×D4×C61—22no—
79C2×D4×C61—24no—
80C2×D4×C61—48no—
81C2×D4×C61—44no—
82C2×D4×C61—432no—
83C2×D4×C61—416no—
84C2×D4×C61—416no—
85C2×D4×C61—416no—
86C2×D4×C61C2×D414yes—
87C2×D4×C61C2×D414yes—
88C22×C4×C61—22no—
89C22×C4×C61—24no—
90C22×C4×C61—24no—
91C22×C4×C61—48no—
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D411yescharacteristic
94SD16×C61—48no—
95M16×C61—42no—
96C2×C8×C61—22no—
97C2×C8×C61—24no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—42no—
102C4⋊C4×C61C2×D414yes—
103C22⋊C4×C61—28no—
104C22⋊C4×C61—24no—
105C22⋊C4×C61—24no—
106C22⋊C4×C61—41no—
107C42×C61—41no—
108C42×C61C2×D411yescharacteristic
109C42×C61C2×D412yes—
110C42×C61C2411yescharacteristic
111C23×C61—28no—
112C23×C61—24no—
113C23×C61—24no—
114C23×C61—44no—
115C23×C61—48no—
116C23×C61—432no—
117C23×C61—44no—
118C23×C61—44no—
119C23×C61—88no—
120C23×C61—832no—
121C23×C61#12011yescharacteristic
122Q8×C61—44no—
123D4×C61—28no—
124D4×C61—24no—
125D4×C61—44no—
126D4×C61—48no—
127D4×C61—432no—
128D4×C61—48no—
129D4×C61—82no—
130D4×C61AGL(1,ℤ/8)18yes—
131C2×C4×C61—22no—
132C2×C4×C61—24no—
133C2×C4×C61—24no—
134C2×C4×C61—48no—
135C2×C4×C61—44no—
136C2×C4×C61—48no—
137C2×C4×C61—42no—
138C2×C4×C61#12011yescharacteristic
139C2×C4×C61#12012yes—
140C2×C4×C61C22×D411yescharacteristic
141C2×C4×C61C22×D412yes—
142C8×C61—42no—
143C8×C61—44no—
144#18529C6111yesmaximal, characteristic, direct product factor, Sylow
145C22×C61—22no—
146C22×C61—28no—
147C22×C61—416no—
148C22×C61—44no—
149C22×C61—44no—
150C22×C61—48no—
151C22×C61—44no—
152C22×C61—88no—
153C22×C61—832no—
154C22×C61#45512yes—
155C22×C61C2×#12011yescharacteristic
156C4×C61—22no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—44no—
160C4×C61—82no—
161C4×C61C2×AGL(1,ℤ/8)14yes—
162C2×D42C2×C6111yescharacteristic
163#3138C2×C6111yescharacteristic, complemented
164#3110C2×C6112yescomplemented
165#1639C2×C6118yescomplemented
166#1555C2×C6111yescharacteristic, complemented
167#1490C2×C6112yescomplemented
168C2×C61—22no—
169C2×C61—48no—
170C2×C61—84no—
171C2×C61#300312yes—
172C2×C61C2×#45511yescharacteristic
173C23×D4—22no—
174D42—22no—
175D42—24no—
176D42C22×C6114yes—
177#533C22×C6111yescharacteristic, complemented
178C2×#127C22×C6111yescharacteristic
179C2×#121—22no—
180C2×#120—22no—
181C2×C4×D4C22×C6112yes—
182#490—22no—
183#486C22×C6114yescomplemented
184#484—22no—
185#474C22×C6114yescomplemented
186#461C22×C6114yescomplemented
187#450—22no—
188#440—24no—
189#432C22×C6112yescomplemented
190#418—22no—
191C4×M16C22×C6111yescharacteristic, complemented
192#331C22×C6114yescomplemented
193D4⋊C8C22×C6118yescomplemented
194C61#1852911yescharacteristic, direct product factor, Sylow
195C25—22no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—416no—
201C22×D4D4×C6112yes—
202C22×D4D4×C6112yes—
203C23×C4—22no—
204C2×SD16—24no—
205C2×M16—21no—
206C4⋊Q8C23×C6112yescomplemented
207#127—22no—
208#127D4×C6112yes—
209#127C23×C6112yes—
210#125—21no—
211C22⋊Q8—22no—
212#121—28no—
213#121—28no—
214#120—28no—
215#120—28no—
216C4×Q8—22no—
217C4×D4—22no—
218C4×D4—22no—
219C4×D4D4×C6114yes—
220C4×D4C23×C6114yes—
221#117—21no—
222C2×C4⋊C4D4×C6112yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42C23×C6111yescharacteristic
226#107—24no—
227C4⋊C8C23×C6114yescomplemented
228#103—24no—
229#102—28no—
230#102—24no—
231C22⋊C8—22no—
232#97—21no—
233#97—22no—
234C4×C8C23×C6112yescomplemented
235C24—22no—
236C24—28no—
237C24—22no—
238C24—416no—
239C24—84no—
240C2×Q8—22no—
241C2×D4—28no—
242C2×D4—216no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—48no—
248C2×D4—44no—
249C2×D4—432no—
250C2×D4—416no—
251C2×D4—416no—
252C2×D4—416no—
253C2×D4C2×D4×C6114yes—
254C2×D4C2×D4×C6114yes—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4C2×D4×C6112yes—
260C22×C4C2×D4×C6111yescharacteristic
261SD16—48no—
262M16—42no—
263C2×C8—22no—
264C2×C8—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4C2×D4×C6114yes—
270C22⋊C4—28no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—41no—
274C42—41no—
275C42C2×D4×C6111yescharacteristic
276C42C2×D4×C6112yes—
277C42C24×C6111yescharacteristic
278C23—28no—
279C23—24no—
280C23—24no—
281C23—44no—
282C23—48no—
283C23—432no—
284C23—44no—
285C23—44no—
286C23—88no—
287C23—832no—
288C23#120×C6111yescharacteristic
289Q8—44no—
290D4—28no—
291D4—24no—
292D4—44no—
293D4—48no—
294D4—432no—
295D4—48no—
296D4—82no—
297D4AGL(1,ℤ/8)×C6118yes—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4#120×C6111yescharacteristic
306C2×C4#120×C6112yes—
307C2×C4C22×D4×C6111yescharacteristic
308C2×C4C22×D4×C6112yes—
309C8—42no—
310C8—44no—
311C22—22no—
312C22—28no—
313C22—416no—
314C22—44no—
315C22—44no—
316C22—48no—
317C22—44no—
318C22—88no—
319C22—832no—
320C22C61×#45512yes—
321C22C2×#120×C6111yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—44no—
326C4—82no—
327C4C2×AGL(1,ℤ/8)×C6114yes—
328C2—22no—
329C2—48no—
330C2—84no—
331C2C61×#300312yes—
332C2C2×C61×#45511yescharacteristic
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC42C24×C61
prime radicalC2×D42×C61C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3138C2semidirect
3. C61×#3110C2semidirect
4. C61×#1639C2semidirect
5. C61×#1555C2semidirect
6. C61×#1490C2semidirect
11. C61×#533C22semidirect
17. C61×#486C22semidirect
19. C61×#474C22semidirect
20. C61×#461C22semidirect
23. C61×#432C22semidirect
25. C4×M16×C61C22semidirect
26. C61×#331C22semidirect
27. C61×D4⋊C8C22semidirect
39. C4⋊Q8×C61C23semidirect
60. C4⋊C8×C61C23semidirect
67. C4×C8×C61C23semidirect
144. #18529C61direct (N × H)
163. #3138C2×C61semidirect
164. #3110C2×C61semidirect
165. #1639C2×C61semidirect
166. #1555C2×C61semidirect
167. #1490C2×C61semidirect
177. #533C22×C61semidirect
183. #486C22×C61semidirect
185. #474C22×C61semidirect
186. #461C22×C61semidirect
189. #432C22×C61semidirect
191. C4×M16C22×C61semidirect
192. #331C22×C61semidirect
193. D4⋊C8C22×C61semidirect
194. C61#18529direct (N × H)
206. C4⋊Q8C23×C61semidirect
227. C4⋊C8C23×C61semidirect
234. C4×C8C23×C61semidirect

Maximal subgroups

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

subgroupindex
C2×D42×C612
C61×#31382
C61×#31102
C61×#16392
C61×#15552
C61×#14902
#1852961