Finite Group Databasebeta

← C61×#13300 #29516733

Subgroups — C61×#13300 #29516733

Order 15616.

492 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#2566C211yesmaximal, characteristic
2C61×#1799C211yesmaximal, characteristic
3C4×#99×C61C211yesmaximal, characteristic
4C61×#1393C212yesmaximal
5C61×#1393C212yesmaximal
6C2×#125×C61C411yescharacteristic, complemented
7C2×C4×D4×C61C2211yescharacteristic
8C2×#99×C61C2212yes—
9C2×#99×C61C2212yes—
10C61×#396—21no—
11C61×#396—21no—
12C61×#390C2211yescharacteristic
13C61×#390C2211yescharacteristic
14C61×#385—21no—
15C61×#385—21no—
16C61×#384C411yescharacteristic, complemented
17C61×#382C411yescharacteristic, complemented
18C61×#380—21no—
19C4×C4⋊C4×C61C411yescharacteristic, complemented
20C4×C22⋊C4×C61—21no—
21C61×#342—21no—
22C61×#342—21no—
23C22×D4×C61C2311yescharacteristic
24C23×C4×C61D411yescharacteristic
25C23×C4×C61D411yescharacteristic
26#125×C61—41no—
27#125×C61—41no—
28C4×D4×C61—41no—
29C4×D4×C61—41no—
30C2×C4⋊C4×C61—21no—
31C2×C4⋊C4×C61—21no—
32C2×C4⋊C4×C61—21no—
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61—21no—
35C2×C4⋊C4×C61—41no—
36C2×C4⋊C4×C61C2×C411yescharacteristic
37C2×C4⋊C4×C61C2×C411yescharacteristic
38C2×C4⋊C4×C61C2×C411yescharacteristic
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61D411yescharacteristic
46C2×C22⋊C4×C61D411yescharacteristic
47C2×C42×C61—41no—
48C2×C42×C61C2×C411yescharacteristic
49C2×C42×C61C2×C411yescharacteristic
50C2×C42×C61C2×C411yescharacteristic
51#99×C61—22no—
52#99×C61—24no—
53#99×C61Q814yes—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—41no—
58#95×C61—41no—
59C24×C61C2×D411yescharacteristic
60C24×C61C2×D411yescharacteristic
61C2×D4×C61—41no—
62C2×D4×C61—41no—
63C2×D4×C61C2×Q812yes—
64C2×D4×C61#4112yes—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—42no—
74C22×C4×C61—42no—
75C22×C4×C61—81no—
76C22×C4×C61—81no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C4⋊C4×C61—41no—
83C4⋊C4×C61—42no—
84C4⋊C4×C61—42no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—81no—
90C4⋊C4×C61—81no—
91C4⋊C4×C61—82no—
92C22⋊C4×C61—24no—
93C22⋊C4×C61—41no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—42no—
96C22⋊C4×C61—42no—
97C22⋊C4×C61—42no—
98C22⋊C4×C61—42no—
99C22⋊C4×C61—42no—
100C22⋊C4×C61—81no—
101C22⋊C4×C61—81no—
102C42×C61—21no—
103C42×C61—21no—
104C42×C61—41no—
105C42×C61—41no—
106C42×C61—82no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61—21no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61C2×C22⋊C411yescharacteristic
116C23×C61C22⋊Q812yes—
117C23×C61#12411yescharacteristic
118C23×C61C4⋊Q811yescharacteristic
119D4×C61—42no—
120D4×C61—42no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—22no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—44no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—81no—
146C2×C4×C61—81no—
147C2×C4×C61—82no—
148C2×C4×C61—82no—
149C2×C4×C61—82no—
150C2×C4×C61—82no—
151C2×C4×C61—82no—
152C2×C4×C61#9911yescharacteristic
153C2×C4×C61#9911yescharacteristic
154C2×C4×C61C4×Q812yes—
155#13300C6111yesmaximal, characteristic, direct product factor, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—22no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—42no—
162C22×C61—41no—
163C22×C61—41no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61#39111yescharacteristic
171C22×C61C2×#9911yescharacteristic
172C22×C61#41011yescharacteristic
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—22no—
176C4×C61—42no—
177C4×C61—41no—
178C4×C61—41no—
179C4×C61—41no—
180C4×C61—81no—
181C4×C61—84no—
182C4×C61—81no—
183C4×C61—81no—
184C4×C61—162no—
185#2566C2×C6111yescharacteristic
186#1799C2×C6111yescharacteristic
187C4×#99C2×C6111yescharacteristic
188#1393C2×C6112yes—
189#1393C2×C6112yes—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—42no—
193C2×C61—41no—
194C2×C61—41no—
195C2×C61#199611yescharacteristic
196C2×C61#212611yescharacteristic
197C2×C61#212711yescharacteristic
198C2×#125C4×C6111yescharacteristic, complemented
199C2×C4×D4C22×C6111yescharacteristic
200C2×#99C22×C6112yes—
201C2×#99C22×C6112yes—
202#396—21no—
203#396—21no—
204#390C22×C6111yescharacteristic
205#390C22×C6111yescharacteristic
206#385—21no—
207#385—21no—
208#384C4×C6111yescharacteristic, complemented
209#382C4×C6111yescharacteristic, complemented
210#380—21no—
211C4×C4⋊C4C4×C6111yescharacteristic, complemented
212C4×C22⋊C4—21no—
213#342—21no—
214#342—21no—
215C61#1330011yescharacteristic, direct product factor, Sylow
216C22×D4C23×C6111yescharacteristic
217C23×C4D4×C6111yescharacteristic
218C23×C4D4×C6111yescharacteristic
219#125—41no—
220#125—41no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4C2×C4×C6111yescharacteristic
230C2×C4⋊C4C2×C4×C6111yescharacteristic
231C2×C4⋊C4C2×C4×C6111yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4D4×C6111yescharacteristic
239C2×C22⋊C4D4×C6111yescharacteristic
240C2×C42—41no—
241C2×C42C2×C4×C6111yescharacteristic
242C2×C42C2×C4×C6111yescharacteristic
243C2×C42C2×C4×C6111yescharacteristic
244#99—22no—
245#99—24no—
246#99Q8×C6114yes—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—41no—
251#95—41no—
252C24C2×D4×C6111yescharacteristic
253C24C2×D4×C6111yescharacteristic
254C2×D4—41no—
255C2×D4—41no—
256C2×D4C2×Q8×C6112yes—
257C2×D4#41×C6112yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4C22⋊C4×C6111yescharacteristic
271C22×C4C22⋊C4×C6111yescharacteristic
272C22×C4C22⋊C4×C6111yescharacteristic
273C22×C4C22⋊C4×C6111yescharacteristic
274C22×C4C22×C4×C6111yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—82no—
285C22⋊C4—24no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C42—21no—
296C42—21no—
297C42—41no—
298C42—41no—
299C42—82no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×C22⋊C4×C6111yescharacteristic
309C23C22⋊Q8×C6112yes—
310C23#124×C6111yescharacteristic
311C23C4⋊Q8×C6111yescharacteristic
312D4—42no—
313D4—42no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4#99×C6111yescharacteristic
346C2×C4#99×C6111yescharacteristic
347C2×C4C4×Q8×C6112yes—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22C61×#39111yescharacteristic
363C22C2×#99×C6111yescharacteristic
364C22C61×#41011yescharacteristic
365C4—21no—
366C4—21no—
367C4—22no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—84no—
374C4—81no—
375C4—81no—
376C4—162no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—41no—
381C2—41no—
382C2C61×#199611yescharacteristic
383C2C61×#212611yescharacteristic
384C2C61×#212711yescharacteristic
subgroupG/H
center, socleC22×C61C2×#99
commutatorC22×C4C22×C4×C61
Frattini, cosocleC22×D4C23×C61
prime radicalC22×D4×C61C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×C4×D4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×#125×C61C4semidirect
16. C61×#384C4semidirect
17. C61×#382C4semidirect
19. C4×C4⋊C4×C61C4semidirect
155. #13300C61direct (N × H)
198. C2×#125C4×C61semidirect
208. #384C4×C61semidirect
209. #382C4×C61semidirect
211. C4×C4⋊C4C4×C61semidirect
215. C61#13300direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#25662
C61×#17992
C4×#99×C612
C61×#13932
#1330061