Finite Group Databasebeta

← C61×#12782 #29516215

Subgroups — C61×#12782 #29516215

Order 15616.

730 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3544C211yesmaximal, characteristic
2C61×#2133C212yesmaximal, complemented
3C2×C61×#356C211yesmaximal, characteristic, complemented
4C61×#1912C212yesmaximal, complemented
5C61×#1886C211yesmaximal, characteristic, complemented
6C61×#563—21no—
7C61×#563C412yescomplemented
8Q82×C61C411yescharacteristic, complemented
9C61×#555C411yescharacteristic, complemented
10C61×#554—21no—
11C61×#552—22no—
12C61×#547—22no—
13C61×#544—22no—
14C61×#542—22no—
15C61×#536—21no—
16C61×#536C2212yes—
17C2×C4⋊Q8×C61C2211yescharacteristic
18C61×#529—21no—
19C61×#519C2211yescharacteristic
20C61×#421—22no—
21C61×#419—21no—
22C2×#101×C61C2212yes—
23C2×#101×C61C2211yescharacteristic
24C61×#356—21no—
25C61×#356—21no—
26C61×#356—22no—
27C61×#355—22no—
28C61×#339—21no—
29C61×H(3,ℤ/4)—22no—
30C2×#41×C61—21no—
31C2×#41×C61D412yes—
32C22×Q8×C61D411yescharacteristic
33C22×Q8×C61C2311yescharacteristic
34C2×M16×C61—22no—
35C2×M16×C61—21no—
36C4⋊Q8×C61—21no—
37C4⋊Q8×C61—21no—
38C4⋊Q8×C61—22no—
39C4⋊Q8×C61—22no—
40C4⋊Q8×C61C2×C411yescharacteristic
41C4⋊Q8×C61C2×C411yescharacteristic
42C4⋊Q8×C61C2×C412yes—
43C4⋊Q8×C61D411yescharacteristic
44C4⋊Q8×C61D411yescharacteristic
45C4⋊Q8×C61D412yes—
46#126×C61—22no—
47#126×C61—22no—
48#126×C61—22no—
49#125×C61—21no—
50#125×C61—22no—
51#125×C61—22no—
52#125×C61—22no—
53#124×C61—21no—
54#124×C61—21no—
55#124×C61—21no—
56#124×C61C2×C412yes—
57#124×C61D412yes—
58#123×C61—22no—
59#123×C61—22no—
60#123×C61—22no—
61#123×C61—22no—
62#123×C61—22no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61—22no—
66C22⋊Q8×C61—21no—
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71#121×C61—21no—
72#121×C61—22no—
73C4×Q8×C61—22no—
74C4×Q8×C61—21no—
75C4×Q8×C61—42no—
76C4×D4×C61—22no—
77C4×D4×C61—22no—
78C4×D4×C61—21no—
79#117×C61—21no—
80#117×C61—22no—
81#117×C61—22no—
82#117×C61—22no—
83#117×C61—21no—
84#117×C61D412yes—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—22no—
87C2×C42×C61D411yescharacteristic
88#104×C61—44no—
89#103×C61—42no—
90#101×C61—22no—
91#101×C61—22no—
92#101×C61—22no—
93#41×C61—42no—
94#41×C61—42no—
95#41×C61—81no—
96#41×C61—81no—
97C2×Q8×C61—21no—
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—41no—
101C2×Q8×C61—41no—
102C2×Q8×C61—41no—
103C2×Q8×C61—41no—
104C2×Q8×C61C22⋊C411yescharacteristic
105C2×Q8×C61C22⋊C412yes—
106C2×Q8×C61C22⋊C411yescharacteristic
107C2×Q8×C61C22×C411yescharacteristic
108C2×Q8×C61C2×D411yescharacteristic
109C2×Q8×C61C2×D412yes—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61C22⋊C412yes—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D412yes—
121M16×C61—42no—
122M16×C61—41no—
123M16×C61—41no—
124M16×C61—42no—
125C2×C8×C61—41no—
126C2×C8×C61—42no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—42no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—41no—
155C42×C61—21no—
156C42×C61—22no—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61—24no—
160C42×C61—41no—
161C42×C61C22⋊C411yescharacteristic
162C42×C61C22⋊C411yescharacteristic
163C42×C61C22⋊C412yes—
164C42×C61C22⋊C412yes—
165C23×C61—22no—
166C23×C61#12011yescharacteristic
167Q8×C61—41no—
168Q8×C61—41no—
169Q8×C61—42no—
170Q8×C61—42no—
171Q8×C61—41no—
172Q8×C61—41no—
173Q8×C61—42no—
174Q8×C61—81no—
175Q8×C61—81no—
176D4×C61—42no—
177D4×C61—42no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—81no—
181C2×C4×C61—21no—
182C2×C4×C61—22no—
183C2×C4×C61—21no—
184C2×C4×C61—22no—
185C2×C4×C61—21no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61—42no—
193C2×C4×C61—44no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—82no—
198C2×C4×C61—82no—
199C2×C4×C61C2×C22⋊C411yescharacteristic
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12012yes—
203C8×C61—82no—
204C8×C61—81no—
205#12782C6111yesmaximal, characteristic, direct product factor, Sylow
206C22×C61—41no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61#37911yescharacteristic
211C22×C61B(4,𝔽2)11yescharacteristic
212C22×C61#45811yescharacteristic
213C4×C61—22no—
214C4×C61—22no—
215C4×C61—21no—
216C4×C61—22no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—41no—
220C4×C61—42no—
221C4×C61—41no—
222C4×C61—41no—
223C4×C61—42no—
224C4×C61—82no—
225#3544C2×C6111yescharacteristic
226#2133C2×C6112yescomplemented
227C2×#356C2×C6111yescharacteristic, complemented
228#1912C2×C6112yescomplemented
229#1886C2×C6111yescharacteristic, complemented
230C2×C61—21no—
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—82no—
234C2×C61#179011yescharacteristic
235#563—21no—
236#563C4×C6112yescomplemented
237Q82C4×C6111yescharacteristic, complemented
238#555C4×C6111yescharacteristic, complemented
239#554—21no—
240#552—22no—
241#547—22no—
242#544—22no—
243#542—22no—
244#536—21no—
245#536C22×C6112yes—
246C2×C4⋊Q8C22×C6111yescharacteristic
247#529—21no—
248#519C22×C6111yescharacteristic
249#421—22no—
250#419—21no—
251C2×#101C22×C6112yes—
252C2×#101C22×C6111yescharacteristic
253#356—21no—
254#356—21no—
255#356—22no—
256#355—22no—
257#339—21no—
258H(3,ℤ/4)—22no—
259C61#1278211yescharacteristic, direct product factor, Sylow
260C2×#41—21no—
261C2×#41D4×C6112yes—
262C22×Q8D4×C6111yescharacteristic
263C22×Q8C23×C6111yescharacteristic
264C2×M16—22no—
265C2×M16—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8C2×C4×C6111yescharacteristic
271C4⋊Q8C2×C4×C6111yescharacteristic
272C4⋊Q8C2×C4×C6112yes—
273C4⋊Q8D4×C6111yescharacteristic
274C4⋊Q8D4×C6111yescharacteristic
275C4⋊Q8D4×C6112yes—
276#126—22no—
277#126—22no—
278#126—22no—
279#125—21no—
280#125—22no—
281#125—22no—
282#125—22no—
283#124—21no—
284#124—21no—
285#124—21no—
286#124C2×C4×C6112yes—
287#124D4×C6112yes—
288#123—22no—
289#123—22no—
290#123—22no—
291#123—22no—
292#123—22no—
293C22⋊Q8—21no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—22no—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301#121—21no—
302#121—22no—
303C4×Q8—21no—
304C4×Q8—22no—
305C4×Q8—42no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—21no—
309#117—22no—
310#117—21no—
311#117—22no—
312#117—22no—
313#117—21no—
314#117D4×C6112yes—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C42D4×C6111yescharacteristic
318#104—44no—
319#103—42no—
320#101—22no—
321#101—22no—
322#101—22no—
323#41—42no—
324#41—42no—
325#41—81no—
326#41—81no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8C22⋊C4×C6111yescharacteristic
335C2×Q8C22⋊C4×C6112yes—
336C2×Q8C22⋊C4×C6111yescharacteristic
337C2×Q8C22×C4×C6111yescharacteristic
338C2×Q8C2×D4×C6111yescharacteristic
339C2×Q8C2×D4×C6112yes—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4C22⋊C4×C6112yes—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4C2×D4×C6111yescharacteristic
350C22×C4C2×D4×C6112yes—
351M16—42no—
352M16—41no—
353M16—41no—
354M16—42no—
355C2×C8—41no—
356C2×C8—42no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—41no—
385C42—21no—
386C42—22no—
387C42—21no—
388C42—21no—
389C42—24no—
390C42—41no—
391C42C22⋊C4×C6111yescharacteristic
392C42C22⋊C4×C6111yescharacteristic
393C42C22⋊C4×C6112yes—
394C42C22⋊C4×C6112yes—
395C23—22no—
396C23#120×C6111yescharacteristic
397Q8—41no—
398Q8—41no—
399Q8—42no—
400Q8—41no—
401Q8—41no—
402Q8—42no—
403Q8—42no—
404Q8—81no—
405Q8—81no—
406D4—42no—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—81no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4C2×C22⋊C4×C6111yescharacteristic
430C2×C4C2×C22⋊C4×C6112yes—
431C2×C4#120×C6111yescharacteristic
432C2×C4#120×C6112yes—
433C8—82no—
434C8—81no—
435C22—41no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22C61×#37911yescharacteristic
440C22C61×B(4,𝔽2)11yescharacteristic
441C22C61×#45811yescharacteristic
442C4—22no—
443C4—22no—
444C4—21no—
445C4—22no—
446C4—21no—
447C4—42no—
448C4—41no—
449C4—42no—
450C4—41no—
451C4—41no—
452C4—42no—
453C4—82no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—82no—
458C2C61×#179011yescharacteristic
subgroupG/H
center, socleC2×C61#1790
commutatorC2×Q8C22×C4×C61
Frattini, cosocleC22×Q8C23×C61
prime radicalC61×#3544C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C61 ◁ C23×C61 ◁ C22×Q8×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2133C2semidirect
3. C2×C61×#356C2semidirect
4. C61×#1912C2semidirect
5. C61×#1886C2semidirect
7. C61×#563C4semidirect
8. Q82×C61C4semidirect
9. C61×#555C4semidirect
205. #12782C61direct (N × H)
226. #2133C2×C61semidirect
227. C2×#356C2×C61semidirect
228. #1912C2×C61semidirect
229. #1886C2×C61semidirect
236. #563C4×C61semidirect
237. Q82C4×C61semidirect
238. #555C4×C61semidirect
259. C61#12782direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#35442
C61×#21332
C2×C61×#3562
C61×#19122
C61×#18862
#1278261