Finite Group Databasebeta

← C61×#10243 #29513676

Subgroups — C61×#10243 #29513676

Order 15616.

578 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3226C211yesmaximal, characteristic
2C61×#3037C211yesmaximal, characteristic
3C61×#2546C211yesmaximal, characteristic, complemented
4C61×#1546C211yesmaximal, characteristic, complemented
5C61×#1502C211yesmaximal, characteristic
6C61×#1479C211yesmaximal, characteristic, complemented
7C61×#1303C211yesmaximal, characteristic, complemented
8C2×#137×C61—21no—
9C61×#532C2211yescharacteristic
10C2×C4⋊Q8×C61—21no—
11C2×C4⋊Q8×C61C2211yescharacteristic
12C2×C4×Q8×C61C2211yescharacteristic
13C61×#500—21no—
14C61×#486—21no—
15C61×#480—22no—
16C61×#474—22no—
17C61×#464—22no—
18C61×#462—21no—
19C61×#460—21no—
20C61×#431C2211yescharacteristic
21C61×#431C2211yescharacteristic
22C61×#423C2211yescharacteristic
23C61×#417—21no—
24C61×#398—22no—
25C61×#391—21no—
26C61×#389—22no—
27C4×C4⋊C4×C61C2211yescharacteristic
28C61×#332—22no—
29C61×#327—22no—
30C61×Q8⋊C8—22no—
31C2×#41×C61—21no—
32C22×Q8×C61—21no—
33C22×Q8×C61D411yescharacteristic
34#137×C61—42no—
35#137×C61—42no—
36C2×Q16×C61—22no—
37C2×SD16×C61—22no—
38C2×M16×C61—21no—
39C4⋊Q8×C61—21no—
40C4⋊Q8×C61—21no—
41C4⋊Q8×C61—21no—
42C4⋊Q8×C61—21no—
43C4⋊Q8×C61—42no—
44C4⋊Q8×C61—42no—
45C4⋊Q8×C61D411yescharacteristic
46#127×C61D411yescharacteristic
47#124×C61—21no—
48#121×C61—21no—
49#121×C61—21no—
50C4×Q8×C61—21no—
51C4×Q8×C61—21no—
52C4×Q8×C61—21no—
53C4×Q8×C61—21no—
54C4×D4×C61—21no—
55C4×D4×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—24no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61D412yes—
61C2×C4⋊C4×C61D411yescharacteristic
62C2×C42×C61—21no—
63C2×C42×C61—21no—
64C2×C42×C61C2311yescharacteristic
65#107×C61—24no—
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C4⋊C8×C61D412yes—
69C4⋊C8×C61D412yes—
70#103×C61—22no—
71#102×C61—22no—
72#102×C61—24no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61—21no—
75C4×C8×C61—21no—
76C4×C8×C61—21no—
77#95×C61—24no—
78#95×C61D412yes—
79#41×C61—41no—
80#41×C61—41no—
81#41×C61—41no—
82#41×C61—41no—
83C2×Q8×C61—21no—
84C2×Q8×C61—21no—
85C2×Q8×C61—21no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61—41no—
90C2×Q8×C61—41no—
91C2×Q8×C61—41no—
92C2×Q8×C61—41no—
93C2×Q8×C61—41no—
94C2×Q8×C61—41no—
95C2×Q8×C61D812yes—
96C2×D4×C61—21no—
97C2×D4×C61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—41no—
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—41no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108Q16×C61—44no—
109SD16×C61—44no—
110M16×C61—41no—
111M16×C61—41no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61—41no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—48no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61—41no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C22⋊C4×C61—41no—
129C42×C61—41no—
130C42×C61—41no—
131C42×C61—41no—
132C42×C61—41no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C23×C61—41no—
138C23×C61#12011yescharacteristic
139Q8×C61—22no—
140Q8×C61—22no—
141Q8×C61—41no—
142Q8×C61—41no—
143Q8×C61—41no—
144Q8×C61—41no—
145Q8×C61—81no—
146Q8×C61—81no—
147Q8×C61—81no—
148Q8×C61—81no—
149D4×C61—41no—
150D4×C61—41no—
151D4×C61—42no—
152D4×C61—81no—
153D4×C61—81no—
154D4×C61—81no—
155C2×C4×C61—22no—
156C2×C4×C61—24no—
157C2×C4×C61—22no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61—42no—
169C2×C4×C61—81no—
170C2×C4×C61—81no—
171C2×C4×C61#12011yescharacteristic
172C2×C4×C61#12011yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12711yescharacteristic
177C2×C4×C61C2×D811yescharacteristic
178C2×C4×C61#13511yescharacteristic
179C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
181C8×C61—42no—
182C8×C61—42no—
183C8×C61—81no—
184#10243C6111yesmaximal, characteristic, direct product factor, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61#39211yescharacteristic
191C22×C61#44711yescharacteristic
192C22×C61#44911yescharacteristic
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—44no—
197C4×C61—42no—
198C4×C61—82no—
199C4×C61—81no—
200C4×C61—81no—
201C4×C61#44711yescharacteristic
202C4×C61#44911yescharacteristic
203C4×C61#45612yes—
204C4×C61#46611yescharacteristic
205C4×C61#46911yescharacteristic
206#3226C2×C6111yescharacteristic
207#3037C2×C6111yescharacteristic
208#2546C2×C6111yescharacteristic, complemented
209#1546C2×C6111yescharacteristic, complemented
210#1502C2×C6111yescharacteristic
211#1479C2×C6111yescharacteristic, complemented
212#1303C2×C6111yescharacteristic, complemented
213C2×C61—21no—
214C2×C61—21no—
215C2×C61—161no—
216C2×C61#199911yescharacteristic
217C2×C61#200511yescharacteristic
218C2×C61#200911yescharacteristic
219C2×#137—21no—
220#532C22×C6111yescharacteristic
221C2×C4⋊Q8—21no—
222C2×C4⋊Q8C22×C6111yescharacteristic
223C2×C4×Q8C22×C6111yescharacteristic
224#500—21no—
225#486—21no—
226#480—22no—
227#474—22no—
228#464—22no—
229#462—21no—
230#460—21no—
231#431C22×C6111yescharacteristic
232#431C22×C6111yescharacteristic
233#423C22×C6111yescharacteristic
234#417—21no—
235#398—22no—
236#391—21no—
237#389—22no—
238C4×C4⋊C4C22×C6111yescharacteristic
239#332—22no—
240#327—22no—
241Q8⋊C8—22no—
242C61#1024311yescharacteristic, direct product factor, Sylow
243C2×#41—21no—
244C22×Q8—21no—
245C22×Q8D4×C6111yescharacteristic
246#137—42no—
247#137—42no—
248C2×Q16—22no—
249C2×SD16—22no—
250C2×M16—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8D4×C6111yescharacteristic
258#127D4×C6111yescharacteristic
259#124—21no—
260#121—21no—
261#121—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×D4—21no—
267C4×D4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4D4×C6111yescharacteristic
273C2×C4⋊C4D4×C6112yes—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C23×C6111yescharacteristic
277#107—24no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8D4×C6112yes—
281C4⋊C8D4×C6112yes—
282#103—22no—
283#102—22no—
284#102—24no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289#95—24no—
290#95D4×C6112yes—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8D8×C6112yes—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—41no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4C2×D4×C6111yescharacteristic
318C22×C4C2×D4×C6111yescharacteristic
319C22×C4C2×D4×C6111yescharacteristic
320Q16—44no—
321SD16—44no—
322M16—41no—
323M16—41no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C4⋊C4—24no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—48no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42C2×D4×C6111yescharacteristic
346C42C2×D4×C6111yescharacteristic
347C42C2×D4×C6111yescharacteristic
348C42#41×C6111yescharacteristic
349C23—41no—
350C23#120×C6111yescharacteristic
351Q8—22no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355Q8—41no—
356Q8—41no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361D4—41no—
362D4—41no—
363D4—42no—
364D4—81no—
365D4—81no—
366D4—81no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4#120×C6111yescharacteristic
384C2×C4#120×C6111yescharacteristic
385C2×C4#121×C6111yescharacteristic
386C2×C4#121×C6111yescharacteristic
387C2×C4#121×C6111yescharacteristic
388C2×C4#127×C6111yescharacteristic
389C2×C4C2×D8×C6111yescharacteristic
390C2×C4#135×C6111yescharacteristic
391C2×C4AGL(1,ℤ/8)×C6111yescharacteristic
392C2×C4AGL(1,ℤ/8)×C6111yescharacteristic
393C8—42no—
394C8—42no—
395C8—81no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C61×#39211yescharacteristic
402C22C61×#44711yescharacteristic
403C22C61×#44911yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—44no—
408C4—42no—
409C4—81no—
410C4—82no—
411C4—81no—
412C4C61×#44711yescharacteristic
413C4C61×#44911yescharacteristic
414C4C61×#45612yes—
415C4C61×#46611yescharacteristic
416C4C61×#46911yescharacteristic
417C2—21no—
418C2—21no—
419C2—161no—
420C2C61×#199911yescharacteristic
421C2C61×#200511yescharacteristic
422C2C61×#200911yescharacteristic
subgroupG/H
center, socleC22×C61#392
commutator, Frattini, cosocleC2×C42C23×C61
prime radicalC61×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
3. C61×#2546C2semidirect
4. C61×#1546C2semidirect
6. C61×#1479C2semidirect
7. C61×#1303C2semidirect
184. #10243C61direct (N × H)
208. #2546C2×C61semidirect
209. #1546C2×C61semidirect
211. #1479C2×C61semidirect
212. #1303C2×C61semidirect
242. C61#10243direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#32262
C61×#30372
C61×#25462
C61×#15462
C61×#15022
C61×#14792
C61×#13032
#1024361