Finite Group Databasebeta

← C61×#10331 #29513764

Subgroups — C61×#10331 #29513764

Order 15616.

672 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C61×#3193C211yesmaximal, characteristic
2C61×#2502C211yesmaximal, characteristic, complemented
3C61×#2501C211yesmaximal, characteristic
4C61×#1821C211yesmaximal, characteristic, complemented
5C61×#1508C211yesmaximal, characteristic, complemented
6C61×#1507C211yesmaximal, characteristic
7C61×#1297C211yesmaximal, characteristic, complemented
8C2×#123×C61—22no—
9C2×C22⋊Q8×C61C2211yescharacteristic
10C2×#121×C61C2211yescharacteristic
11C22×C4⋊C4×C61C2211yescharacteristic
12C61×#484—21no—
13C61×#480—21no—
14C61×#468—22no—
15C61×#450—21no—
16C61×#448—21no—
17C61×#407C2211yescharacteristic
18C61×#407C2211yescharacteristic
19C61×#399—22no—
20C61×#398—22no—
21C61×#391—21no—
22C61×#390—21no—
23C61×#385—22no—
24C61×#384—21no—
25C61×#384—21no—
26C4×C22⋊C4×C61C2211yescharacteristic
27C4×C22⋊C4×C61C2211yescharacteristic
28C61×#343—22no—
29C61×#328—22no—
30C61×#327—22no—
31C22×Q8×C61D411yescharacteristic
32C22×D4×C61—21no—
33C22×D4×C61D411yescharacteristic
34C23×C4×C61D412yes—
35C23×C4×C61C2311yescharacteristic
36C2×SD16×C61—22no—
37C2×M16×C61—21no—
38C2×M16×C61D412yes—
39#123×C61—48no—
40C22⋊Q8×C61—21no—
41C22⋊Q8×C61—21no—
42C22⋊Q8×C61—21no—
43C22⋊Q8×C61—21no—
44#121×C61—21no—
45#121×C61—21no—
46#121×C61—21no—
47#121×C61—21no—
48C2×C4⋊C4×C61—24no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61—21no—
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61—24no—
57C2×C4⋊C4×C61D411yescharacteristic
58C2×C4⋊C4×C61D411yescharacteristic
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61—24no—
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C42×C61—21no—
65C2×C42×C61—21no—
66#107×C61—22no—
67#103×C61—22no—
68#102×C61—22no—
69C22⋊C8×C61—21no—
70C22⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73#95×C61—24no—
74#95×C61D412yes—
75#95×C61D412yes—
76C24×C61—21no—
77C24×C61C2×D411yescharacteristic
78C2×Q8×C61—21no—
79C2×Q8×C61—21no—
80C2×Q8×C61—42no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—42no—
84C2×D4×C61—44no—
85C2×D4×C61—41no—
86C2×D4×C61—41no—
87C22×C4×C61—22no—
88C22×C4×C61—22no—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—21no—
93C22×C4×C61—21no—
94C22×C4×C61—22no—
95C22×C4×C61—42no—
96C22×C4×C61—44no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61#4111yescharacteristic
103SD16×C61—82no—
104M16×C61—42no—
105M16×C61—42no—
106C2×C8×C61—22no—
107C2×C8×C61—41no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—48no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C4⋊C4×C61—84no—
120C22⋊C4×C61—24no—
121C22⋊C4×C61—24no—
122C22⋊C4×C61—48no—
123C22⋊C4×C61—41no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C42×C61—42no—
128C42×C61—42no—
129C23×C61—22no—
130C23×C61—22no—
131C23×C61—41no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—41no—
135C23×C61—42no—
136C23×C61#12011yescharacteristic
137C23×C61#12011yescharacteristic
138C23×C61#12111yescharacteristic
139Q8×C61—42no—
140Q8×C61—81no—
141D4×C61—42no—
142D4×C61—81no—
143D4×C61—82no—
144D4×C61—82no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—24no—
149C2×C4×C61—24no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—44no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—44no—
166C2×C4×C61—82no—
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12711yescharacteristic
171C8×C61—42no—
172C8×C61—81no—
173#10331C6111yesmaximal, characteristic, direct product factor, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—44no—
182C22×C61—84no—
183C22×C61—81no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61#39211yescharacteristic
187C22×C61#45312yes—
188C22×C61#45612yes—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—44no—
193C4×C61—44no—
194C4×C61—81no—
195C4×C61—82no—
196C4×C61—81no—
197C4×C61—81no—
198C4×C61#45812yes—
199#3193C2×C6111yescharacteristic
200#2502C2×C6111yescharacteristic, complemented
201#2501C2×C6111yescharacteristic
202#1821C2×C6111yescharacteristic, complemented
203#1508C2×C6111yescharacteristic, complemented
204#1507C2×C6111yescharacteristic
205#1297C2×C6111yescharacteristic, complemented
206C2×C61—22no—
207C2×C61—22no—
208C2×C61—41no—
209C2×C61—161no—
210C2×C61#200211yescharacteristic
211C2×C61#200511yescharacteristic
212C2×C61#202211yescharacteristic
213C2×#123—22no—
214C2×C22⋊Q8C22×C6111yescharacteristic
215C2×#121C22×C6111yescharacteristic
216C22×C4⋊C4C22×C6111yescharacteristic
217#484—21no—
218#480—21no—
219#468—22no—
220#450—21no—
221#448—21no—
222#407C22×C6111yescharacteristic
223#407C22×C6111yescharacteristic
224#399—22no—
225#398—22no—
226#391—21no—
227#390—21no—
228#385—22no—
229#384—21no—
230#384—21no—
231C4×C22⋊C4C22×C6111yescharacteristic
232C4×C22⋊C4C22×C6111yescharacteristic
233#343—22no—
234#328—22no—
235#327—22no—
236C61#1033111yescharacteristic, direct product factor, Sylow
237C22×Q8D4×C6111yescharacteristic
238C22×D4—21no—
239C22×D4D4×C6111yescharacteristic
240C23×C4D4×C6112yes—
241C23×C4C23×C6111yescharacteristic
242C2×SD16—22no—
243C2×M16—21no—
244C2×M16D4×C6112yes—
245#123—48no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250#121—21no—
251#121—21no—
252#121—21no—
253#121—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4D4×C6111yescharacteristic
264C2×C4⋊C4D4×C6111yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C42—21no—
271C2×C42—21no—
272#107—22no—
273#103—22no—
274#102—22no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279#95—24no—
280#95D4×C6112yes—
281#95D4×C6112yes—
282C24—21no—
283C24C2×D4×C6111yescharacteristic
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—42no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4—41no—
292C2×D4—41no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4C2×D4×C6111yescharacteristic
304C22×C4C2×D4×C6111yescharacteristic
305C22×C4C2×D4×C6111yescharacteristic
306C22×C4C2×D4×C6111yescharacteristic
307C22×C4C2×D4×C6111yescharacteristic
308C22×C4#41×C6111yescharacteristic
309SD16—82no—
310M16—42no—
311M16—42no—
312C2×C8—22no—
313C2×C8—41no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—48no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—84no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—48no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C42—42no—
334C42—42no—
335C23—22no—
336C23—22no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—41no—
341C23—42no—
342C23#120×C6111yescharacteristic
343C23#120×C6111yescharacteristic
344C23#121×C6111yescharacteristic
345Q8—42no—
346Q8—81no—
347D4—42no—
348D4—81no—
349D4—82no—
350D4—82no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—82no—
373C2×C4#120×C6111yescharacteristic
374C2×C4#121×C6111yescharacteristic
375C2×C4#121×C6111yescharacteristic
376C2×C4#127×C6111yescharacteristic
377C8—42no—
378C8—81no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22—84no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22C61×#39211yescharacteristic
392C22C61×#45312yes—
393C22C61×#45612yes—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—44no—
398C4—44no—
399C4—81no—
400C4—81no—
401C4—82no—
402C4—81no—
403C4C61×#45812yes—
404C2—22no—
405C2—22no—
406C2—41no—
407C2—161no—
408C2C61×#200211yescharacteristic
409C2C61×#200511yescharacteristic
410C2C61×#202211yescharacteristic
subgroupG/H
center, socleC22×C61#392
commutator, Frattini, cosocleC23×C4C23×C61
prime radicalC2×#121×C61C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2502C2semidirect
4. C61×#1821C2semidirect
5. C61×#1508C2semidirect
7. C61×#1297C2semidirect
173. #10331C61direct (N × H)
200. #2502C2×C61semidirect
202. #1821C2×C61semidirect
203. #1508C2×C61semidirect
205. #1297C2×C61semidirect
236. C61#10331direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#31932
C61×#25022
C61×#25012
C61×#18212
C61×#15082
C61×#15072
C61×#12972
#1033161