Finite Group Databasebeta

← C61×#10186 #29513619

Subgroups — C61×#10186 #29513619

Order 15616.

528 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3229C211yesmaximal, characteristic, complemented
2C61×#3077C211yesmaximal, characteristic
3C2×C61×#461C211yesmaximal, characteristic
4C61×#1546C211yesmaximal, characteristic, complemented
5C61×#1481C211yesmaximal, characteristic, complemented
6C61×#1460C211yesmaximal, characteristic
7C61×#1301C211yesmaximal, characteristic, complemented
8C22×SD16×C61—21no—
9C61×#533C2211yescharacteristic
10C2×C4⋊Q8×C61C2211yescharacteristic
11C2×C4×D4×C61C2211yescharacteristic
12C61×#498—21no—
13C61×#487—21no—
14C61×#483—22no—
15C61×#478—22no—
16C61×#476—21no—
17C61×#475—22no—
18C61×#474—21no—
19C61×#461—22no—
20C61×#461—22no—
21C61×#431C2211yescharacteristic
22C61×#431C2211yescharacteristic
23C61×#428—21no—
24C61×#423C2211yescharacteristic
25C2×C4⋊C8×C61C2211yescharacteristic
26C61×#418—21no—
27C2×#103×C61—21no—
28C61×#332—22no—
29C61×Q8⋊C8—22no—
30C61×#323—22no—
31C22×Q8×C61—21no—
32C22×D4×C61D411yescharacteristic
33C23×C4×C61—21no—
34C2×SD16×C61—24no—
35C2×SD16×C61—42no—
36C2×SD16×C61—42no—
37C2×M16×C61—21no—
38C22×C8×C61—21no—
39C4⋊Q8×C61—21no—
40C4⋊Q8×C61—21no—
41C4⋊Q8×C61—21no—
42C4⋊Q8×C61D412yes—
43C4⋊Q8×C61D411yescharacteristic
44C4⋊Q8×C61D411yescharacteristic
45#125×C61—21no—
46C22⋊Q8×C61—21no—
47C22⋊Q8×C61—21no—
48C4×Q8×C61—21no—
49C4×Q8×C61—21no—
50C4×D4×C61—21no—
51C4×D4×C61—21no—
52C4×D4×C61—21no—
53C4×D4×C61—21no—
54#117×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61D411yescharacteristic
58C2×C22⋊C4×C61—21no—
59C2×C42×C61C2311yescharacteristic
60#107×C61—22no—
61#106×C61—22no—
62#106×C61—24no—
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61—21no—
66C4⋊C8×C61—21no—
67C4⋊C8×C61D412yes—
68C4⋊C8×C61D412yes—
69#103×C61—24no—
70#103×C61—24no—
71#102×C61—22no—
72#102×C61—22no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61D412yes—
75C4×C8×C61—21no—
76C4×C8×C61—21no—
77C24×C61—21no—
78C2×Q8×C61—21no—
79C2×Q8×C61—22no—
80C2×Q8×C61—21no—
81C2×Q8×C61—41no—
82C2×Q8×C61—41no—
83C2×Q8×C61—41no—
84C2×Q8×C61—41no—
85C2×Q8×C61—41no—
86C2×D4×C61—21no—
87C2×D4×C61—21no—
88C2×D4×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61SD1612yes—
92C22×C4×C61—21no—
93C22×C4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—82no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100SD16×C61—48no—
101M16×C61—41no—
102M16×C61—41no—
103C2×C8×C61—22no—
104C2×C8×C61—22no—
105C2×C8×C61—22no—
106C2×C8×C61—41no—
107C2×C8×C61—41no—
108C2×C8×C61—41no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—41no—
119C4⋊C4×C61—41no—
120C4⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—41no—
124C42×C61—41no—
125C42×C61C2×D411yescharacteristic
126C42×C61C2×D411yescharacteristic
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C23×C61—22no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—81no—
133C23×C61—81no—
134C23×C61#12111yescharacteristic
135Q8×C61—42no—
136Q8×C61—42no—
137Q8×C61—42no—
138Q8×C61—42no—
139Q8×C61—81no—
140Q8×C61—81no—
141D4×C61—22no—
142D4×C61—22no—
143D4×C61—41no—
144D4×C61—41no—
145C2×C4×C61—22no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—82no—
152C2×C4×C61—81no—
153C2×C4×C61—81no—
154C2×C4×C61—82no—
155C2×C4×C61—81no—
156C2×C4×C61#12011yescharacteristic
157C2×C4×C61#12011yescharacteristic
158C2×C4×C61#12011yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12711yescharacteristic
162C2×C4×C61C2×SD1611yescharacteristic
163C2×C4×C61#13511yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61#13711yescharacteristic
166C8×C61—42no—
167C8×C61—42no—
168C8×C61—42no—
169C8×C61—81no—
170#10186C6111yesmaximal, characteristic, direct product factor, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—42no—
174C22×C61—82no—
175C22×C61—82no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61#39211yescharacteristic
179C22×C61#46511yescharacteristic
180C22×C61#46811yescharacteristic
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—21no—
184C4×C61—82no—
185C4×C61—81no—
186C4×C61—81no—
187C4×C61—81no—
188C4×C61#44811yescharacteristic
189C4×C61#44911yescharacteristic
190C4×C61#45011yescharacteristic
191C4×C61#45211yescharacteristic
192C4×C61#47112yes—
193#3229C2×C6111yescharacteristic, complemented
194#3077C2×C6111yescharacteristic
195C2×#461C2×C6111yescharacteristic
196#1546C2×C6111yescharacteristic, complemented
197#1481C2×C6111yescharacteristic, complemented
198#1460C2×C6111yescharacteristic
199#1301C2×C6111yescharacteristic, complemented
200C2×C61—21no—
201C2×C61—21no—
202C2×C61—82no—
203C2×C61#200011yescharacteristic
204C2×C61#200911yescharacteristic
205C2×C61#201911yescharacteristic
206C22×SD16—21no—
207#533C22×C6111yescharacteristic
208C2×C4⋊Q8C22×C6111yescharacteristic
209C2×C4×D4C22×C6111yescharacteristic
210#498—21no—
211#487—21no—
212#483—22no—
213#478—22no—
214#476—21no—
215#475—22no—
216#474—21no—
217#461—22no—
218#461—22no—
219#431C22×C6111yescharacteristic
220#431C22×C6111yescharacteristic
221#428—21no—
222#423C22×C6111yescharacteristic
223C2×C4⋊C8C22×C6111yescharacteristic
224#418—21no—
225C2×#103—21no—
226#332—22no—
227Q8⋊C8—22no—
228#323—22no—
229C61#1018611yescharacteristic, direct product factor, Sylow
230C22×Q8—21no—
231C22×D4D4×C6111yescharacteristic
232C23×C4—21no—
233C2×SD16—24no—
234C2×SD16—42no—
235C2×SD16—42no—
236C2×M16—21no—
237C22×C8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8D4×C6112yes—
242C4⋊Q8D4×C6111yescharacteristic
243C4⋊Q8D4×C6111yescharacteristic
244#125—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253#117—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4D4×C6111yescharacteristic
257C2×C22⋊C4—21no—
258C2×C42C23×C6111yescharacteristic
259#107—22no—
260#106—22no—
261#106—24no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8D4×C6112yes—
267C4⋊C8D4×C6112yes—
268#103—24no—
269#103—24no—
270#102—22no—
271#102—22no—
272C22⋊C8—21no—
273C22⋊C8D4×C6112yes—
274C4×C8—21no—
275C4×C8—21no—
276C24—21no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4SD16×C6112yes—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—82no—
296C22×C4C2×D4×C6111yescharacteristic
297C22×C4C2×D4×C6111yescharacteristic
298C22×C4#41×C6111yescharacteristic
299SD16—48no—
300M16—41no—
301M16—41no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—41no—
324C42C2×D4×C6111yescharacteristic
325C42C2×D4×C6111yescharacteristic
326C42C2×D4×C6111yescharacteristic
327C42C2×D4×C6111yescharacteristic
328C23—22no—
329C23—41no—
330C23—41no—
331C23—81no—
332C23—81no—
333C23#121×C6111yescharacteristic
334Q8—42no—
335Q8—42no—
336Q8—42no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340D4—22no—
341D4—22no—
342D4—41no—
343D4—41no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4#120×C6111yescharacteristic
356C2×C4#120×C6111yescharacteristic
357C2×C4#120×C6111yescharacteristic
358C2×C4#121×C6111yescharacteristic
359C2×C4#121×C6111yescharacteristic
360C2×C4#127×C6111yescharacteristic
361C2×C4C2×SD16×C6111yescharacteristic
362C2×C4#135×C6111yescharacteristic
363C2×C4AGL(1,ℤ/8)×C6111yescharacteristic
364C2×C4#137×C6111yescharacteristic
365C8—42no—
366C8—42no—
367C8—42no—
368C8—81no—
369C22—21no—
370C22—21no—
371C22—42no—
372C22—82no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22C61×#39211yescharacteristic
377C22C61×#46511yescharacteristic
378C22C61×#46811yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—82no—
386C4C61×#44811yescharacteristic
387C4C61×#44911yescharacteristic
388C4C61×#45011yescharacteristic
389C4C61×#45211yescharacteristic
390C4C61×#47112yes—
391C2—21no—
392C2—21no—
393C2—82no—
394C2C61×#200011yescharacteristic
395C2C61×#200911yescharacteristic
396C2C61×#201911yescharacteristic
subgroupG/H
center, socleC22×C61#392
commutator, Frattini, cosocleC2×C42C23×C61
prime radicalC22×D4×C61D4
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
1. C61×#3229C2semidirect
4. C61×#1546C2semidirect
5. C61×#1481C2semidirect
7. C61×#1301C2semidirect
170. #10186C61direct (N × H)
193. #3229C2×C61semidirect
196. #1546C2×C61semidirect
197. #1481C2×C61semidirect
199. #1301C2×C61semidirect
229. C61#10186direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#32292
C61×#30772
C2×C61×#4612
C61×#15462
C61×#14812
C61×#14602
C61×#13012
#1018661