Finite Group Databasebeta

← C61×#12176 #29515609

Subgroups — C61×#12176 #29515609

Order 15616.

486 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2677C211yesmaximal, characteristic, complemented
2C61×#2044C211yesmaximal, characteristic
3C61×#1613C211yesmaximal, characteristic
4C61×#1601C211yesmaximal, characteristic
5C61×#1352C211yesmaximal, characteristic, complemented
6C61×#1351C211yesmaximal, characteristic, complemented
7C61×#1326C211yesmaximal, characteristic, complemented
8C61×#534C2211yescharacteristic
9C2×C4⋊Q8×C61—21no—
10C2×C22⋊Q8×C61—21no—
11C61×#481—21no—
12C61×#448—21no—
13C2×#99×C61—21no—
14C61×#399C2211yescharacteristic
15C61×#398—21no—
16C61×#395—21no—
17C61×#393C2211yescharacteristic
18C61×#388—21no—
19C61×#387—21no—
20C61×#386—21no—
21C61×#385C2211yescharacteristic
22C61×#342C2211yescharacteristic
23C61×#327—21no—
24C61×#327—21no—
25C61×#323C2211yescharacteristic
26C61×#323C2211yescharacteristic
27C22×Q8×C61—21no—
28C22×Q8×C61D411yescharacteristic
29C22×D4×C61—21no—
30C23×C4×C61D411yescharacteristic
31C2×SD16×C61—41no—
32C4⋊Q8×C61—44no—
33#123×C61—41no—
34C22⋊Q8×C61—44no—
35#121×C61—21no—
36#121×C61—21no—
37#120×C61—41no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61—21no—
43C2×C4⋊C4×C61—21no—
44C2×C4⋊C4×C61D411yescharacteristic
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—21no—
48C2×C22⋊C4×C61—41no—
49C2×C22⋊C4×C61D411yescharacteristic
50C2×C22⋊C4×C61C2311yescharacteristic
51C2×C42×C61—21no—
52#106×C61—41no—
53#103×C61—41no—
54#102×C61—41no—
55#99×C61—42no—
56C22⋊C8×C61—21no—
57C22⋊C8×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61—21no—
62#95×C61—41no—
63#95×C61D411yescharacteristic
64#95×C61D411yescharacteristic
65C24×C61—21no—
66C24×C61C2×D411yescharacteristic
67C2×Q8×C61—42no—
68C2×Q8×C61—42no—
69C2×Q8×C61—42no—
70C2×Q8×C61—41no—
71C2×Q8×C61—81no—
72C2×D4×C61—21no—
73C2×D4×C61—21no—
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C2×D4×C61—41no—
78C2×D4×C61—41no—
79C2×D4×C61—41no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61—41no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92SD16×C61—82no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—42no—
99C4⋊C4×C61—42no—
100C4⋊C4×C61—42no—
101C4⋊C4×C61—42no—
102C4⋊C4×C61—42no—
103C4⋊C4×C61—41no—
104C22⋊C4×C61—21no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—42no—
107C22⋊C4×C61—42no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—82no—
113C22⋊C4×C61SD1612yescomplemented
114C42×C61—42no—
115C23×C61—21no—
116C23×C61—21no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—81no—
123C23×C61—81no—
124C23×C61#12011yescharacteristic
125C23×C61C2×SD1611yescharacteristic
126C23×C61#13711yescharacteristic
127Q8×C61—42no—
128Q8×C61—81no—
129Q8×C61—84no—
130D4×C61—81no—
131D4×C61—81no—
132D4×C61—81no—
133D4×C61—81no—
134D4×C61—81no—
135D4×C61—81no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—81no—
163C2×C4×C61—81no—
164C2×C4×C61—81no—
165C2×C4×C61—81no—
166C8×C61—81no—
167C8×C61—81no—
168#12176C6111yesmaximal, characteristic, direct product factor, Sylow
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—41no—
176C22×C61—41no—
177C22×C61—81no—
178C22×C61—82no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—161no—
183C22×C61#44811yescharacteristic
184C22×C61#45411yescharacteristic
185C22×C61B(4,𝔽2)11yescharacteristic
186C4×C61—42no—
187C4×C61—42no—
188C4×C61—41no—
189C4×C61—42no—
190C4×C61—42no—
191C4×C61—41no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61—81no—
195C4×C61—161no—
196#2677C2×C6111yescharacteristic, complemented
197#2044C2×C6111yescharacteristic
198#1613C2×C6111yescharacteristic
199#1601C2×C6111yescharacteristic
200#1352C2×C6111yescharacteristic, complemented
201#1351C2×C6111yescharacteristic, complemented
202#1326C2×C6111yescharacteristic, complemented
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—41no—
206C2×C61—41no—
207C2×C61—161no—
208C2×C61#159911yescharacteristic
209C2×C61#219611yescharacteristic
210C2×C61#220111yescharacteristic
211#534C22×C6111yescharacteristic
212C2×C4⋊Q8—21no—
213C2×C22⋊Q8—21no—
214#481—21no—
215#448—21no—
216C2×#99—21no—
217#399C22×C6111yescharacteristic
218#398—21no—
219#395—21no—
220#393C22×C6111yescharacteristic
221#388—21no—
222#387—21no—
223#386—21no—
224#385C22×C6111yescharacteristic
225#342C22×C6111yescharacteristic
226#327—21no—
227#327—21no—
228#323C22×C6111yescharacteristic
229#323C22×C6111yescharacteristic
230C61#1217611yescharacteristic, direct product factor, Sylow
231C22×Q8—21no—
232C22×Q8D4×C6111yescharacteristic
233C22×D4—21no—
234C23×C4D4×C6111yescharacteristic
235C2×SD16—41no—
236C4⋊Q8—44no—
237#123—41no—
238C22⋊Q8—44no—
239#121—21no—
240#121—21no—
241#120—41no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4D4×C6111yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4D4×C6111yescharacteristic
254C2×C22⋊C4C23×C6111yescharacteristic
255C2×C42—21no—
256#106—41no—
257#103—41no—
258#102—41no—
259#99—42no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—41no—
267#95D4×C6111yescharacteristic
268#95D4×C6111yescharacteristic
269C24—21no—
270C24C2×D4×C6111yescharacteristic
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—81no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×D4×C6111yescharacteristic
295C22×C4C2×D4×C6111yescharacteristic
296SD16—82no—
297C2×C8—41no—
298C2×C8—41no—
299C4⋊C4—24no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C22⋊C4—21no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—82no—
317C22⋊C4SD16×C6112yescomplemented
318C42—42no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—81no—
327C23—81no—
328C23#120×C6111yescharacteristic
329C23C2×SD16×C6111yescharacteristic
330C23#137×C6111yescharacteristic
331Q8—42no—
332Q8—81no—
333Q8—84no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C8—81no—
371C8—81no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—82no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—161no—
386C22C61×#44811yescharacteristic
387C22C61×#45411yescharacteristic
388C22C61×B(4,𝔽2)11yescharacteristic
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—161no—
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—161no—
404C2C61×#159911yescharacteristic
405C2C61×#219611yescharacteristic
406C2C61×#220111yescharacteristic
subgroupG/H
center, socleC22×C61B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C61
prime radicalC61×#534C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C61 ◁ C2×C22⋊C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#2677C2semidirect
5. C61×#1352C2semidirect
6. C61×#1351C2semidirect
7. C61×#1326C2semidirect
113. C22⋊C4×C61SD16semidirect
168. #12176C61direct (N × H)
196. #2677C2×C61semidirect
200. #1352C2×C61semidirect
201. #1351C2×C61semidirect
202. #1326C2×C61semidirect
230. C61#12176direct (N × H)
317. C22⋊C4SD16×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#26772
C61×#20442
C61×#16132
C61×#16012
C61×#13522
C61×#13512
C61×#13262
#1217661