Finite Group Databasebeta

← C61×#12268 #29515701

Subgroups — C61×#12268 #29515701

Order 15616.

458 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2735C211yesmaximal, characteristic
2C61×#2029C211yesmaximal, characteristic
3C61×#1614C211yesmaximal, characteristic, complemented
4C61×#1608C211yesmaximal, characteristic, complemented
5C61×#1352C211yesmaximal, characteristic, complemented
6C61×#1344C211yesmaximal, characteristic, complemented
7C61×#1316C211yesmaximal, characteristic
8C61×#543C2211yescharacteristic
9C61×#484—21no—
10C61×#483—21no—
11C61×#399C2211yescharacteristic
12C61×#397—21no—
13C61×#397C2211yescharacteristic
14C61×#396—21no—
15C61×#388—21no—
16C61×#388—21no—
17C61×#388—21no—
18C61×#379—21no—
19C61×#379—21no—
20C61×#379C2211yescharacteristic
21C61×#342—21no—
22C61×#342C2211yescharacteristic
23C61×#328—21no—
24C61×#328—21no—
25C61×#323C2211yescharacteristic
26C61×#323C2211yescharacteristic
27C25×C61D411yescharacteristic
28C22⋊Q8×C61—21no—
29C22⋊Q8×C61—21no—
30C22⋊Q8×C61—41no—
31C22⋊Q8×C61—41no—
32C2×C4⋊C4×C61—21no—
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61D411yescharacteristic
35C2×C4⋊C4×C61D411yescharacteristic
36C2×C22⋊C4×C61—21no—
37C2×C22⋊C4×C61—21no—
38C2×C22⋊C4×C61—21no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—41no—
47C2×C22⋊C4×C61—41no—
48C2×C22⋊C4×C61D411yescharacteristic
49C2×C22⋊C4×C61C2311yescharacteristic
50C2×C42×C61—21no—
51#107×C61—41no—
52#106×C61—41no—
53#103×C61—41no—
54#103×C61—41no—
55C22⋊C8×C61—21no—
56C22⋊C8×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61D411yescharacteristic
61#95×C61D411yescharacteristic
62C24×C61—21no—
63C24×C61—21no—
64C24×C61—21no—
65C24×C61—41no—
66C24×C61—41no—
67C24×C61—42no—
68C24×C61—81no—
69C24×C61C2×D411yescharacteristic
70C2×Q8×C61—21no—
71C2×Q8×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—41no—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C2×C8×C61—41no—
83C2×C8×C61—41no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—81no—
91C4⋊C4×C61—81no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—42no—
94C22⋊C4×C61—42no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C22⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C22⋊C4×C61—41no—
101C22⋊C4×C61—41no—
102C22⋊C4×C61—81no—
103C22⋊C4×C61—81no—
104C22⋊C4×C61—81no—
105C22⋊C4×C61—81no—
106C22⋊C4×C61—81no—
107C22⋊C4×C61—82no—
108C22⋊C4×C61—82no—
109C42×C61—81no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—22no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—42no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—81no—
125C23×C61—82no—
126C23×C61—81no—
127C23×C61—84no—
128C23×C61—82no—
129C23×C61—81no—
130C23×C61#12011yescharacteristic
131C23×C61#13511yescharacteristic
132C23×C61#13711yescharacteristic
133Q8×C61—81no—
134Q8×C61—81no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
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—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C8×C61—81no—
162C8×C61—81no—
163#12268C6111yesmaximal, characteristic, direct product factor, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—44no—
175C22×C61—42no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—82no—
180C22×C61—82no—
181C22×C61—82no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61#45211yescharacteristic
185C22×C61#45511yescharacteristic
186C22×C61B(4,𝔽2)11yescharacteristic
187C4×C61—41no—
188C4×C61—41no—
189C4×C61—81no—
190C4×C61—81no—
191C4×C61—81no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61—81no—
195C4×C61—81no—
196C4×C61—161no—
197#2735C2×C6111yescharacteristic
198#2029C2×C6111yescharacteristic
199#1614C2×C6111yescharacteristic, complemented
200#1608C2×C6111yescharacteristic, complemented
201#1352C2×C6111yescharacteristic, complemented
202#1344C2×C6111yescharacteristic, complemented
203#1316C2×C6111yescharacteristic
204C2×C61—21no—
205C2×C61—21no—
206C2×C61—41no—
207C2×C61—41no—
208C2×C61—82no—
209C2×C61#160611yescharacteristic
210C2×C61#219711yescharacteristic
211C2×C61#220011yescharacteristic
212#543C22×C6111yescharacteristic
213#484—21no—
214#483—21no—
215#399C22×C6111yescharacteristic
216#397—21no—
217#397C22×C6111yescharacteristic
218#396—21no—
219#388—21no—
220#388—21no—
221#388—21no—
222#379—21no—
223#379—21no—
224#379C22×C6111yescharacteristic
225#342—21no—
226#342C22×C6111yescharacteristic
227#328—21no—
228#328—21no—
229#323C22×C6111yescharacteristic
230#323C22×C6111yescharacteristic
231C61#1226811yescharacteristic, direct product factor, Sylow
232C25D4×C6111yescharacteristic
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—41no—
236C22⋊Q8—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4D4×C6111yescharacteristic
240C2×C4⋊C4D4×C6111yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4D4×C6111yescharacteristic
254C2×C22⋊C4C23×C6111yescharacteristic
255C2×C42—21no—
256#107—41no—
257#106—41no—
258#103—41no—
259#103—41no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95D4×C6111yescharacteristic
266#95D4×C6111yescharacteristic
267C24—21no—
268C24—21no—
269C24—21no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—81no—
274C24C2×D4×C6111yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4C2×D4×C6111yescharacteristic
286C22×C4C2×D4×C6111yescharacteristic
287C2×C8—41no—
288C2×C8—41no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C22⋊C4—81no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42—81no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—81no—
330C23—82no—
331C23—81no—
332C23—82no—
333C23—84no—
334C23—81no—
335C23#120×C6111yescharacteristic
336C23#135×C6111yescharacteristic
337C23#137×C6111yescharacteristic
338Q8—81no—
339Q8—81no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
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—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—44no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—81no—
387C22—81no—
388C22C61×#45211yescharacteristic
389C22C61×#45511yescharacteristic
390C22C61×B(4,𝔽2)11yescharacteristic
391C4—41no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—161no—
401C2—21no—
402C2—21no—
403C2—41no—
404C2—41no—
405C2—82no—
406C2C61×#160611yescharacteristic
407C2C61×#219711yescharacteristic
408C2C61×#220011yescharacteristic
subgroupG/H
center, socleC22×C61B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C61
prime radicalC25×C61D4
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
3. C61×#1614C2semidirect
4. C61×#1608C2semidirect
5. C61×#1352C2semidirect
6. C61×#1344C2semidirect
163. #12268C61direct (N × H)
199. #1614C2×C61semidirect
200. #1608C2×C61semidirect
201. #1352C2×C61semidirect
202. #1344C2×C61semidirect
231. C61#12268direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#27352
C61×#20292
C61×#16142
C61×#16082
C61×#13522
C61×#13442
C61×#13162
#1226861