Finite Group Databasebeta

← C61×#11820 #29515253

Subgroups — C61×#11820 #29515253

Order 15616.

626 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2388C211yesmaximal, characteristic
2C61×#1799C211yesmaximal, characteristic, complemented
3C61×#1466C211yesmaximal, characteristic, complemented
4C61×#1351C212yesmaximal, complemented
5C61×#1343C212yesmaximal, complemented
6C22×#41×C61C411yescharacteristic, complemented
7C2×C4×D4×C61—21no—
8C2×C4×D4×C61C2211yescharacteristic
9C61×#432—21no—
10C2×#99×C61—21no—
11C61×#395C411yescharacteristic, complemented
12C61×#394—21no—
13C61×#394C411yescharacteristic, complemented
14C61×#393C2211yescharacteristic
15C61×#392—21no—
16C61×#392C2211yescharacteristic
17C61×#380—21no—
18C4×C22⋊C4×C61—21no—
19C4×C22⋊C4×C61C411yescharacteristic, complemented
20C61×#342C2212yes—
21C61×#323—21no—
22C61×#323C2212yes—
23C2×#41×C61—44no—
24C2×#41×C61—41no—
25C2×#41×C61—41no—
26C22×Q8×C61C2×C411yescharacteristic
27C22×D4×C61—21no—
28C22×D4×C61—21no—
29C22×D4×C61C2×C411yescharacteristic
30C22×D4×C61D411yescharacteristic
31C23×C4×C61—21no—
32C23×C4×C61C2×C411yescharacteristic
33C23×C4×C61D411yescharacteristic
34C4×D4×C61—41no—
35C4×D4×C61—41no—
36C4×D4×C61—44no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—41no—
39C2×C4⋊C4×C61C2×C411yescharacteristic
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—22no—
46C2×C22⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61C2311yescharacteristic
48C2×C42×C61—21no—
49C2×C42×C61D411yescharacteristic
50C4⋊C8×C61—41no—
51#99×C61—42no—
52C22⋊C8×C61—22no—
53#97×C61—41no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—41no—
57#95×C61C2×C411yescharacteristic
58#95×C61C2×C411yescharacteristic
59C24×C61—21no—
60C24×C61C22⋊C411yescharacteristic
61C24×C61C2×D411yescharacteristic
62#41×C61—44no—
63#41×C61—44no—
64#41×C61—44no—
65#41×C61—44no—
66C2×Q8×C61—42no—
67C2×Q8×C61—41no—
68C2×D4×C61—21no—
69C2×D4×C61—21no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61—42no—
73C2×D4×C61—42no—
74C2×D4×C61—41no—
75C2×D4×C61—42no—
76C2×D4×C61—42no—
77C2×D4×C61—42no—
78C2×D4×C61—41no—
79C2×D4×C61—41no—
80C2×D4×C61—42no—
81C22×C4×C61—24no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—21no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—42no—
90C22×C4×C61—42no—
91C22×C4×C61—42no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61—42no—
95C22×C4×C61C22⋊C411yescharacteristic
96C22×C4×C61C22⋊C411yescharacteristic
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C22×C411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C2×C8×C61—42no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—42no—
103C4⋊C4×C61—82no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—42no—
107C22⋊C4×C61—42no—
108C22⋊C4×C61—42no—
109C22⋊C4×C61—42no—
110C22⋊C4×C61—42no—
111C22⋊C4×C61—42no—
112C22⋊C4×C61—42no—
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—42no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—22no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61C2×C22⋊C411yescharacteristic
133C23×C61AGL(1,ℤ/8)11yescharacteristic
134C23×C61#13711yescharacteristic
135Q8×C61—42no—
136Q8×C61—42no—
137D4×C61—42no—
138D4×C61—42no—
139D4×C61—42no—
140D4×C61—44no—
141D4×C61—44no—
142D4×C61—48no—
143D4×C61—42no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—21no—
148C2×C4×C61—21no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—44no—
154C2×C4×C61—42no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—42no—
160C2×C4×C61—44no—
161C2×C4×C61—42no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—81no—
169C2×C4×C61—82no—
170C2×C4×C61—82no—
171C2×C4×C61—82no—
172C2×C4×C61#9911yescharacteristic
173C2×C4×C61#9911yescharacteristic
174C8×C61—82no—
175#11820C6111yesmaximal, characteristic, direct product factor, Sylow
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61—42no—
183C22×C61—42no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—42no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61—81no—
194C22×C61—81no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61C2×#9911yescharacteristic
198C22×C61#41711yescharacteristic
199C22×C61#42111yescharacteristic
200C4×C61—21no—
201C4×C61—42no—
202C4×C61—42no—
203C4×C61—41no—
204C4×C61—41no—
205C4×C61—41no—
206C4×C61—41no—
207C4×C61—81no—
208C4×C61—162no—
209C4×C61AΣL(1,𝔽16)12yes—
210#2388C2×C6111yescharacteristic
211#1799C2×C6111yescharacteristic, complemented
212#1466C2×C6111yescharacteristic, complemented
213#1351C2×C6112yescomplemented
214#1343C2×C6112yescomplemented
215C2×C61—21no—
216C2×C61—21no—
217C2×C61—41no—
218C2×C61—41no—
219C2×C61—81no—
220C2×C61—82no—
221C2×C61#151311yescharacteristic
222C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
223C2×C61#212611yescharacteristic
224C22×#41C4×C6111yescharacteristic, complemented
225C2×C4×D4—21no—
226C2×C4×D4C22×C6111yescharacteristic
227#432—21no—
228C2×#99—21no—
229#395C4×C6111yescharacteristic, complemented
230#394—21no—
231#394C4×C6111yescharacteristic, complemented
232#393C22×C6111yescharacteristic
233#392—21no—
234#392C22×C6111yescharacteristic
235#380—21no—
236C4×C22⋊C4—21no—
237C4×C22⋊C4C4×C6111yescharacteristic, complemented
238#342C22×C6112yes—
239#323—21no—
240#323C22×C6112yes—
241C61#1182011yescharacteristic, direct product factor, Sylow
242C2×#41—44no—
243C2×#41—41no—
244C2×#41—41no—
245C22×Q8C2×C4×C6111yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2×C4×C6111yescharacteristic
249C22×D4D4×C6111yescharacteristic
250C23×C4—21no—
251C23×C4C2×C4×C6111yescharacteristic
252C23×C4D4×C6111yescharacteristic
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—44no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4C2×C4×C6111yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4D4×C6111yescharacteristic
266C2×C22⋊C4C23×C6111yescharacteristic
267C2×C42—21no—
268C2×C42D4×C6111yescharacteristic
269C4⋊C8—41no—
270#99—42no—
271C22⋊C8—22no—
272#97—41no—
273#95—21no—
274#95—21no—
275#95—41no—
276#95C2×C4×C6111yescharacteristic
277#95C2×C4×C6111yescharacteristic
278C24—21no—
279C24C22⋊C4×C6111yescharacteristic
280C24C2×D4×C6111yescharacteristic
281#41—44no—
282#41—44no—
283#41—44no—
284#41—44no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C22×C4—24no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4C22⋊C4×C6111yescharacteristic
315C22×C4C22⋊C4×C6111yescharacteristic
316C22×C4C22⋊C4×C6111yescharacteristic
317C22×C4C22×C4×C6111yescharacteristic
318C22×C4C2×D4×C6111yescharacteristic
319C2×C8—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C42—21no—
333C42—21no—
334C42—42no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23C2×C22⋊C4×C6111yescharacteristic
352C23AGL(1,ℤ/8)×C6111yescharacteristic
353C23#137×C6111yescharacteristic
354Q8—42no—
355Q8—42no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—44no—
360D4—44no—
361D4—48no—
362D4—42no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4#99×C6111yescharacteristic
392C2×C4#99×C6111yescharacteristic
393C8—82no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—41no—
399C22—41no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—42no—
407C22—41no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22C2×#99×C6111yescharacteristic
416C22C61×#41711yescharacteristic
417C22C61×#42111yescharacteristic
418C4—21no—
419C4—42no—
420C4—41no—
421C4—42no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—162no—
427C4C61×AΣL(1,𝔽16)12yes—
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—82no—
433C2—81no—
434C2C61×#151311yescharacteristic
435C2C2×C61×AΣL(1,𝔽16)11yescharacteristic
436C2C61×#212611yescharacteristic
subgroupG/H
center, socleC22×C61C2×#99
commutatorC22×C4C22×C4×C61
Frattini, cosocleC2×C22⋊C4C23×C61
prime radicalC61×#2388C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×C4×D4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#1799C2semidirect
3. C61×#1466C2semidirect
4. C61×#1351C2semidirect
5. C61×#1343C2semidirect
6. C22×#41×C61C4semidirect
11. C61×#395C4semidirect
13. C61×#394C4semidirect
19. C4×C22⋊C4×C61C4semidirect
175. #11820C61direct (N × H)
211. #1799C2×C61semidirect
212. #1466C2×C61semidirect
213. #1351C2×C61semidirect
214. #1343C2×C61semidirect
224. C22×#41C4×C61semidirect
229. #395C4×C61semidirect
231. #394C4×C61semidirect
237. C4×C22⋊C4C4×C61semidirect
241. C61#11820direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#23882
C61×#17992
C61×#14662
C61×#13512
C61×#13432
#1182061