Finite Group Databasebeta

← C61×#7213 #29510646

Subgroups — C61×#7213 #29510646

Order 15616.

560 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 82 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×#1819C211yesmaximal, characteristic
2C61×#1462C211yesmaximal, characteristic
3C61×#1437C211yesmaximal, characteristic
4C22×C22⋊C4×C61—21no—
5C22×C22⋊C4×C61C2211yescharacteristic
6C22×C42×C61C411yescharacteristic, complemented
7C61×#407—21no—
8C61×#379—21no—
9C2×#95×C61—21no—
10C2×#95×C61C411yescharacteristic, complemented
11C61×#342—21no—
12C61×#342C412yescomplemented
13C61×#323—21no—
14C61×#323C412yescomplemented
15C25×C61D411yescharacteristic
16C23×C4×C61—21no—
17C23×C4×C61—21no—
18C23×C4×C61C2×C411yescharacteristic
19C23×C4×C61D411yescharacteristic
20C2×M16×C61—41no—
21C2×C22⋊C4×C61—24no—
22C2×C22⋊C4×C61—22no—
23C2×C22⋊C4×C61—41no—
24C2×C22⋊C4×C61—41no—
25C2×C22⋊C4×C61—42no—
26C2×C22⋊C4×C61—42no—
27C2×C22⋊C4×C61—41no—
28C2×C22⋊C4×C61—41no—
29C2×C22⋊C4×C61C2×C411yescharacteristic
30C2×C22⋊C4×C61C2×C411yescharacteristic
31C2×C22⋊C4×C61D411yescharacteristic
32C2×C22⋊C4×C61Q811yescharacteristic
33C2×C42×C61—41no—
34C2×C42×C61—42no—
35C22⋊C8×C61—22no—
36C22⋊C8×C61—41no—
37#95×C61—41no—
38#95×C61—42no—
39C24×C61—22no—
40C24×C61—21no—
41C24×C61—21no—
42C24×C61—41no—
43C24×C61—41no—
44C24×C61—41no—
45C24×C61—41no—
46C24×C61—41no—
47C24×C61C22⋊C411yescharacteristic
48C24×C61C22⋊C411yescharacteristic
49C24×C61C4⋊C411yescharacteristic
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—22no—
58C22×C4×C61—41no—
59C22×C4×C61—41no—
60C22×C4×C61—41no—
61C22×C4×C61—41no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61—41no—
66C22×C4×C61—41no—
67C22×C4×C61—41no—
68C22×C4×C61—41no—
69C22×C4×C61—41no—
70C22×C4×C61—41no—
71C22×C4×C61—41no—
72C22×C4×C61—42no—
73C22×C4×C61C4211yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C4⋊C411yescharacteristic
76C22×C4×C61C4⋊C411yescharacteristic
77M16×C61—81no—
78M16×C61—81no—
79C2×C8×C61—42no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—41no—
84C22⋊C4×C61—48no—
85C22⋊C4×C61—42no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—81no—
88C22⋊C4×C61—81no—
89C22⋊C4×C61—81no—
90C22⋊C4×C61—81no—
91C42×C61—42no—
92C42×C61—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—81no—
119C23×C61—81no—
120C23×C61—81no—
121C23×C61—81no—
122C23×C61—81no—
123C23×C61—81no—
124C23×C61#9511yescharacteristic
125C23×C61#9911yescharacteristic
126C23×C61#9911yescharacteristic
127C23×C61#10412yes—
128C2×C4×C61—24no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
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—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—81no—
152C2×C4×C61—81no—
153C2×C4×C61—81no—
154C2×C4×C61—81no—
155C2×C4×C61—81no—
156C2×C4×C61—82no—
157C2×C4×C61—82no—
158C2×C4×C61—82no—
159C8×C61—82no—
160#7213C6111yesmaximal, characteristic, direct product factor, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—21no—
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—22no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61—41no—
179C22×C61—41no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61—41no—
183C22×C61—81no—
184C22×C61—82no—
185C22×C61—82no—
186C22×C61—82no—
187C22×C61—82no—
188C22×C61—81no—
189C22×C61H(3,ℤ/4)11yescharacteristic
190C22×C61#33911yescharacteristic
191C22×C61#34211yescharacteristic
192C22×C61AΣL(1,𝔽16)11yescharacteristic
193C22×C61#35211yescharacteristic
194C22×C61#35311yescharacteristic
195C22×C61#35411yescharacteristic
196C4×C61—44no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—82no—
202C4×C61—162no—
203#1819C2×C6111yescharacteristic
204#1462C2×C6111yescharacteristic
205#1437C2×C6111yescharacteristic
206C2×C61—21no—
207C2×C61—21no—
208C2×C61—21no—
209C2×C61—21no—
210C2×C61—21no—
211C2×C61—21no—
212C2×C61—42no—
213C2×C61—81no—
214C2×C61#128411yescharacteristic
215C2×C61#139011yescharacteristic
216C2×C61#139311yescharacteristic
217C22×C22⋊C4—21no—
218C22×C22⋊C4C22×C6111yescharacteristic
219C22×C42C4×C6111yescharacteristic, complemented
220#407—21no—
221#379—21no—
222C2×#95—21no—
223C2×#95C4×C6111yescharacteristic, complemented
224#342—21no—
225#342C4×C6112yescomplemented
226#323—21no—
227#323C4×C6112yescomplemented
228C61#721311yescharacteristic, direct product factor, Sylow
229C25D4×C6111yescharacteristic
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C2×C4×C6111yescharacteristic
233C23×C4D4×C6111yescharacteristic
234C2×M16—41no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4C2×C4×C6111yescharacteristic
244C2×C22⋊C4C2×C4×C6111yescharacteristic
245C2×C22⋊C4D4×C6111yescharacteristic
246C2×C22⋊C4Q8×C6111yescharacteristic
247C2×C42—41no—
248C2×C42—42no—
249C22⋊C8—22no—
250C22⋊C8—41no—
251#95—41no—
252#95—42no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24C22⋊C4×C6111yescharacteristic
262C24C22⋊C4×C6111yescharacteristic
263C24C4⋊C4×C6111yescharacteristic
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4C42×C6111yescharacteristic
288C22×C4C22⋊C4×C6111yescharacteristic
289C22×C4C4⋊C4×C6111yescharacteristic
290C22×C4C4⋊C4×C6111yescharacteristic
291M16—81no—
292M16—81no—
293C2×C8—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—48no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C22⋊C4—81no—
305C42—42no—
306C42—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—42no—
319C23—42no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—81no—
333C23—81no—
334C23—81no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23#95×C6111yescharacteristic
339C23#99×C6111yescharacteristic
340C23#99×C6111yescharacteristic
341C23#104×C6112yes—
342C2×C4—24no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C8—82no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—22no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—82no—
401C22—81no—
402C22C61×H(3,ℤ/4)11yescharacteristic
403C22C61×#33911yescharacteristic
404C22C61×#34211yescharacteristic
405C22C61×AΣL(1,𝔽16)11yescharacteristic
406C22C61×#35211yescharacteristic
407C22C61×#35311yescharacteristic
408C22C61×#35411yescharacteristic
409C4—44no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—82no—
415C4—162no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—81no—
424C2C61×#128411yescharacteristic
425C2C61×#139011yescharacteristic
426C2C61×#139311yescharacteristic
subgroupG/H
center, socleC22×C61#342
commutatorC22×C4C42×C61
Frattini, cosocleC22×C22⋊C4C22×C61
prime radicalC25×C61D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C24×C61 ◁ C22×C22⋊C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
6. C22×C42×C61C4semidirect
10. C2×#95×C61C4semidirect
12. C61×#342C4semidirect
14. C61×#323C4semidirect
160. #7213C61direct (N × H)
219. C22×C42C4×C61semidirect
223. C2×#95C4×C61semidirect
225. #342C4×C61semidirect
227. #323C4×C61semidirect
228. C61#7213direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#18192
C61×#14622
C61×#14372
#721361