Finite Group Databasebeta

← C61×#9516 #29512949

Subgroups — C61×#9516 #29512949

Order 15616.

626 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2975C211yesmaximal, characteristic, complemented
2C61×#2534C211yesmaximal, characteristic
3C61×#1465C211yesmaximal, characteristic, complemented
4C61×#1462C212yesmaximal, complemented
5C61×#1305C212yesmaximal, complemented
6C2×#123×C61C411yescharacteristic
7C2×#120×C61C411yescharacteristic
8C2×C4×D4×C61C2211yescharacteristic
9C22×C22⋊C4×C61C2211yescharacteristic
10C61×#436—24no—
11C61×#431—21no—
12C61×#407—21no—
13C61×#407C2212yes—
14C61×#407C2212yes—
15C61×#396—21no—
16C61×#394—21no—
17C61×#388—21no—
18C61×#386—21no—
19C61×#381C411yescharacteristic
20C61×#380C2211yescharacteristic
21C61×#379C411yescharacteristic
22C61×#324—21no—
23C61×#323—21no—
24C61×#323—24no—
25C25×C61C2×C411yescharacteristic
26C22×D4×C61—21no—
27C22×D4×C61C2×C411yescharacteristic
28C23×C4×C61C2×C411yescharacteristic
29C23×C4×C61D411yescharacteristic
30C23×C4×C61C2311yescharacteristic
31C2×M16×C61—22no—
32C2×M16×C61—22no—
33C2×M16×C61—22no—
34#123×C61—42no—
35#120×C61—42no—
36C4×D4×C61—22no—
37C4×D4×C61—22no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—22no—
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61C2×C411yescharacteristic
51C2×C22⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61D411yescharacteristic
53C2×C42×C61D411yescharacteristic
54C4⋊C8×C61—22no—
55C22⋊C8×C61—22no—
56C22⋊C8×C61—22no—
57C22⋊C8×C61—22no—
58C22⋊C8×C61—22no—
59C22⋊C8×C61D414yescomplemented
60C4×C8×C61—22no—
61#95×C61—21no—
62#95×C61—21no—
63#95×C61—21no—
64C24×C61—21no—
65C24×C61—22no—
66C24×C61—42no—
67C24×C61—41no—
68C24×C61—41no—
69C24×C61—81no—
70C24×C61C22⋊C411yescharacteristic
71C24×C61C22×C411yescharacteristic
72C2×D4×C61—22no—
73C2×D4×C61—22no—
74C2×D4×C61—41no—
75C2×D4×C61—42no—
76C2×D4×C61—42no—
77C2×D4×C61—81no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—22no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—81no—
87C22×C4×C61—81no—
88C22×C4×C61C22⋊C411yescharacteristic
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22⋊C411yescharacteristic
91C22×C4×C61C2×D412yes—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61#4112yes—
95M16×C61—82no—
96M16×C61—82no—
97M16×C61—82no—
98C2×C8×C61—24no—
99C2×C8×C61—42no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—42no—
102C4⋊C4×C61—81no—
103C22⋊C4×C61—22no—
104C22⋊C4×C61—22no—
105C22⋊C4×C61—42no—
106C22⋊C4×C61—42no—
107C22⋊C4×C61—42no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—42no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C22⋊C4×C61—81no—
114C42×C61—21no—
115C42×C61—21no—
116C23×C61—21no—
117C23×C61—21no—
118C23×C61—22no—
119C23×C61—24no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61—41no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61—41no—
129C23×C61—41no—
130C23×C61—42no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—82no—
135C23×C61—82no—
136C23×C61—84no—
137C23×C61—84no—
138C23×C61C2×C22⋊C411yescharacteristic
139C23×C61C4×D412yes—
140D4×C61—44no—
141D4×C61—84no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
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—82no—
164C2×C4×C61#10012yes—
165C2×C4×C61#12011yescharacteristic
166C2×C4×C61#12112yes—
167C2×C4×C61#12311yescharacteristic
168C8×C61—44no—
169C8×C61—82no—
170#9516C6111yesmaximal, characteristic, direct product factor, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—22no—
174C22×C61—22no—
175C22×C61—22no—
176C22×C61—42no—
177C22×C61—42no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—42no—
181C22×C61—42no—
182C22×C61—41no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—84no—
189C22×C61—84no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61#38611yescharacteristic
193C22×C61C2×#10011yescharacteristic
194C22×C61#41311yescharacteristic
195C4×C61—22no—
196C4×C61—21no—
197C4×C61—22no—
198C4×C61—21no—
199C4×C61—41no—
200C4×C61—41no—
201C4×C61—81no—
202C4×C61—81no—
203C4×C61—81no—
204#2975C2×C6111yescharacteristic, complemented
205#2534C2×C6111yescharacteristic
206#1465C2×C6111yescharacteristic, complemented
207#1462C2×C6112yescomplemented
208#1305C2×C6112yescomplemented
209C2×C61—21no—
210C2×C61—21no—
211C2×C61—42no—
212C2×C61—81no—
213C2×C61—82no—
214C2×C61#189111yescharacteristic
215C2×C61#190011yescharacteristic
216C2×C61#191311yescharacteristic
217C2×#123C4×C6111yescharacteristic
218C2×#120C4×C6111yescharacteristic
219C2×C4×D4C22×C6111yescharacteristic
220C22×C22⋊C4C22×C6111yescharacteristic
221#436—24no—
222#431—21no—
223#407—21no—
224#407C22×C6112yes—
225#407C22×C6112yes—
226#396—21no—
227#394—21no—
228#388—21no—
229#386—21no—
230#381C4×C6111yescharacteristic
231#380C22×C6111yescharacteristic
232#379C4×C6111yescharacteristic
233#324—21no—
234#323—21no—
235#323—24no—
236C61#951611yescharacteristic, direct product factor, Sylow
237C25C2×C4×C6111yescharacteristic
238C22×D4—21no—
239C22×D4C2×C4×C6111yescharacteristic
240C23×C4C2×C4×C6111yescharacteristic
241C23×C4D4×C6111yescharacteristic
242C23×C4C23×C6111yescharacteristic
243C2×M16—22no—
244C2×M16—22no—
245C2×M16—22no—
246#123—42no—
247#120—42no—
248C4×D4—22no—
249C4×D4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4D4×C6111yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4C2×C4×C6111yescharacteristic
262C2×C22⋊C4C2×C4×C6111yescharacteristic
263C2×C22⋊C4C2×C4×C6111yescharacteristic
264C2×C22⋊C4D4×C6111yescharacteristic
265C2×C42D4×C6111yescharacteristic
266C4⋊C8—22no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8D4×C6114yescomplemented
272C4×C8—22no—
273#95—21no—
274#95—21no—
275#95—21no—
276C24—21no—
277C24—22no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—81no—
282C24C22⋊C4×C6111yescharacteristic
283C24C22×C4×C6111yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—81no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C22⋊C4×C6111yescharacteristic
301C22×C4C22⋊C4×C6111yescharacteristic
302C22×C4C22⋊C4×C6111yescharacteristic
303C22×C4C2×D4×C6112yes—
304C22×C4C2×D4×C6111yescharacteristic
305C22×C4C2×D4×C6111yescharacteristic
306C22×C4#41×C6112yes—
307M16—82no—
308M16—82no—
309M16—82no—
310C2×C8—24no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—81no—
326C42—21no—
327C42—21no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—24no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—82no—
347C23—82no—
348C23—84no—
349C23—84no—
350C23C2×C22⋊C4×C6111yescharacteristic
351C23C4×D4×C6112yes—
352D4—44no—
353D4—84no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4#100×C6112yes—
377C2×C4#120×C6111yescharacteristic
378C2×C4#121×C6112yes—
379C2×C4#123×C6111yescharacteristic
380C8—44no—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—84no—
400C22—84no—
401C22—81no—
402C22—81no—
403C22C61×#38611yescharacteristic
404C22C2×#100×C6111yescharacteristic
405C22C61×#41311yescharacteristic
406C4—22no—
407C4—21no—
408C4—22no—
409C4—21no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C4—81no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—81no—
419C2—82no—
420C2C61×#189111yescharacteristic
421C2C61×#190011yescharacteristic
422C2C61×#191311yescharacteristic
subgroupG/H
center, socleC22×C61#386
commutatorC24C22×C4×C61
Frattini, cosocleC23×C4C23×C61
prime radicalC2×#120×C61C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#2975C2semidirect
3. C61×#1465C2semidirect
4. C61×#1462C2semidirect
5. C61×#1305C2semidirect
59. C22⋊C8×C61D4semidirect
170. #9516C61direct (N × H)
204. #2975C2×C61semidirect
206. #1465C2×C61semidirect
207. #1462C2×C61semidirect
208. #1305C2×C61semidirect
236. C61#9516direct (N × H)
271. C22⋊C8D4×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#29752
C61×#25342
C61×#14652
C61×#14622
C61×#13052
#951661