Finite Group Databasebeta

← C61×#9047 #29512480

Subgroups — C61×#9047 #29512480

Order 15616.

492 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3103C211yesmaximal, characteristic
2C61×#2932C211yesmaximal, characteristic
3C61×#1985C211yesmaximal, characteristic, complemented
4C61×#1540C211yesmaximal, characteristic
5C61×#1487C211yesmaximal, characteristic, complemented
6C61×#1301C211yesmaximal, characteristic, complemented
7C61×#1281C211yesmaximal, characteristic, complemented
8C2×#131×C61—21no—
9C61×#532C2211yescharacteristic
10C2×C4×Q8×C61C2211yescharacteristic
11C61×#481C412yes—
12C61×#475—21no—
13C61×#460—21no—
14C61×#448C412yes—
15C4×SD16×C61—22no—
16C61×#435—22no—
17C61×#435—22no—
18C61×#431C2211yescharacteristic
19C61×#431C2211yescharacteristic
20C61×#423C2211yescharacteristic
21C2×C4⋊C8×C61C2211yescharacteristic
22C61×#408—21no—
23C2×#97×C61C2211yescharacteristic
24C61×#336—21no—
25C61×#336—21no—
26C61×#332—21no—
27C61×#331—21no—
28C61×Q8⋊C8—22no—
29C2×#41×C61—21no—
30C22×Q8×C61C2×C411yescharacteristic
31C2×SD16×C61—22no—
32#131×C61—42no—
33#131×C61—42no—
34C2×M16×C61—22no—
35C2×M16×C61—21no—
36C22×C8×C61—22no—
37C22×C8×C61—21no—
38C22×C8×C61D411yescharacteristic
39C22×C8×C61D411yescharacteristic
40C4⋊Q8×C61D411yescharacteristic
41#127×C61D411yescharacteristic
42#124×C61—21no—
43#121×C61—21no—
44#121×C61C2×C412yes—
45C4×Q8×C61—21no—
46C4×Q8×C61—21no—
47C4×Q8×C61—21no—
48C4×Q8×C61—21no—
49C4×D4×C61—21no—
50C4×D4×C61D412yes—
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61C2×C411yescharacteristic
53C2×C42×C61—21no—
54C2×C42×C61C2311yescharacteristic
55#106×C61—22no—
56C4⋊C8×C61—21no—
57C4⋊C8×C61—21no—
58C4⋊C8×C61—21no—
59C4⋊C8×C61—21no—
60C4⋊C8×C61—21no—
61C4⋊C8×C61D412yes—
62#103×C61—22no—
63#102×C61—22no—
64C22⋊C8×C61—24no—
65C22⋊C8×C61—21no—
66C22⋊C8×C61C2×C412yes—
67#97×C61—21no—
68#97×C61—21no—
69C4×C8×C61—21no—
70C4×C8×C61—21no—
71#41×C61—41no—
72#41×C61—41no—
73#41×C61—41no—
74#41×C61—41no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77C2×Q8×C61—21no—
78C2×Q8×C61—42no—
79C2×D4×C61—22no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61C22⋊C412yes—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—21no—
86C22×C4×C61C22×C411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89SD16×C61—44no—
90M16×C61—44no—
91M16×C61—41no—
92M16×C61—41no—
93C2×C8×C61—21no—
94C2×C8×C61—21no—
95C2×C8×C61—22no—
96C2×C8×C61—21no—
97C2×C8×C61—22no—
98C2×C8×C61—21no—
99C2×C8×C61—21no—
100C2×C8×C61—44no—
101C2×C8×C61—41no—
102C2×C8×C61—41no—
103C2×C8×C61—41no—
104C2×C8×C61—41no—
105C2×C8×C61SD1612yescomplemented
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61C22⋊C412yes—
112C22⋊C4×C61—22no—
113C22⋊C4×C61—22no—
114C42×C61—41no—
115C42×C61—41no—
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C23×C61—22no—
121C23×C61C4×D411yescharacteristic
122Q8×C61—22no—
123Q8×C61—22no—
124Q8×C61—41no—
125Q8×C61—41no—
126Q8×C61—41no—
127Q8×C61—41no—
128D4×C61—42no—
129D4×C61—42no—
130D4×C61—42no—
131D4×C61—41no—
132D4×C61—41no—
133D4×C61—81no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—41no—
144C2×C4×C61C2×C22⋊C411yescharacteristic
145C2×C4×C61C4×D411yescharacteristic
146C2×C4×C61#12011yescharacteristic
147C2×C4×C61#12111yescharacteristic
148C2×C4×C61#12111yescharacteristic
149C2×C4×C61#12311yescharacteristic
150C2×C4×C61C2×SD1611yescharacteristic
151C2×C4×C61#13511yescharacteristic
152C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C61#13711yescharacteristic
154C8×C61—22no—
155C8×C61—42no—
156C8×C61—42no—
157C8×C61—41no—
158C8×C61—81no—
159C8×C61—81no—
160#9047C6111yesmaximal, characteristic, direct product factor, Sylow
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61—42no—
166C22×C61#38611yescharacteristic
167C22×C61C4×SD1611yescharacteristic
168C22×C61#44011yescharacteristic
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—42no—
174C4×C61—42no—
175C4×C61—81no—
176C4×C61#45011yescharacteristic
177C4×C61#45211yescharacteristic
178C4×C61#46111yescharacteristic
179C4×C61#46311yescharacteristic
180#3103C2×C6111yescharacteristic
181#2932C2×C6111yescharacteristic
182#1985C2×C6111yescharacteristic, complemented
183#1540C2×C6111yescharacteristic
184#1487C2×C6111yescharacteristic, complemented
185#1301C2×C6111yescharacteristic, complemented
186#1281C2×C6111yescharacteristic, complemented
187C2×C61—21no—
188C2×C61—21no—
189C2×C61—82no—
190C2×C61#187711yescharacteristic
191C2×C61#188111yescharacteristic
192C2×C61#189111yescharacteristic
193C2×#131—21no—
194#532C22×C6111yescharacteristic
195C2×C4×Q8C22×C6111yescharacteristic
196#481C4×C6112yes—
197#475—21no—
198#460—21no—
199#448C4×C6112yes—
200C4×SD16—22no—
201#435—22no—
202#435—22no—
203#431C22×C6111yescharacteristic
204#431C22×C6111yescharacteristic
205#423C22×C6111yescharacteristic
206C2×C4⋊C8C22×C6111yescharacteristic
207#408—21no—
208C2×#97C22×C6111yescharacteristic
209#336—21no—
210#336—21no—
211#332—21no—
212#331—21no—
213Q8⋊C8—22no—
214C61#904711yescharacteristic, direct product factor, Sylow
215C2×#41—21no—
216C22×Q8C2×C4×C6111yescharacteristic
217C2×SD16—22no—
218#131—42no—
219#131—42no—
220C2×M16—22no—
221C2×M16—21no—
222C22×C8—22no—
223C22×C8—21no—
224C22×C8D4×C6111yescharacteristic
225C22×C8D4×C6111yescharacteristic
226C4⋊Q8D4×C6111yescharacteristic
227#127D4×C6111yescharacteristic
228#124—21no—
229#121—21no—
230#121C2×C4×C6112yes—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×D4—21no—
236C4×D4D4×C6112yes—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C2×C4×C6111yescharacteristic
239C2×C42—21no—
240C2×C42C23×C6111yescharacteristic
241#106—22no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8D4×C6112yes—
248#103—22no—
249#102—22no—
250C22⋊C8—24no—
251C22⋊C8—21no—
252C22⋊C8C2×C4×C6112yes—
253#97—21no—
254#97—21no—
255C4×C8—21no—
256C4×C8—21no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—42no—
265C2×D4—22no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4C22⋊C4×C6112yes—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4C22×C4×C6111yescharacteristic
273C22×C4C2×D4×C6111yescharacteristic
274C22×C4#41×C6111yescharacteristic
275SD16—44no—
276M16—44no—
277M16—41no—
278M16—41no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—44no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8SD16×C6112yescomplemented
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4C22⋊C4×C6112yes—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C42—41no—
301C42—41no—
302C42C2×D4×C6111yescharacteristic
303C42C2×D4×C6111yescharacteristic
304C42C2×D4×C6111yescharacteristic
305C42#41×C6111yescharacteristic
306C23—22no—
307C23C4×D4×C6111yescharacteristic
308Q8—22no—
309Q8—22no—
310Q8—41no—
311Q8—41no—
312Q8—41no—
313Q8—41no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—41no—
318D4—41no—
319D4—81no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4C2×C22⋊C4×C6111yescharacteristic
331C2×C4C4×D4×C6111yescharacteristic
332C2×C4#120×C6111yescharacteristic
333C2×C4#121×C6111yescharacteristic
334C2×C4#121×C6111yescharacteristic
335C2×C4#123×C6111yescharacteristic
336C2×C4C2×SD16×C6111yescharacteristic
337C2×C4#135×C6111yescharacteristic
338C2×C4AGL(1,ℤ/8)×C6111yescharacteristic
339C2×C4#137×C6111yescharacteristic
340C8—22no—
341C8—42no—
342C8—42no—
343C8—41no—
344C8—81no—
345C8—81no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22C61×#38611yescharacteristic
352C22C4×SD16×C6111yescharacteristic
353C22C61×#44011yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—42no—
360C4—81no—
361C4C61×#45011yescharacteristic
362C4C61×#45211yescharacteristic
363C4C61×#46111yescharacteristic
364C4C61×#46311yescharacteristic
365C2—21no—
366C2—21no—
367C2—82no—
368C2C61×#187711yescharacteristic
369C2C61×#188111yescharacteristic
370C2C61×#189111yescharacteristic
subgroupG/H
centerC2×C4×C61#120
commutatorC22×C4C22×C4×C61
Frattini, cosocleC2×C42C23×C61
socleC22×C61#386
prime radicalC61×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
3. C61×#1985C2semidirect
5. C61×#1487C2semidirect
6. C61×#1301C2semidirect
7. C61×#1281C2semidirect
105. C2×C8×C61SD16semidirect
160. #9047C61direct (N × H)
182. #1985C2×C61semidirect
184. #1487C2×C61semidirect
185. #1301C2×C61semidirect
186. #1281C2×C61semidirect
214. C61#9047direct (N × H)
291. C2×C8SD16×C61semidirect

Maximal subgroups

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

subgroupindex
C61×#31032
C61×#29322
C61×#19852
C61×#15402
C61×#14872
C61×#13012
C61×#12812
#904761