Finite Group Databasebeta

← C61×#8717 #29512150

Subgroups — C61×#8717 #29512150

Order 15616.

588 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#2285C211yesmaximal, characteristic
2C61×#1817C211yesmaximal, characteristic
3C61×#1434C211yesmaximal, characteristic
4C61×#1294C214yesmaximal
5C2×#117×C61—21no—
6C22×C42×C61C2211yescharacteristic
7C2×C22⋊C8×C61C2212yes—
8C61×#384C411yescharacteristic, complemented
9C61×#384C411yescharacteristic, complemented
10C61×#382—21no—
11C61×#380—21no—
12C61×#380C2212yes—
13C4×C4⋊C4×C61—21no—
14C4×C4⋊C4×C61C412yescomplemented
15C2×#95×C61—21no—
16C2×#95×C61—21no—
17C2×#95×C61C2212yes—
18C61×#336—21no—
19C61×#336—21no—
20C23×C4×C61—22no—
21C23×C4×C61D411yescharacteristic
22C23×C4×C61D411yescharacteristic
23C23×C4×C61C2311yescharacteristic
24C22×C8×C61—22no—
25#117×C61—44no—
26C2×C4⋊C4×C61—21no—
27C2×C4⋊C4×C61—21no—
28C2×C4⋊C4×C61—21no—
29C2×C4⋊C4×C61—21no—
30C2×C4⋊C4×C61C2×C412yes—
31C2×C4⋊C4×C61C2×C412yes—
32C2×C22⋊C4×C61—22no—
33C2×C42×C61—24no—
34C2×C42×C61—22no—
35C2×C42×C61C2×C411yescharacteristic
36C2×C42×C61C2×C411yescharacteristic
37C2×C42×C61D411yescharacteristic
38C2×C42×C61D411yescharacteristic
39C22⋊C8×C61—24no—
40#95×C61—24no—
41#95×C61—22no—
42#95×C61—22no—
43#95×C61—24no—
44#95×C61—42no—
45C24×C61C2×D411yescharacteristic
46C22×C4×C61—22no—
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—24no—
51C22×C4×C61—22no—
52C22×C4×C61—22no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61—41no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61—42no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61—42no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61SD1612yes—
67C22×C4×C61Q1612yes—
68C22×C4×C61C22×C411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61#4114yes—
71C2×C8×C61—42no—
72C2×C8×C61—42no—
73C2×C8×C61—42no—
74C4⋊C4×C61—42no—
75C4⋊C4×C61—42no—
76C4⋊C4×C61—42no—
77C4⋊C4×C61—42no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—42no—
80C22⋊C4×C61—44no—
81C42×C61—22no—
82C42×C61—22no—
83C42×C61—22no—
84C42×C61—22no—
85C42×C61—44no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61C2×C22⋊C411yescharacteristic
91C23×C61#12312yes—
92C23×C61C2×SD1611yescharacteristic
93C23×C61C2×Q1611yescharacteristic
94C23×C61AGL(1,ℤ/8)12yes—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—24no—
106C2×C4×C61—24no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—44no—
111C2×C4×C61—44no—
112C2×C4×C61—44no—
113C2×C4×C61—44no—
114C2×C4×C61—44no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—82no—
122C2×C4×C61—82no—
123C2×C4×C61—82no—
124C2×C4×C61—82no—
125C2×C4×C61#10312yes—
126C2×C4×C61#10312yes—
127C2×C4×C61#10412yes—
128C2×C4×C61#11712yes—
129C2×C4×C61#12312yes—
130C8×C61—82no—
131#8717C6111yesmaximal, characteristic, direct product factor, Sylow
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61#38111yescharacteristic
141C22×C61#41011yescharacteristic
142C22×C61#41011yescharacteristic
143C22×C61C2×#10311yescharacteristic
144C22×C61#41811yescharacteristic
145C22×C61C2×#10411yescharacteristic
146C22×C61#42111yescharacteristic
147C22×C61#48112yes—
148C22×C61#48412yes—
149C4×C61—22no—
150C4×C61—22no—
151C4×C61—22no—
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—41no—
155C4×C61—84no—
156C4×C61—82no—
157C4×C61—82no—
158#2285C2×C6111yescharacteristic
159#1817C2×C6111yescharacteristic
160#1434C2×C6111yescharacteristic
161#1294C2×C6114yes—
162C2×C61—22no—
163C2×C61—22no—
164C2×C61#150311yescharacteristic
165C2×C61#150511yescharacteristic
166C2×C61#151411yescharacteristic
167C2×C61#151811yescharacteristic
168C2×C61#182011yescharacteristic
169C2×C61#182411yescharacteristic
170C2×C61#182611yescharacteristic
171C2×#117—21no—
172C22×C42C22×C6111yescharacteristic
173C2×C22⋊C8C22×C6112yes—
174#384C4×C6111yescharacteristic, complemented
175#384C4×C6111yescharacteristic, complemented
176#382—21no—
177#380—21no—
178#380C22×C6112yes—
179C4×C4⋊C4—21no—
180C4×C4⋊C4C4×C6112yescomplemented
181C2×#95—21no—
182C2×#95—21no—
183C2×#95C22×C6112yes—
184#336—21no—
185#336—21no—
186C61#871711yescharacteristic, direct product factor, Sylow
187C23×C4—22no—
188C23×C4D4×C6111yescharacteristic
189C23×C4D4×C6111yescharacteristic
190C23×C4C23×C6111yescharacteristic
191C22×C8—22no—
192#117—44no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×C4×C6112yes—
198C2×C4⋊C4C2×C4×C6112yes—
199C2×C22⋊C4—22no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42C2×C4×C6111yescharacteristic
203C2×C42C2×C4×C6111yescharacteristic
204C2×C42D4×C6111yescharacteristic
205C2×C42D4×C6111yescharacteristic
206C22⋊C8—24no—
207#95—24no—
208#95—22no—
209#95—22no—
210#95—24no—
211#95—42no—
212C24C2×D4×C6111yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C22⋊C4×C6111yescharacteristic
230C22×C4C22⋊C4×C6111yescharacteristic
231C22×C4C22⋊C4×C6111yescharacteristic
232C22×C4C22⋊C4×C6111yescharacteristic
233C22×C4SD16×C6112yes—
234C22×C4Q16×C6112yes—
235C22×C4C22×C4×C6111yescharacteristic
236C22×C4C2×D4×C6111yescharacteristic
237C22×C4#41×C6114yes—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—44no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—44no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23C2×C22⋊C4×C6111yescharacteristic
258C23#123×C6112yes—
259C23C2×SD16×C6111yescharacteristic
260C23C2×Q16×C6111yescharacteristic
261C23AGL(1,ℤ/8)×C6112yes—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#103×C6112yes—
293C2×C4#103×C6112yes—
294C2×C4#104×C6112yes—
295C2×C4#117×C6112yes—
296C2×C4#123×C6112yes—
297C8—82no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—22no—
306C22C61×#38111yescharacteristic
307C22C61×#41011yescharacteristic
308C22C61×#41011yescharacteristic
309C22C2×#103×C6111yescharacteristic
310C22C61×#41811yescharacteristic
311C22C2×#104×C6111yescharacteristic
312C22C61×#42111yescharacteristic
313C22C61×#48112yes—
314C22C61×#48412yes—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—84no—
322C4—82no—
323C4—82no—
324C2—22no—
325C2—22no—
326C2C61×#150311yescharacteristic
327C2C61×#150511yescharacteristic
328C2C61×#151411yescharacteristic
329C2C61×#151811yescharacteristic
330C2C61×#182011yescharacteristic
331C2C61×#182411yescharacteristic
332C2C61×#182611yescharacteristic
subgroupG/H
center, socleC23×C61C2×C22⋊C4
commutatorC22×C4C22×C4×C61
Frattini, cosocleC23×C4C23×C61
prime radicalC24×C61C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C61 ◁ C22×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
8. C61×#384C4semidirect
9. C61×#384C4semidirect
14. C4×C4⋊C4×C61C4semidirect
131. #8717C61direct (N × H)
174. #384C4×C61semidirect
175. #384C4×C61semidirect
180. C4×C4⋊C4C4×C61semidirect
186. C61#8717direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#22852
C61×#18172
C61×#14342
C61×#12942
#871761