Finite Group Databasebeta

← C61×#8540 #29511973

Subgroups — C61×#8540 #29511973

Order 15616.

718 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 142 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×#3491C211yesmaximal, characteristic
2C61×#1822C211yesmaximal, characteristic, complemented
3C61×#1818C211yesmaximal, characteristic, complemented
4C61×#1468C214yesmaximal, complemented
5C61×#552—24no—
6C61×#551—24no—
7C61×#543C411yescharacteristic
8C61×#539—22no—
9C61×#534C411yescharacteristic
10C61×#529—22no—
11C2×#123×C61C2212yes—
12C61×#518C412yes—
13C2×#117×C61C2211yescharacteristic
14C61×#407C2212yes—
15C2×C22⋊C8×C61C2212yes—
16C61×#343—22no—
17C61×#336—22no—
18C61×#324—24no—
19C61×#323—24no—
20C22×D4×C61C2×C412yes—
21C23×C4×C61C2311yescharacteristic
22C2×M16×C61—22no—
23C22×C8×C61—22no—
24#126×C61—24no—
25#126×C61—24no—
26#125×C61—24no—
27#125×C61—22no—
28#124×C61—22no—
29#123×C61—24no—
30#123×C61—22no—
31#123×C61—22no—
32#123×C61—24no—
33#123×C61—22no—
34C22⋊Q8×C61—24no—
35C22⋊Q8×C61—22no—
36C22⋊Q8×C61—24no—
37#121×C61—22no—
38#120×C61—22no—
39C4×Q8×C61—22no—
40C4×D4×C61—22no—
41C4×D4×C61—24no—
42C4×D4×C61—24no—
43#117×C61—24no—
44#117×C61—22no—
45#117×C61—24no—
46C2×C4⋊C4×C61D412yes—
47C2×C4⋊C4×C61D414yes—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61C2×C412yes—
51C2×C22⋊C4×C61C2×C411yescharacteristic
52C2×C22⋊C4×C61D414yes—
53C2×C42×C61D412yes—
54C22⋊C8×C61—24no—
55C22⋊C8×C61—24no—
56C24×C61C22⋊C412yes—
57C24×C61C22×C411yescharacteristic
58C2×Q8×C61—22no—
59C2×D4×C61—22no—
60C2×D4×C61—22no—
61C2×D4×C61—22no—
62C2×D4×C61—44no—
63C22×C4×C61—24no—
64C22×C4×C61—42no—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22⋊C414yes—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D414yes—
72M16×C61—44no—
73C2×C8×C61—44no—
74C2×C8×C61—42no—
75C2×C8×C61—42no—
76C4⋊C4×C61—24no—
77C4⋊C4×C61—24no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—22no—
80C4⋊C4×C61—24no—
81C4⋊C4×C61—24no—
82C4⋊C4×C61—24no—
83C22⋊C4×C61—24no—
84C22⋊C4×C61—24no—
85C22⋊C4×C61—22no—
86C22⋊C4×C61—22no—
87C22⋊C4×C61—22no—
88C22⋊C4×C61—22no—
89C22⋊C4×C61—24no—
90C22⋊C4×C61—24no—
91C42×C61—24no—
92C42×C61—22no—
93C42×C61—22no—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—44no—
99C23×C61C2×C22⋊C412yes—
100C23×C61C2×C22⋊C411yescharacteristic
101Q8×C61—82no—
102D4×C61—44no—
103D4×C61—44no—
104D4×C61—82no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—24no—
111C2×C4×C61—44no—
112C2×C4×C61—42no—
113C2×C4×C61—44no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—84no—
117C2×C4×C61#12012yes—
118C2×C4×C61#12012yes—
119C8×C61—82no—
120C8×C61—82no—
121#8540C6111yesmaximal, characteristic, direct product factor, Sylow
122C22×C61—22no—
123C22×C61—22no—
124C22×C61—24no—
125C22×C61—42no—
126C22×C61—44no—
127C22×C61—82no—
128C22×C61—162no—
129C22×C61#37911yescharacteristic
130C22×C61#41012yes—
131C4×C61—22no—
132C4×C61—22no—
133C4×C61—42no—
134C4×C61—42no—
135C4×C61—44no—
136C4×C61—42no—
137#3491C2×C6111yescharacteristic
138#1822C2×C6111yescharacteristic, complemented
139#1818C2×C6111yescharacteristic, complemented
140#1468C2×C6114yescomplemented
141C2×C61—22no—
142C2×C61—42no—
143C2×C61—82no—
144C2×C61#178211yescharacteristic
145C2×C61#179112yes—
146#552—24no—
147#551—24no—
148#543C4×C6111yescharacteristic
149#539—22no—
150#534C4×C6111yescharacteristic
151#529—22no—
152C2×#123C22×C6112yes—
153#518C4×C6112yes—
154C2×#117C22×C6111yescharacteristic
155#407C22×C6112yes—
156C2×C22⋊C8C22×C6112yes—
157#343—22no—
158#336—22no—
159#324—24no—
160#323—24no—
161C61#854011yescharacteristic, direct product factor, Sylow
162C22×D4C2×C4×C6112yes—
163C23×C4C23×C6111yescharacteristic
164C2×M16—22no—
165C22×C8—22no—
166#126—24no—
167#126—24no—
168#125—22no—
169#125—24no—
170#124—22no—
171#123—24no—
172#123—22no—
173#123—22no—
174#123—22no—
175#123—24no—
176C22⋊Q8—24no—
177C22⋊Q8—22no—
178C22⋊Q8—24no—
179#121—22no—
180#120—22no—
181C4×Q8—22no—
182C4×D4—22no—
183C4×D4—24no—
184C4×D4—24no—
185#117—24no—
186#117—22no—
187#117—24no—
188C2×C4⋊C4D4×C6112yes—
189C2×C4⋊C4D4×C6114yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4C2×C4×C6111yescharacteristic
192C2×C22⋊C4C2×C4×C6111yescharacteristic
193C2×C22⋊C4C2×C4×C6112yes—
194C2×C22⋊C4D4×C6114yes—
195C2×C42D4×C6112yes—
196C22⋊C8—24no—
197C22⋊C8—24no—
198C24C22⋊C4×C6112yes—
199C24C22×C4×C6111yescharacteristic
200C2×Q8—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—44no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4C22⋊C4×C6112yes—
208C22×C4C22⋊C4×C6112yes—
209C22×C4C22⋊C4×C6114yes—
210C22×C4C22⋊C4×C6112yes—
211C22×C4C2×D4×C6111yescharacteristic
212C22×C4C2×D4×C6111yescharacteristic
213C22×C4C2×D4×C6114yes—
214M16—44no—
215C2×C8—44no—
216C2×C8—42no—
217C2×C8—42no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C42—24no—
234C42—22no—
235C42—22no—
236C23—22no—
237C23—22no—
238C23—42no—
239C23—42no—
240C23—44no—
241C23C2×C22⋊C4×C6112yes—
242C23C2×C22⋊C4×C6111yescharacteristic
243Q8—82no—
244D4—44no—
245D4—44no—
246D4—82no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4#120×C6112yes—
260C2×C4#120×C6112yes—
261C8—82no—
262C8—82no—
263C22—22no—
264C22—22no—
265C22—24no—
266C22—42no—
267C22—44no—
268C22—82no—
269C22—162no—
270C22C61×#37911yescharacteristic
271C22C61×#41012yes—
272C4—22no—
273C4—22no—
274C4—42no—
275C4—42no—
276C4—44no—
277C4—42no—
278C2—22no—
279C2—42no—
280C2—82no—
281C2C61×#178211yescharacteristic
282C2C61×#179112yes—
subgroupG/H
center, socleC22×C61#379
commutatorC24C22×C4×C61
Frattini, cosocleC23×C4C23×C61
prime radicalC61×#534C4
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
2. C61×#1822C2semidirect
3. C61×#1818C2semidirect
4. C61×#1468C2semidirect
121. #8540C61direct (N × H)
138. #1822C2×C61semidirect
139. #1818C2×C61semidirect
140. #1468C2×C61semidirect
161. C61#8540direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#34912
C61×#18222
C61×#18182
C61×#14682
#854061