Finite Group Databasebeta

← C61×#12697 #29516130

Subgroups — C61×#12697 #29516130

Order 15616.

1182 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×C61×#535C211yesmaximal, characteristic
2C2×C61×#353C214yesmaximal, complemented
3C61×#1782C212yesmaximal, complemented
4C22×#41×C61C2211yescharacteristic
5C23×D4×C61C411yescharacteristic, complemented
6C61×#535—28no—
7C2×#127×C61C2212yes—
8C2×#124×C61C412yescomplemented
9C2×#121×C61—22no—
10C2×#120×C61—22no—
11C2×#117×C61C411yescharacteristic, complemented
12C2×#99×C61—22no—
13C2×#99×C61C2214yescomplemented
14C61×#381—22no—
15C61×#353—28no—
16C25×C61—21no—
17C2×#41×C61—24no—
18C2×#41×C61—42no—
19C2×#41×C61—42no—
20C22×Q8×C61C2×C411yescharacteristic
21C22×D4×C61—24no—
22C22×D4×C61—24no—
23C22×D4×C61—21no—
24C22×D4×C61—42no—
25C22×D4×C61—42no—
26C22×D4×C61C2×C412yes—
27C22×D4×C61D412yes—
28C22×D4×C61C2311yescharacteristic
29C23×C4×C61C2×C411yescharacteristic
30C23×C4×C61D412yes—
31#127×C61—24no—
32#127×C61D418yes—
33#124×C61—24no—
34#124×C61—24no—
35#121×C61—216no—
36#120×C61—216no—
37#117×C61—24no—
38C2×C4⋊C4×C61—21no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61—24no—
41C2×C22⋊C4×C61—24no—
42C2×C42×C61C2×C412yes—
43#99×C61—28no—
44#99×C61—44no—
45#95×C61—44no—
46C24×C61—24no—
47C24×C61—22no—
48C24×C61—42no—
49C24×C61—42no—
50C24×C61—84no—
51C24×C61C22⋊C411yescharacteristic
52C24×C61C2×D412yes—
53#41×C61—416no—
54C2×Q8×C61—22no—
55C2×Q8×C61—41no—
56C2×Q8×C61—41no—
57C2×D4×C61—24no—
58C2×D4×C61—24no—
59C2×D4×C61—24no—
60C2×D4×C61—216no—
61C2×D4×C61—22no—
62C2×D4×C61—42no—
63C2×D4×C61—41no—
64C2×D4×C61—41no—
65C2×D4×C61—416no—
66C2×D4×C61—42no—
67C2×D4×C61—42no—
68C2×D4×C61—42no—
69C2×D4×C61—42no—
70C2×D4×C61—88no—
71C2×D4×C61C22⋊C414yes—
72C2×D4×C61C2×D414yes—
73C22×C4×C61—22no—
74C22×C4×C61—22no—
75C22×C4×C61—22no—
76C22×C4×C61—22no—
77C22×C4×C61—24no—
78C22×C4×C61—24no—
79C22×C4×C61—42no—
80C22×C4×C61—44no—
81C22×C4×C61C22⋊C412yes—
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22×C411yescharacteristic
84C4⋊C4×C61—24no—
85C22⋊C4×C61—24no—
86C22⋊C4×C61—216no—
87C22⋊C4×C61—44no—
88C22⋊C4×C61—44no—
89C42×C61—22no—
90C42×C61C22⋊C414yes—
91C23×C61—21no—
92C23×C61—22no—
93C23×C61—21no—
94C23×C61—22no—
95C23×C61—24no—
96C23×C61—42no—
97C23×C61—44no—
98C23×C61—48no—
99C23×C61—48no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—88no—
103C23×C61—88no—
104C23×C61—88no—
105C23×C61#9912yes—
106C23×C61C2×C22⋊C411yescharacteristic
107C23×C61#12014yes—
108Q8×C61—44no—
109D4×C61—44no—
110D4×C61—44no—
111D4×C61—48no—
112D4×C61—44no—
113D4×C61—48no—
114D4×C61—816no—
115C2×C4×C61—28no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—24no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—48no—
125C2×C4×C61—44no—
126C2×C4×C61—84no—
127C2×C4×C61—88no—
128C2×C4×C61#9912yes—
129C2×C4×C61C2×C22⋊C412yes—
130#12697C6111yesmaximal, characteristic, direct product factor, Sylow
131C22×C61—22no—
132C22×C61—22no—
133C22×C61—24no—
134C22×C61—44no—
135C22×C61—42no—
136C22×C61—42no—
137C22×C61—48no—
138C22×C61—44no—
139C22×C61—42no—
140C22×C61—44no—
141C22×C61—82no—
142C22×C61—84no—
143C22×C61—816no—
144C22×C61—84no—
145C22×C61#37911yescharacteristic
146C22×C61C2×#9911yescharacteristic
147C22×C61C2×#9911yescharacteristic
148C4×C61—22no—
149C4×C61—22no—
150C4×C61—44no—
151C4×C61—42no—
152C4×C61—164no—
153C2×#535C2×C6111yescharacteristic
154C2×#353C2×C6114yescomplemented
155#1782C2×C6112yescomplemented
156C2×C61—21no—
157C2×C61—21no—
158C2×C61—44no—
159C2×C61—42no—
160C2×C61—84no—
161C2×C61#178111yescharacteristic
162C2×C61#212812yes—
163C22×#41C22×C6111yescharacteristic
164C23×D4C4×C6111yescharacteristic, complemented
165#535—28no—
166C2×#127C22×C6112yes—
167C2×#124C4×C6112yescomplemented
168C2×#121—22no—
169C2×#120—22no—
170C2×#117C4×C6111yescharacteristic, complemented
171C2×#99—22no—
172C2×#99C22×C6114yescomplemented
173#381—22no—
174#353—28no—
175C61#1269711yescharacteristic, direct product factor, Sylow
176C25—21no—
177C2×#41—24no—
178C2×#41—42no—
179C2×#41—42no—
180C22×Q8C2×C4×C6111yescharacteristic
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4C2×C4×C6112yes—
187C22×D4D4×C6112yes—
188C22×D4C23×C6111yescharacteristic
189C23×C4C2×C4×C6111yescharacteristic
190C23×C4D4×C6112yes—
191#127—24no—
192#127D4×C6118yes—
193#124—24no—
194#124—24no—
195#121—216no—
196#120—216no—
197#117—24no—
198C2×C4⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C42C2×C4×C6112yes—
203#99—28no—
204#99—44no—
205#95—44no—
206C24—24no—
207C24—22no—
208C24—42no—
209C24—42no—
210C24—84no—
211C24C22⋊C4×C6111yescharacteristic
212C24C2×D4×C6112yes—
213#41—416no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—216no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—416no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—88no—
231C2×D4C22⋊C4×C6114yes—
232C2×D4C2×D4×C6114yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4C22⋊C4×C6112yes—
242C22×C4C22⋊C4×C6111yescharacteristic
243C22×C4C22×C4×C6111yescharacteristic
244C4⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—216no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—22no—
250C42C22⋊C4×C6114yes—
251C23—21no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—24no—
256C23—42no—
257C23—44no—
258C23—48no—
259C23—48no—
260C23—42no—
261C23—42no—
262C23—88no—
263C23—88no—
264C23—88no—
265C23#99×C6112yes—
266C23C2×C22⋊C4×C6111yescharacteristic
267C23#120×C6114yes—
268Q8—44no—
269D4—44no—
270D4—44no—
271D4—48no—
272D4—44no—
273D4—48no—
274D4—816no—
275C2×C4—28no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—84no—
287C2×C4—88no—
288C2×C4#99×C6112yes—
289C2×C4C2×C22⋊C4×C6112yes—
290C22—22no—
291C22—22no—
292C22—24no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22—48no—
297C22—44no—
298C22—42no—
299C22—44no—
300C22—82no—
301C22—84no—
302C22—816no—
303C22—84no—
304C22C61×#37911yescharacteristic
305C22C2×#99×C6111yescharacteristic
306C22C2×#99×C6111yescharacteristic
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—164no—
312C2—21no—
313C2—21no—
314C2—44no—
315C2—42no—
316C2—84no—
317C2C61×#178111yescharacteristic
318C2C61×#212812yes—
subgroupG/H
center, socleC22×C61C2×#99
commutatorC22×C4C22×C4×C61
Frattini, cosocleC22×D4C23×C61
prime radicalC2×C61×#535C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C22×#41×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C61×#353C2semidirect
3. C61×#1782C2semidirect
5. C23×D4×C61C4semidirect
8. C2×#124×C61C4semidirect
11. C2×#117×C61C4semidirect
13. C2×#99×C61C22semidirect
130. #12697C61direct (N × H)
154. C2×#353C2×C61semidirect
155. #1782C2×C61semidirect
164. C23×D4C4×C61semidirect
167. C2×#124C4×C61semidirect
170. C2×#117C4×C61semidirect
172. C2×#99C22×C61semidirect
175. C61#12697direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C61×#5352
C2×C61×#3532
C61×#17822
#1269761