Finite Group Databasebeta

← C61×#13224 #29516657

Subgroups — C61×#13224 #29516657

Order 15616.

552 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2547C211yesmaximal, characteristic
2C2×C61×#354C214yesmaximal
3C61×#1799C211yesmaximal, characteristic
4C4×#99×C61C211yesmaximal, characteristic
5C2×C4⋊Q8×C61C411yescharacteristic, complemented
6C2×#124×C61C2212yes—
7C2×C4×D4×C61C2211yescharacteristic
8C2×#99×C61C2212yes—
9C2×#99×C61C2212yes—
10C61×#397—21no—
11C61×#397—21no—
12C61×#391C412yescomplemented
13C61×#380—21no—
14C61×#380—21no—
15C61×#380—21no—
16C4×C4⋊C4×C61C411yescharacteristic, complemented
17C4×C22⋊C4×C61—21no—
18C61×#354—28no—
19C61×#342—21no—
20C61×#342—21no—
21C22×Q8×C61C2×C412yes—
22C22×D4×C61C2311yescharacteristic
23C23×C4×C61D411yescharacteristic
24C23×C4×C61D411yescharacteristic
25C4⋊Q8×C61—41no—
26C4⋊Q8×C61—41no—
27#124×C61—24no—
28#124×C61—24no—
29C4×D4×C61—41no—
30C4×D4×C61—41no—
31C2×C4⋊C4×C61—21no—
32C2×C4⋊C4×C61—21no—
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61—41no—
35C2×C4⋊C4×C61C2×C411yescharacteristic
36C2×C22⋊C4×C61—22no—
37C2×C22⋊C4×C61—22no—
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61D411yescharacteristic
41C2×C22⋊C4×C61D411yescharacteristic
42C2×C42×C61—41no—
43C2×C42×C61C2×C412yes—
44C2×C42×C61C2×C411yescharacteristic
45#99×C61—22no—
46#99×C61—24no—
47#99×C61D414yes—
48#95×C61—21no—
49#95×C61—22no—
50#95×C61—22no—
51#95×C61—41no—
52#95×C61—41no—
53C24×C61C2×D411yescharacteristic
54C24×C61C2×D411yescharacteristic
55C2×Q8×C61—24no—
56C2×Q8×C61—42no—
57C2×Q8×C61—42no—
58C2×D4×C61—41no—
59C2×D4×C61—41no—
60C2×D4×C61C2×D412yes—
61C2×D4×C61#4112yes—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C4⋊C4×C61—41no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—42no—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—41no—
82C4⋊C4×C61—82no—
83C22⋊C4×C61—24no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—44no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—42no—
90C22⋊C4×C61—42no—
91C22⋊C4×C61—42no—
92C42×C61—21no—
93C42×C61—21no—
94C42×C61—24no—
95C42×C61—82no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61#12112yes—
106C23×C61#12411yescharacteristic
107C23×C61#12711yescharacteristic
108Q8×C61—84no—
109D4×C61—42no—
110D4×C61—42no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—22no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—44no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—81no—
130C2×C4×C61—81no—
131C2×C4×C61—82no—
132C2×C4×C61—82no—
133C2×C4×C61—82no—
134C2×C4×C61—82no—
135C2×C4×C61—82no—
136C2×C4×C61#9911yescharacteristic
137C2×C4×C61#9911yescharacteristic
138C2×C4×C61C4×D412yes—
139#13224C6111yesmaximal, characteristic, direct product factor, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—22no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61—42no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—81no—
153C22×C61—81no—
154C22×C61#39011yescharacteristic
155C22×C61C2×#9911yescharacteristic
156C22×C61#41011yescharacteristic
157C4×C61—21no—
158C4×C61—22no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—41no—
162C4×C61—82no—
163C4×C61—84no—
164C4×C61—82no—
165C4×C61—162no—
166C4×C61#35412yescomplemented
167#2547C2×C6111yescharacteristic
168C2×#354C2×C6114yes—
169#1799C2×C6111yescharacteristic
170C4×#99C2×C6111yescharacteristic
171C2×C61—21no—
172C2×C61—21no—
173C2×C61—42no—
174C2×C61—41no—
175C2×C61—41no—
176C2×C61#196411yescharacteristic
177C2×C61C2×#35411yescharacteristic
178C2×C61#212911yescharacteristic
179C2×C4⋊Q8C4×C6111yescharacteristic, complemented
180C2×#124C22×C6112yes—
181C2×C4×D4C22×C6111yescharacteristic
182C2×#99C22×C6112yes—
183C2×#99C22×C6112yes—
184#397—21no—
185#397—21no—
186#391C4×C6112yescomplemented
187#380—21no—
188#380—21no—
189#380—21no—
190C4×C4⋊C4C4×C6111yescharacteristic, complemented
191C4×C22⋊C4—21no—
192#354—28no—
193#342—21no—
194#342—21no—
195C61#1322411yescharacteristic, direct product factor, Sylow
196C22×Q8C2×C4×C6112yes—
197C22×D4C23×C6111yescharacteristic
198C23×C4D4×C6111yescharacteristic
199C23×C4D4×C6111yescharacteristic
200C4⋊Q8—41no—
201C4⋊Q8—41no—
202#124—24no—
203#124—24no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4C2×C4×C6111yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4D4×C6111yescharacteristic
216C2×C22⋊C4D4×C6111yescharacteristic
217C2×C42—41no—
218C2×C42C2×C4×C6112yes—
219C2×C42C2×C4×C6111yescharacteristic
220#99—22no—
221#99—24no—
222#99D4×C6114yes—
223#95—21no—
224#95—22no—
225#95—22no—
226#95—41no—
227#95—41no—
228C24C2×D4×C6111yescharacteristic
229C24C2×D4×C6111yescharacteristic
230C2×Q8—24no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4C2×D4×C6112yes—
236C2×D4#41×C6112yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4C22⋊C4×C6111yescharacteristic
248C22×C4C22⋊C4×C6111yescharacteristic
249C22×C4C22⋊C4×C6111yescharacteristic
250C22×C4C22⋊C4×C6111yescharacteristic
251C22×C4C22×C4×C6111yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—82no—
258C22⋊C4—24no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C42—21no—
268C42—21no—
269C42—24no—
270C42—82no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×C22⋊C4×C6111yescharacteristic
280C23#121×C6112yes—
281C23#124×C6111yescharacteristic
282C23#127×C6111yescharacteristic
283Q8—84no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4#99×C6111yescharacteristic
312C2×C4#99×C6111yescharacteristic
313C2×C4C4×D4×C6112yes—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22C61×#39011yescharacteristic
329C22C2×#99×C6111yescharacteristic
330C22C61×#41011yescharacteristic
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—84no—
338C4—82no—
339C4—162no—
340C4C61×#35412yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—41no—
345C2—41no—
346C2C61×#196411yescharacteristic
347C2C2×C61×#35411yescharacteristic
348C2C61×#212911yescharacteristic
subgroupG/H
center, socleC22×C61C2×#99
commutatorC22×C4C22×C4×C61
Frattini, cosocleC22×D4C23×C61
prime radicalC22×D4×C61C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×C4×D4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8×C61C4semidirect
12. C61×#391C4semidirect
16. C4×C4⋊C4×C61C4semidirect
139. #13224C61direct (N × H)
166. C4×C61#354semidirect
179. C2×C4⋊Q8C4×C61semidirect
186. #391C4×C61semidirect
190. C4×C4⋊C4C4×C61semidirect
195. C61#13224direct (N × H)
340. C4C61×#354semidirect

Maximal subgroups

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

subgroupindex
C61×#25472
C2×C61×#3542
C61×#17992
C4×#99×C612
#1322461