Finite Group Databasebeta

← C61×#9498 #29512931

Subgroups — C61×#9498 #29512931

Order 15616.

900 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2405C211yesmaximal, characteristic
2C61×#2312C211yesmaximal, characteristic, complemented
3C61×#1819C211yesmaximal, characteristic, complemented
4C61×#1781C212yesmaximal
5C61×#1304C212yesmaximal, complemented
6C23×D4×C61C2211yescharacteristic
7C2×#123×C61—21no—
8C22×C22⋊C4×C61C2211yescharacteristic
9C2×#99×C61—24no—
10C61×#394—22no—
11C61×#388—24no—
12C61×#386—21no—
13C61×#385C412yescomplemented
14C61×#380—21no—
15C61×#379C412yescomplemented
16C61×#379C2212yes—
17C61×#379C2211yescharacteristic
18C61×#379C2212yes—
19C4×C22⋊C4×C61—21no—
20C61×#342—22no—
21C25×C61C2×C411yescharacteristic
22C25×C61C2311yescharacteristic
23C22×D4×C61—21no—
24C22×D4×C61—22no—
25C22×D4×C61—21no—
26C22×D4×C61—22no—
27C22×D4×C61—44no—
28C23×C4×C61C2×C411yescharacteristic
29C23×C4×C61D411yescharacteristic
30C23×C4×C61D411yescharacteristic
31#123×C61—44no—
32C2×C4⋊C4×C61—22no—
33C2×C4⋊C4×C61—22no—
34C2×C22⋊C4×C61—22no—
35C2×C22⋊C4×C61—22no—
36C2×C22⋊C4×C61—22no—
37C2×C22⋊C4×C61—21no—
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—22no—
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61—22no—
45C2×C22⋊C4×C61C2×C412yes—
46C2×C22⋊C4×C61C2×C412yes—
47C2×C22⋊C4×C61D412yes—
48C2×C22⋊C4×C61D414yes—
49C2×C42×C61—22no—
50#99×C61—48no—
51#95×C61—22no—
52#95×C61—22no—
53#95×C61—22no—
54C24×C61—21no—
55C24×C61—22no—
56C24×C61—42no—
57C24×C61—42no—
58C24×C61—42no—
59C24×C61—42no—
60C24×C61—81no—
61C24×C61—81no—
62C24×C61C22×C411yescharacteristic
63C24×C61C2×D411yescharacteristic
64C24×C61C2×D411yescharacteristic
65C24×C61C2×D412yes—
66C24×C61#4112yes—
67C2×D4×C61—28no—
68C2×D4×C61—24no—
69C2×D4×C61—42no—
70C2×D4×C61—44no—
71C2×D4×C61—42no—
72C2×D4×C61—44no—
73C2×D4×C61—42no—
74C2×D4×C61—44no—
75C2×D4×C61—41no—
76C2×D4×C61—42no—
77C2×D4×C61—41no—
78C22×C4×C61—21no—
79C22×C4×C61—22no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—21no—
84C22×C4×C61—22no—
85C22×C4×C61—24no—
86C22×C4×C61—42no—
87C22×C4×C61—42no—
88C22×C4×C61—42no—
89C22×C4×C61—81no—
90C22×C4×C61C22⋊C412yes—
91C22×C4×C61C22⋊C412yes—
92C4⋊C4×C61—44no—
93C4⋊C4×C61—82no—
94C22⋊C4×C61—24no—
95C22⋊C4×C61—24no—
96C22⋊C4×C61—44no—
97C22⋊C4×C61—44no—
98C22⋊C4×C61—42no—
99C22⋊C4×C61—44no—
100C22⋊C4×C61—42no—
101C22⋊C4×C61—44no—
102C22⋊C4×C61—44no—
103C22⋊C4×C61—82no—
104C22⋊C4×C61—82no—
105C22⋊C4×C61—81no—
106C22⋊C4×C61—82no—
107C42×C61—82no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61—24no—
113C23×C61—22no—
114C23×C61—24no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—42no—
119C23×C61—41no—
120C23×C61—42no—
121C23×C61—41no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61—42no—
125C23×C61—42no—
126C23×C61—41no—
127C23×C61—42no—
128C23×C61—41no—
129C23×C61—42no—
130C23×C61—82no—
131C23×C61—82no—
132C23×C61—88no—
133C23×C61—88no—
134C23×C61C2×C22⋊C411yescharacteristic
135C23×C61C4×D412yes—
136C23×C61#12011yescharacteristic
137C23×C61#12112yes—
138C23×C61#12311yescharacteristic
139D4×C61—48no—
140D4×C61—44no—
141D4×C61—44no—
142C2×C4×C61—22no—
143C2×C4×C61—24no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—42no—
153C2×C4×C61—44no—
154C2×C4×C61—42no—
155C2×C4×C61—44no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—44no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61—82no—
162C2×C4×C61—82no—
163C2×C4×C61—82no—
164#9498C6111yesmaximal, characteristic, direct product factor, Sylow
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—22no—
170C22×C61—22no—
171C22×C61—21no—
172C22×C61—22no—
173C22×C61—22no—
174C22×C61—21no—
175C22×C61—22no—
176C22×C61—21no—
177C22×C61—22no—
178C22×C61—44no—
179C22×C61—44no—
180C22×C61—44no—
181C22×C61—42no—
182C22×C61—84no—
183C22×C61—82no—
184C22×C61—82no—
185C22×C61—84no—
186C22×C61—84no—
187C22×C61—84no—
188C22×C61—81no—
189C22×C61—81no—
190C22×C61#38611yescharacteristic
191C22×C61#41012yes—
192C22×C61B(4,𝔽2)12yes—
193C4×C61—42no—
194C4×C61—42no—
195C4×C61—42no—
196C4×C61—84no—
197C4×C61—82no—
198C4×C61—162no—
199#2405C2×C6111yescharacteristic
200#2312C2×C6111yescharacteristic, complemented
201#1819C2×C6111yescharacteristic, complemented
202#1781C2×C6112yes—
203#1304C2×C6112yescomplemented
204C2×C61—22no—
205C2×C61—22no—
206C2×C61—42no—
207C2×C61—41no—
208C2×C61—42no—
209C2×C61—82no—
210C2×C61#188911yescharacteristic
211C2×C61#189911yescharacteristic
212C2×C61#191311yescharacteristic
213C23×D4C22×C6111yescharacteristic
214C2×#123—21no—
215C22×C22⋊C4C22×C6111yescharacteristic
216C2×#99—24no—
217#394—22no—
218#388—24no—
219#386—21no—
220#385C4×C6112yescomplemented
221#380—21no—
222#379C4×C6112yescomplemented
223#379C22×C6112yes—
224#379C22×C6111yescharacteristic
225#379C22×C6112yes—
226C4×C22⋊C4—21no—
227#342—22no—
228C61#949811yescharacteristic, direct product factor, Sylow
229C25C2×C4×C6111yescharacteristic
230C25C23×C6111yescharacteristic
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—44no—
236C23×C4C2×C4×C6111yescharacteristic
237C23×C4D4×C6111yescharacteristic
238C23×C4D4×C6111yescharacteristic
239#123—44no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4C2×C4×C6112yes—
254C2×C22⋊C4C2×C4×C6112yes—
255C2×C22⋊C4D4×C6112yes—
256C2×C22⋊C4D4×C6114yes—
257C2×C42—22no—
258#99—48no—
259#95—22no—
260#95—22no—
261#95—22no—
262C24—21no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—81no—
269C24—81no—
270C24C22×C4×C6111yescharacteristic
271C24C2×D4×C6111yescharacteristic
272C24C2×D4×C6111yescharacteristic
273C24C2×D4×C6112yes—
274C24#41×C6112yes—
275C2×D4—28no—
276C2×D4—24no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—41no—
284C2×D4—42no—
285C2×D4—41no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4C22⋊C4×C6112yes—
299C22×C4C22⋊C4×C6112yes—
300C4⋊C4—44no—
301C4⋊C4—82no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—81no—
314C22⋊C4—82no—
315C42—82no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—24no—
321C23—22no—
322C23—24no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—42no—
329C23—41no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23—82no—
339C23—82no—
340C23—88no—
341C23—88no—
342C23C2×C22⋊C4×C6111yescharacteristic
343C23C4×D4×C6112yes—
344C23#120×C6111yescharacteristic
345C23#121×C6112yes—
346C23#123×C6111yescharacteristic
347D4—48no—
348D4—44no—
349D4—44no—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—22no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—22no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—42no—
389C22—84no—
390C22—82no—
391C22—82no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—81no—
396C22—81no—
397C22C61×#38611yescharacteristic
398C22C61×#41012yes—
399C22C61×B(4,𝔽2)12yes—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—84no—
404C4—82no—
405C4—162no—
406C2—22no—
407C2—22no—
408C2—42no—
409C2—42no—
410C2—41no—
411C2—82no—
412C2C61×#188911yescharacteristic
413C2C61×#189911yescharacteristic
414C2C61×#191311yescharacteristic
subgroupG/H
center, socleC22×C61#386
commutatorC24C22×C4×C61
Frattini, cosocleC25C23×C61
prime radicalC23×D4×C61C22
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C25×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2312C2semidirect
3. C61×#1819C2semidirect
5. C61×#1304C2semidirect
13. C61×#385C4semidirect
15. C61×#379C4semidirect
164. #9498C61direct (N × H)
200. #2312C2×C61semidirect
201. #1819C2×C61semidirect
203. #1304C2×C61semidirect
220. #385C4×C61semidirect
222. #379C4×C61semidirect
228. C61#9498direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#24052
C61×#23122
C61×#18192
C61×#17812
C61×#13042
#949861