Finite Group Databasebeta

← C61×#12263 #29515696

Subgroups — C61×#12263 #29515696

Order 15616.

438 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2760C211yesmaximal, characteristic
2C61×#2030C211yesmaximal, characteristic, complemented
3C61×#1602C211yesmaximal, characteristic, complemented
4C61×#1596C211yesmaximal, characteristic
5C61×#1351C211yesmaximal, characteristic, complemented
6C61×#1343C211yesmaximal, characteristic
7C61×#1317C211yesmaximal, characteristic, complemented
8C61×#543C2211yescharacteristic
9C61×#451—21no—
10C61×#450—21no—
11C61×#401—21no—
12C61×#393—21no—
13C61×#393C2211yescharacteristic
14C61×#392C2211yescharacteristic
15C61×#386—21no—
16C61×#386—21no—
17C61×#386—21no—
18C61×#381—21no—
19C61×#381—21no—
20C61×#381C2211yescharacteristic
21C61×#342—21no—
22C61×#342C2211yescharacteristic
23C61×#328—21no—
24C61×#328—21no—
25C61×#323C2211yescharacteristic
26C61×#323C2211yescharacteristic
27C22×Q8×C61D411yescharacteristic
28C22×D4×C61—21no—
29C22×D4×C61D411yescharacteristic
30C23×C4×C61—21no—
31C23×C4×C61D411yescharacteristic
32C2×Q16×C61—41no—
33C2×SD16×C61—41no—
34C22⋊Q8×C61—21no—
35C22⋊Q8×C61—21no—
36C22⋊Q8×C61—41no—
37C22⋊Q8×C61—41no—
38C2×C22⋊C4×C61—21no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61D411yescharacteristic
46C2×C22⋊C4×C61C2311yescharacteristic
47#103×C61—41no—
48#102×C61—41no—
49C22⋊C8×C61—21no—
50C22⋊C8×C61—21no—
51#95×C61—21no—
52#95×C61—21no—
53#95×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—41no—
59#95×C61—41no—
60#95×C61D411yescharacteristic
61#95×C61D411yescharacteristic
62C24×C61—21no—
63C24×C61C2×D411yescharacteristic
64C2×Q8×C61—21no—
65C2×Q8×C61—21no—
66C2×Q8×C61—41no—
67C2×Q8×C61—81no—
68C2×D4×C61—41no—
69C2×D4×C61—42no—
70C2×D4×C61—42no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—81no—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61—42no—
86C22×C4×C61—41no—
87C22×C4×C61—81no—
88C22×C4×C61—81no—
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91Q16×C61—82no—
92SD16×C61—82no—
93C2×C8×C61—41no—
94C2×C8×C61—41no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C22⋊C4×C61—41no—
101C22⋊C4×C61—41no—
102C22⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—81no—
106C22⋊C4×C61—81no—
107C22⋊C4×C61—81no—
108C22⋊C4×C61—81no—
109C22⋊C4×C61—81no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—22no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61#12011yescharacteristic
122C23×C61#13511yescharacteristic
123C23×C61#13711yescharacteristic
124Q8×C61—42no—
125Q8×C61—81no—
126Q8×C61—81no—
127Q8×C61—81no—
128D4×C61—42no—
129D4×C61—81no—
130D4×C61—82no—
131D4×C61—82no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—42no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—84no—
155C2×C4×C61—81no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C8×C61—81no—
162C8×C61—81no—
163#12263C6111yesmaximal, characteristic, direct product factor, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—42no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61#45211yescharacteristic
181C22×C61#45511yescharacteristic
182C22×C61B(4,𝔽2)11yescharacteristic
183C4×C61—41no—
184C4×C61—41no—
185C4×C61—81no—
186C4×C61—81no—
187C4×C61—81no—
188C4×C61—81no—
189C4×C61—81no—
190C4×C61—81no—
191C4×C61—81no—
192C4×C61—161no—
193#2760C2×C6111yescharacteristic
194#2030C2×C6111yescharacteristic, complemented
195#1602C2×C6111yescharacteristic, complemented
196#1596C2×C6111yescharacteristic
197#1351C2×C6111yescharacteristic, complemented
198#1343C2×C6111yescharacteristic
199#1317C2×C6111yescharacteristic, complemented
200C2×C61—21no—
201C2×C61—21no—
202C2×C61—41no—
203C2×C61—41no—
204C2×C61—82no—
205C2×C61#160611yescharacteristic
206C2×C61#219711yescharacteristic
207C2×C61#220011yescharacteristic
208#543C22×C6111yescharacteristic
209#451—21no—
210#450—21no—
211#401—21no—
212#393—21no—
213#393C22×C6111yescharacteristic
214#392C22×C6111yescharacteristic
215#386—21no—
216#386—21no—
217#386—21no—
218#381—21no—
219#381—21no—
220#381C22×C6111yescharacteristic
221#342—21no—
222#342C22×C6111yescharacteristic
223#328—21no—
224#328—21no—
225#323C22×C6111yescharacteristic
226#323C22×C6111yescharacteristic
227C61#1226311yescharacteristic, direct product factor, Sylow
228C22×Q8D4×C6111yescharacteristic
229C22×D4—21no—
230C22×D4D4×C6111yescharacteristic
231C23×C4—21no—
232C23×C4D4×C6111yescharacteristic
233C2×Q16—41no—
234C2×SD16—41no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—41no—
238C22⋊Q8—41no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4D4×C6111yescharacteristic
247C2×C22⋊C4C23×C6111yescharacteristic
248#103—41no—
249#102—41no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261#95D4×C6111yescharacteristic
262#95D4×C6111yescharacteristic
263C24—21no—
264C24C2×D4×C6111yescharacteristic
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—81no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—81no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4C2×D4×C6111yescharacteristic
291C22×C4C2×D4×C6111yescharacteristic
292Q16—82no—
293SD16—82no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23#120×C6111yescharacteristic
323C23#135×C6111yescharacteristic
324C23#137×C6111yescharacteristic
325Q8—42no—
326Q8—81no—
327Q8—81no—
328Q8—81no—
329D4—42no—
330D4—81no—
331D4—82no—
332D4—82no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—84no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C8—81no—
363C8—81no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—42no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22C61×#45211yescharacteristic
381C22C61×#45511yescharacteristic
382C22C61×B(4,𝔽2)11yescharacteristic
383C4—41no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—161no—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—82no—
398C2C61×#160611yescharacteristic
399C2C61×#219711yescharacteristic
400C2C61×#220011yescharacteristic
subgroupG/H
center, socleC22×C61B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C61
prime radicalC61×#392C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C61 ◁ C2×C22⋊C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2030C2semidirect
3. C61×#1602C2semidirect
5. C61×#1351C2semidirect
7. C61×#1317C2semidirect
163. #12263C61direct (N × H)
194. #2030C2×C61semidirect
195. #1602C2×C61semidirect
197. #1351C2×C61semidirect
199. #1317C2×C61semidirect
227. C61#12263direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#27602
C61×#20302
C61×#16022
C61×#15962
C61×#13512
C61×#13432
C61×#13172
#1226361