Finite Group Databasebeta

← C61×#12150 #29515583

Subgroups — C61×#12150 #29515583

Order 15616.

432 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2722C211yesmaximal, characteristic, complemented
2C61×#2044C211yesmaximal, characteristic
3C61×#1607C211yesmaximal, characteristic
4C61×#1607C211yesmaximal, characteristic
5C61×#1344C211yesmaximal, characteristic, complemented
6C61×#1344C211yesmaximal, characteristic, complemented
7C61×#1326C211yesmaximal, characteristic, complemented
8C61×#534C2211yescharacteristic
9C2×#126×C61—21no—
10C61×#480—21no—
11C61×#480—21no—
12C2×#99×C61—21no—
13C61×#398—21no—
14C61×#398—21no—
15C61×#397C2211yescharacteristic
16C61×#397C2211yescharacteristic
17C61×#389—21no—
18C61×#387—21no—
19C61×#386—21no—
20C61×#385C2211yescharacteristic
21C61×#380—21no—
22C61×#342C2211yescharacteristic
23C61×#327—21no—
24C61×#327—21no—
25C61×#323C2211yescharacteristic
26C61×#323C2211yescharacteristic
27C22×D4×C61—21no—
28C23×C4×C61D411yescharacteristic
29#126×C61—42no—
30#126×C61—42no—
31#123×C61—41no—
32#121×C61—21no—
33#121×C61—21no—
34#120×C61—41no—
35C2×C4⋊C4×C61—21no—
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—41no—
47C2×C22⋊C4×C61D411yescharacteristic
48C2×C22⋊C4×C61C2311yescharacteristic
49C2×C42×C61—21no—
50#107×C61—41no—
51#107×C61—41no—
52#102×C61—41no—
53#102×C61—41no—
54#99×C61—42no—
55C22⋊C8×C61—21no—
56C22⋊C8×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61—21no—
62#95×C61—41no—
63#95×C61D411yescharacteristic
64#95×C61D411yescharacteristic
65C24×C61—21no—
66C24×C61C2×D411yescharacteristic
67C2×D4×C61—21no—
68C2×D4×C61—21no—
69C2×D4×C61—41no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
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—42no—
83C22×C4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C2×C8×C61—41no—
88C2×C8×C61—41no—
89C4⋊C4×C61—21no—
90C4⋊C4×C61—21no—
91C4⋊C4×C61—42no—
92C4⋊C4×C61—42no—
93C4⋊C4×C61—42no—
94C4⋊C4×C61—42no—
95C4⋊C4×C61—42no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—81no—
99C22⋊C4×C61—21no—
100C22⋊C4×C61—41no—
101C22⋊C4×C61—41no—
102C22⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—82no—
110C22⋊C4×C61D812yes—
111C42×C61—42no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—81no—
120C23×C61—81no—
121C23×C61#12011yescharacteristic
122C23×C61C2×D811yescharacteristic
123C23×C61AGL(1,ℤ/8)11yescharacteristic
124D4×C61—81no—
125D4×C61—81no—
126D4×C61—81no—
127D4×C61—81no—
128D4×C61—81no—
129D4×C61—81no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
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—42no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
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—41no—
155C2×C4×C61—41no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C8×C61—81no—
161C8×C61—81no—
162#12150C6111yesmaximal, characteristic, direct product factor, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—81no—
172C22×C61—82no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—161no—
177C22×C61#44711yescharacteristic
178C22×C61#45411yescharacteristic
179C22×C61B(4,𝔽2)11yescharacteristic
180C4×C61—42no—
181C4×C61—41no—
182C4×C61—42no—
183C4×C61—41no—
184C4×C61—81no—
185C4×C61—81no—
186C4×C61—81no—
187C4×C61—81no—
188C4×C61—81no—
189C4×C61—161no—
190#2722C2×C6111yescharacteristic, complemented
191#2044C2×C6111yescharacteristic
192#1607C2×C6111yescharacteristic
193#1607C2×C6111yescharacteristic
194#1344C2×C6111yescharacteristic, complemented
195#1344C2×C6111yescharacteristic, complemented
196#1326C2×C6111yescharacteristic, complemented
197C2×C61—21no—
198C2×C61—21no—
199C2×C61—41no—
200C2×C61—41no—
201C2×C61—161no—
202C2×C61#159811yescharacteristic
203C2×C61#219711yescharacteristic
204C2×C61#220111yescharacteristic
205#534C22×C6111yescharacteristic
206C2×#126—21no—
207#480—21no—
208#480—21no—
209C2×#99—21no—
210#398—21no—
211#398—21no—
212#397C22×C6111yescharacteristic
213#397C22×C6111yescharacteristic
214#389—21no—
215#387—21no—
216#386—21no—
217#385C22×C6111yescharacteristic
218#380—21no—
219#342C22×C6111yescharacteristic
220#327—21no—
221#327—21no—
222#323C22×C6111yescharacteristic
223#323C22×C6111yescharacteristic
224C61#1215011yescharacteristic, direct product factor, Sylow
225C22×D4—21no—
226C23×C4D4×C6111yescharacteristic
227#126—42no—
228#126—42no—
229#123—41no—
230#121—21no—
231#121—21no—
232#120—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4D4×C6111yescharacteristic
240C2×C4⋊C4D4×C6111yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4D4×C6111yescharacteristic
246C2×C22⋊C4C23×C6111yescharacteristic
247C2×C42—21no—
248#107—41no—
249#107—41no—
250#102—41no—
251#102—41no—
252#99—42no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—41no—
261#95D4×C6111yescharacteristic
262#95D4×C6111yescharacteristic
263C24—21no—
264C24C2×D4×C6111yescharacteristic
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4C2×D4×C6111yescharacteristic
284C22×C4C2×D4×C6111yescharacteristic
285C2×C8—41no—
286C2×C8—41no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—81no—
297C22⋊C4—21no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—82no—
308C22⋊C4D8×C6112yes—
309C42—42no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—81no—
318C23—81no—
319C23#120×C6111yescharacteristic
320C23C2×D8×C6111yescharacteristic
321C23AGL(1,ℤ/8)×C6111yescharacteristic
322D4—81no—
323D4—81no—
324D4—81no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
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—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C8—81no—
359C8—81no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22—82no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—161no—
374C22C61×#44711yescharacteristic
375C22C61×#45411yescharacteristic
376C22C61×B(4,𝔽2)11yescharacteristic
377C4—42no—
378C4—41no—
379C4—42no—
380C4—41no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—161no—
387C2—21no—
388C2—21no—
389C2—41no—
390C2—41no—
391C2—161no—
392C2C61×#159811yescharacteristic
393C2C61×#219711yescharacteristic
394C2C61×#220111yescharacteristic
subgroupG/H
center, socleC22×C61B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C61
prime radicalC61×#534C22
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
1. C61×#2722C2semidirect
5. C61×#1344C2semidirect
6. C61×#1344C2semidirect
7. C61×#1326C2semidirect
162. #12150C61direct (N × H)
190. #2722C2×C61semidirect
194. #1344C2×C61semidirect
195. #1344C2×C61semidirect
196. #1326C2×C61semidirect
224. C61#12150direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#27222
C61×#20442
C61×#16072
C61×#13442
C61×#13262
#1215061