Finite Group Databasebeta

← C61×#12189 #29515622

Subgroups — C61×#12189 #29515622

Order 15616.

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

normal subgroups only

This group has 432 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×#2771C211yesmaximal, characteristic, complemented
2C61×#2022C211yesmaximal, characteristic
3C61×#1613C211yesmaximal, characteristic
4C61×#1607C211yesmaximal, characteristic
5C61×#1352C211yesmaximal, characteristic, complemented
6C61×#1344C211yesmaximal, characteristic, complemented
7C61×#1317C211yesmaximal, characteristic, complemented
8C61×#534C2211yescharacteristic
9C2×#123×C61—21no—
10C61×#481—21no—
11C61×#480—21no—
12C2×#99×C61—21no—
13C61×#400—21no—
14C61×#400—21no—
15C61×#399C2211yescharacteristic
16C61×#397C2211yescharacteristic
17C61×#389—21no—
18C61×#388—21no—
19C61×#387—21no—
20C61×#385—21no—
21C61×#381C2211yescharacteristic
22C61×#342C2211yescharacteristic
23C61×#327—21no—
24C61×#327—21no—
25C61×#323C2211yescharacteristic
26C61×#323C2211yescharacteristic
27C22×D4×C61—21no—
28C23×C4×C61D411yescharacteristic
29#123×C61—42no—
30#123×C61—41no—
31#123×C61—42no—
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—41no—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61—21no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—41no—
46C2×C22⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61C2311yescharacteristic
48C2×C42×C61—21no—
49#107×C61—41no—
50#106×C61—41no—
51#102×C61—41no—
52#102×C61—41no—
53#99×C61—42no—
54C22⋊C8×C61—21no—
55C22⋊C8×C61—21no—
56#95×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—21no—
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—21no—
83C22×C4×C61—21no—
84C22×C4×C61—41no—
85C22×C4×C61—41no—
86C22×C4×C61—41no—
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C2×C8×C61—41no—
90C2×C8×C61—41no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—42no—
94C4⋊C4×C61—42no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C4⋊C4×C61—44no—
98C4⋊C4×C61—81no—
99C4⋊C4×C61—81no—
100C22⋊C4×C61—22no—
101C22⋊C4×C61—41no—
102C22⋊C4×C61—42no—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C22⋊C4×C61—44no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C42×C61—81no—
111C23×C61—21no—
112C23×C61—21no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—81no—
119C23×C61—81no—
120C23×C61#12011yescharacteristic
121C23×C61#13511yescharacteristic
122C23×C61AGL(1,ℤ/8)11yescharacteristic
123D4×C61—81no—
124D4×C61—81no—
125D4×C61—81no—
126D4×C61—81no—
127D4×C61—81no—
128D4×C61—81no—
129C2×C4×C61—22no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
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—
159C8×C61—81no—
160C8×C61—81no—
161#12189C6111yesmaximal, characteristic, direct product factor, Sylow
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—81no—
171C22×C61—82no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61—161no—
176C22×C61#44911yescharacteristic
177C22×C61#45311yescharacteristic
178C22×C61B(4,𝔽2)11yescharacteristic
179C4×C61—42no—
180C4×C61—41no—
181C4×C61—41no—
182C4×C61—81no—
183C4×C61—81no—
184C4×C61—82no—
185C4×C61—81no—
186C4×C61—81no—
187C4×C61—81no—
188C4×C61—81no—
189#2771C2×C6111yescharacteristic, complemented
190#2022C2×C6111yescharacteristic
191#1613C2×C6111yescharacteristic
192#1607C2×C6111yescharacteristic
193#1352C2×C6111yescharacteristic, complemented
194#1344C2×C6111yescharacteristic, complemented
195#1317C2×C6111yescharacteristic, complemented
196C2×C61—21no—
197C2×C61—21no—
198C2×C61—41no—
199C2×C61—41no—
200C2×C61—161no—
201C2×C61#160011yescharacteristic
202C2×C61#219811yescharacteristic
203C2×C61#220111yescharacteristic
204#534C22×C6111yescharacteristic
205C2×#123—21no—
206#481—21no—
207#480—21no—
208C2×#99—21no—
209#400—21no—
210#400—21no—
211#399C22×C6111yescharacteristic
212#397C22×C6111yescharacteristic
213#389—21no—
214#388—21no—
215#387—21no—
216#385—21no—
217#381C22×C6111yescharacteristic
218#342C22×C6111yescharacteristic
219#327—21no—
220#327—21no—
221#323C22×C6111yescharacteristic
222#323C22×C6111yescharacteristic
223C61#1218911yescharacteristic, direct product factor, Sylow
224C22×D4—21no—
225C23×C4D4×C6111yescharacteristic
226#123—42no—
227#123—41no—
228#123—42no—
229#121—21no—
230#121—21no—
231#120—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4D4×C6111yescharacteristic
238C2×C4⋊C4D4×C6111yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4D4×C6111yescharacteristic
244C2×C22⋊C4C23×C6111yescharacteristic
245C2×C42—21no—
246#107—41no—
247#106—41no—
248#102—41no—
249#102—41no—
250#99—42no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95D4×C6111yescharacteristic
261#95D4×C6111yescharacteristic
262C24—21no—
263C24C2×D4×C6111yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4C2×D4×C6111yescharacteristic
285C22×C4C2×D4×C6111yescharacteristic
286C2×C8—41no—
287C2×C8—41no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C22⋊C4—22no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—44no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—81no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—81no—
316C23—81no—
317C23#120×C6111yescharacteristic
318C23#135×C6111yescharacteristic
319C23AGL(1,ℤ/8)×C6111yescharacteristic
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324D4—81no—
325D4—81no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
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—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—81no—
354C2×C4—81no—
355C2×C4—81no—
356C8—81no—
357C8—81no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22—82no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—161no—
372C22C61×#44911yescharacteristic
373C22C61×#45311yescharacteristic
374C22C61×B(4,𝔽2)11yescharacteristic
375C4—42no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—82no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—161no—
390C2C61×#160011yescharacteristic
391C2C61×#219811yescharacteristic
392C2C61×#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×#2771C2semidirect
5. C61×#1352C2semidirect
6. C61×#1344C2semidirect
7. C61×#1317C2semidirect
161. #12189C61direct (N × H)
189. #2771C2×C61semidirect
193. #1352C2×C61semidirect
194. #1344C2×C61semidirect
195. #1317C2×C61semidirect
223. C61#12189direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#27712
C61×#20222
C61×#16132
C61×#16072
C61×#13522
C61×#13442
C61×#13172
#1218961