Finite Group Databasebeta

← C61×#12240 #29515673

Subgroups — C61×#12240 #29515673

Order 15616.

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

normal subgroups only

This group has 350 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×#2792C211yesmaximal, characteristic
2C61×#2060C211yesmaximal, characteristic
3C61×#1611C211yesmaximal, characteristic
4C61×#1605C211yesmaximal, characteristic
5C61×#1354C211yesmaximal, characteristic
6C61×#1346C211yesmaximal, characteristic
7C61×#1325C211yesmaximal, characteristic
8C61×#544C2211yescharacteristic
9C61×#483—21no—
10C61×#451—21no—
11C61×#428—21no—
12C61×#398—21no—
13C61×#398C2211yescharacteristic
14C61×#395—21no—
15C61×#395C2211yescharacteristic
16C61×#389—21no—
17C61×#389—21no—
18C61×#387—21no—
19C61×#387—21no—
20C61×#384C2211yescharacteristic
21C61×#343C2211yescharacteristic
22C61×#339—21no—
23C61×#328—21no—
24C61×#328—21no—
25C61×#324C2211yescharacteristic
26C61×#324C2211yescharacteristic
27C22×Q8×C61D411yescharacteristic
28C2×Q16×C61—41no—
29C2×M16×C61—21no—
30C4⋊Q8×C61—41no—
31#125×C61—41no—
32C22⋊Q8×C61—21no—
33C22⋊Q8×C61—21no—
34#117×C61—21no—
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×C61—21no—
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C4⋊C4×C61D411yescharacteristic
44C2×C4⋊C4×C61C2311yescharacteristic
45C2×C42×C61—21no—
46C2×C42×C61D411yescharacteristic
47#107×C61—41no—
48#106×C61—41no—
49#106×C61—41no—
50#103×C61—41no—
51#103×C61—41no—
52C22⋊C8×C61—21no—
53C22⋊C8×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61D411yescharacteristic
61#95×C61D411yescharacteristic
62C2×Q8×C61—21no—
63C2×Q8×C61—21no—
64C2×Q8×C61—41no—
65C2×Q8×C61—81no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61C2×D411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75Q16×C61—82no—
76M16×C61—22no—
77M16×C61—41no—
78C2×C8×C61—41no—
79C2×C8×C61—41no—
80C2×C8×C61—41no—
81C4⋊C4×C61—24no—
82C4⋊C4×C61—21no—
83C4⋊C4×C61—21no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C4⋊C4×C61—81no—
94C4⋊C4×C61—81no—
95C4⋊C4×C61—81no—
96C4⋊C4×C61—81no—
97C4⋊C4×C61—81no—
98C22⋊C4×C61—41no—
99C42×C61—21no—
100C42×C61—21no—
101C42×C61—41no—
102C42×C61—81no—
103C23×C61#12011yescharacteristic
104Q8×C61—42no—
105Q8×C61—81no—
106Q8×C61—81no—
107Q8×C61—81no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—22no—
111C2×C4×C61—22no—
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61#13511yescharacteristic
135C2×C4×C61#13711yescharacteristic
136C8×C61—42no—
137C8×C61—81no—
138C8×C61—81no—
139#12240C6111yesmaximal, characteristic, direct product factor, Sylow
140C22×C61—21no—
141C22×C61—21no—
142C22×C61#45211yescharacteristic
143C22×C61#45611yescharacteristic
144C22×C61B(4,𝔽2)11yescharacteristic
145C4×C61—21no—
146C4×C61—21no—
147C4×C61—42no—
148C4×C61—42no—
149C4×C61—42no—
150C4×C61—41no—
151C4×C61—41no—
152C4×C61—81no—
153C4×C61—81no—
154C4×C61—81no—
155C4×C61—81no—
156C4×C61—81no—
157#2792C2×C6111yescharacteristic
158#2060C2×C6111yescharacteristic
159#1611C2×C6111yescharacteristic
160#1605C2×C6111yescharacteristic
161#1354C2×C6111yescharacteristic
162#1346C2×C6111yescharacteristic
163#1325C2×C6111yescharacteristic
164C2×C61—21no—
165C2×C61—21no—
166C2×C61#160311yescharacteristic
167C2×C61#220011yescharacteristic
168C2×C61#220511yescharacteristic
169#544C22×C6111yescharacteristic
170#483—21no—
171#451—21no—
172#428—21no—
173#398—21no—
174#398C22×C6111yescharacteristic
175#395—21no—
176#395C22×C6111yescharacteristic
177#389—21no—
178#389—21no—
179#387—21no—
180#387—21no—
181#384C22×C6111yescharacteristic
182#343C22×C6111yescharacteristic
183#339—21no—
184#328—21no—
185#328—21no—
186#324C22×C6111yescharacteristic
187#324C22×C6111yescharacteristic
188C61#1224011yescharacteristic, direct product factor, Sylow
189C22×Q8D4×C6111yescharacteristic
190C2×Q16—41no—
191C2×M16—21no—
192C4⋊Q8—41no—
193#125—41no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196#117—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4D4×C6111yescharacteristic
205C2×C4⋊C4D4×C6111yescharacteristic
206C2×C4⋊C4C23×C6111yescharacteristic
207C2×C42—21no—
208C2×C42D4×C6111yescharacteristic
209#107—41no—
210#106—41no—
211#106—41no—
212#103—41no—
213#103—41no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95D4×C6111yescharacteristic
223#95D4×C6111yescharacteristic
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—81no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4C2×D4×C6111yescharacteristic
235C22×C4C2×D4×C6111yescharacteristic
236C22×C4C2×D4×C6111yescharacteristic
237Q16—82no—
238M16—22no—
239M16—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C4⋊C4—24no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C22⋊C4—41no—
261C42—21no—
262C42—21no—
263C42—41no—
264C42—81no—
265C23#120×C6111yescharacteristic
266Q8—42no—
267Q8—81no—
268Q8—81no—
269Q8—81no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4#135×C6111yescharacteristic
297C2×C4#137×C6111yescharacteristic
298C8—42no—
299C8—81no—
300C8—81no—
301C22—21no—
302C22—21no—
303C22C61×#45211yescharacteristic
304C22C61×#45611yescharacteristic
305C22C61×B(4,𝔽2)11yescharacteristic
306C4—21no—
307C4—21no—
308C4—42no—
309C4—42no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4—81no—
318C2—21no—
319C2—21no—
320C2C61×#160311yescharacteristic
321C2C61×#220011yescharacteristic
322C2C61×#220511yescharacteristic
subgroupG/H
center, socleC22×C61B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C61
prime radicalC23×C61#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C61 ◁ C2×C4⋊C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
139. #12240C61direct (N × H)
188. C61#12240direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#27922
C61×#20602
C61×#16112
C61×#16052
C61×#13542
C61×#13462
C61×#13252
#1224061