Finite Group Databasebeta

← C61×#19056 #29522489

Subgroups — C61×#19056 #29522489

Order 15616.

992 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 222 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×#3549C211yesmaximal, characteristic
2C61×#3216C212yesmaximal, complemented
3C61×#3135C211yesmaximal, characteristic, complemented
4C61×#1659C218yesmaximal, complemented
5C61×#1536C211yesmaximal, characteristic, complemented
6C61×#1530C212yesmaximal, complemented
7C61×H(3,𝔽4)C2214yes—
8C61×#555—22no—
9C61×#555—21no—
10D42×C61—22no—
11D42×C61—21no—
12C61×#535—22no—
13C61×#535—24no—
14C61×#533C2211yescharacteristic, complemented
15C61×#532C2212yes—
16C61×#532C2211yescharacteristic
17C61×#497C2214yescomplemented
18C61×#491—21no—
19C61×#490—21no—
20C61×#489C2214yescomplemented
21C61×#489C2214yescomplemented
22C61×#462—22no—
23C61×#460—22no—
24C61×#452—24no—
25C61×#432C2212yescomplemented
26C61×#416—22no—
27C2×#97×C61C2211yescharacteristic, complemented
28C61×#329C2218yescomplemented
29C61×#329C2214yescomplemented
30C2×#41×C61D412yes—
31C2×#41×C61D414yes—
32C22×D4×C61—24no—
33C22×D4×C61—22no—
34C2×Q16×C61—24no—
35C2×SD16×C61—24no—
36C22×C8×C61—21no—
37C4⋊Q8×C61D412yes—
38C4⋊Q8×C61D411yescharacteristic
39C4⋊Q8×C61C2312yes—
40#127×C61—24no—
41#127×C61D412yes—
42#127×C61D411yescharacteristic
43#125×C61—21no—
44#124×C61—24no—
45#124×C61—28no—
46#124×C61C2312yes—
47#124×C61C2314yes—
48C22⋊Q8×C61—22no—
49#121×C61—24no—
50#121×C61—24no—
51#121×C61—24no—
52#121×C61—22no—
53#121×C61—22no—
54#121×C61—22no—
55#120×C61—24no—
56#120×C61—24no—
57#120×C61—28no—
58C4×Q8×C61—22no—
59C4×D4×C61—22no—
60C4×D4×C61—22no—
61C4×D4×C61—22no—
62#117×C61—24no—
63#117×C61—22no—
64#117×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—22no—
67#103×C61—24no—
68#103×C61—24no—
69#102×C61—24no—
70#102×C61—24no—
71C22⋊C8×C61—22no—
72#97×C61—21no—
73#97×C61C2312yes—
74#97×C61C2314yes—
75C24×C61—24no—
76#41×C61—44no—
77#41×C61—84no—
78C2×Q8×C61—22no—
79C2×Q8×C61C2×D412yes—
80C2×Q8×C61C2×D414yes—
81C2×D4×C61—24no—
82C2×D4×C61—28no—
83C2×D4×C61—22no—
84C2×D4×C61—24no—
85C2×D4×C61—22no—
86C2×D4×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—88no—
89C2×D4×C61—84no—
90C2×D4×C61C2×D412yes—
91C2×D4×C61C2×D414yes—
92C22×C4×C61—24no—
93C22×C4×C61—22no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D412yes—
96Q16×C61—84no—
97SD16×C61—84no—
98C2×C8×C61—22no—
99C2×C8×C61—24no—
100C2×C8×C61—42no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—42no—
107C22⋊C4×C61—28no—
108C22⋊C4×C61—24no—
109C22⋊C4×C61—24no—
110C22⋊C4×C61—24no—
111C22⋊C4×C61—24no—
112C22⋊C4×C61—41no—
113C42×C61—24no—
114C42×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D412yes—
117C42×C61C2411yescharacteristic
118C23×C61—22no—
119C23×C61—24no—
120C23×C61—84no—
121C23×C61—88no—
122C23×C61#12011yescharacteristic
123Q8×C61—44no—
124Q8×C61—42no—
125Q8×C61—42no—
126Q8×C61—42no—
127D4×C61—44no—
128D4×C61—42no—
129D4×C61—44no—
130D4×C61—44no—
131D4×C61—44no—
132D4×C61—82no—
133D4×C61—84no—
134D4×C61—84no—
135D4×C61—88no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—24no—
139C2×C4×C61—42no—
140C2×C4×C61—48no—
141C2×C4×C61—44no—
142C2×C4×C61—42no—
143C2×C4×C61#12012yes—
144C2×C4×C61#12011yescharacteristic
145C2×C4×C61C22×D411yescharacteristic
146C2×C4×C61C22×D412yes—
147C8×C61—42no—
148C8×C61—44no—
149#19056C6111yesmaximal, characteristic, direct product factor, Sylow
150C22×C61—21no—
151C22×C61—22no—
152C22×C61—42no—
153C22×C61—44no—
154C22×C61—44no—
155C22×C61—44no—
156C22×C61—44no—
157C22×C61—168no—
158C22×C61C2×#12011yescharacteristic
159C4×C61—21no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61—42no—
163C4×C61—44no—
164C4×C61—82no—
165C4×C61#45512yes—
166#3549C2×C6111yescharacteristic
167#3216C2×C6112yescomplemented
168#3135C2×C6111yescharacteristic, complemented
169#1659C2×C6118yescomplemented
170#1536C2×C6111yescharacteristic, complemented
171#1530C2×C6112yescomplemented
172C2×C61—41no—
173C2×C61—82no—
174C2×C61—84no—
175C2×C61C2×#45511yescharacteristic
176C2×C61#301912yes—
177H(3,𝔽4)C22×C6114yes—
178#555—22no—
179#555—21no—
180D42—22no—
181D42—21no—
182#535—22no—
183#535—24no—
184#533C22×C6111yescharacteristic, complemented
185#532C22×C6111yescharacteristic
186#532C22×C6112yes—
187#497C22×C6114yescomplemented
188#491—21no—
189#490—21no—
190#489C22×C6114yescomplemented
191#489C22×C6114yescomplemented
192#462—22no—
193#460—22no—
194#452—24no—
195#432C22×C6112yescomplemented
196#416—22no—
197C2×#97C22×C6111yescharacteristic, complemented
198#329C22×C6114yescomplemented
199#329C22×C6118yescomplemented
200C61#1905611yescharacteristic, direct product factor, Sylow
201C2×#41D4×C6112yes—
202C2×#41D4×C6114yes—
203C22×D4—24no—
204C22×D4—22no—
205C2×Q16—24no—
206C2×SD16—24no—
207C22×C8—21no—
208C4⋊Q8D4×C6112yes—
209C4⋊Q8D4×C6111yescharacteristic
210C4⋊Q8C23×C6112yes—
211#127—24no—
212#127D4×C6112yes—
213#127D4×C6111yescharacteristic
214#125—21no—
215#124—24no—
216#124—28no—
217#124C23×C6112yes—
218#124C23×C6114yes—
219C22⋊Q8—22no—
220#121—24no—
221#121—24no—
222#121—24no—
223#121—22no—
224#121—22no—
225#121—22no—
226#120—24no—
227#120—24no—
228#120—28no—
229C4×Q8—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233#117—24no—
234#117—22no—
235#117—21no—
236C2×C42C23×C6111yescharacteristic
237C4⋊C8—22no—
238#103—24no—
239#103—24no—
240#102—24no—
241#102—24no—
242C22⋊C8—22no—
243#97—21no—
244#97C23×C6112yes—
245#97C23×C6114yes—
246C24—24no—
247#41—44no—
248#41—84no—
249C2×Q8—22no—
250C2×Q8C2×D4×C6112yes—
251C2×Q8C2×D4×C6114yes—
252C2×D4—24no—
253C2×D4—28no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—88no—
260C2×D4—84no—
261C2×D4C2×D4×C6112yes—
262C2×D4C2×D4×C6114yes—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4C2×D4×C6111yescharacteristic
266C22×C4C2×D4×C6112yes—
267Q16—84no—
268SD16—84no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—42no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—42no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—41no—
284C42—24no—
285C42—41no—
286C42C2×D4×C6111yescharacteristic
287C42C2×D4×C6112yes—
288C42C24×C6111yescharacteristic
289C23—22no—
290C23—24no—
291C23—84no—
292C23—88no—
293C23#120×C6111yescharacteristic
294Q8—44no—
295Q8—42no—
296Q8—42no—
297Q8—42no—
298D4—44no—
299D4—42no—
300D4—44no—
301D4—44no—
302D4—44no—
303D4—82no—
304D4—84no—
305D4—84no—
306D4—88no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—48no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4#120×C6112yes—
315C2×C4#120×C6111yescharacteristic
316C2×C4C22×D4×C6111yescharacteristic
317C2×C4C22×D4×C6112yes—
318C8—42no—
319C8—44no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—168no—
328C22C2×#120×C6111yescharacteristic
329C4—21no—
330C4—22no—
331C4—22no—
332C4—42no—
333C4—44no—
334C4—82no—
335C4C61×#45512yes—
336C2—41no—
337C2—82no—
338C2—84no—
339C2C2×C61×#45511yescharacteristic
340C2C61×#301912yes—
subgroupG/H
center, socleC22×C61C2×#120
commutator, Frattini, cosocleC42C24×C61
prime radicalC61×#3549C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#3216C2semidirect
3. C61×#3135C2semidirect
4. C61×#1659C2semidirect
5. C61×#1536C2semidirect
6. C61×#1530C2semidirect
14. C61×#533C22semidirect
17. C61×#497C22semidirect
20. C61×#489C22semidirect
21. C61×#489C22semidirect
25. C61×#432C22semidirect
27. C2×#97×C61C22semidirect
28. C61×#329C22semidirect
29. C61×#329C22semidirect
149. #19056C61direct (N × H)
167. #3216C2×C61semidirect
168. #3135C2×C61semidirect
169. #1659C2×C61semidirect
170. #1536C2×C61semidirect
171. #1530C2×C61semidirect
184. #533C22×C61semidirect
187. #497C22×C61semidirect
190. #489C22×C61semidirect
191. #489C22×C61semidirect
195. #432C22×C61semidirect
197. C2×#97C22×C61semidirect
198. #329C22×C61semidirect
199. #329C22×C61semidirect
200. C61#19056direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#35492
C61×#32162
C61×#31352
C61×#16592
C61×#15362
C61×#15302
#1905661