Finite Group Databasebeta

← C61×#33738 #29537171

Subgroups — C61×#33738 #29537171

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3170C211yesmaximal, characteristic, complemented
2C61×#3168C211yesmaximal, characteristic
3C61×#2255C211yesmaximal, characteristic, complemented
4C61×#2251C211yesmaximal, characteristic, complemented
5C61×#2245C211yesmaximal, characteristic, complemented
6C61×#2243C211yesmaximal, characteristic, complemented
7C61×#2220C212yesmaximal
8C61×#2219C212yesmaximal, complemented
9C4×Q32×C61C212yesmaximal, complemented
10C4×SD32×C61C212yesmaximal, complemented
11C4×M32×C61C211yesmaximal, characteristic, complemented
12C61×#533—21no—
13C61×#529—21no—
14C2×Q32×C61C2212yes—
15C2×SD32×C61C2212yes—
16C2×M32×C61C2212yes—
17C61×#500C2211yescharacteristic
18C61×#499C2211yescharacteristic
19C61×#495C2211yescharacteristic
20C61×#494C2211yescharacteristic
21C61×#488—21no—
22C61×#487—21no—
23C61×#467—21no—
24C61×#467C2211yescharacteristic
25C61×#467C2212yes—
26C61×#466C2211yescharacteristic
27C61×#465—21no—
28C4×Q16×C61—21no—
29C4×Q16×C61C2211yescharacteristic
30C4×Q16×C61C2212yes—
31C4×SD16×C61—21no—
32C4×D8×C61C2211yescharacteristic
33C2×C4×C8×C61C2211yescharacteristic
34C61×#367C2212yes—
35C61×#366C2212yes—
36C61×#358C2212yes—
37C61×#358C2212yes—
38C61×#358C2212yes—
39C61×#357C2212yes—
40C61×#346C2211yescharacteristic
41C61×#346C2211yescharacteristic
42C4×C16×C61C2212yes—
43C2×Q16×C61—21no—
44C2×Q16×C61C2312yes—
45C2×Q16×C61C2311yescharacteristic
46C2×SD16×C61—21no—
47C2×D8×C61C2311yescharacteristic
48C22×C8×C61D411yescharacteristic
49C22×C8×C61C2311yescharacteristic
50C4⋊Q8×C61—21no—
51C4⋊Q8×C61—21no—
52#126×C61—41no—
53#125×C61—21no—
54#125×C61—41no—
55#124×C61—21no—
56#123×C61—41no—
57C22⋊Q8×C61—22no—
58C22⋊Q8×C61—21no—
59C22⋊Q8×C61—41no—
60#121×C61—21no—
61C4×Q8×C61—22no—
62C4×Q8×C61—21no—
63C4×Q8×C61—41no—
64C4×D4×C61—21no—
65C4×D4×C61—41no—
66#117×C61—41no—
67#117×C61—41no—
68C2×C42×C61D411yescharacteristic
69Q32×C61—24no—
70SD32×C61—24no—
71M32×C61—24no—
72C2×C16×C61C2312yes—
73C2×C16×C61C2312yes—
74#107×C61—21no—
75#107×C61C2312yes—
76#107×C61C2311yescharacteristic
77#107×C61C2311yescharacteristic
78#106×C61—21no—
79#103×C61—21no—
80#103×C61—22no—
81#103×C61—21no—
82#103×C61—22no—
83#102×C61—21no—
84#102×C61—21no—
85C4×C8×C61D411yescharacteristic
86C4×C8×C61D411yescharacteristic
87C4×C8×C61C2311yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—21no—
90C2×Q8×C61—22no—
91C2×Q8×C61—41no—
92C2×D4×C61—21no—
93C2×D4×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61D811yescharacteristic
96C22×C4×C61D811yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98Q16×C61—22no—
99Q16×C61—22no—
100Q16×C61—22no—
101Q16×C61—42no—
102SD16×C61—42no—
103D8×C61—22no—
104C2×C8×C61—22no—
105C2×C8×C61—22no—
106C2×C8×C61C2×D411yescharacteristic
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61C2411yescharacteristic
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61—42no—
119C4⋊C4×C61—41no—
120C4⋊C4×C61—42no—
121C4⋊C4×C61—41no—
122C22⋊C4×C61—42no—
123C22⋊C4×C61—41no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C42×C61—41no—
129C42×C61—42no—
130C42×C61D811yescharacteristic
131C42×C61D811yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61C2×D411yescharacteristic
134C16×C61—22no—
135C16×C61#4114yes—
136C23×C61—41no—
137C23×C61C2×D811yescharacteristic
138Q8×C61—42no—
139Q8×C61—41no—
140Q8×C61—41no—
141Q8×C61—42no—
142Q8×C61—82no—
143D4×C61—41no—
144D4×C61—41no—
145D4×C61—82no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—82no—
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61C2×D811yescharacteristic
159C2×C4×C61C2×D811yescharacteristic
160C2×C4×C61C2×D811yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C8×C61—21no—
163C8×C61—21no—
164C8×C61#13512yes—
165C8×C61C2×#4112yes—
166#33738C6111yesmaximal, characteristic, direct product factor, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61C22×D811yescharacteristic
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—81no—
179C4×C61—81no—
180C4×C61—82no—
181C4×C61—81no—
182C4×C61—82no—
183C4×C61#51012yes—
184C4×C61#53211yescharacteristic
185C4×C61C2×#13511yescharacteristic
186#3170C2×C6111yescharacteristic, complemented
187#3168C2×C6111yescharacteristic
188#2255C2×C6111yescharacteristic, complemented
189#2251C2×C6111yescharacteristic, complemented
190#2245C2×C6111yescharacteristic, complemented
191#2243C2×C6111yescharacteristic, complemented
192#2220C2×C6112yes—
193#2219C2×C6112yescomplemented
194C4×Q32C2×C6112yescomplemented
195C4×SD32C2×C6112yescomplemented
196C4×M32C2×C6111yescharacteristic, complemented
197C2×C61—41no—
198C2×C61—161no—
199C2×C61#316911yescharacteristic
200C2×C61C2×#51011yescharacteristic
201C2×C61#341411yescharacteristic
202#533—21no—
203#529—21no—
204C2×Q32C22×C6112yes—
205C2×SD32C22×C6112yes—
206C2×M32C22×C6112yes—
207#500C22×C6111yescharacteristic
208#499C22×C6111yescharacteristic
209#495C22×C6111yescharacteristic
210#494C22×C6111yescharacteristic
211#488—21no—
212#487—21no—
213#467—21no—
214#467C22×C6111yescharacteristic
215#467C22×C6112yes—
216#466C22×C6111yescharacteristic
217#465—21no—
218C4×Q16—21no—
219C4×Q16C22×C6111yescharacteristic
220C4×Q16C22×C6112yes—
221C4×SD16—21no—
222C4×D8C22×C6111yescharacteristic
223C2×C4×C8C22×C6111yescharacteristic
224#367C22×C6112yes—
225#366C22×C6112yes—
226#358C22×C6112yes—
227#358C22×C6112yes—
228#358C22×C6112yes—
229#357C22×C6112yes—
230#346C22×C6111yescharacteristic
231#346C22×C6111yescharacteristic
232C4×C16C22×C6112yes—
233C61#3373811yescharacteristic, direct product factor, Sylow
234C2×Q16—21no—
235C2×Q16C23×C6112yes—
236C2×Q16C23×C6111yescharacteristic
237C2×SD16—21no—
238C2×D8C23×C6111yescharacteristic
239C22×C8D4×C6111yescharacteristic
240C22×C8C23×C6111yescharacteristic
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243#126—41no—
244#125—21no—
245#125—41no—
246#124—21no—
247#123—41no—
248C22⋊Q8—21no—
249C22⋊Q8—22no—
250C22⋊Q8—41no—
251#121—21no—
252C4×Q8—21no—
253C4×Q8—22no—
254C4×Q8—41no—
255C4×D4—21no—
256C4×D4—41no—
257#117—41no—
258#117—41no—
259C2×C42D4×C6111yescharacteristic
260Q32—24no—
261SD32—24no—
262M32—24no—
263C2×C16C23×C6112yes—
264C2×C16C23×C6112yes—
265#107—21no—
266#107C23×C6112yes—
267#107C23×C6111yescharacteristic
268#107C23×C6111yescharacteristic
269#106—21no—
270#103—21no—
271#103—22no—
272#103—21no—
273#103—22no—
274#102—21no—
275#102—21no—
276C4×C8D4×C6111yescharacteristic
277C4×C8D4×C6111yescharacteristic
278C4×C8C23×C6111yescharacteristic
279C4×C8C23×C6111yescharacteristic
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×D4—21no—
284C2×D4—41no—
285C22×C4—41no—
286C22×C4D8×C6111yescharacteristic
287C22×C4D8×C6111yescharacteristic
288C22×C4C2×D4×C6111yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—22no—
292Q16—42no—
293SD16—42no—
294D8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×D4×C6111yescharacteristic
298C2×C8C2×D4×C6111yescharacteristic
299C2×C8C2×D4×C6111yescharacteristic
300C2×C8C24×C6111yescharacteristic
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C42—41no—
320C42—42no—
321C42D8×C6111yescharacteristic
322C42D8×C6111yescharacteristic
323C42C2×D4×C6111yescharacteristic
324C42C2×D4×C6111yescharacteristic
325C16—22no—
326C16#41×C6114yes—
327C23—41no—
328C23C2×D8×C6111yescharacteristic
329Q8—42no—
330Q8—41no—
331Q8—41no—
332Q8—42no—
333Q8—82no—
334D4—41no—
335D4—41no—
336D4—82no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—82no—
347C2×C4C2×D8×C6111yescharacteristic
348C2×C4C2×D8×C6111yescharacteristic
349C2×C4C2×D8×C6111yescharacteristic
350C2×C4C2×D8×C6111yescharacteristic
351C2×C4C2×D8×C6111yescharacteristic
352C2×C4C22×D4×C6111yescharacteristic
353C8—21no—
354C8—21no—
355C8#135×C6112yes—
356C8C2×#41×C6112yes—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22C22×D8×C6111yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—82no—
371C4—81no—
372C4—82no—
373C4C61×#51012yes—
374C4C61×#53211yescharacteristic
375C4C2×#135×C6111yescharacteristic
376C2—41no—
377C2—161no—
378C2C61×#316911yescharacteristic
379C2C2×C61×#51011yescharacteristic
380C2C61×#341411yescharacteristic
subgroupG/H
centerC2×C4×C61C2×D8
commutator, Frattini, cosocleC2×C8C24×C61
socleC22×C61C22×D8
prime radicalC61×#466C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ C2×C4×C8×C61 ◁ G

Complements

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

NH ≅ G/Ntype
1. C61×#3170C2semidirect
3. C61×#2255C2semidirect
4. C61×#2251C2semidirect
5. C61×#2245C2semidirect
6. C61×#2243C2semidirect
8. C61×#2219C2semidirect
9. C4×Q32×C61C2semidirect
10. C4×SD32×C61C2semidirect
11. C4×M32×C61C2semidirect
166. #33738C61direct (N × H)
186. #3170C2×C61semidirect
188. #2255C2×C61semidirect
189. #2251C2×C61semidirect
190. #2245C2×C61semidirect
191. #2243C2×C61semidirect
193. #2219C2×C61semidirect
194. C4×Q32C2×C61semidirect
195. C4×SD32C2×C61semidirect
196. C4×M32C2×C61semidirect
233. C61#33738direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#31702
C61×#31682
C61×#22552
C61×#22512
C61×#22452
C61×#22432
C61×#22202
C61×#22192
C4×Q32×C612
C4×SD32×C612
C4×M32×C612
#3373861