Finite Group Databasebeta

← C61×#9316 #29512749

Subgroups — C61×#9316 #29512749

Order 15616.

512 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C61×#3039C211yesmaximal, characteristic
2C61×#2988C211yesmaximal, characteristic
3C61×#1988C211yesmaximal, characteristic, complemented
4C61×#1806C211yesmaximal, characteristic
5C61×#1546C211yesmaximal, characteristic, complemented
6C61×#1303C211yesmaximal, characteristic, complemented
7C61×#1292C211yesmaximal, characteristic, complemented
8C2×#137×C61—21no—
9C2×C4⋊Q8×C61C2211yescharacteristic
10C4×#41×C61C2211yescharacteristic
11C61×#462—21no—
12C61×#462C412yescomplemented
13C61×#461—21no—
14C61×#461C412yescomplemented
15C8×Q8×C61—21no—
16C8×D4×C61—22no—
17C8×D4×C61—21no—
18C61×#431—21no—
19C61×#431C2211yescharacteristic
20C61×#423C2211yescharacteristic
21C61×#423C2211yescharacteristic
22C2×C4⋊C8×C61—21no—
23C2×C4⋊C8×C61C2211yescharacteristic
24C2×#104×C61—21no—
25C61×#419—21no—
26C4×C4⋊C4×C61C2211yescharacteristic
27C61×#340—22no—
28C61×#339—21no—
29C61×#332—21no—
30C61×#332—21no—
31C2×#41×C61D411yescharacteristic
32C22×Q8×C61—21no—
33#137×C61—42no—
34#137×C61—42no—
35C2×Q16×C61—22no—
36C2×SD16×C61—22no—
37C2×M16×C61—21no—
38C2×M16×C61—21no—
39C22×C8×C61—22no—
40C22×C8×C61D412yes—
41C4⋊Q8×C61—21no—
42C4⋊Q8×C61—21no—
43C4⋊Q8×C61—21no—
44C4⋊Q8×C61C2×C412yes—
45C4×Q8×C61C2×C411yescharacteristic
46C4×Q8×C61D411yescharacteristic
47C4×D4×C61—21no—
48C4×D4×C61—21no—
49C4×D4×C61C2×C411yescharacteristic
50C4×D4×C61D411yescharacteristic
51#117×C61—21no—
52#117×C61D411yescharacteristic
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61—21no—
56C2×C42×C61—21no—
57C2×C42×C61—21no—
58C2×C42×C61C2311yescharacteristic
59C4⋊C8×C61—22no—
60C4⋊C8×C61—21no—
61C4⋊C8×C61—21no—
62C4⋊C8×C61—21no—
63C4⋊C8×C61—21no—
64C4⋊C8×C61—21no—
65C4⋊C8×C61C2×C412yes—
66C4⋊C8×C61D411yescharacteristic
67C4⋊C8×C61D411yescharacteristic
68#104×C61—42no—
69#103×C61—22no—
70#103×C61—22no—
71C22⋊C8×C61—22no—
72C22⋊C8×C61—21no—
73C4×C8×C61—22no—
74C4×C8×C61—21no—
75#95×C61—21no—
76#41×C61—21no—
77#41×C61—21no—
78#41×C61D812yes—
79#41×C61SD1612yes—
80C2×Q8×C61—22no—
81C2×Q8×C61—41no—
82C2×Q8×C61—41no—
83C2×Q8×C61—41no—
84C2×Q8×C61—41no—
85C2×Q8×C61—41no—
86C2×Q8×C61C22⋊C411yescharacteristic
87C2×D4×C61—21no—
88C2×D4×C61C22⋊C411yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—41no—
93C22×C4×C61—41no—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97Q16×C61—44no—
98SD16×C61—44no—
99M16×C61—41no—
100M16×C61—41no—
101M16×C61—41no—
102M16×C61—41no—
103C2×C8×C61—22no—
104C2×C8×C61—21no—
105C2×C8×C61—22no—
106C2×C8×C61—21no—
107C2×C8×C61—42no—
108C2×C8×C61—41no—
109C2×C8×C61—82no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61C22⋊C411yescharacteristic
119C4⋊C4×C61C22⋊C411yescharacteristic
120C22⋊C4×C61—21no—
121C22⋊C4×C61—21no—
122C22⋊C4×C61—21no—
123C42×C61—21no—
124C42×C61—21no—
125C42×C61—21no—
126C42×C61—41no—
127C42×C61—41no—
128C42×C61C22×C411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C23×C61—21no—
133C23×C61#12111yescharacteristic
134Q8×C61—21no—
135Q8×C61—42no—
136Q8×C61—42no—
137Q8×C61—81no—
138Q8×C61—81no—
139Q8×C61#10212yes—
140D4×C61—22no—
141D4×C61—22no—
142D4×C61—21no—
143D4×C61#10212yes—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—21no—
150C2×C4×C61—21no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—81no—
157C2×C4×C61—81no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61—81no—
161C2×C4×C61C2×C22⋊C411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61C4×D411yescharacteristic
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12311yescharacteristic
167C2×C4×C61C2×D811yescharacteristic
168C2×C4×C61C2×SD1611yescharacteristic
169C2×C4×C61#13511yescharacteristic
170C2×C4×C61#13511yescharacteristic
171C8×C61—42no—
172C8×C61—42no—
173C8×C61—41no—
174C8×C61—41no—
175C8×C61—81no—
176#9316C6111yesmaximal, characteristic, direct product factor, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—22no—
180C22×C61—41no—
181C22×C61—41no—
182C22×C61#38611yescharacteristic
183C22×C61#46511yescharacteristic
184C22×C61#46611yescharacteristic
185C4×C61—21no—
186C4×C61—22no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—41no—
190C4×C61—41no—
191C4×C61—82no—
192C4×C61—82no—
193C4×C61C2×#10211yescharacteristic
194C4×C61#41611yescharacteristic
195C4×C61#45511yescharacteristic
196C4×C61#45611yescharacteristic
197C4×C61#46511yescharacteristic
198C4×C61#46611yescharacteristic
199#3039C2×C6111yescharacteristic
200#2988C2×C6111yescharacteristic
201#1988C2×C6111yescharacteristic, complemented
202#1806C2×C6111yescharacteristic
203#1546C2×C6111yescharacteristic, complemented
204#1303C2×C6111yescharacteristic, complemented
205#1292C2×C6111yescharacteristic, complemented
206C2×C61—21no—
207C2×C61—21no—
208C2×C61—42no—
209C2×C61#188211yescharacteristic
210C2×C61#189311yescharacteristic
211C2×C61#190211yescharacteristic
212C2×#137—21no—
213C2×C4⋊Q8C22×C6111yescharacteristic
214C4×#41C22×C6111yescharacteristic
215#462—21no—
216#462C4×C6112yescomplemented
217#461—21no—
218#461C4×C6112yescomplemented
219C8×Q8—21no—
220C8×D4—22no—
221C8×D4—21no—
222#431—21no—
223#431C22×C6111yescharacteristic
224#423C22×C6111yescharacteristic
225#423C22×C6111yescharacteristic
226C2×C4⋊C8—21no—
227C2×C4⋊C8C22×C6111yescharacteristic
228C2×#104—21no—
229#419—21no—
230C4×C4⋊C4C22×C6111yescharacteristic
231#340—22no—
232#339—21no—
233#332—21no—
234#332—21no—
235C61#931611yescharacteristic, direct product factor, Sylow
236C2×#41D4×C6111yescharacteristic
237C22×Q8—21no—
238#137—42no—
239#137—42no—
240C2×Q16—22no—
241C2×SD16—22no—
242C2×M16—21no—
243C2×M16—21no—
244C22×C8—22no—
245C22×C8D4×C6112yes—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C2×C4×C6112yes—
250C4×Q8C2×C4×C6111yescharacteristic
251C4×Q8D4×C6111yescharacteristic
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C2×C4×C6111yescharacteristic
255C4×D4D4×C6111yescharacteristic
256#117—21no—
257#117D4×C6111yescharacteristic
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42C23×C6111yescharacteristic
264C4⋊C8—22no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8C2×C4×C6112yes—
271C4⋊C8D4×C6111yescharacteristic
272C4⋊C8D4×C6111yescharacteristic
273#104—42no—
274#103—22no—
275#103—22no—
276C22⋊C8—22no—
277C22⋊C8—21no—
278C4×C8—22no—
279C4×C8—21no—
280#95—21no—
281#41—21no—
282#41—21no—
283#41D8×C6112yes—
284#41SD16×C6112yes—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8C22⋊C4×C6111yescharacteristic
292C2×D4—21no—
293C2×D4C22⋊C4×C6111yescharacteristic
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C2×D4×C6111yescharacteristic
300C22×C4C2×D4×C6111yescharacteristic
301C22×C4#41×C6111yescharacteristic
302Q16—44no—
303SD16—44no—
304M16—41no—
305M16—41no—
306M16—41no—
307M16—41no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—82no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4C22⋊C4×C6111yescharacteristic
324C4⋊C4C22⋊C4×C6111yescharacteristic
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—41no—
332C42—41no—
333C42C22×C4×C6111yescharacteristic
334C42C2×D4×C6111yescharacteristic
335C42C2×D4×C6111yescharacteristic
336C42#41×C6111yescharacteristic
337C23—21no—
338C23#121×C6111yescharacteristic
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—81no—
343Q8—81no—
344Q8#102×C6112yes—
345D4—22no—
346D4—22no—
347D4—21no—
348D4#102×C6112yes—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4C2×C22⋊C4×C6111yescharacteristic
367C2×C4C4×D4×C6111yescharacteristic
368C2×C4C4×D4×C6111yescharacteristic
369C2×C4#120×C6111yescharacteristic
370C2×C4#121×C6111yescharacteristic
371C2×C4#123×C6111yescharacteristic
372C2×C4C2×D8×C6111yescharacteristic
373C2×C4C2×SD16×C6111yescharacteristic
374C2×C4#135×C6111yescharacteristic
375C2×C4#135×C6111yescharacteristic
376C8—42no—
377C8—42no—
378C8—41no—
379C8—41no—
380C8—81no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22C61×#38611yescharacteristic
387C22C61×#46511yescharacteristic
388C22C61×#46611yescharacteristic
389C4—21no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—82no—
396C4—82no—
397C4C2×#102×C6111yescharacteristic
398C4C61×#41611yescharacteristic
399C4C61×#45511yescharacteristic
400C4C61×#45611yescharacteristic
401C4C61×#46511yescharacteristic
402C4C61×#46611yescharacteristic
403C2—21no—
404C2—21no—
405C2—42no—
406C2C61×#188211yescharacteristic
407C2C61×#189311yescharacteristic
408C2C61×#190211yescharacteristic
subgroupG/H
center, socleC22×C61#386
commutatorC42C22×C4×C61
Frattini, cosocleC2×C42C23×C61
prime radicalC2×#41×C61D4
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
3. C61×#1988C2semidirect
5. C61×#1546C2semidirect
6. C61×#1303C2semidirect
7. C61×#1292C2semidirect
12. C61×#462C4semidirect
14. C61×#461C4semidirect
176. #9316C61direct (N × H)
201. #1988C2×C61semidirect
203. #1546C2×C61semidirect
204. #1303C2×C61semidirect
205. #1292C2×C61semidirect
216. #462C4×C61semidirect
218. #461C4×C61semidirect
235. C61#9316direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#30392
C61×#29882
C61×#19882
C61×#18062
C61×#15462
C61×#13032
C61×#12922
#931661