Finite Group Databasebeta

← C61×#10332 #29513765

Subgroups — C61×#10332 #29513765

Order 15616.

600 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3193C211yesmaximal, characteristic
2C61×#2504C211yesmaximal, characteristic
3C61×#2503C211yesmaximal, characteristic, complemented
4C61×#1821C211yesmaximal, characteristic, complemented
5C61×#1508C211yesmaximal, characteristic, complemented
6C61×#1507C211yesmaximal, characteristic
7C61×#1297C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C61C2211yescharacteristic
9C2×#121×C61C2211yescharacteristic
10C22×C4⋊C4×C61C2211yescharacteristic
11C61×#484—21no—
12C61×#480—21no—
13C61×#468—22no—
14C61×#450—21no—
15C61×#448—21no—
16C61×#407C2211yescharacteristic
17C61×#407C2211yescharacteristic
18C61×#400—22no—
19C61×#397—22no—
20C61×#391—21no—
21C61×#390—21no—
22C61×#386—22no—
23C61×#382—21no—
24C61×#382—21no—
25C61×#381—22no—
26C4×C22⋊C4×C61C2211yescharacteristic
27C4×C22⋊C4×C61C2211yescharacteristic
28C61×#343—22no—
29C61×#328—22no—
30C61×#327—22no—
31C22×Q8×C61D411yescharacteristic
32C22×D4×C61—21no—
33C22×D4×C61D411yescharacteristic
34C23×C4×C61D412yes—
35C23×C4×C61C2311yescharacteristic
36C2×SD16×C61—22no—
37C2×M16×C61—21no—
38C2×M16×C61D412yes—
39C22⋊Q8×C61—21no—
40C22⋊Q8×C61—21no—
41C22⋊Q8×C61—21no—
42C22⋊Q8×C61—21no—
43#121×C61—21no—
44#121×C61—21no—
45#121×C61—21no—
46#121×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61—21no—
51C2×C4⋊C4×C61—21no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61D411yescharacteristic
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61—21no—
58C2×C22⋊C4×C61—21no—
59C2×C22⋊C4×C61—21no—
60C2×C22⋊C4×C61—24no—
61C2×C42×C61—21no—
62C2×C42×C61—21no—
63#107×C61—22no—
64#103×C61—22no—
65#102×C61—22no—
66C22⋊C8×C61—21no—
67C22⋊C8×C61—21no—
68C22⋊C8×C61—21no—
69C22⋊C8×C61—21no—
70#95×C61—24no—
71#95×C61—24no—
72#95×C61—24no—
73#95×C61D412yes—
74#95×C61D412yes—
75C24×C61—21no—
76C24×C61C2×D411yescharacteristic
77C2×Q8×C61—21no—
78C2×Q8×C61—21no—
79C2×Q8×C61—42no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—42no—
83C2×D4×C61—44no—
84C2×D4×C61—41no—
85C2×D4×C61—41no—
86C22×C4×C61—22no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—22no—
91C22×C4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—42no—
95C22×C4×C61—44no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102SD16×C61—82no—
103M16×C61—42no—
104M16×C61—42no—
105C2×C8×C61—22no—
106C2×C8×C61—41no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C4⋊C4×C61—41no—
117C22⋊C4×C61—42no—
118C22⋊C4×C61—42no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—84no—
124C42×C61—42no—
125C42×C61—42no—
126C23×C61—22no—
127C23×C61—22no—
128C23×C61—41no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61—41no—
132C23×C61—42no—
133C23×C61#12011yescharacteristic
134C23×C61#12011yescharacteristic
135C23×C61#12111yescharacteristic
136Q8×C61—42no—
137Q8×C61—81no—
138D4×C61—42no—
139D4×C61—81no—
140D4×C61—82no—
141D4×C61—82no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61—44no—
153C2×C4×C61—42no—
154C2×C4×C61—42no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61—42no—
161C2×C4×C61—42no—
162C2×C4×C61—44no—
163C2×C4×C61—82no—
164C2×C4×C61#12011yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12711yescharacteristic
168C8×C61—42no—
169C8×C61—81no—
170#10332C6111yesmaximal, characteristic, direct product factor, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—44no—
179C22×C61—84no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61#39211yescharacteristic
184C22×C61#45412yes—
185C22×C61#45512yes—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—82no—
190C4×C61—82no—
191C4×C61—81no—
192C4×C61—81no—
193C4×C61—82no—
194C4×C61—81no—
195C4×C61#45812yes—
196#3193C2×C6111yescharacteristic
197#2504C2×C6111yescharacteristic
198#2503C2×C6111yescharacteristic, complemented
199#1821C2×C6111yescharacteristic, complemented
200#1508C2×C6111yescharacteristic, complemented
201#1507C2×C6111yescharacteristic
202#1297C2×C6111yescharacteristic, complemented
203C2×C61—22no—
204C2×C61—22no—
205C2×C61—41no—
206C2×C61—161no—
207C2×C61#200311yescharacteristic
208C2×C61#200411yescharacteristic
209C2×C61#202211yescharacteristic
210C2×C22⋊Q8C22×C6111yescharacteristic
211C2×#121C22×C6111yescharacteristic
212C22×C4⋊C4C22×C6111yescharacteristic
213#484—21no—
214#480—21no—
215#468—22no—
216#450—21no—
217#448—21no—
218#407C22×C6111yescharacteristic
219#407C22×C6111yescharacteristic
220#400—22no—
221#397—22no—
222#391—21no—
223#390—21no—
224#386—22no—
225#382—21no—
226#382—21no—
227#381—22no—
228C4×C22⋊C4C22×C6111yescharacteristic
229C4×C22⋊C4C22×C6111yescharacteristic
230#343—22no—
231#328—22no—
232#327—22no—
233C61#1033211yescharacteristic, direct product factor, Sylow
234C22×Q8D4×C6111yescharacteristic
235C22×D4—21no—
236C22×D4D4×C6111yescharacteristic
237C23×C4D4×C6112yes—
238C23×C4C23×C6111yescharacteristic
239C2×SD16—22no—
240C2×M16—21no—
241C2×M16D4×C6112yes—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D4×C6111yescharacteristic
258C2×C4⋊C4D4×C6111yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—24no—
264C2×C42—21no—
265C2×C42—21no—
266#107—22no—
267#103—22no—
268#102—22no—
269C22⋊C8—21no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95D4×C6112yes—
277#95D4×C6112yes—
278C24—21no—
279C24C2×D4×C6111yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—42no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—42no—
286C2×D4—44no—
287C2×D4—41no—
288C2×D4—41no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4C2×D4×C6111yescharacteristic
300C22×C4C2×D4×C6111yescharacteristic
301C22×C4C2×D4×C6111yescharacteristic
302C22×C4C2×D4×C6111yescharacteristic
303C22×C4C2×D4×C6111yescharacteristic
304C22×C4#41×C6111yescharacteristic
305SD16—82no—
306M16—42no—
307M16—42no—
308C2×C8—22no—
309C2×C8—41no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—84no—
327C42—42no—
328C42—42no—
329C23—22no—
330C23—22no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—42no—
336C23#120×C6111yescharacteristic
337C23#120×C6111yescharacteristic
338C23#121×C6111yescharacteristic
339Q8—42no—
340Q8—81no—
341D4—42no—
342D4—81no—
343D4—82no—
344D4—82no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—82no—
367C2×C4#120×C6111yescharacteristic
368C2×C4#121×C6111yescharacteristic
369C2×C4#121×C6111yescharacteristic
370C2×C4#127×C6111yescharacteristic
371C8—42no—
372C8—81no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—44no—
381C22—84no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C61×#39211yescharacteristic
386C22C61×#45412yes—
387C22C61×#45512yes—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—82no—
397C4C61×#45812yes—
398C2—22no—
399C2—22no—
400C2—41no—
401C2—161no—
402C2C61×#200311yescharacteristic
403C2C61×#200411yescharacteristic
404C2C61×#202211yescharacteristic
subgroupG/H
center, socleC22×C61#392
commutator, Frattini, cosocleC23×C4C23×C61
prime radicalC2×#121×C61C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
3. C61×#2503C2semidirect
4. C61×#1821C2semidirect
5. C61×#1508C2semidirect
7. C61×#1297C2semidirect
170. #10332C61direct (N × H)
198. #2503C2×C61semidirect
199. #1821C2×C61semidirect
200. #1508C2×C61semidirect
202. #1297C2×C61semidirect
233. C61#10332direct (N × H)

Maximal subgroups

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

subgroupindex
C61×#31932
C61×#25042
C61×#25032
C61×#18212
C61×#15082
C61×#15072
C61×#12972
#1033261