Finite Group Databasebeta

← #29591009

Subgroups — #29591009

Order 15616.

364 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 73 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
1#21455472C211yesmaximal, characteristic
2#21447438C211yesmaximal, characteristic
3#21445864C211yesmaximal, characteristic
4#21443808C211yesmaximal, characteristic, complemented
5#21443807C211yesmaximal, characteristic, complemented
6#21443806C211yesmaximal, characteristic, complemented
7C61×#1340C211yesmaximal, characteristic, complemented
8C2×Q16×D61C411yescharacteristic, complemented
9C2×#14519785C2211yescharacteristic
10#16840335—21no—
11M32×D61—21no—
12#16840021C411yescharacteristic, complemented
13#16840020C2211yescharacteristic
14#16840017C411yescharacteristic, complemented
15#16840002—21no—
16#16840001—21no—
17#16839798—21no—
18#16839797C411yescharacteristic, complemented
19#16839769—21no—
20#16839735—21no—
21#16839544C2211yescharacteristic
22#16839543—21no—
23#16839519C2211yescharacteristic
24C61×#470C2211yescharacteristic
25C61×#349C2211yescharacteristic
26C61×H(3,ℤ/4)C2211yescharacteristic
27C2×#11758222D411yescharacteristic
28C2×Q8×D61—21no—
29C2×Dic244C2×C411yescharacteristic
30C2×#11758202—21no—
31Q16×D61—41no—
32Q16×D61—41no—
33#14519785—41no—
34#14519785—41no—
35C2×#11758189C2×C411yescharacteristic
36C2×#11758189D411yescharacteristic
37C22×#92418D411yescharacteristic
38C2×C8×D61C2×C411yescharacteristic
39C2×C8×D61D411yescharacteristic
40#14519776—41no—
41#14519763—41no—
42#14519762—41no—
43#14519759—41no—
44#14519757—21no—
45#14519756—21no—
46#14519753—21no—
47#14519736—21no—
48#14519735—21no—
49#14519717—21no—
50#14519716—41no—
51C16×D61—41no—
52#14519710—21no—
53#14519710—41no—
54#14519709C2×C411yescharacteristic
55#14519708—41no—
56#14519702—21no—
57#14519701—21no—
58#14519700—21no—
59#14519691—41no—
60#14519691—41no—
61C2×Q16×C61C2×C411yescharacteristic
62C2×M16×C61C2311yescharacteristic
63C22⋊Q8×C61—21no—
64#117×C61—21no—
65M32×C61—21no—
66#106×C61C2×C411yescharacteristic
67#103×C61—21no—
68C2×Q8×C61—21no—
69C22×C4×C61C2×D411yescharacteristic
70C2×C22⋊D61D811yescharacteristic
71C2×C22⋊D61D811yescharacteristic
72Q8×D61—42no—
73Q8×D61—42no—
74#11758222—41no—
75#11758222—41no—
76C2×D244SD1611yescharacteristic
77C2×C4×D61—41no—
78C2×C4×D61C22⋊C411yescharacteristic
79C2×C4×D61C22⋊C411yescharacteristic
80C2×Dic122—21no—
81C2×Dic122—41no—
82C2×Dic122SD1611yescharacteristic
83Q16×C61—41no—
84M16×C61—41no—
85C2×C8×C61C22×C411yescharacteristic
86C2×C8×C61C2×D411yescharacteristic
87C4⋊C4×C61—21no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C42×C61—41no—
93C22⋊Dic61—41no—
94C22⋊Dic61—41no—
95#11758202—41no—
96#11758202—41no—
97#11758198—41no—
98#11758198—41no—
99#11758198—41no—
100C4⋊Dic61—21no—
101C4⋊Dic61—41no—
102C4⋊Dic61—41no—
103#11758196—41no—
104#11758196—41no—
105C4×Dic61—41no—
106C4×Dic61—41no—
107C2×#92418—21no—
108C2×#92418—21no—
109C2×#92418C22⋊C411yescharacteristic
110C2×#92418C22⋊C411yescharacteristic
111Dic244—41no—
112#11758189—41no—
113#11758189—41no—
114C8×D61—21no—
115C8×D61—21no—
116C8×D61—21no—
117C8×D61—21no—
118C16×C61—41no—
119#11758185—41no—
120C23×C61C2×D811yescharacteristic
121C22×D61#10211yescharacteristic
122C22×D61#10211yescharacteristic
123Q8×C61—41no—
124Q8×C61—41no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—81no—
130C2×C4×C61C2×C22⋊C411yescharacteristic
131C2×C4×C61C2×SD1611yescharacteristic
132C22⋊D61—41no—
133C22⋊D61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—81no—
137C2×Dic61#10211yescharacteristic
138C2×Dic61#10211yescharacteristic
139D244—41no—
140C4×D61—21no—
141C4×D61—21no—
142C4×D61—21no—
143C4×D61—21no—
144C4×D61—82no—
145Dic122—41no—
146Dic122—41no—
147Dic122—41no—
148Dic122—82no—
149C8×C61—21no—
150C8×C61—21no—
151#92418—21no—
152#92418#9911yescharacteristic
153#92418#9911yescharacteristic
154#11869—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—41no—
157C22×C61C2×#10211yescharacteristic
158C2×D61—21no—
159C2×D61—21no—
160C2×D61—41no—
161C2×D61—41no—
162C4×C61—21no—
163C4×C61—81no—
164C4×C61—81no—
165C4×C61—161no—
166C4×C61C2×#9911yescharacteristic
167C4×C61#42111yescharacteristic
168Dic61—21no—
169Dic61—21no—
170Dic61—81no—
171Dic61—81no—
172Dic61—161no—
173#3059—611no—
174#2115—611no—
175#1812—611no—
176#1340—611no—
177#1340—611no—
178#1340—611no—
179#1340D6111yescharacteristic, complemented
180C2×C61—21no—
181C2×C61—41no—
182C2×C61#151911yescharacteristic
183D61—41no—
184D61—41no—
185C22×Q16—611no—
186C2×#131—611no—
187#536—1221no—
188C2×M32—1221no—
189#470—611no—
190#470—611no—
191#470—611no—
192#470C2×D6111yescharacteristic
193#467—1221no—
194#467—1221no—
195#427—611no—
196#427—1221no—
197#419—1221no—
198#410—1221no—
199#349—611no—
200#349—1221no—
201#349C2×D6111yescharacteristic
202H(3,ℤ/4)—611no—
203H(3,ℤ/4)C2×D6111yescharacteristic
204C61#1186911yescharacteristic, complemented, Sylow
205C2×#41—611no—
206C22×Q8—1221no—
207C2×Q16—611no—
208C2×Q16—1221no—
209C2×Q16—2441no—
210C2×Q16—2441no—
211C2×Q16C4×D6111yescharacteristic
212#131—2441no—
213#131—2441no—
214C2×M16—611no—
215C2×M16—611no—
216C2×M16C22×D6111yescharacteristic
217C22×C8—611no—
218C22×C8—611no—
219C22×C8—611no—
220C4⋊Q8—2441no—
221#124—2441no—
222#123—2441no—
223#123—2441no—
224C22⋊Q8—21no—
225C22⋊Q8—1221no—
226C22⋊Q8—1221no—
227C22⋊Q8—1221no—
228#117—21no—
229#117—1221no—
230#117—1221no—
231M32—21no—
232M32—1221no—
233M32—2441no—
234C2×C16—2441no—
235#107—1221no—
236#107—2441no—
237#106—611no—
238#106—2441no—
239#106C4×D6111yescharacteristic
240#103—21no—
241#103—1221no—
242#103—1221no—
243#103—1221no—
244#99—2441no—
245#99—2441no—
246#41—2441no—
247#41—2441no—
248C2×Q8—21no—
249C2×Q8—611no—
250C2×Q8—1221no—
251C2×Q8—2441no—
252C2×Q8—2442no—
253C2×Q8—2442no—
254C2×D4—611no—
255C2×D4—611no—
256C2×D4—611no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—2441no—
260C22×C4D4×D6111yescharacteristic
261Q16—41no—
262Q16—2441no—
263Q16—2441no—
264Q16—2441no—
265M16—41no—
266M16—2441no—
267M16—2441no—
268C2×C8—611no—
269C2×C8—611no—
270C2×C8—1221no—
271C2×C8—1221no—
272C2×C8—1221no—
273C2×C8—1221no—
274C2×C8—1221no—
275C2×C8—1221no—
276C2×C8C2×C4×D6111yescharacteristic
277C2×C8D4×D6111yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—1221no—
282C4⋊C4—2441no—
283C4⋊C4—2441no—
284C4⋊C4—2441no—
285C4⋊C4—2441no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—2441no—
289C22⋊C4—2441no—
290C22⋊C4—2441no—
291C22⋊C4—2441no—
292C22⋊C4—2441no—
293C42—41no—
294C42—2441no—
295C42—2441no—
296C16—41no—
297C16—2441no—
298C23—611no—
299C23—611no—
300C23D8×D6111yescharacteristic
301Q8—41no—
302Q8—41no—
303Q8—4882no—
304Q8—2441no—
305Q8—2441no—
306Q8—2441no—
307D4—2441no—
308D4—2441no—
309D4—2441no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—81no—
315C2×C4—611no—
316C2×C4—611no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—1221no—
320C2×C4—1221no—
321C2×C4—4882no—
322C2×C4—4881no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4C22⋊C4×D6111yescharacteristic
326C2×C4SD16×D6111yescharacteristic
327C8—21no—
328C8—21no—
329C8—611no—
330C8—611no—
331C8—1221no—
332C22—21no—
333C22—41no—
334C22—1221no—
335C22—1221no—
336C22—2441no—
337C22—2441no—
338C22#102×D6111yescharacteristic
339C4—21no—
340C4—81no—
341C4—81no—
342C4—161no—
343C4—1221no—
344C4—1221no—
345C4—9761no—
346C4—4881no—
347C4—4881no—
348C4#99×D6111yescharacteristic
349C4#1683977711yescharacteristic
350C2—21no—
351C2—41no—
352C2—2441no—
353C2—2441no—
354C2#2144438511yescharacteristic
subgroupG/H
centerC2#21444385
commutatorC2×C8×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#1340C2
socleC2×C61#1519
prime radicalC2×#11758222D4
cosocleC2×M16×C61C23
hypercenter#1340D61
hypocenterC61#11869
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1340

Complements

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

NH ≅ G/Ntype
4. #21443808C2semidirect
5. #21443807C2semidirect
6. #21443806C2semidirect
7. C61×#1340C2semidirect
8. C2×Q16×D61C4semidirect
12. #16840021C4semidirect
14. #16840017C4semidirect
18. #16839797C4semidirect
179. #1340D61semidirect
204. C61#11869semidirect

Maximal subgroups

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

subgroupindex
#214554722
#214474382
#214458642
#214438082
#214438072
#214438062
C61×#13402
#1186961