Finite Group Databasebeta

← #29591017

Subgroups — #29591017

Order 15616.

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

normal subgroups only

This group has 360 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#21445868C211yesmaximal, characteristic
4#21443814C211yesmaximal, characteristic, complemented
5#21443813C211yesmaximal, characteristic, complemented
6#21443812C211yesmaximal, characteristic, complemented
7C61×#1342C211yesmaximal, characteristic, complemented
8C2×Q16×D61C411yescharacteristic
9C2×#14519785C2211yescharacteristic
10#16840335—21no—
11M32×D61—21no—
12#16840021C411yescharacteristic
13#16840020C2211yescharacteristic
14#16840017C411yescharacteristic
15#16840002—21no—
16#16840001—21no—
17C2×#14519712—21no—
18#16839797C411yescharacteristic
19#16839769—21no—
20#16839744—21no—
21#16839544C2211yescharacteristic
22#16839543—21no—
23#16839520C2211yescharacteristic
24C61×#470C2211yescharacteristic
25C61×#349C2211yescharacteristic
26C61×#338C2211yescharacteristic
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×#11758194—21no—
36C2×#11758189C2×C411yescharacteristic
37C2×#11758189D411yescharacteristic
38C22×#92418D411yescharacteristic
39C2×C8×D61C2×C411yescharacteristic
40C2×C8×D61D411yescharacteristic
41#14519776—41no—
42#14519763—41no—
43#14519762—41no—
44#14519759—41no—
45#14519757—21no—
46#14519756—21no—
47#14519753—21no—
48#14519735—21no—
49#14519717—21no—
50#14519716—41no—
51C16×D61—41no—
52#14519712—41no—
53#14519712—41no—
54#14519710—21no—
55#14519709C2×C411yescharacteristic
56#14519702—21no—
57#14519701—21no—
58#14519700—21no—
59#14519695—41no—
60#14519693—41no—
61C2×Q16×C61C2×C411yescharacteristic
62C2×M16×C61—21no—
63C2×M16×C61C2311yescharacteristic
64C22⋊Q8×C61—21no—
65M32×C61—21no—
66#106×C61C2×C411yescharacteristic
67#103×C61—21no—
68C2×Q8×C61—21no—
69C22×C4×C61C2×D411yescharacteristic
70C2×C22⋊D61SD1611yescharacteristic
71C2×C22⋊D61SD1611yescharacteristic
72Q8×D61—42no—
73Q8×D61—42no—
74#11758222—41no—
75#11758222—41no—
76C2×D244D811yescharacteristic
77C2×C4×D61—41no—
78C2×C4×D61C22⋊C411yescharacteristic
79C2×C4×D61C22⋊C411yescharacteristic
80C2×Dic122—21no—
81C2×Dic122—41no—
82C2×Dic122D811yescharacteristic
83Q16×C61—41no—
84M16×C61—41no—
85M16×C61—41no—
86M16×C61—41no—
87C2×C8×C61—41no—
88C2×C8×C61C22×C411yescharacteristic
89C2×C8×C61C2×D411yescharacteristic
90C4⋊C4×C61—21no—
91C4⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊Dic61—41no—
94#11758202—41no—
95#11758202—41no—
96#11758198—41no—
97#11758198—41no—
98#11758198—41no—
99C4⋊Dic61—21no—
100C4⋊Dic61—41no—
101#11758196—41no—
102#11758196—41no—
103C4×Dic61—41no—
104#11758194—41no—
105#11758194—41no—
106C2×#92418—21no—
107C2×#92418—21no—
108C2×#92418—41no—
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×SD1611yescharacteristic
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×C61C2×C22⋊C411yescharacteristic
130C2×C4×C61C2×D811yescharacteristic
131C22⋊D61—41no—
132C22⋊D61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61#10211yescharacteristic
136C2×Dic61#10211yescharacteristic
137D244—41no—
138C4×D61—21no—
139C4×D61—21no—
140C4×D61—21no—
141C4×D61—21no—
142C4×D61—82no—
143Dic122—41no—
144Dic122—41no—
145Dic122—41no—
146Dic122—82no—
147C8×C61—21no—
148C8×C61—21no—
149C8×C61—81no—
150#92418—21no—
151#92418—81no—
152#92418#9911yescharacteristic
153#92418#9911yescharacteristic
154#11871—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×C61C2×#9911yescharacteristic
166C4×C61#42111yescharacteristic
167Dic61—21no—
168Dic61—21no—
169Dic61—81no—
170Dic61—81no—
171#3059—611no—
172#2115—611no—
173#1813—611no—
174#1342—611no—
175#1342—611no—
176#1342—611no—
177#1342D6111yescharacteristic, complemented
178C2×C61—21no—
179C2×C61—41no—
180C2×C61#151911yescharacteristic
181D61—41no—
182D61—41no—
183C22×Q16—611no—
184C2×#131—611no—
185#536—1221no—
186C2×M32—1221no—
187#470—611no—
188#470—611no—
189#470—611no—
190#470C2×D6111yescharacteristic
191#467—1221no—
192#467—1221no—
193C2×#108—1221no—
194#427—611no—
195#419—1221no—
196#413—1221no—
197#349—611no—
198#349—1221no—
199#349C2×D6111yescharacteristic
200#338—611no—
201#338C2×D6111yescharacteristic
202C61#1187111yescharacteristic, complemented, Sylow
203C2×#41—611no—
204C22×Q8—1221no—
205C2×Q16—611no—
206C2×Q16—1221no—
207C2×Q16—2441no—
208C2×Q16—2441no—
209C2×Q16C4×D6111yescharacteristic
210#131—2441no—
211#131—2441no—
212C2×M16—21no—
213C2×M16—611no—
214C2×M16—611no—
215C2×M16—1221no—
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—1221no—
229M32—21no—
230M32—1221no—
231M32—2441no—
232C2×C16—2441no—
233#108—2441no—
234#108—2441no—
235#107—1221no—
236#106—611no—
237#106C4×D6111yescharacteristic
238#103—21no—
239#103—1221no—
240#103—1221no—
241#103—1221no—
242#101—2441no—
243#100—2441no—
244#41—2441no—
245#41—2441no—
246C2×Q8—21no—
247C2×Q8—611no—
248C2×Q8—1221no—
249C2×Q8—2441no—
250C2×Q8—2442no—
251C2×Q8—2442no—
252C2×D4—611no—
253C2×D4—611no—
254C2×D4—611no—
255C22×C4—611no—
256C22×C4—611no—
257C22×C4—2441no—
258C22×C4D4×D6111yescharacteristic
259Q16—41no—
260Q16—2441no—
261Q16—2441no—
262Q16—2441no—
263M16—41no—
264M16—41no—
265M16—41no—
266M16—2441no—
267M16—2441no—
268M16—2441no—
269M16—2441no—
270C2×C8—41no—
271C2×C8—611no—
272C2×C8—611no—
273C2×C8—1221no—
274C2×C8—1221no—
275C2×C8—1221no—
276C2×C8—1221no—
277C2×C8—1221no—
278C2×C8—1221no—
279C2×C8—2441no—
280C2×C8C2×C4×D6111yescharacteristic
281C2×C8D4×D6111yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4—1221no—
285C4⋊C4—2441no—
286C4⋊C4—2441no—
287C4⋊C4—2441no—
288C22⋊C4—41no—
289C22⋊C4—2441no—
290C22⋊C4—2441no—
291C22⋊C4—2441no—
292C22⋊C4—2441no—
293C42—2441no—
294C16—41no—
295C16—2441no—
296C23—611no—
297C23—611no—
298C23SD16×D6111yescharacteristic
299Q8—41no—
300Q8—41no—
301Q8—4882no—
302Q8—2441no—
303Q8—2441no—
304Q8—2441no—
305D4—2441no—
306D4—2441no—
307D4—2441no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—611no—
313C2×C4—611no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—4882no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4C22⋊C4×D6111yescharacteristic
322C2×C4D8×D6111yescharacteristic
323C8—21no—
324C8—21no—
325C8—81no—
326C8—611no—
327C8—611no—
328C8—1221no—
329C8—4881no—
330C22—21no—
331C22—41no—
332C22—1221no—
333C22—1221no—
334C22—2441no—
335C22—2441no—
336C22#102×D6111yescharacteristic
337C4—21no—
338C4—81no—
339C4—81no—
340C4—1221no—
341C4—1221no—
342C4—4881no—
343C4—4881no—
344C4#99×D6111yescharacteristic
345C4#1683977711yescharacteristic
346C2—21no—
347C2—41no—
348C2—2441no—
349C2—2441no—
350C2#2144438511yescharacteristic
subgroupG/H
centerC2#21444385
commutatorC2×C8×C61C22×C4
FrattiniC2×M16C22×D61
FittingC61×#1342C2
socleC2×C61#1519
prime radicalC2×#11758222D4
cosocleC2×M16×C61C23
hypercenter#1342D61
hypocenterC61#11871
LCSG ▷ C2×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1342

Complements

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

NH ≅ G/Ntype
4. #21443814C2semidirect
5. #21443813C2semidirect
6. #21443812C2semidirect
7. C61×#1342C2semidirect
177. #1342D61semidirect
202. C61#11871semidirect

Maximal subgroups

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

subgroupindex
#214554722
#214474382
#214458682
#214438142
#214438132
#214438122
C61×#13422
#1187161