Finite Group Databasebeta

← #29591539

Subgroups — #29591539

Order 15616.

280 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 69 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#21456449C211yesmaximal, characteristic
2#21447634C211yesmaximal, characteristic, complemented
3#21447583C211yesmaximal, characteristic, complemented
4#21443880C211yesmaximal, characteristic, complemented
5#21443859C211yesmaximal, characteristic, complemented
6#21443859C211yesmaximal, characteristic, complemented
7C61×#1366C211yesmaximal, characteristic, complemented
8#16840375C411yescharacteristic
9#16840368C411yescharacteristic
10M32×D61—21no—
11#16840135C2211yescharacteristic
12#16840132C411yescharacteristic
13#16840128C411yescharacteristic
14#16840009—21no—
15#16839992—21no—
16#16839861—21no—
17#16839849—21no—
18#16839716C2211yescharacteristic
19#16839583C2211yescharacteristic
20#16839583C2211yescharacteristic
21#16839582—21no—
22#16839543—21no—
23#16839539—21no—
24C61×#499C2211yescharacteristic
25C61×#364C2211yescharacteristic
26C61×#364C2211yescharacteristic
27C2×#11758190—21no—
28C2×#11758190—21no—
29C2×#11758189D411yescharacteristic
30C2×C8×D61D411yescharacteristic
31#14519777C2×C411yescharacteristic
32#14519774C2×C411yescharacteristic
33#14519772—41no—
34#14519768—41no—
35#14519762—41no—
36#14519757—21no—
37#14519752—21no—
38#14519752—41no—
39#14519742—21no—
40#14519739—21no—
41#14519734C2×C411yescharacteristic
42#14519717—21no—
43#14519717—21no—
44#14519716—41no—
45C16×D61—41no—
46#14519700—21no—
47#14519700—21no—
48#14519697—21no—
49#14519697—21no—
50#14519688C2×C411yescharacteristic
51#14519687D411yescharacteristic
52C8×Dic61D411yescharacteristic
53#125×C61C2×C411yescharacteristic
54#125×C61C2×C411yescharacteristic
55M32×C61—21no—
56M32×C61—21no—
57#107×C61—21no—
58#106×C61—21no—
59C4×C8×C61C2311yescharacteristic
60C2×D244—21no—
61C2×D244—41no—
62C2×D244—41no—
63C2×C4×D61—41no—
64C2×C4×D61C22⋊C411yescharacteristic
65C2×Dic122—21no—
66C2×C8×C61C2×D411yescharacteristic
67C2×C8×C61C2×D411yescharacteristic
68C4⋊C4×C61—21no—
69C4⋊C4×C61—21no—
70C4⋊C4×C61—41no—
71C4⋊C4×C61—41no—
72C42×C61C22×C411yescharacteristic
73#11758198—21no—
74#11758198—41no—
75#11758198—41no—
76#11758198—41no—
77C4⋊Dic61—41no—
78C4⋊Dic61—41no—
79#11758196—21no—
80#11758196—41no—
81C4×Dic61—41no—
82C4×Dic61C22⋊C411yescharacteristic
83C2×#92418C22⋊C411yescharacteristic
84C2×#92418C22⋊C411yescharacteristic
85#11758190—41no—
86#11758190—41no—
87#11758190—41no—
88#11758190—41no—
89#11758189—21no—
90#11758189—21no—
91C8×D61—21no—
92C8×D61—21no—
93C16×C61—41no—
94C16×C61—41no—
95#11758185—41no—
96#11758185—41no—
97C22×D61—41no—
98C22×D61#10011yescharacteristic
99C2×C4×C61—21no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61C2×C22⋊C411yescharacteristic
103C2×Dic61—21no—
104C2×Dic61—41no—
105C2×Dic61#10011yescharacteristic
106D244—41no—
107D244—41no—
108D244—161no—
109D244—161no—
110C4×D61—21no—
111C4×D61—21no—
112C4×D61—161no—
113Dic122—41no—
114Dic122—41no—
115C8×C61#13511yescharacteristic
116C8×C61#13511yescharacteristic
117C8×C61AGL(1,ℤ/8)11yescharacteristic
118C8×C61#13711yescharacteristic
119#92418—21no—
120#92418—21no—
121#11991—611nomaximal, Sylow
122C22×C61C2×#10011yescharacteristic
123C2×D61—21no—
124C2×D61—41no—
125C2×D61—81no—
126C2×D61—81no—
127C2×D61—81no—
128C4×C61—41no—
129C4×C61—81no—
130C4×C61—81no—
131C4×C61#41611yescharacteristic
132C4×C61#41711yescharacteristic
133Dic61—21no—
134Dic61—41no—
135Dic61—81no—
136#3174—611no—
137#2166—611no—
138#2153—611no—
139#1366—611no—
140#1366D6111yescharacteristic, complemented
141#1359—611no—
142#1359—611no—
143C2×C61—21no—
144C2×C61#155111yescharacteristic
145D61—41no—
146D61—161no—
147#541—611no—
148#540—611no—
149C2×M32—1221no—
150#499C2×D6111yescharacteristic
151#492—611no—
152#491—611no—
153#490—611no—
154#468—1221no—
155#465—1221no—
156#440—1221no—
157C4×SD16—1221no—
158#405—611no—
159#364—611no—
160#364—611no—
161#364—1221no—
162#364C2×D6111yescharacteristic
163#364C2×D6111yescharacteristic
164#349—1221no—
165#347—1221no—
166C61#1199111yescharacteristic, complemented, Sylow
167C2×SD16—1221no—
168C2×SD16—1221no—
169C2×M16—611no—
170C22×C8—611no—
171C4⋊Q8—611no—
172#127—611no—
173#126—2441no—
174#125—2441no—
175#125C4×D6111yescharacteristic
176#125C4×D6111yescharacteristic
177#123—2441no—
178C22⋊Q8—1221no—
179#121—1221no—
180#121—2441no—
181C4×Q8—1221no—
182C4×D4—1221no—
183#117—611no—
184M32—21no—
185M32—21no—
186M32—1221no—
187M32—1221no—
188M32—2441no—
189C2×C16—2441no—
190#107—21no—
191#106—21no—
192#103—1221no—
193#103—1221no—
194#102—1221no—
195#102—1221no—
196#97—611no—
197#97—611no—
198C4×C8—611no—
199C4×C8C22×D6111yescharacteristic
200C2×Q8—1221no—
201C2×D4—1221no—
202C2×D4—2441no—
203C2×D4—2441no—
204C22×C4—611no—
205C22×C4—2441no—
206SD16—2441no—
207SD16—2441no—
208SD16—2441no—
209SD16—2441no—
210M16—1221no—
211M16—1221no—
212C2×C8—611no—
213C2×C8—611no—
214C2×C8—1221no—
215C2×C8—1221no—
216C2×C8D4×D6111yescharacteristic
217C2×C8D4×D6111yescharacteristic
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—1221no—
223C4⋊C4—2441no—
224C4⋊C4—2441no—
225C4⋊C4—2441no—
226C22⋊C4—1221no—
227C22⋊C4—2441no—
228C22⋊C4—2441no—
229C22⋊C4—2441no—
230C42—611no—
231C42—2441no—
232C42C2×C4×D6111yescharacteristic
233C16—41no—
234C16—41no—
235C16—2441no—
236C16—2441no—
237C23—611no—
238C23—2441no—
239Q8—2441no—
240Q8—2441no—
241D4—9761no—
242D4—9761no—
243D4—2441no—
244D4—2441no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—611no—
249C2×C4—1221no—
250C2×C4—1221no—
251C2×C4—1221no—
252C2×C4—9761no—
253C2×C4—2441no—
254C2×C4C22⋊C4×D6111yescharacteristic
255C8—1221no—
256C8—1221no—
257C8#1451979811yescharacteristic
258C8#1451980011yescharacteristic
259C8#1451980411yescharacteristic
260C8#1451980911yescharacteristic
261C22—1221no—
262C22—4881no—
263C22—4881no—
264C22—4881no—
265C22—2441no—
266C22#100×D6111yescharacteristic
267C4—41no—
268C4—81no—
269C4—81no—
270C4—1221no—
271C4—4881no—
272C4—2441no—
273C4#1683975711yescharacteristic
274C4#1683976211yescharacteristic
275C2—21no—
276C2—9761no—
277C2—2441no—
278C2#2144451311yescharacteristic
subgroupG/H
centerC2#21444513
commutatorC42×C61C22×C4
FrattiniC4×C8C22×D61
FittingC61×#1366C2
socleC2×C61#1551
prime radical#16840368C4
cosocleC4×C8×C61C23
hypercenter#1366D61
hypocenterC61#11991
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1366

Complements

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

NH ≅ G/Ntype
2. #21447634C2semidirect
3. #21447583C2semidirect
4. #21443880C2semidirect
5. #21443859C2semidirect
6. #21443859C2semidirect
7. C61×#1366C2semidirect
140. #1366D61semidirect
166. C61#11991semidirect

Maximal subgroups

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

subgroupindex
#214564492
#214476342
#214475832
#214438802
#214438592
C61×#13662
#1199161