Finite Group Databasebeta

← #29591547

Subgroups — #29591547

Order 15616.

360 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 79 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#21456278C211yesmaximal, characteristic
2#21447631C211yesmaximal, characteristic, complemented
3#21443859C212yesmaximal, complemented
4#21443857C212yesmaximal, complemented
5C61×#2165C211yesmaximal, characteristic, complemented
6C2×#14519810—21no—
7C2×#14519810—21no—
8#16840312C411yescharacteristic
9C2×#14519777C411yescharacteristic
10#16840156—21no—
11#16840156—21no—
12#16840142C411yescharacteristic
13#16840142C2211yescharacteristic
14#16840135C411yescharacteristic
15#16840135C2211yescharacteristic
16#16839583C2212yes—
17#16839544—21no—
18#16839540—21no—
19C4×M16×C61C2211yescharacteristic
20C61×#364C2212yes—
21C61×#349—21no—
22C61×#347—21no—
23C2×#11758222—21no—
24C22×Dic122—21no—
25#14519810—44no—
26#14519810—44no—
27C2×Dic244—21no—
28C2×Dic244—21no—
29C2×Dic244—21no—
30C2×Dic244—21no—
31C2×#11758190—21no—
32C2×#11758190—21no—
33C2×#11758190—21no—
34C2×#11758190—21no—
35#14519777—21no—
36#14519777—41no—
37#14519777C2×C411yescharacteristic
38#14519777C2×C411yescharacteristic
39#14519777C2×C411yescharacteristic
40#14519774C2×C411yescharacteristic
41#14519765—21no—
42#14519751—41no—
43C4×D244—41no—
44C2×C4⋊Dic61—41no—
45#14519717—22no—
46C2×M16×C61D411yescharacteristic
47C2×M16×C61D411yescharacteristic
48C2×C42×C61C2×C411yescharacteristic
49M32×C61—22no—
50#97×C61D411yescharacteristic
51#97×C61D411yescharacteristic
52C4×C8×C61C2×C411yescharacteristic
53C4×C8×C61C2311yescharacteristic
54C22×C4×C61—21no—
55C22×C4×C61C22⋊C411yescharacteristic
56C2×C22⋊D61—41no—
57C22×Dic61—41no—
58#11758222—42no—
59#11758222—42no—
60C2×D244—21no—
61C2×D244—41no—
62C2×C4×D61—41no—
63C2×Dic122—21no—
64C2×Dic122—21no—
65C2×Dic122—21no—
66C2×Dic122—42no—
67C2×Dic122—42no—
68C2×Dic122—41no—
69M16×C61—22no—
70M16×C61—22no—
71C2×C8×C61C22⋊C411yescharacteristic
72C2×C8×C61C22⋊C411yescharacteristic
73C2×C8×C61C2×D411yescharacteristic
74C2×C8×C61C2×D411yescharacteristic
75C42×C61—21no—
76C42×C61C22⋊C411yescharacteristic
77C42×C61C22×C411yescharacteristic
78#11758198—42no—
79C4⋊Dic61—42no—
80C4⋊Dic61—41no—
81C4⋊Dic61—41no—
82C4⋊Dic61—41no—
83Dic244—42no—
84Dic244—42no—
85Dic244—42no—
86Dic244—42no—
87#11758190—42no—
88#11758190—42no—
89#11758190—42no—
90#11758190—42no—
91C16×C61—42no—
92#11758185—42no—
93C23×C61#10011yescharacteristic
94C22×D61—41no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—42no—
100C2×C4×C61#10011yescharacteristic
101C2×C4×C61C2×C22⋊C411yescharacteristic
102C22⋊D61—82no—
103C2×Dic61—41no—
104C2×Dic61—41no—
105C2×Dic61—41no—
106C2×Dic61—82no—
107D244—41no—
108D244—41no—
109D244—82no—
110C4×D61—82no—
111Dic122—41no—
112Dic122—41no—
113Dic122—41no—
114Dic122—41no—
115Dic122—41no—
116Dic122—41no—
117Dic122—82no—
118C8×C61—21no—
119C8×C61—21no—
120C8×C61AGL(1,ℤ/8)12yes—
121C8×C61AGL(1,ℤ/8)12yes—
122#11993—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—41no—
125C22×C61C2×#10011yescharacteristic
126C2×D61—82no—
127C2×D61—81no—
128C4×C61—22no—
129C4×C61—21no—
130C4×C61—41no—
131C4×C61#41811yescharacteristic
132C4×C61#41811yescharacteristic
133Dic61—81no—
134Dic61—81no—
135Dic61—81no—
136#3152—611no—
137#2165—611no—
138#2165D6111yescharacteristic, complemented
139#1359—612no—
140#1358—612no—
141C2×C61—21no—
142C2×C61—41no—
143C2×C61#155511yescharacteristic
144D61—161no—
145C2×#137—1221no—
146C2×#137—1221no—
147#532—611no—
148C2×C4⋊Q8—611no—
149#497—1221no—
150#497—1221no—
151#494—611no—
152#494—611no—
153#492—611no—
154#492—611no—
155C4×M16C2×D6111yescharacteristic
156#364—612no—
157#364C2×D6112yes—
158#349—21no—
159#349—1221no—
160#347—21no—
161#347—1221no—
162C61#1199311yescharacteristic, complemented, Sylow
163C2×#41—1221no—
164C22×Q8—1221no—
165#137—2444no—
166#137—2444no—
167C2×Q16—1221no—
168C2×Q16—1221no—
169C2×Q16—1221no—
170C2×Q16—1221no—
171C2×SD16—1221no—
172C2×SD16—1221no—
173C2×SD16—1221no—
174C2×SD16—1221no—
175C2×M16D24411yescharacteristic
176C2×M16C22⋊D6111yescharacteristic
177C4⋊Q8—611no—
178C4⋊Q8—611no—
179C4⋊Q8—611no—
180C4⋊Q8—1221no—
181C4⋊Q8—2441no—
182#127—611no—
183#124—1221no—
184#121—2441no—
185C4×D4—2441no—
186C2×C4⋊C4—2441no—
187C2×C42C4×D6111yescharacteristic
188M32—22no—
189M32—1222no—
190#97D24411yescharacteristic
191#97C22⋊D6111yescharacteristic
192C4×C8C4×D6111yescharacteristic
193C4×C8C22×D6111yescharacteristic
194#41—2442no—
195#41—2442no—
196C2×Q8—1221no—
197C2×Q8—1221no—
198C2×Q8—1221no—
199C2×Q8—2442no—
200C2×Q8—2442no—
201C2×Q8—2441no—
202C2×D4—1221no—
203C2×D4—2441no—
204C2×D4—2441no—
205C22×C4—21no—
206C22×C4—2441no—
207C22×C4—2441no—
208C22×C4#1175819811yescharacteristic
209Q16—2442no—
210Q16—2442no—
211Q16—2442no—
212Q16—2442no—
213SD16—2442no—
214SD16—2442no—
215SD16—2442no—
216SD16—2442no—
217M16—22no—
218M16—22no—
219C2×C8#1175819811yescharacteristic
220C2×C8#1175819811yescharacteristic
221C2×C8C2×D24411yescharacteristic
222C2×C8C2×C22⋊D6111yescharacteristic
223C4⋊C4—2441no—
224C4⋊C4—2441no—
225C4⋊C4—2441no—
226C4⋊C4—2442no—
227C22⋊C4—2442no—
228C42—21no—
229C42#1175819811yescharacteristic
230C42C2×C4×D6111yescharacteristic
231C16—42no—
232C16—2442no—
233C23—2441no—
234C23#1451969311yescharacteristic
235Q8—4882no—
236Q8—2441no—
237Q8—2441no—
238Q8—2441no—
239Q8—2441no—
240Q8—2441no—
241Q8—2441no—
242D4—4882no—
243D4—4882no—
244D4—2441no—
245D4—2441no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—4882no—
252C2×C4—4882no—
253C2×C4—2441no—
254C2×C4—2441no—
255C2×C4—2441no—
256C2×C4#1451969311yescharacteristic
257C2×C4C2×#1175819811yescharacteristic
258C8—21no—
259C8—21no—
260C8#1451980212yes—
261C8#1451980512yes—
262C22—21no—
263C22—41no—
264C22—4882no—
265C22—4881no—
266C22C2×#1451969311yescharacteristic
267C4—22no—
268C4—21no—
269C4—41no—
270C4—4881no—
271C4—4881no—
272C4—4881no—
273C4#1683976611yescharacteristic
274C4#1683976811yescharacteristic
275C2—21no—
276C2—41no—
277C2—9761no—
278C2#2144453211yescharacteristic
subgroupG/H
centerC2#21444532
commutatorC42×C61C22×C4
FrattiniC4×C8C22×D61
FittingC61×#2165C2
socleC2×C61#1555
prime radical#16840312C4
cosocleC4×C8×C61C23
hypercenter#2165D61
hypocenterC61#11993
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2165

Complements

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

NH ≅ G/Ntype
2. #21447631C2semidirect
3. #21443859C2semidirect
4. #21443857C2semidirect
5. C61×#2165C2semidirect
138. #2165D61semidirect
162. C61#11993semidirect

Maximal subgroups

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

subgroupindex
#214562782
#214476312
#214438592
#214438572
C61×#21652
#1199361