Finite Group Databasebeta

← #29592583

Subgroups — #29592583

Order 15616.

472 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 59 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#21450848C211yesmaximal, characteristic, complemented
2#21444788C212yesmaximal
3#21443848C212yesmaximal, complemented
4#21443765C211yesmaximal, characteristic, complemented
5C61×#2039C211yesmaximal, characteristic
6C2×#14519776—22no—
7#16839900—22no—
8#16839682—21no—
9#16839682C2212yes—
10#16839658—22no—
11C4×C4⋊Dic61—21no—
12#16839496—22no—
13#16839488C2212yes—
14C61×#537C2211yescharacteristic
15C2×#104×C61—21no—
16C61×#417—21no—
17C61×#384C2211yescharacteristic
18C61×#343C2211yescharacteristic
19C22×Dic122—21no—
20C22×Dic122D412yes—
21C2×#11758200—42no—
22#14519776—48no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×#11758196—22no—
26C2×#11758196—22no—
27C2×C4×Dic61—22no—
28#14519700—42no—
29C22⋊#92418—22no—
30#14519684—21no—
31#14519684—22no—
32#14519684D412yes—
33C2×#41×C61—21no—
34C2×M16×C61—21no—
35C22⋊Q8×C61—41no—
36#121×C61—22no—
37#121×C61—41no—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61D411yescharacteristic
40C2×C4⋊C4×C61C2311yescharacteristic
41C2×C42×C61D411yescharacteristic
42#104×C61—42no—
43#103×C61—41no—
44#102×C61—41no—
45#41×C61—41no—
46#41×C61—41no—
47#41×C61—41no—
48#41×C61—41no—
49C2×Q8×C61—21no—
50C2×D4×C61—21no—
51C2×D4×C61—22no—
52C2×D4×C61—41no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—41no—
56C22×C4×C61C2×D411yescharacteristic
57C22×C4×C61C2×D412yes—
58C22×Dic61—22no—
59C22×Dic61—22no—
60C2×Dic122—42no—
61C2×Dic122—44no—
62C2×Dic122—44no—
63C2×Dic122—81no—
64M16×C61—41no—
65M16×C61—41no—
66C2×C8×C61—41no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—41no—
69C4⋊C4×C61—44no—
70C4⋊C4×C61—41no—
71C22⋊C4×C61—42no—
72C42×C61—21no—
73C42×C61SD1612yescomplemented
74#11758200—84no—
75C4⋊Dic61—24no—
76C4⋊Dic61—24no—
77#11758196—44no—
78#11758196—44no—
79C4×Dic61—44no—
80C2×#92418—42no—
81C23×C61—41no—
82C23×C61#12011yescharacteristic
83Q8×C61—41no—
84Q8×C61—41no—
85D4×C61—41no—
86D4×C61—41no—
87D4×C61—82no—
88D4×C61—81no—
89D4×C61—81no—
90C2×C4×C61—22no—
91C2×C4×C61—22no—
92C2×C4×C61—22no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—22no—
96C2×C4×C61—42no—
97C2×C4×C61—42no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—81no—
101C2×C4×C61—81no—
102C2×C4×C61C2×SD1611yescharacteristic
103C2×C4×C61#13711yescharacteristic
104C2×Dic61—24no—
105C2×Dic61—24no—
106C2×Dic61—42no—
107C2×Dic61—42no—
108C2×Dic61—42no—
109C2×Dic61—42no—
110Dic122—44no—
111Dic122—82no—
112Dic122—84no—
113C8×C61—81no—
114#92418—82no—
115#12180—611nomaximal, Sylow
116C22×C61—21no—
117C22×C61—21no—
118C22×C61—81no—
119C22×C61—81no—
120C22×C61—81no—
121C22×C61#44811yescharacteristic
122C22×C61#45411yescharacteristic
123C22×C61B(4,𝔽2)11yescharacteristic
124C4×C61—24no—
125C4×C61—21no—
126C4×C61—21no—
127C4×C61—42no—
128C4×C61—81no—
129C4×C61—81no—
130Dic61—44no—
131Dic61—44no—
132#2551—611no—
133#2039D6111yescharacteristic
134#1604—612no—
135#1354—612no—
136#1325—611no—
137C2×C61—21no—
138C2×C61—21no—
139C2×C61—161no—
140C2×C61#159911yescharacteristic
141C2×C61#219611yescharacteristic
142C2×C61#220511yescharacteristic
143#537C2×D6111yescharacteristic
144C2×C4⋊Q8—1222no—
145#448—1222no—
146C2×#104—21no—
147#417—21no—
148#395—612no—
149#395—1221no—
150#389—1222no—
151#384C2×D6111yescharacteristic
152C4×C4⋊C4—1221no—
153#343C2×D6111yescharacteristic
154#327—1222no—
155#324—612no—
156C61#1218011yescharacteristic, complemented, Sylow
157C2×#41—21no—
158C22×Q8—612no—
159C22×Q8—1221no—
160C2×SD16—2442no—
161C2×M16—21no—
162C4⋊Q8—2448no—
163C22⋊Q8—41no—
164#121—22no—
165#121—41no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—1221no—
168C2×C4⋊C4—1221no—
169C2×C4⋊C4—1222no—
170C2×C4⋊C4—1222no—
171C2×C4⋊C4C22⋊D6111yescharacteristic
172C2×C4⋊C4C22×D6111yescharacteristic
173C2×C42—1222no—
174C2×C42C22⋊D6111yescharacteristic
175#104—42no—
176#103—41no—
177#103—2442no—
178#102—41no—
179C22⋊C8—1222no—
180#95—612no—
181#95—1221no—
182#95—1222no—
183#41—41no—
184#41—41no—
185#41—41no—
186#41—41no—
187C2×Q8—21no—
188C2×Q8—4881no—
189C2×Q8—2442no—
190C2×Q8—2444no—
191C2×Q8—2444no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—41no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—1222no—
199C22×C4—1222no—
200C22×C4D4×D6112yes—
201C22×C4C2×C22⋊D6111yescharacteristic
202SD16—4884no—
203M16—41no—
204M16—41no—
205C2×C8—41no—
206C2×C8—2442no—
207C4⋊C4—22no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—44no—
211C4⋊C4—1224no—
212C4⋊C4—1224no—
213C4⋊C4—2444no—
214C4⋊C4—2444no—
215C22⋊C4—42no—
216C42—21no—
217C42—2444no—
218C42#1175820012yescomplemented
219C23—41no—
220C23#1451974511yescharacteristic
221Q8—41no—
222Q8—41no—
223Q8—4882no—
224Q8—4884no—
225Q8—2444no—
226D4—41no—
227D4—41no—
228D4—82no—
229D4—81no—
230D4—81no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—1224no—
244C2×C4—1224no—
245C2×C4—2442no—
246C2×C4—2442no—
247C2×C4—2442no—
248C2×C4—2442no—
249C2×C4C2×#1175820011yescharacteristic
250C2×C4#1451980711yescharacteristic
251C8—81no—
252C8—4882no—
253C22—21no—
254C22—21no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#1683990011yescharacteristic
259C22#1683993411yescharacteristic
260C22#1683994811yescharacteristic
261C4—24no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—81no—
266C4—81no—
267C4—2444no—
268C4—2444no—
269C2—21no—
270C2—21no—
271C2—161no—
272C2#2144475811yescharacteristic
273C2#2144780211yescharacteristic
274C2#2144784811yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2039C2
socleC22×C61B(4,𝔽2)
prime radicalC61×#537C22
hypercenter#2039D61
hypocenterC61#12180
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #21450848C2semidirect
3. #21443848C2semidirect
4. #21443765C2semidirect
73. C42×C61SD16semidirect
156. C61#12180semidirect
218. C42#11758200semidirect

Maximal subgroups

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

subgroupindex
#214508482
#214447882
#214438482
#214437652
C61×#20392
#1218061