Finite Group Databasebeta

← #29592612

Subgroups — #29592612

Order 15616.

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

normal subgroups only

This group has 360 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#21454091C211yesmaximal, characteristic, complemented
2#21444856C212yesmaximal
3#21443845C212yesmaximal, complemented
4#21443763C211yesmaximal, characteristic, complemented
5C61×#2040C211yesmaximal, characteristic
6#16840078—22no—
7#16839704—22no—
8#16839701C2212yes—
9#16839683—22no—
10#16839682—21no—
11#16839628—21no—
12#16839496—22no—
13#16839488C2212yes—
14C61×#537C2211yescharacteristic
15C2×#104×C61—21no—
16C61×#417—21no—
17C61×#382C2211yescharacteristic
18C61×#343C2211yescharacteristic
19C22×Dic122—21no—
20C2×C4⋊Dic61—21no—
21C2×C4⋊Dic61—21no—
22C2×C4⋊Dic61D412yes—
23C2×#11758196—22no—
24#14519708—42no—
25#14519698—42no—
26C22⋊#92418—22no—
27#14519684—21no—
28#14519684—22no—
29#14519684—22no—
30#14519684—22no—
31#14519684D412yes—
32C2×#41×C61—21no—
33C2×M16×C61—21no—
34C22⋊Q8×C61—41no—
35#121×C61—22no—
36#121×C61—41no—
37C2×C4⋊C4×C61D411yescharacteristic
38C2×C4⋊C4×C61C2311yescharacteristic
39C2×C42×C61D411yescharacteristic
40#104×C61—42no—
41#103×C61—41no—
42#102×C61—41no—
43#95×C61—22no—
44#41×C61—41no—
45#41×C61—41no—
46#41×C61—41no—
47#41×C61—41no—
48C2×Q8×C61—21no—
49C2×D4×C61—21no—
50C2×D4×C61—22no—
51C2×D4×C61—41no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—41no—
55C22×C4×C61C2×D411yescharacteristic
56C22×C4×C61C2×D412yes—
57C22×Dic61—22no—
58C22×Dic61—22no—
59C2×Dic122—82no—
60C2×Dic122—81no—
61M16×C61—41no—
62M16×C61—41no—
63C2×C8×C61—41no—
64C4⋊C4×C61—22no—
65C4⋊C4×C61—41no—
66C4⋊C4×C61—41no—
67C22⋊C4×C61—42no—
68C42×C61—21no—
69C42×C61SD1612yes—
70C4⋊Dic61—42no—
71C4⋊Dic61—81no—
72C4⋊Dic61—81no—
73#11758196—82no—
74C2×#92418—42no—
75C23×C61—41no—
76C23×C61#12011yescharacteristic
77Q8×C61—41no—
78Q8×C61—41no—
79D4×C61—41no—
80D4×C61—41no—
81D4×C61—82no—
82D4×C61—81no—
83D4×C61—81no—
84C2×C4×C61—21no—
85C2×C4×C61—21no—
86C2×C4×C61—22no—
87C2×C4×C61—22no—
88C2×C4×C61—41no—
89C2×C4×C61—41no—
90C2×C4×C61—42no—
91C2×C4×C61—42no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61—81no—
95C2×C4×C61—81no—
96C2×C4×C61C2×SD1611yescharacteristic
97C2×C4×C61#13711yescharacteristic
98C2×Dic61—42no—
99C2×Dic61—42no—
100C2×Dic61—42no—
101C2×Dic61—42no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104Dic122—84no—
105C8×C61—81no—
106#92418—82no—
107#12185—611nomaximal, Sylow
108C22×C61—21no—
109C22×C61—21no—
110C22×C61—81no—
111C22×C61—81no—
112C22×C61—81no—
113C22×C61#44811yescharacteristic
114C22×C61#45411yescharacteristic
115C22×C61B(4,𝔽2)11yescharacteristic
116C4×C61—21no—
117C4×C61—21no—
118C4×C61—42no—
119C4×C61—42no—
120C4×C61—81no—
121C4×C61—81no—
122Dic61—82no—
123Dic61—82no—
124#2844—611no—
125#2040D6111yescharacteristic
126#1616—612no—
127#1353—612no—
128#1324—611no—
129C2×C61—21no—
130C2×C61—21no—
131C2×C61—161no—
132C2×C61#159911yescharacteristic
133C2×C61#220111yescharacteristic
134C2×C61#220211yescharacteristic
135#537C2×D6111yescharacteristic
136#481—1222no—
137C2×#104—21no—
138#417—21no—
139#400—612no—
140#400—1222no—
141#395—1222no—
142#395—1221no—
143#382—1221no—
144#382C2×D6111yescharacteristic
145#343C2×D6111yescharacteristic
146#327—1222no—
147#324—612no—
148C61#1218511yescharacteristic, complemented, Sylow
149C2×#41—21no—
150C22×Q8—1221no—
151C2×M16—21no—
152C22⋊Q8—41no—
153#121—22no—
154#121—41no—
155C2×C4⋊C4—612no—
156C2×C4⋊C4—1221no—
157C2×C4⋊C4—1221no—
158C2×C4⋊C4—1222no—
159C2×C4⋊C4C22⋊D6111yescharacteristic
160C2×C4⋊C4C22×D6111yescharacteristic
161C2×C42C22⋊D6111yescharacteristic
162#106—2442no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—2442no—
167C22⋊C8—1222no—
168#95—22no—
169#95—612no—
170#95—1222no—
171#95—1221no—
172#95—1222no—
173#95—1222no—
174#41—41no—
175#41—41no—
176#41—41no—
177#41—41no—
178C2×Q8—21no—
179C2×Q8—4882no—
180C2×Q8—4881no—
181C2×D4—21no—
182C2×D4—22no—
183C2×D4—41no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—1222no—
188C22×C4—1222no—
189C22×C4D4×D6112yes—
190C22×C4C2×C22⋊D6111yescharacteristic
191M16—41no—
192M16—41no—
193C2×C8—41no—
194C2×C8—2442no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—4882no—
199C4⋊C4—4881no—
200C4⋊C4—4881no—
201C4⋊C4—2442no—
202C22⋊C4—42no—
203C42—21no—
204C42#1175820012yes—
205C23—41no—
206C23#1451974511yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—4884no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—81no—
214D4—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—2442no—
228C2×C4—2442no—
229C2×C4—2442no—
230C2×C4—2442no—
231C2×C4—2442no—
232C2×C4—2442no—
233C2×C4C2×#1175820011yescharacteristic
234C2×C4#1451980711yescharacteristic
235C8—81no—
236C8—4882no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#1683990011yescharacteristic
243C22#1683993411yescharacteristic
244C22#1683994811yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—4882no—
252C4—4882no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2#2144475811yescharacteristic
257C2#2144783111yescharacteristic
258C2#2144783511yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2040C2
socleC22×C61B(4,𝔽2)
prime radicalC61×#537C22
hypercenter#2040D61
hypocenterC61#12185
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #21454091C2semidirect
3. #21443845C2semidirect
4. #21443763C2semidirect
148. C61#12185semidirect

Maximal subgroups

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

subgroupindex
#214540912
#214448562
#214438452
#214437632
C61×#20402
#1218561