Finite Group Databasebeta

← #29593068

Subgroups — #29593068

Order 15616.

442 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 63 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#21447164C211yesmaximal, characteristic
2#21444792C212yesmaximal
3#21443849C212yesmaximal
4#21443765C211yesmaximal, characteristic
5C61×#2551C211yesmaximal, characteristic
6#16840393C2211yescharacteristic
7#16839918—22no—
8#16839802—21no—
9#16839531C2211yescharacteristic
10#16839522—21no—
11#16839498—22no—
12#16839488C2212yes—
13C2×C4⋊Q8×C61—22no—
14C61×#395—21no—
15C61×#395C2212yes—
16C61×#389—22no—
17C61×#384C2211yescharacteristic
18C4×C4⋊C4×C61—21no—
19C2×#11758202—42no—
20C2×#11758194—21no—
21#14519776—41no—
22#14519768—41no—
23#14519756—22no—
24#14519736—21no—
25#14519710—41no—
26#14519708—41no—
27#14519701—42no—
28C22⋊#92418—22no—
29C22×Q8×C61—21no—
30C22×Q8×C61D412yes—
31C4⋊Q8×C61—48no—
32C2×C4⋊C4×C61—21no—
33C2×C4⋊C4×C61—21no—
34C2×C4⋊C4×C61—22no—
35C2×C4⋊C4×C61—22no—
36C2×C4⋊C4×C61—22no—
37C2×C4⋊C4×C61D411yescharacteristic
38C2×C4⋊C4×C61C2311yescharacteristic
39C2×C42×C61—22no—
40C2×C42×C61D411yescharacteristic
41#95×C61—21no—
42#95×C61—22no—
43#95×C61D412yes—
44C2×Q8×C61—44no—
45C2×Q8×C61—44no—
46C2×Q8×C61—42no—
47C2×Q8×C61—81no—
48C22×C4×C61—21no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—21no—
52C22×C4×C61C2×D411yescharacteristic
53C22×C4×C61C2×D412yes—
54C2×Dic122—22no—
55C4⋊C4×C61—24no—
56C4⋊C4×C61—24no—
57C4⋊C4×C61—41no—
58C4⋊C4×C61—44no—
59C4⋊C4×C61—44no—
60C4⋊C4×C61—44no—
61C4⋊C4×C61—41no—
62C42×C61—21no—
63C42×C61—44no—
64C42×C61Q1612yescomplemented
65C22⋊Dic61—41no—
66#11758202—84no—
67C4⋊Dic61—22no—
68C4⋊Dic61—21no—
69C4⋊Dic61—21no—
70#11758196—42no—
71C4×Dic61—41no—
72#11758194—22no—
73#11758194—41no—
74C2×#92418—41no—
75C2×#92418—42no—
76C23×C61#12011yescharacteristic
77Q8×C61—44no—
78Q8×C61—82no—
79Q8×C61—84no—
80C2×C4×C61—22no—
81C2×C4×C61—22no—
82C2×C4×C61—22no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—24no—
86C2×C4×C61—24no—
87C2×C4×C61—22no—
88C2×C4×C61—42no—
89C2×C4×C61—42no—
90C2×C4×C61—42no—
91C2×C4×C61—42no—
92C2×C4×C61—42no—
93C2×C4×C61—42no—
94C2×C4×C61—41no—
95C2×C4×C61C2×Q1611yescharacteristic
96C2×C4×C61#13711yescharacteristic
97C2×Dic61—41no—
98C2×Dic61—41no—
99Dic122—82no—
100#92418—42no—
101#92418—82no—
102#12256—611nomaximal, Sylow
103C22×C61—21no—
104C22×C61—21no—
105C22×C61#45111yescharacteristic
106C22×C61#45611yescharacteristic
107C22×C61B(4,𝔽2)11yescharacteristic
108C4×C61—24no—
109C4×C61—21no—
110C4×C61—21no—
111C4×C61—44no—
112C4×C61—42no—
113C4×C61—44no—
114C4×C61—81no—
115Dic61—81no—
116Dic61—81no—
117#2551D6111yescharacteristic
118#2060—611no—
119#1605—612no—
120#1354—612no—
121#1325—611no—
122C2×C61—21no—
123C2×C61—21no—
124C2×C61#160511yescharacteristic
125C2×C61#219611yescharacteristic
126C2×C61#220511yescharacteristic
127#544—611no—
128C2×C4⋊Q8—22no—
129#451—1222no—
130#428—1221no—
131#395—21no—
132#395C2×D6112yes—
133#389—22no—
134#384C2×D6111yescharacteristic
135C4×C4⋊C4—21no—
136#343—611no—
137#339—1221no—
138#328—1222no—
139#324—612no—
140C61#1225611yescharacteristic, complemented, Sylow
141C22×Q8—21no—
142C22×Q8C22⋊D6112yes—
143C2×Q16—2442no—
144C2×M16—1221no—
145C4⋊Q8—48no—
146C4⋊Q8—2441no—
147#125—2441no—
148C22⋊Q8—1222no—
149#117—1221no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊D6111yescharacteristic
156C2×C4⋊C4C22×D6111yescharacteristic
157C2×C42—22no—
158C2×C42C22⋊D6111yescharacteristic
159#107—2441no—
160#106—2441no—
161#103—2442no—
162C22⋊C8—1222no—
163#95—21no—
164#95—22no—
165#95C22⋊D6112yes—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—81no—
170C2×Q8—1222no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×C22⋊D6111yescharacteristic
176C22×C4C2×C22⋊D6112yes—
177Q16—4884no—
178M16—1222no—
179M16—2441no—
180C2×C8—2442no—
181C2×C8—2441no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—1222no—
190C4⋊C4—1221no—
191C4⋊C4—1221no—
192C4⋊C4—2442no—
193C22⋊C4—2441no—
194C42—21no—
195C42—44no—
196C42—2441no—
197C42#1175820212yescomplemented
198C23#1451974611yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—4882no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—2441no—
219C2×C4—2441no—
220C2×C4C2×#1175820211yescharacteristic
221C2×C4#1451980811yescharacteristic
222C8—4882no—
223C8—2442no—
224C22—21no—
225C22—21no—
226C22#1683991811yescharacteristic
227C22#1683994511yescharacteristic
228C22#1683994911yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—4881no—
237C4—4881no—
238C2—21no—
239C2—21no—
240C2#2144479311yescharacteristic
241C2#2144780311yescharacteristic
242C2#2144784911yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2551C2
socleC22×C61B(4,𝔽2)
prime radicalC23×C61#120
hypercenter#2551D61
hypocenterC61#12256
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
64. C42×C61Q16semidirect
140. C61#12256semidirect
197. C42#11758202semidirect

Maximal subgroups

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

subgroupindex
#214471642
#214447922
#214438492
#214437652
C61×#25512
#1225661