Finite Group Databasebeta

← #30595190

Subgroups — #30595190

Order 15616.

290 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 46 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#21462112C211yesmaximal, characteristic
2#21462092C211yesmaximal, characteristic
3#21446647C211yesmaximal, characteristic
4#16840748—21no—
5#16840741—21no—
6#16840673C412yescomplemented
7#16840671—21no—
8#16840669—21no—
9#16840667C412yescomplemented
10#16840312C2211yescharacteristic
11#16839786C411yescharacteristic, complemented
12C2×#14519705—21no—
13C2×#14519705—21no—
14C43×C61C411yescharacteristic, complemented
15D61⋊M16—41no—
16#14519856—41no—
17#14519838—22no—
18#14519837—41no—
19#14519836—22no—
20#14519835—41no—
21C2×#11758222D411yescharacteristic
22C2×#11758222D411yescharacteristic
23C2×#11758194—21no—
24#14519777C2×C411yescharacteristic
25#14519774C2×C411yescharacteristic
26#14519765D411yescharacteristic
27#14519765Q811yescharacteristic
28#14519751—41no—
29C4×D244—41no—
30C4⋊#92418—41no—
31#14519705—42no—
32#14519705—42no—
33C2×C42×C61—21no—
34C2×C42×C61—41no—
35C2×C42×C61C2×C411yescharacteristic
36C22×C4×C61—21no—
37C22×C4×C61—41no—
38C22×C4×C61C22⋊C411yescharacteristic
39C2×C22⋊D61—21no—
40C2×C22⋊D61—21no—
41#11758222—41no—
42#11758222—41no—
43#11758222—41no—
44#11758222—41no—
45C2×D244—41no—
46C2×D244C22⋊C411yescharacteristic
47C2×D244C4⋊C411yescharacteristic
48C2×C4×D61—21no—
49C2×C4×D61—21no—
50C2×Dic122C22⋊C411yescharacteristic
51C2×Dic122C4⋊C411yescharacteristic
52C22⋊#6923—42no—
53#11758217—42no—
54C4⋊#6923—81no—
55C4×#6923—81no—
56#11758213—81no—
57D61⋊C8—81no—
58C42×C61—21no—
59C42×C61—22no—
60C42×C61—41no—
61C42×C61—44no—
62C42×C61C4211yescharacteristic
63C42×C61C4⋊C411yescharacteristic
64#11758198—41no—
65#11758198—41no—
66C4⋊Dic61—41no—
67#11758194—41no—
68#11758194—41no—
69C2×#92418—41no—
70C23×C61#9911yescharacteristic
71C22×D61—21no—
72C22×D61—21no—
73C2×#6923—82no—
74C2×C4×C61—21no—
75C2×C4×C61—21no—
76C2×C4×C61—21no—
77C2×C4×C61—21no—
78C2×C4×C61—42no—
79C2×C4×C61—42no—
80C2×C4×C61—44no—
81C2×C4×C61#9511yescharacteristic
82C2×C4×C61#9911yescharacteristic
83C22⋊D61—81no—
84C22⋊D61—81no—
85C2×Dic61—21no—
86C2×Dic61—21no—
87D244—41no—
88D244—41no—
89D244—82no—
90C4×D61—41no—
91C4×D61—41no—
92C4×D61—41no—
93C4×D61—41no—
94Dic122—41no—
95Dic122—41no—
96#92420—82no—
97#92418—81no—
98#7497—611nomaximal, Sylow
99C22×C61—21no—
100C22×C61—41no—
101C22×C61#34211yescharacteristic
102C2×D61—41no—
103C2×D61—41no—
104C2×D61—81no—
105C2×D61—81no—
106#6923—162no—
107C4×C61—22no—
108C4×C61—21no—
109C4×C61—41no—
110C4×C61—44no—
111C4×C61#35311yescharacteristic
112C4×C61#35411yescharacteristic
113Dic61—41no—
114Dic61—41no—
115#2132—611no—
116#2126—611no—
117#1962—611no—
118C2×C61—21no—
119C2×C61—41no—
120C2×C61#139311yescharacteristic
121D61—81no—
122D61—81no—
123#532—611no—
124#423—611no—
125C2×#104—1221no—
126C2×#104—1221no—
127#413—1221no—
128#410—1221no—
129C43#692311yescharacteristic, complemented
130#356—612no—
131#355—1221no—
132#354—1221no—
133#353—612no—
134C61#749711yescharacteristic, complemented, Sylow
135C2×#41—611no—
136C2×#41—611no—
137C2×M16—1221no—
138C2×M16—2441no—
139C4⋊Q8—611no—
140#127—611no—
141#124—611no—
142#124—611no—
143#121—2441no—
144C4×D4—2441no—
145#117—2441no—
146C2×C42—21no—
147C2×C42—41no—
148C2×C42C2×#692311yescharacteristic
149C4⋊C8—2441no—
150#104—2442no—
151#104—2442no—
152#101—1222no—
153#100—2441no—
154#99—1222no—
155#99—2441no—
156#41—2441no—
157#41—2441no—
158#41—2441no—
159#41—2441no—
160C2×Q8—611no—
161C2×Q8—611no—
162C2×D4—611no—
163C2×D4—611no—
164C2×D4—1221no—
165C2×D4—1221no—
166C2×D4—2441no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—1221no—
170C22×C4—1221no—
171C22×C4C22⋊#692311yescharacteristic
172M16—4881no—
173M16—2441no—
174M16—2441no—
175M16—2442no—
176C2×C8—4881no—
177C2×C8—2441no—
178C4⋊C4—4881no—
179C4⋊C4—2441no—
180C22⋊C4—2441no—
181C22⋊C4—2441no—
182C22⋊C4—2442no—
183C42—21no—
184C42—22no—
185C42—41no—
186C42—44no—
187C42—4881no—
188C42C4×#692311yescharacteristic
189C42C4⋊#692311yescharacteristic
190C23—1221no—
191C23—1221no—
192C23#1451983511yescharacteristic
193Q8—2441no—
194Q8—2441no—
195D4—4882no—
196D4—4881no—
197D4—4881no—
198D4—2441no—
199D4—2441no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—1221no—
208C2×C4—1221no—
209C2×C4—4882no—
210C2×C4—2441no—
211C2×C4—2441no—
212C2×C4—2441no—
213C2×C4—2441no—
214C2×C4(C2×#6923)⋊C411yescharacteristic
215C2×C4#1451983611yescharacteristic
216C8—4882no—
217C8—4881no—
218C22—21no—
219C22—41no—
220C22—4881no—
221C22—4881no—
222C22—2441no—
223C22—2441no—
224C22#1684064011yescharacteristic
225C4—22no—
226C4—21no—
227C4—41no—
228C4—44no—
229C4—9762no—
230C4—2441no—
231C4—2441no—
232C4#1684066711yescharacteristic
233C4#1684066911yescharacteristic
234C2—21no—
235C2—41no—
236C2—4881no—
237C2—4881no—
238C2#2146091311yescharacteristic
subgroupG/H
centerC4#16840667
commutatorC42×C61C42
FrattiniC2×C42C2×#6923
FittingC43×C61C4
socleC2×C61#1393
prime radical, cosocle#16840312C22
hypercenterC43#6923
hypocenterC61#7497
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C43

Complements

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

NH ≅ G/Ntype
6. #16840673C4semidirect
9. #16840667C4semidirect
11. #16839786C4semidirect
14. C43×C61C4semidirect
129. C43#6923semidirect
134. C61#7497semidirect

Maximal subgroups

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

subgroupindex
#214621122
#214620922
#214466472
#749761