Finite Group Databasebeta

← #29577647

Subgroups — #29577647

Order 15616.

456 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 99 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#21456222C211yesmaximal, characteristic
2#21444446C211yesmaximal, characteristic, complemented
3#21444326C212yesmaximal
4#21443712C212yesmaximal, complemented
5C61×#2371C211yesmaximal, characteristic, complemented
6#16840312C2211yescharacteristic
7#16840132C412yescomplemented
8#16840128C412yescomplemented
9#16840119—22no—
10#16839819C2212yes—
11#16839761—22no—
12C4×#11758194C2211yescharacteristic
13#16839504—22no—
14#16839495—22no—
15#16839495—22no—
16C2×#125×C61C2211yescharacteristic
17C61×#387—22no—
18C61×#387—22no—
19C4×C4⋊C4×C61—22no—
20C61×#376C2212yes—
21C2×#11758222—21no—
22C2×#11758194—21no—
23#14519777C2×C411yescharacteristic
24#14519774C2×C411yescharacteristic
25#14519765—21no—
26#14519751—22no—
27C4×D244—22no—
28C4⋊#92418—22no—
29#14519700—24no—
30#14519697—24no—
31C22⋊#92418—22no—
32#14519688—22no—
33#14519688C2×C412yes—
34C4×#92418—21no—
35#125×C61—22no—
36#125×C61C2×C412yes—
37#125×C61D412yes—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61D412yes—
41C2×C42×C61—22no—
42C2×C42×C61—22no—
43C2×C42×C61C2311yescharacteristic
44#95×C61—22no—
45#95×C61—22no—
46#95×C61—22no—
47#95×C61D412yes—
48#95×C61D412yes—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61C2×D411yescharacteristic
53C22×C4×C61C2×D412yes—
54C2×C22⋊D61—41no—
55#11758222—44no—
56C2×D244—21no—
57C2×D244—22no—
58C2×C4×D61—41no—
59C2×Dic122—21no—
60C4⋊C4×C61—24no—
61C4⋊C4×C61—24no—
62C4⋊C4×C61—22no—
63C4⋊C4×C61—44no—
64C4⋊C4×C61C22⋊C414yes—
65C42×C61—44no—
66C42×C61—44no—
67C42×C61C22×C411yescharacteristic
68C42×C61C2×D411yescharacteristic
69C42×C61#4112yes—
70#11758198—42no—
71C4⋊Dic61—22no—
72#11758194—42no—
73C2×#92418—22no—
74C2×#92418—42no—
75C23×C61#12011yescharacteristic
76C22×D61—41no—
77C2×C4×C61—22no—
78C2×C4×C61—24no—
79C2×C4×C61—24no—
80C2×C4×C61—42no—
81C2×C4×C61—42no—
82C2×C4×C61—42no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61C2×C22⋊C411yescharacteristic
89C2×C4×C61C4×D412yes—
90C2×C4×C61#12112yes—
91C2×C4×C61#12311yescharacteristic
92C22⋊D61—82no—
93C2×Dic61—41no—
94D244—42no—
95D244—82no—
96C4×D61—82no—
97Dic122—42no—
98#92418—42no—
99#92418—82no—
100#9442—611nomaximal, Sylow
101C22×C61—22no—
102C22×C61#38611yescharacteristic
103C22×C61#45412yes—
104C2×D61—81no—
105C2×D61—82no—
106C4×C61—21no—
107C4×C61—21no—
108C4×C61—22no—
109C4×C61—22no—
110C4×C61—44no—
111C4×C61—44no—
112C4×C61—82no—
113Dic61—81no—
114#3142—611no—
115#2371D6111yescharacteristic, complemented
116#1534—611no—
117#1505—612no—
118#1300—612no—
119C2×C61—22no—
120C2×C61#188811yescharacteristic
121C2×C61#190612yes—
122D61—161no—
123#532—611no—
124C2×#125C2×D6111yescharacteristic
125#491—612no—
126#490—612no—
127#488—1222no—
128#432—612no—
129#417—1222no—
130C4×M16—611no—
131#387—22no—
132#387—22no—
133C4×C4⋊C4—22no—
134#376C2×D6112yes—
135#330—1222no—
136#327—1222no—
137#327—1222no—
138C61#944211yescharacteristic, complemented, Sylow
139C2×#41—1221no—
140C2×M16—1221no—
141C4⋊Q8—611no—
142#127—611no—
143#125—22no—
144#125C4×D6112yes—
145#125D24412yes—
146#124—1221no—
147#121—1222no—
148C4×D4—1222no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊D6112yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C22×D6111yescharacteristic
155C4⋊C8—1222no—
156#103—1224no—
157#102—1224no—
158C22⋊C8—1222no—
159#97—612no—
160#97—1222no—
161C4×C8—1221no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95C22⋊D6112yes—
166#95C22⋊D6112yes—
167#41—2444no—
168C2×Q8—1221no—
169C2×D4—1221no—
170C2×D4—1222no—
171C2×D4—2441no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—2441no—
176C22×C4C2×C22⋊D6111yescharacteristic
177C22×C4C2×C22⋊D6112yes—
178M16—2442no—
179C2×C8—1222no—
180C2×C8—2442no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C4⋊C4—1222no—
186C4⋊C4#1175819814yes—
187C22⋊C4—2442no—
188C42—44no—
189C42—44no—
190C42C2×C4×D6111yescharacteristic
191C42C2×D24411yescharacteristic
192C42#1175822212yes—
193C23—2441no—
194C23#1451974611yescharacteristic
195Q8—2442no—
196D4—4882no—
197D4—4882no—
198D4—2442no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—4882no—
211C2×C4—2441no—
212C2×C4C2×#1175819811yescharacteristic
213C2×C4C4×C22⋊D6112yes—
214C2×C4#1451975112yes—
215C2×C4#1451976011yescharacteristic
216C8—4882no—
217C8—2442no—
218C22—22no—
219C22—4881no—
220C22—4882no—
221C22#1683964111yescharacteristic
222C22#1683993512yes—
223C4—21no—
224C4—21no—
225C4—22no—
226C4—22no—
227C4—44no—
228C4—44no—
229C4—82no—
230C4—4881no—
231C2—22no—
232C2—9761no—
233C2#2144622211yescharacteristic
234C2#2144632012yes—
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2371C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#2371D61
hypocenterC61#9442
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
2. #21444446C2semidirect
4. #21443712C2semidirect
5. C61×#2371C2semidirect
7. #16840132C4semidirect
8. #16840128C4semidirect
115. #2371D61semidirect
138. C61#9442semidirect

Maximal subgroups

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

subgroupindex
#214562222
#214444462
#214443262
#214437122
C61×#23712
#944261