Finite Group Databasebeta

← #30604147

Subgroups — #30604147

Order 15616.

330 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 66 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#21462189C212yesmaximal, complemented
2#21460765C212yesmaximal, complemented
3#21460752C212yesmaximal
4#21454451C211yesmaximal, characteristic
5#16840684—22no—
6#16840658C2214yes—
7#16840655—22no—
8C2×C42×D61C411yescharacteristic
9#16839818—22no—
10#16839814—22no—
11#16839786C2211yescharacteristic
12#16839725C411yescharacteristic
13#16839725C2211yescharacteristic
14M16×Dic61C411yescharacteristic
15M16×Dic61C2211yescharacteristic
16C61×#423C411yescharacteristic
17#14519851—24no—
18C22×C4×D61—21no—
19C22×C4×D61C2×C411yescharacteristic
20C2×#11758194C2×C411yescharacteristic
21C2×#11758194C2311yescharacteristic
22C2×C4×Dic61—21no—
23C2×C4×Dic61C2×C411yescharacteristic
24C42×D61—21no—
25C42×D61—21no—
26C42×D61—21no—
27C42×D61—21no—
28C4⋊#92418—22no—
29#14519689—22no—
30#14519689—22no—
31#14519687—21no—
32#14519687D412yes—
33C8×Dic61—21no—
34C8×Dic61D412yes—
35C2×M16×C61C2×C411yescharacteristic
36C2×M16×C61C2×C411yescharacteristic
37C2×C42×C61C2×C411yescharacteristic
38C4⋊C8×C61—22no—
39C23×D61M1611yescharacteristic
40C22×C4×C61—21no—
41C22×C4×C61C22×C411yescharacteristic
42C22×Dic61—21no—
43C22×Dic61M1611yescharacteristic
44C2×C4×D61—21no—
45C2×C4×D61—21no—
46C2×C4×D61—21no—
47C2×C4×D61—21no—
48C2×C4×D61—21no—
49C2×C4×D61—21no—
50C2×C4×D61—44no—
51C2×C4×D61—41no—
52C2×C4×D61—41no—
53M16×C61—41no—
54M16×C61—41no—
55C2×C8×C61—21no—
56C2×C8×C61C22⋊C412yes—
57C42×C61—21no—
58C42×C61M1611yescharacteristic
59C42×C61M1611yescharacteristic
60C4×Dic61—21no—
61C4×Dic61—21no—
62C4×Dic61—21no—
63C4×Dic61—21no—
64C4×Dic61—21no—
65C4×Dic61C22⋊C411yescharacteristic
66C4×Dic61C22⋊C411yescharacteristic
67#11758194—41no—
68#11758194—41no—
69C2×#92418—21no—
70C2×#92418C2×D412yes—
71#11758187—44no—
72C23×C61C2×M1611yescharacteristic
73C22×D61—21no—
74C22×D61—42no—
75C22×D61—41no—
76C2×C4×C61—21no—
77C2×C4×C61—21no—
78C2×C4×C61—21no—
79C2×C4×C61—21no—
80C2×C4×C61—42no—
81C2×C4×C61C2×C22⋊C411yescharacteristic
82C2×C4×C61C2×M1611yescharacteristic
83C2×Dic61—21no—
84C2×Dic61—21no—
85C2×Dic61—21no—
86C2×Dic61—42no—
87C2×Dic61—41no—
88C4×D61—24no—
89C4×D61—42no—
90C4×D61—42no—
91C4×D61—42no—
92C4×D61—42no—
93C4×D61—41no—
94C4×D61—41no—
95C8×C61—22no—
96C8×C61—41no—
97#92418—22no—
98#92418—41no—
99#11521—611nomaximal, Sylow
100C22×C61—21no—
101C22×C61—41no—
102C22×C61#40711yescharacteristic
103C2×D61—22no—
104C2×D61—41no—
105C2×D61—42no—
106C2×D61—41no—
107C2×D61—42no—
108C4×C61—22no—
109C4×C61—21no—
110C4×C61—41no—
111C4×C61#41011yescharacteristic
112C4×C61#41311yescharacteristic
113Dic61—22no—
114Dic61—41no—
115Dic61—41no—
116#2920—611no—
117#2165—612no—
118#1327—612no—
119#1320—612no—
120C2×C61—21no—
121C2×C61—41no—
122C2×C61#146411yescharacteristic
123D61—42no—
124C22×C42—611no—
125#432—1222no—
126#431—1222no—
127#423—611no—
128#423#692311yescharacteristic
129#407—611no—
130#407—611no—
131C4×M16—611no—
132C4×M16—611no—
133#364—1222no—
134#349—614no—
135#347—1222no—
136C61#1152111yescharacteristic, complemented, Sylow
137C23×C4—611no—
138C23×C4—1221no—
139C2×M16—611no—
140C2×M16—611no—
141C2×M16C2×#692311yescharacteristic
142C2×M16C2×#692311yescharacteristic
143C2×C42—611no—
144C2×C42—1221no—
145C2×C42—1221no—
146C2×C42—1221no—
147C2×C42—1221no—
148C2×C42—1221no—
149C2×C42C2×#692311yescharacteristic
150M32—1224no—
151C4⋊C8—22no—
152C4⋊C8—1222no—
153C22⋊C8—1222no—
154C22⋊C8—1222no—
155#97—612no—
156#97—1221no—
157C4×C8—612no—
158C4×C8—1221no—
159C24—611no—
160C22×C4—21no—
161C22×C4—611no—
162C22×C4—1221no—
163C22×C4—1221no—
164C22×C4—1221no—
165C22×C4—1221no—
166C22×C4—1221no—
167C22×C4—1221no—
168C22×C4—1221no—
169C22×C4—2441no—
170C22×C4—2441no—
171C22×C4—2444no—
172C22×C4C22×#692311yescharacteristic
173M16—41no—
174M16—41no—
175M16—2441no—
176M16—2441no—
177C2×C8—21no—
178C2×C8—612no—
179C2×C8—1221no—
180C2×C8C22⋊#692312yes—
181C42—21no—
182C42—611no—
183C42—611no—
184C42—1221no—
185C42—1221no—
186C42—1221no—
187C42—1221no—
188C42—1221no—
189C42#1175821311yescharacteristic
190C42#1175821311yescharacteristic
191C16—2444no—
192C23—1221no—
193C23—2441no—
194C23—2442no—
195C23D61⋊M1611yescharacteristic
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4—1224no—
202C2×C4—1221no—
203C2×C4—1221no—
204C2×C4—1221no—
205C2×C4—2442no—
206C2×C4—2442no—
207C2×C4—2442no—
208C2×C4—2442no—
209C2×C4—2442no—
210C2×C4—2441no—
211C2×C4—2441no—
212C2×C4—2441no—
213C2×C4C2×C22⋊#692311yescharacteristic
214C2×C4C2×#1175821311yescharacteristic
215C8—22no—
216C8—41no—
217C8—1222no—
218C8—2441no—
219C22—21no—
220C22—41no—
221C22—1222no—
222C22—2441no—
223C22—2442no—
224C22—2442no—
225C22—2441no—
226C22#1684073311yescharacteristic
227C4—22no—
228C4—21no—
229C4—41no—
230C4—1222no—
231C4—2441no—
232C4—2441no—
233C4#1684074311yescharacteristic
234C4#1684074911yescharacteristic
235C2—21no—
236C2—41no—
237C2—2442no—
238C2#2146107311yescharacteristic
subgroupG/H
centerC4#16840743
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#423C4
socleC2×C61#1464
prime radicalC23×D61M16
cosocleC2×#11758194C23
hypercenter#423#6923
hypocenterC61#11521
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #21462189C2semidirect
2. #21460765C2semidirect
136. C61#11521semidirect

Maximal subgroups

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

subgroupindex
#214621892
#214607652
#214607522
#214544512
#1152161