Finite Group Databasebeta

← #30595180

Subgroups — #30595180

Order 15616.

270 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 42 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#21462091C211yesmaximal, characteristic
2#21461820C211yesmaximal, characteristic
3#21444452C211yesmaximal, characteristic
4#16840840C411yescharacteristic, complemented
5#16840761—21no—
6#16840743—21no—
7#16840742C411yescharacteristic, complemented
8#16840669—21no—
9#16840666C412yescomplemented
10#16840633—21no—
11#16840311C2211yescharacteristic
12#16839817—21no—
13#16839502C411yescharacteristic, complemented
14#16839501—21no—
15C61×#329C411yescharacteristic, complemented
16#14519864—42no—
17D61⋊M16—21no—
18C2×#11758217—41no—
19#14519856—21no—
20#14519856—41no—
21#14519842—42no—
22#14519836—22no—
23#14519836—41no—
24#14519835—41no—
25C2×#11758226D411yescharacteristic
26C2×#11758224C2×C411yescharacteristic
27#14519775Q811yescharacteristic
28#14519773C2×C411yescharacteristic
29#14519765D411yescharacteristic
30#14519750—41no—
31#14519737—41no—
32C42×D61D411yescharacteristic
33#14519706—41no—
34#14519689—41no—
35#14519688—21no—
36#124×C61C2×C411yescharacteristic
37#97×C61—21no—
38C2×Q8×C61C4⋊C411yescharacteristic
39C2×D4×C61C4211yescharacteristic
40C2×C22⋊D61—21no—
41C2×C22⋊D61—41no—
42C22×Dic61—21no—
43#11758226—41no—
44#11758226—41no—
45#11758224—41no—
46#11758224—41no—
47C2×D244—21no—
48C2×D244C22⋊C411yescharacteristic
49C2×C4×D61—21no—
50C2×C4×D61—21no—
51C2×C4×D61C22⋊C411yescharacteristic
52C2×Dic122C4⋊C411yescharacteristic
53C22⋊#6923—42no—
54C22⋊#6923—41no—
55#11758217—41no—
56#11758217—42no—
57C2×#92420—42no—
58C4⋊#6923—41no—
59C4⋊#6923—81no—
60C4×#6923—41no—
61C4×#6923—81no—
62#11758213—22no—
63D61⋊C8—41no—
64C2×C8×C61—41no—
65C22⋊C4×C61—41no—
66C42×C61C22⋊C411yescharacteristic
67#11758198—41no—
68#11758196—41no—
69C4×Dic61—21no—
70C4×Dic61C4⋊C411yescharacteristic
71C2×#92418—41no—
72C23×C61—21no—
73C22×D61—21no—
74C22×D61#9911yescharacteristic
75C2×#6923—81no—
76C2×#6923—82no—
77Q8×C61—41no—
78D4×C61—41no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61#9511yescharacteristic
82C22⋊D61—42no—
83C22⋊D61—82no—
84C2×Dic61—21no—
85C2×Dic61—21no—
86C2×Dic61—41no—
87C2×Dic61—41no—
88C2×Dic61#9911yescharacteristic
89D244—41no—
90D244—81no—
91C4×D61—21no—
92C4×D61—21no—
93C4×D61—41no—
94C4×D61—41no—
95C4×D61—81no—
96Dic122—41no—
97#92420—42no—
98C8×C61—81no—
99#92418—81no—
100#7494—611nomaximal, Sylow
101C22×C61—41no—
102C22×C61—81no—
103C22×C61#34211yescharacteristic
104C2×D61—21no—
105C2×D61—41no—
106C2×D61—41no—
107C2×D61—81no—
108#6923—161no—
109#6923—162no—
110C4×C61—21no—
111C4×C61—41no—
112C4×C61—41no—
113Dic61—41no—
114Dic61—41no—
115Dic61AΣL(1,𝔽16)11yescharacteristic
116Dic61#35211yescharacteristic
117#2126—611no—
118#1884—611no—
119#1536—611no—
120C2×C61—21no—
121C2×C61—81no—
122C2×C61#139011yescharacteristic
123D61—41no—
124D61—81no—
125#568—611no—
126#532—611no—
127#432—1221no—
128#421—1221no—
129#410—611no—
130#410—1221no—
131#354—1221no—
132#353—612no—
133#339—1221no—
134#329—611no—
135#329—1221no—
136#329#692311yescharacteristic, complemented
137C61#749411yescharacteristic, complemented, Sylow
138C2×#41—611no—
139C2×#41—611no—
140#131—2442no—
141C2×M16—1221no—
142C2×M16—2441no—
143C4⋊Q8—611no—
144#127—611no—
145#124—611no—
146#124C2×#692311yescharacteristic
147#121—2441no—
148C4×D4—2441no—
149#117—1221no—
150#117—2441no—
151C2×C42—611no—
152C4⋊C8—2441no—
153#104—2442no—
154#99—1222no—
155#99—2441no—
156#99—2441no—
157C22⋊C8—2441no—
158#97—21no—
159#97—1221no—
160#41—2441no—
161#41—2441no—
162#41—2441no—
163#41—2441no—
164C2×Q8—611no—
165C2×Q8C4⋊#692311yescharacteristic
166C2×D4—611no—
167C2×D4—1221no—
168C2×D4—1221no—
169C2×D4—2441no—
170C2×D4C4×#692311yescharacteristic
171C22×C4—611no—
172C22×C4—1221no—
173C22×C4—1221no—
174C22×C4—1221no—
175M16—1222no—
176M16—2441no—
177M16—2442no—
178C2×C8—41no—
179C2×C8—2441no—
180C2×C8—2441no—
181C2×C8—2442no—
182C4⋊C4—4881no—
183C4⋊C4—2441no—
184C4⋊C4—2441no—
185C22⋊C4—41no—
186C22⋊C4—2441no—
187C22⋊C4—2441no—
188C22⋊C4—2442no—
189C42—611no—
190C42—1221no—
191C42—4881no—
192C42—2441no—
193C42C22⋊#692311yescharacteristic
194C23—21no—
195C23—611no—
196C23—1221no—
197Q8—41no—
198Q8—2441no—
199D4—41no—
200D4—4881no—
201D4—4882no—
202D4—2441no—
203D4—2442no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—611no—
207C2×C4—1221no—
208C2×C4—1221no—
209C2×C4—1221no—
210C2×C4—1221no—
211C2×C4—4881no—
212C2×C4—4881no—
213C2×C4—4882no—
214C2×C4—2441no—
215C2×C4—2441no—
216C2×C4—2441no—
217C2×C4—2441no—
218C2×C4(C2×#6923)⋊C411yescharacteristic
219C8—81no—
220C8—4881no—
221C8—2442no—
222C22—41no—
223C22—81no—
224C22—1221no—
225C22—4881no—
226C22—2441no—
227C22—2441no—
228C22#1684063911yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—611no—
233C4—611no—
234C4—9762no—
235C4—9761no—
236C4—2441no—
237C4—2441no—
238C2—21no—
239C2—81no—
240C2—4881no—
241C2—2441no—
242C2#2146090511yescharacteristic
subgroupG/H
centerC2#21460905
commutatorC2×D4×C61C42
Frattini#124C2×#6923
FittingC61×#329C4
socleC2×C61#1390
prime radical, cosocle#16840311C22
hypercenter#329#6923
hypocenterC61#7494
LCSG ▷ C2×D4×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #329

Complements

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

NH ≅ G/Ntype
4. #16840840C4semidirect
7. #16840742C4semidirect
9. #16840666C4semidirect
13. #16839502C4semidirect
15. C61×#329C4semidirect
136. #329#6923semidirect
137. C61#7494semidirect

Maximal subgroups

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

subgroupindex
#214620912
#214618202
#214444522
#749461