Finite Group Databasebeta

← #44506374

Subgroups — #44506374

Order 30848.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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#29507867C211yesmaximal, characteristic, complemented
2#29507850C211yesmaximal, characteristic
3#29507634C211yesmaximal, characteristic, complemented
4#29507454C211yesmaximal, characteristic, complemented
5#29507451C211yesmaximal, characteristic, complemented
6#29507443C211yesmaximal, characteristic, complemented
7#539×C241C211yesmaximal, characteristic, complemented
8C22×D964C2211yescharacteristic
9C2×#16838757—21no—
10C2×#16838755—21no—
11C2×#16838747—21no—
12C2×#16838747—21no—
13#21440562—21no—
14#21440562—21no—
15C22⋊#14515349C2211yescharacteristic
16C22⋊#14515349C2211yescharacteristic
17#21440549C2211yescharacteristic
18#126×C241—21no—
19#125×C241—21no—
20#123×C241—21no—
21#123×C241—21no—
22C22⋊Q8×C241—21no—
23C22⋊Q8×C241C2211yescharacteristic
24#121×C241C2211yescharacteristic
25C4×D4×C241—21no—
26C2×C4⋊C4×C241C2211yescharacteristic
27C23×D241—21no—
28C2×D964—21no—
29C2×D964—21no—
30C2×D964—42no—
31C22×Dic241—21no—
32#16838757—42no—
33#16838755—42no—
34C2×#14515349—21no—
35C2×#14515349—21no—
36#16838747—42no—
37#16838747—42no—
38C2×Q8×C241D411yescharacteristic
39C2×D4×C241—21no—
40C2×D4×C241D411yescharacteristic
41C22×C4×C241—21no—
42C22×C4×C241D411yescharacteristic
43C22×C4×C241D411yescharacteristic
44C22×C4×C241C2311yescharacteristic
45C4⋊C4×C241—21no—
46C4⋊C4×C241—21no—
47C4⋊C4×C241—21no—
48C4⋊C4×C241—21no—
49C4⋊C4×C241—21no—
50C4⋊C4×C241—21no—
51C4⋊C4×C241D411yescharacteristic
52C4⋊C4×C241D411yescharacteristic
53C22⋊C4×C241—21no—
54C22⋊C4×C241—21no—
55C22⋊C4×C241—21no—
56C22⋊C4×C241—21no—
57C22⋊C4×C241—21no—
58C42×C241—21no—
59C22×D241—22no—
60C22×D241—41no—
61C22×D241—41no—
62C22×D241—41no—
63C22×D241—41no—
64C22×D241—41no—
65C22×D241—41no—
66D964—22no—
67D964—22no—
68D964—41no—
69D964—41no—
70C2×Dic241—41no—
71C2×Dic241—41no—
72C2×Dic241—41no—
73#14515349—41no—
74#14515349—41no—
75C23×C241—21no—
76C23×C241C2×D411yescharacteristic
77Q8×C241—41no—
78D4×C241—41no—
79D4×C241—41no—
80D4×C241—41no—
81C2×C4×C241—21no—
82C2×C4×C241—21no—
83C2×C4×C241—21no—
84C2×C4×C241—21no—
85C2×C4×C241—21no—
86C2×C4×C241—21no—
87C2×C4×C241—21no—
88C2×C4×C241—41no—
89C2×C4×C241—41no—
90C2×C4×C241—81no—
91C2×C4×C241C2×D411yescharacteristic
92C2×C4×C241C2×D411yescharacteristic
93C22×C241—21no—
94C22×C241—21no—
95C22×C241—41no—
96C22×C241—41no—
97C22×C241—41no—
98C22×C241#12011yescharacteristic
99C22×C241#13511yescharacteristic
100C22×C241AGL(1,ℤ/8)11yescharacteristic
101C2×D241—42no—
102C2×D241—42no—
103C2×D241—42no—
104C2×D241—81no—
105C2×D241—81no—
106C2×D241—81no—
107C2×D241—81no—
108C4×C241—21no—
109C4×C241—21no—
110C4×C241—41no—
111C4×C241—41no—
112C4×C241—41no—
113C4×C241—41no—
114C4×C241—41no—
115Dic241—81no—
116C2×C241—21no—
117C2×C241—21no—
118C2×C241—81no—
119C2×C241#44911yescharacteristic
120C2×C241#45411yescharacteristic
121C2×C241B(4,𝔽2)11yescharacteristic
122D241—82no—
123C241#159711yescharacteristic, complemented, Sylow
124#1597—2411nomaximal, Sylow
125#539D24111yescharacteristic, complemented
126#450—2411no—
127#447—2411no—
128#394—2411no—
129#328—2411no—
130#327—2411no—
131#324—2411no—
132C22×D4—2411no—
133C2×SD16—4821no—
134C2×D8—4821no—
135#126—21no—
136#125—21no—
137#123—21no—
138#123—21no—
139C22⋊Q8—21no—
140C22⋊Q8C2×D24111yescharacteristic
141#121C2×D24111yescharacteristic
142C4×D4—21no—
143C2×C4⋊C4C2×D24111yescharacteristic
144C2×C22⋊C4—4821no—
145C2×C22⋊C4—4821no—
146#102—4821no—
147#102—4821no—
148C22⋊C8—2411no—
149C22⋊C8—2411no—
150#95—2411no—
151C24—4821no—
152C2×Q8C22⋊D24111yescharacteristic
153C2×D4—21no—
154C2×D4—9642no—
155C2×D4—4821no—
156C2×D4—4821no—
157C2×D4C22⋊D24111yescharacteristic
158C22×C4—21no—
159C22×C4—4821no—
160C22×C4C22⋊D24111yescharacteristic
161C22×C4C22⋊D24111yescharacteristic
162C22×C4C22×D24111yescharacteristic
163SD16—9642no—
164D8—9642no—
165C2×C8—4821no—
166C2×C8—4821no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22⋊D24111yescharacteristic
174C4⋊C4C22⋊D24111yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—9642no—
181C22⋊C4—9642no—
182C42—21no—
183C23—21no—
184C23—9641no—
185C23—9641no—
186C23—9641no—
187C23—9641no—
188C23—9641no—
189C23—9641no—
190C23—4822no—
191C23C2×C22⋊D24111yescharacteristic
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—9641no—
197D4—9641no—
198D4—4822no—
199D4—4822no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—9641no—
211C2×C4—9641no—
212C2×C4—9641no—
213C2×C4C2×C22⋊D24111yescharacteristic
214C2×C4C2×C22⋊D24111yescharacteristic
215C8—9641no—
216C8—9641no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—19281no—
223C22—19281no—
224C22—19281no—
225C22—19281no—
226C22—9642no—
227C22—9642no—
228C22—9642no—
229C22#2144061111yescharacteristic
230C22#2144066211yescharacteristic
231C22#2144066711yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—19281no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—19282no—
244C2#2950786211yescharacteristic
245C2#2950789011yescharacteristic
246C2#2950790411yescharacteristic
subgroupG/H
centerC22#21440611
commutator, cosocleC22×C4×C241C23
FrattiniC22×C4C22×D241
Fitting#539×C241C2
socleC22×C241#120
prime radical#29507850C2
hypercenter#539D241
hypocenterC241#1597
LCSG ▷ C22×C4×C241 ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #29507867C2semidirect
3. #29507634C2semidirect
4. #29507454C2semidirect
5. #29507451C2semidirect
6. #29507443C2semidirect
7. #539×C241C2semidirect
123. C241#1597semidirect
125. #539D241semidirect

Maximal subgroups

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

subgroupindex
#295078672
#295078502
#295076342
#295074542
#295074512
#295074432
#539×C2412
#1597241