Finite Group Databasebeta

← #44400792

Subgroups — #44400792

Order 29056.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#28409830C211yesmaximal, characteristic
2#28409606C211yesmaximal, characteristic, complemented
3#28409048C211yesmaximal, characteristic, complemented
4#28409041C211yesmaximal, characteristic, complemented
5#28409006C211yesmaximal, characteristic, complemented
6#28408804C211yesmaximal, characteristic, complemented
7C2×#137×C227C211yesmaximal, characteristic, complemented
8C2×#16813884C2211yescharacteristic
9#20300319—21no—
10#20300318—21no—
11#20300314—21no—
12#20300313—21no—
13#20300308—22no—
14C2×#16813868C2211yescharacteristic
15#20300286—21no—
16#20300277—21no—
17#20300271—22no—
18#20300264—22no—
19#20300247C2211yescharacteristic
20#20300247C2211yescharacteristic
21#20300214—22no—
22#20300212—22no—
23#20300203—22no—
24C2×#41×C227C2211yescharacteristic
25C22×Q8×C227C2211yescharacteristic
26#137×C227—22no—
27#137×C227—22no—
28C2×Q16×C227—21no—
29C2×SD16×C227—21no—
30C2×M16×C227C2211yescharacteristic
31#16813884—21no—
32#16813884—21no—
33#16813884—21no—
34#16813884—21no—
35C2×Dic454D411yescharacteristic
36C2×D908D411yescharacteristic
37C2×C22⋊D227—21no—
38C2×C4×D227—21no—
39#16813874—22no—
40#16813872—22no—
41#16813871—22no—
42#16813869—22no—
43#16813868—21no—
44#16813868—21no—
45C2×C227⋊C8D412yes—
46C4⋊Dic227—21no—
47C4⋊Dic227—21no—
48Dic227⋊C4—24no—
49C22⋊Dic227—21no—
50C22⋊Dic227D412yes—
51(C2×D227)⋊C4—24no—
52C4×Dic227—21no—
53C4×Dic227—21no—
54#41×C227—21no—
55#41×C227—21no—
56#41×C227—21no—
57#41×C227—21no—
58C2×Q8×C227—21no—
59C2×Q8×C227—21no—
60C2×Q8×C227—21no—
61C2×Q8×C227—21no—
62C2×Q8×C227—21no—
63C2×Q8×C227D411yescharacteristic
64C2×Q8×C227D412yes—
65C2×D4×C227—21no—
66C2×D4×C227D411yescharacteristic
67C22×C4×C227—21no—
68C22×C4×C227—21no—
69C22×C4×C227C2311yescharacteristic
70Q16×C227—24no—
71SD16×C227—24no—
72M16×C227—21no—
73M16×C227D412yes—
74C2×C8×C227—21no—
75C22×D227—21no—
76Dic454—21no—
77Dic454—21no—
78D908—21no—
79D908—21no—
80C22⋊D227—41no—
81C22⋊D227—41no—
82C2×Dic227—22no—
83C2×Dic227—21no—
84C2×Dic227—41no—
85C4×D227—41no—
86C4×D227—41no—
87C227⋊C8—22no—
88C23×C227—21no—
89C23×C227C2×D411yescharacteristic
90Q8×C227—22no—
91Q8×C227—21no—
92Q8×C227—21no—
93Q8×C227—22no—
94Q8×C227—41no—
95Q8×C227—41no—
96D4×C227—21no—
97D4×C227—21no—
98D4×C227—41no—
99D4×C227—41no—
100C2×C4×C227—22no—
101C2×C4×C227—21no—
102C2×C4×C227—41no—
103C2×C4×C227—41no—
104C2×C4×C227—41no—
105C2×C4×C227—41no—
106C2×C4×C227C2×D411yescharacteristic
107C2×C4×C227C2×D411yescharacteristic
108C2×C4×C227C2×D411yescharacteristic
109C2×C4×C227C2×D411yescharacteristic
110C2×C4×C227C2×D411yescharacteristic
111C2×C4×C227#4111yescharacteristic
112C8×C227—22no—
113C22×C227—41no—
114C22×C227—41no—
115C22×C227—81no—
116C22×C227#12011yescharacteristic
117C22×C227#12111yescharacteristic
118C22×C227#12711yescharacteristic
119C2×D227—42no—
120C2×D227—41no—
121C4×C227—41no—
122C4×C227—42no—
123C4×C227#12011yescharacteristic
124C4×C227#12011yescharacteristic
125C4×C227#12111yescharacteristic
126C4×C227#12111yescharacteristic
127Dic227—42no—
128Dic227—41no—
129Dic227—81no—
130C2×C227—21no—
131C2×C227—21no—
132C2×C227—21no—
133C2×C227—81no—
134C2×C227#39211yescharacteristic
135D227—81no—
136C227#200911yescharacteristic, complemented, Sylow
137#2009—2271nomaximal, Sylow
138#575—2271no—
139C2×#137D22711yescharacteristic, complemented
140#536—2271no—
141#421—2271no—
142#419—2271no—
143#410—2271no—
144#344—2271no—
145C2×#41—2271no—
146C2×#41C2×D22711yescharacteristic
147C22×Q8C2×D22711yescharacteristic
148#137—22no—
149#137—22no—
150#137—4541no—
151#137—4541no—
152AGL(1,ℤ/8)—4541no—
153AGL(1,ℤ/8)—4541no—
154#135—4542no—
155C2×Q16—21no—
156C2×SD16—21no—
157C2×M16—2271no—
158C2×M16C2×D22711yescharacteristic
159C4⋊Q8—4541no—
160#124—4541no—
161#123—4542no—
162C22⋊Q8—4542no—
163#117—2271no—
164#117—2271no—
165#104—4542no—
166#103—4542no—
167#99—4542no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—4541no—
173#41—4541no—
174#41—4541no—
175#41—4541no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—2271no—
182C2×Q8C22⋊D22712yes—
183C2×Q8C22⋊D22711yescharacteristic
184C2×D4—21no—
185C2×D4—4541no—
186C2×D4—2271no—
187C2×D4C22⋊D22711yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—4541no—
191C22×C4C22×D22711yescharacteristic
192Q16—24no—
193Q16—4542no—
194SD16—24no—
195SD16—4542no—
196SD16—4542no—
197D8—4542no—
198M16—21no—
199M16—4541no—
200M16—4541no—
201M16C22⋊D22712yes—
202C2×C8—21no—
203C2×C8—2272no—
204C4⋊C4—4544no—
205C4⋊C4—4541no—
206C4⋊C4—4541no—
207C22⋊C4—4544no—
208C22⋊C4—4541no—
209C22⋊C4—2272no—
210C42—4541no—
211C42—4541no—
212C23—21no—
213C23—4541no—
214C23D4×D22711yescharacteristic
215Q8—21no—
216Q8—22no—
217Q8—21no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—4541no—
222Q8—4541no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—9081no—
228D4—9081no—
229D4—4541no—
230D4—4541no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—9081no—
238C2×C4—9081no—
239C2×C4—9081no—
240C2×C4—4542no—
241C2×C4—4541no—
242C2×C4D4×D22711yescharacteristic
243C2×C4D4×D22711yescharacteristic
244C2×C4C2×C22⋊D22711yescharacteristic
245C2×C4C2×C22⋊D22711yescharacteristic
246C2×C4C2×C22⋊D22711yescharacteristic
247C2×C4#1681388311yescharacteristic
248C8—22no—
249C8—4542no—
250C22—41no—
251C22—41no—
252C22—81no—
253C22—9082no—
254C22—9081no—
255C22#2030025611yescharacteristic
256C22#2030026011yescharacteristic
257C22#2030028411yescharacteristic
258C4—41no—
259C4—42no—
260C4—18161no—
261C4—9082no—
262C4—9081no—
263C4#2030025611yescharacteristic
264C4#2030025711yescharacteristic
265C4#2030025911yescharacteristic
266C4#2030026011yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—18161no—
272C2#2840894011yescharacteristic
subgroupG/H
centerC2#28408940
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
FittingC2×#137×C227C2
socleC2×C227#392
prime radical#28409830C2
hypercenterC2×#137D227
hypocenterC227#2009
LCSG ▷ C22×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #28409606C2semidirect
3. #28409048C2semidirect
4. #28409041C2semidirect
5. #28409006C2semidirect
6. #28408804C2semidirect
7. C2×#137×C227C2semidirect
136. C227#2009semidirect
139. C2×#137D227semidirect

Maximal subgroups

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

subgroupindex
#284098302
#284096062
#284090482
#284090412
#284090062
#284088042
C2×#137×C2272
#2009227