Finite Group Databasebeta

← #44400763

Subgroups — #44400763

Order 29056.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20300319C211yesmaximal, characteristic, complemented
2#28409599C211yesmaximal, characteristic
3C2×#20300216C211yesmaximal, characteristic, complemented
4#28409041C211yesmaximal, characteristic, complemented
5C2×#20300205C211yesmaximal, characteristic, complemented
6#28408792C211yesmaximal, characteristic, complemented
7C2×#124×C227C211yesmaximal, characteristic, complemented
8C2×#16813884C2211yescharacteristic
9#20300319—22no—
10#20300319—22no—
11C2×#16813874—21no—
12C2×#16813871—21no—
13C2×#16813868C2211yescharacteristic
14C2×#16813868C2211yescharacteristic
15#20300284—21no—
16#20300275—21no—
17#20300259—22no—
18#20300257—22no—
19#20300247C2211yescharacteristic
20#20300216—22no—
21#20300212—22no—
22#20300205—22no—
23C22×Q8×C227C2211yescharacteristic
24C22×D4×C227C2211yescharacteristic
25#124×C227—22no—
26#124×C227—22no—
27C2×C22⋊C4×C227—22no—
28C2×C42×C227C2211yescharacteristic
29#16813884—21no—
30#16813884—21no—
31#16813884—21no—
32#16813884—21no—
33C2×Dic454D411yescharacteristic
34C2×D908D411yescharacteristic
35C2×C22⋊D227—24no—
36C2×C22⋊D227—21no—
37C2×C4×D227—21no—
38#16813874—24no—
39#16813871—24no—
40#16813868—21no—
41#16813868—21no—
42#16813868—21no—
43#16813868D412yescomplemented
44C2×C227⋊C8—21no—
45C2×C227⋊C8—21no—
46C4⋊Dic227D412yes—
47C22⋊Dic227—24no—
48C22⋊Dic227—21no—
49C4×Dic227—21no—
50C24×C227D412yes—
51C2×Q8×C227—21no—
52C2×Q8×C227—21no—
53C2×Q8×C227—21no—
54C2×Q8×C227—21no—
55C2×Q8×C227—21no—
56C2×Q8×C227D412yescomplemented
57C2×D4×C227—21no—
58C2×D4×C227—21no—
59C2×D4×C227—21no—
60C2×D4×C227—21no—
61C2×D4×C227—21no—
62C2×D4×C227—21no—
63C22×C4×C227—21no—
64C22×C4×C227—21no—
65C22×C4×C227C2311yescharacteristic
66C22⋊C4×C227—28no—
67C42×C227—21no—
68C42×C227D412yescomplemented
69C22×D227—21no—
70Dic454—21no—
71Dic454—21no—
72D908—21no—
73D908—21no—
74C22⋊D227—41no—
75C22⋊D227—41no—
76C22⋊D227—84no—
77C2×Dic227—22no—
78C2×Dic227—21no—
79C4×D227—41no—
80C4×D227—41no—
81C227⋊C8—22no—
82C227⋊C8—41no—
83C23×C227—22no—
84C23×C227—22no—
85C23×C227—22no—
86C23×C227—44no—
87C23×C227C2×D411yescharacteristic
88Q8×C227—22no—
89Q8×C227—22no—
90Q8×C227—41no—
91Q8×C227—41no—
92D4×C227—41no—
93D4×C227—41no—
94D4×C227—41no—
95D4×C227—41no—
96C2×C4×C227—22no—
97C2×C4×C227—22no—
98C2×C4×C227—42no—
99C2×C4×C227—41no—
100C2×C4×C227—41no—
101C2×C4×C227C2×D411yescharacteristic
102C2×C4×C227C2×D411yescharacteristic
103C2×C4×C227C2×D411yescharacteristic
104C2×C4×C227C2×D411yescharacteristic
105C2×C4×C227C2×D411yescharacteristic
106C2×C4×C227#4111yescharacteristic
107C22×C227—22no—
108C22×C227—44no—
109C22×C227—42no—
110C22×C227—44no—
111C22×C227—82no—
112C22×C227#12011yescharacteristic
113C22×C227#12011yescharacteristic
114C22×C227#12111yescharacteristic
115C2×D227—41no—
116C2×D227—42no—
117C4×C227—42no—
118C4×C227—42no—
119C4×C227#12011yescharacteristic
120C4×C227#12111yescharacteristic
121C4×C227#12111yescharacteristic
122C4×C227#12711yescharacteristic
123Dic227—41no—
124Dic227—42no—
125C2×C227—21no—
126C2×C227—21no—
127C2×C227—82no—
128C2×C227#39211yescharacteristic
129C2×C227#45512yes—
130D227—81no—
131C227#200411yescharacteristic, complemented, Sylow
132#2004—2271nomaximal, Sylow
133C2×#137—2271no—
134#535—2271no—
135C2×#124D22711yescharacteristic, complemented
136C2×#104—2271no—
137#419—2271no—
138C2×#100—2271no—
139#339—2271no—
140C2×#41—2271no—
141C22×Q8C2×D22711yescharacteristic
142C22×D4C2×D22711yescharacteristic
143#137—4542no—
144#137—4542no—
145C2×Q16—4541no—
146C2×SD16—4541no—
147C2×M16—2271no—
148C2×M16—2271no—
149#127—4541no—
150#124—22no—
151#124—22no—
152#124—4541no—
153#121—4542no—
154#120—4542no—
155#117—2271no—
156C2×C22⋊C4—22no—
157C2×C42C2×D22711yescharacteristic
158#104—4542no—
159#103—4542no—
160#100—4542no—
161C24C22⋊D22712yes—
162#41—4541no—
163#41—4541no—
164#41—4541no—
165#41—4541no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—2271no—
172C2×Q8C22⋊D22712yescomplemented
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—4544no—
180C2×D4—4541no—
181C2×D4—2271no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—4541no—
185C22×C4C22×D22711yescharacteristic
186Q16—4544no—
187SD16—4544no—
188M16—4541no—
189M16—4541no—
190M16—4541no—
191M16—2272no—
192C2×C8—4541no—
193C2×C8—4541no—
194C4⋊C4—2272no—
195C22⋊C4—28no—
196C22⋊C4—4544no—
197C22⋊C4—4541no—
198C42—21no—
199C42—4541no—
200C42C22⋊D22712yescomplemented
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—4541no—
206C23C2×C22⋊D22711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211Q8—4541no—
212Q8—4541no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—18164no—
218D4—9081no—
219D4—9081no—
220D4—4541no—
221D4—4541no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—9081no—
228C2×C4—9081no—
229C2×C4—4542no—
230C2×C4—4541no—
231C2×C4D4×D22711yescharacteristic
232C2×C4D4×D22711yescharacteristic
233C2×C4D4×D22711yescharacteristic
234C2×C4C2×C22⋊D22711yescharacteristic
235C2×C4C2×C22⋊D22711yescharacteristic
236C2×C4#1681388311yescharacteristic
237C8—9081no—
238C8—4542no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—9081no—
245C22—9082no—
246C22#2030025611yescharacteristic
247C22#2030025711yescharacteristic
248C22#2030026011yescharacteristic
249C4—42no—
250C4—42no—
251C4—9081no—
252C4—9082no—
253C4#2030025611yescharacteristic
254C4#2030025911yescharacteristic
255C4#2030026011yescharacteristic
256C4#2030028411yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—18161no—
261C2#2840894011yescharacteristic
262C2#2840920912yes—
subgroupG/H
centerC22#20300256
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
FittingC2×#124×C227C2
socleC22×C227#120
prime radical#28409599C2
hypercenterC2×#124D227
hypocenterC227#2004
LCSG ▷ C22×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#20300319C2semidirect
3. C2×#20300216C2semidirect
4. #28409041C2semidirect
5. C2×#20300205C2semidirect
6. #28408792C2semidirect
7. C2×#124×C227C2semidirect
43. #16813868D4semidirect
56. C2×Q8×C227D4semidirect
68. C42×C227D4semidirect
131. C227#2004semidirect
135. C2×#124D227semidirect
172. C2×Q8C22⋊D227semidirect
200. C42C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003192
#284095992
C2×#203002162
#284090412
C2×#203002052
#284087922
C2×#124×C2272
#2004227