Finite Group Databasebeta

← #44400765

Subgroups — #44400765

Order 29056.

394 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20300321C211yesmaximal, characteristic, complemented
2#28409599C211yesmaximal, characteristic
3C2×#20300275C211yesmaximal, characteristic, complemented
4C2×#20300214C211yesmaximal, characteristic, complemented
5#28409042C211yesmaximal, characteristic, complemented
6#28408793C211yesmaximal, characteristic, complemented
7C2×#100×C227C211yesmaximal, characteristic, complemented
8C2×#16813884C2211yescharacteristic
9C22×Dic454C2211yescharacteristic
10#20300321—22no—
11#20300321—22no—
12C2×Dic908—21no—
13C2×#16813873—21no—
14#20300284—21no—
15#20300275—21no—
16#20300275—22no—
17#20300275—22no—
18#20300259—22no—
19#20300257—22no—
20#20300247C2211yescharacteristic
21C2×C22⋊Dic227—22no—
22C2×C4×Dic227C2211yescharacteristic
23#20300214—22no—
24#20300213—22no—
25C22×D4×C227C2211yescharacteristic
26C2×M16×C227C2211yescharacteristic
27C2×M16×C227C2211yescharacteristic
28#100×C227—22no—
29#16813884—21no—
30#16813884—21no—
31#16813884—21no—
32#16813884—21no—
33C2×Dic454—21no—
34C2×Dic454—21no—
35C2×Dic454—21no—
36C2×Dic454—21no—
37C2×Dic454—21no—
38C2×Dic454D411yescharacteristic
39C2×Dic454D412yescomplemented
40C2×D908D411yescharacteristic
41C2×C22⋊D227—24no—
42C2×C22⋊D227—21no—
43C22×Dic227—21no—
44C22×Dic227—21no—
45C2×C4×D227—21no—
46Dic908—24no—
47#16813873—24no—
48C4⋊Dic227D412yes—
49C22⋊Dic227—28no—
50C22⋊Dic227—24no—
51C22⋊Dic227—21no—
52C4×Dic227—21no—
53C4×Dic227—21no—
54C4×Dic227D412yescomplemented
55C24×C227D412yes—
56C2×D4×C227—21no—
57C2×D4×C227—21no—
58C2×D4×C227—21no—
59C2×D4×C227—21no—
60C2×D4×C227—21no—
61C2×D4×C227—21no—
62C22×C4×C227C2311yescharacteristic
63M16×C227—21no—
64M16×C227—21no—
65M16×C227—21no—
66M16×C227D412yescomplemented
67C2×C8×C227—21no—
68C2×C8×C227—21no—
69C22×D227—21no—
70Dic454—22no—
71Dic454—21no—
72Dic454—22no—
73Dic454—21no—
74Dic454—41no—
75Dic454—41no—
76D908—21no—
77D908—21no—
78C22⋊D227—41no—
79C22⋊D227—41no—
80C22⋊D227—84no—
81C2×Dic227—22no—
82C2×Dic227—22no—
83C2×Dic227—21no—
84C2×Dic227—22no—
85C2×Dic227—42no—
86C2×Dic227—41no—
87C2×Dic227—41no—
88C4×D227—41no—
89C4×D227—41no—
90C23×C227—22no—
91C23×C227—22no—
92C23×C227—22no—
93C23×C227—44no—
94C23×C227C2×D411yescharacteristic
95D4×C227—41no—
96D4×C227—41no—
97D4×C227—41no—
98D4×C227—41no—
99C2×C4×C227C2×D411yescharacteristic
100C2×C4×C227C2×D411yescharacteristic
101C2×C4×C227C2×D411yescharacteristic
102C2×C4×C227C2×D411yescharacteristic
103C2×C4×C227C2×D411yescharacteristic
104C2×C4×C227#4111yescharacteristic
105C8×C227—22no—
106C8×C227—41no—
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#12011yescharacteristic
118C4×C227#12111yescharacteristic
119C4×C227#12111yescharacteristic
120C4×C227#12711yescharacteristic
121Dic227—42no—
122Dic227—41no—
123Dic227—42no—
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×#124—2271no—
136C2×#104—2271no—
137#419—2271no—
138C2×#100D22711yescharacteristic, complemented
139#339—2271no—
140C2×#41—2271no—
141C22×Q8—2271no—
142C22×D4C2×D22711yescharacteristic
143#137—4542no—
144#137—4542no—
145C2×Q16—4541no—
146C2×SD16—4541no—
147C2×M16C2×D22711yescharacteristic
148C2×M16C2×D22711yescharacteristic
149#127—4541no—
150#124—4542no—
151#124—4541no—
152#124—4542no—
153#121—4542no—
154#120—4542no—
155#117—2271no—
156C2×C22⋊C4—4542no—
157C2×C42—2271no—
158#104—4542no—
159#103—4542no—
160#100—22no—
161C24D90812yes—
162#41—4541no—
163#41—4541no—
164#41—4541no—
165#41—4541no—
166C2×Q8—4541no—
167C2×Q8—4541no—
168C2×Q8—4541no—
169C2×Q8—4541no—
170C2×Q8—4541no—
171C2×Q8—2271no—
172C2×Q8—2272no—
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—4541no—
183C22×C4—4541no—
184C22×C4—4541no—
185C22×C4C22×D22711yescharacteristic
186Q16—4544no—
187SD16—4544no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16C22⋊D22712yescomplemented
192C2×C8—21no—
193C2×C8—21no—
194C4⋊C4—2272no—
195C22⋊C4—4548no—
196C22⋊C4—4544no—
197C22⋊C4—4541no—
198C42—4541no—
199C42—4541no—
200C42—2272no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—44no—
205C23—4541no—
206C23C2×D90811yescharacteristic
207Q8—9081no—
208Q8—9081no—
209Q8—4542no—
210Q8—4541no—
211Q8—4542no—
212Q8—4541no—
213D4—41no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—18164no—
218D4—9081no—
219D4—9081no—
220D4—4541no—
221D4—4541no—
222C2×C4—9082no—
223C2×C4—9081no—
224C2×C4—9081no—
225C2×C4—9081no—
226C2×C4—9081no—
227C2×C4—4542no—
228C2×C4—4542no—
229C2×C4—4542no—
230C2×C4—4541no—
231C2×C4D4×D22711yescharacteristic
232C2×C4D4×D22711yescharacteristic
233C2×C4D4×D22711yescharacteristic
234C2×C4D4×D22711yescharacteristic
235C2×C4C2×C22⋊D22711yescharacteristic
236C2×C4#1681388411yescharacteristic
237C8—22no—
238C8—41no—
239C22—22no—
240C22—44no—
241C22—42no—
242C22—44no—
243C22—82no—
244C22—9081no—
245C22—9082no—
246C22#2030025811yescharacteristic
247C22#2030025811yescharacteristic
248C22#2030026211yescharacteristic
249C4—9082no—
250C4—9081no—
251C4—9082no—
252C4—9082no—
253C4#2030025611yescharacteristic
254C4#2030026111yescharacteristic
255C4#2030026111yescharacteristic
256C4#2030028411yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—18161no—
261C2#2840894111yescharacteristic
262C2#2840921212yes—
subgroupG/H
centerC22#20300258
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
FittingC2×#100×C227C2
socleC22×C227#120
prime radical#28409599C2
hypercenterC2×#100D227
hypocenterC227#2004
LCSG ▷ C22×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#20300321C2semidirect
3. C2×#20300275C2semidirect
4. C2×#20300214C2semidirect
5. #28409042C2semidirect
6. #28408793C2semidirect
7. C2×#100×C227C2semidirect
39. C2×Dic454D4semidirect
54. C4×Dic227D4semidirect
66. M16×C227D4semidirect
131. C227#2004semidirect
138. C2×#100D227semidirect
191. M16C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003212
#284095992
C2×#203002752
C2×#203002142
#284090422
#284087932
C2×#100×C2272
#2004227