Finite Group Databasebeta

← #44401739

Subgroups — #44401739

Order 29056.

260 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 41 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#28409771C211yesmaximal, characteristic, complemented
2#28409222C211yesmaximal, characteristic, complemented
3#28409200C211yesmaximal, characteristic, complemented
4#28408824C211yesmaximal, characteristic, complemented
5#28408821C211yesmaximal, characteristic, complemented
6#28408819C211yesmaximal, characteristic, complemented
7B(4,𝔽2)×C227C211yesmaximal, characteristic, complemented
8#20300313—21no—
9#20300284C2211yescharacteristic
10#20300281—21no—
11#20300270—21no—
12#20300269—21no—
13#20300269C2211yescharacteristic
14#20300261—21no—
15#20300260—21no—
16#20300259—21no—
17#20300257—21no—
18#20300215—21no—
19#20300205C2211yescharacteristic
20#20300203—21no—
21#20300203C2211yescharacteristic
22H(5,𝔽2)×C227C2211yescharacteristic
23#120×C227—21no—
24#120×C227C2211yescharacteristic
25#99×C227—21no—
26#99×C227C2211yescharacteristic
27C2×D908D411yescharacteristic, complemented
28C2×C22⋊D227—21no—
29C2×C22⋊D227—21no—
30C2×C22⋊D227—21no—
31C2×C22⋊D227—21no—
32C22×Dic227D411yescharacteristic, complemented
33C2×C4×D227—21no—
34#16813871—41no—
35#16813869—41no—
36#16813868—21no—
37C4⋊Dic227—21no—
38Dic227⋊C4—21no—
39Dic227⋊C4—21no—
40Dic227⋊C4—41no—
41C22⋊Dic227—21no—
42C22⋊Dic227—21no—
43C22⋊Dic227—21no—
44C22⋊Dic227—21no—
45C22⋊Dic227—41no—
46C22⋊Dic227D411yescharacteristic, complemented
47(C2×D227)⋊C4—21no—
48C4×Dic227D411yescharacteristic, complemented
49C24×C227D411yescharacteristic
50#41×C227—21no—
51#41×C227—41no—
52C2×D4×C227—21no—
53C2×D4×C227—21no—
54C2×D4×C227—21no—
55C2×D4×C227—41no—
56C2×D4×C227—41no—
57C2×D4×C227C2311yescharacteristic
58C22⋊C4×C227—21no—
59C22⋊C4×C227—21no—
60C22⋊C4×C227—41no—
61C22⋊C4×C227D411yescharacteristic
62C22×D227—21no—
63D908—41no—
64C22⋊D227—81no—
65C22⋊D227—81no—
66C22⋊D227—81no—
67C22⋊D227—81no—
68C2×Dic227—21no—
69C2×Dic227—21no—
70C2×Dic227—21no—
71C2×Dic227—41no—
72C2×Dic227—41no—
73C4×D227—81no—
74C227⋊C8—41no—
75C23×C227—21no—
76C23×C227—21no—
77C23×C227—21no—
78C23×C227—41no—
79C23×C227—81no—
80C23×C227C2×D411yescharacteristic
81C23×C227C2×D411yescharacteristic
82Q8×C227—21no—
83D4×C227—21no—
84D4×C227—41no—
85D4×C227—41no—
86D4×C227—41no—
87D4×C227—41no—
88D4×C227—81no—
89D4×C227—82no—
90C2×C4×C227—21no—
91C2×C4×C227—21no—
92C2×C4×C227—21no—
93C2×C4×C227—41no—
94C2×C4×C227—41no—
95C2×C4×C227C2×D411yescharacteristic
96C22×C227—21no—
97C22×C227—21no—
98C22×C227—41no—
99C22×C227—41no—
100C22×C227—41no—
101C22×C227—41no—
102C22×C227—42no—
103C22×C227—41no—
104C22×C227—41no—
105C22×C227—81no—
106C22×C227—81no—
107C22×C227—81no—
108C22×C227#12011yescharacteristic
109C2×D227—41no—
110C2×D227—41no—
111C2×D227—41no—
112C4×C227—21no—
113C4×C227—41no—
114C4×C227—41no—
115C4×C227—81no—
116Dic227—41no—
117Dic227—41no—
118Dic227—41no—
119Dic227—81no—
120C2×C227—21no—
121C2×C227—41no—
122C2×C227—41no—
123C2×C227—81no—
124C2×C227—81no—
125C2×C227B(4,𝔽2)11yescharacteristic
126D227—81no—
127C227#220011yescharacteristic, complemented, Sylow
128#2200—2271nomaximal, Sylow
129#560—2271no—
130#458—2271no—
131B(4,𝔽2)D22711yescharacteristic, complemented
132#453—2271no—
133#353—2271no—
134#352—2271no—
135AΣL(1,𝔽16)—2271no—
136H(5,𝔽2)C2×D22711yescharacteristic
137AGL(1,ℤ/8)—4541no—
138#127—2271no—
139#126—4541no—
140#123—4541no—
141#123—4541no—
142#123—2271no—
143#121—4541no—
144#121—4541no—
145#121—4541no—
146#120—21no—
147#120—4541no—
148#120C2×D22711yescharacteristic
149#104—4541no—
150#100—2271no—
151#99—21no—
152#99—4541no—
153#99—2271no—
154#99C2×D22711yescharacteristic
155C24C22⋊D22711yescharacteristic
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—4541no—
164C2×D4—4541no—
165C2×D4—4541no—
166C2×D4—4541no—
167C2×D4—2271no—
168C2×D4C22×D22711yescharacteristic
169C22×C4—4541no—
170C22×C4—2271no—
171SD16—9081no—
172D8—9081no—
173M16—4541no—
174C4⋊C4—9081no—
175C4⋊C4—4541no—
176C4⋊C4—4541no—
177C4⋊C4—4541no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—41no—
181C22⋊C4—9081no—
182C22⋊C4—4541no—
183C22⋊C4—4541no—
184C22⋊C4—4541no—
185C22⋊C4—4541no—
186C22⋊C4—4541no—
187C22⋊C4—2271no—
188C22⋊C4C22⋊D22711yescharacteristic
189C42—2271no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—4541no—
196C23D4×D22711yescharacteristic
197C23C2×C22⋊D22711yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—82no—
206D4—18161no—
207D4—18161no—
208D4—18161no—
209D4—18161no—
210D4—9081no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—18161no—
217C2×C4—9081no—
218C2×C4—9081no—
219C2×C4—4541no—
220C2×C4—4541no—
221C2×C4—4541no—
222C2×C4D4×D22711yescharacteristic
223C8—9081no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—9081no—
237C22—9081no—
238C22—9081no—
239C22#2030025611yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—81no—
244C4—18161no—
245C4—9081no—
246C4—9081no—
247C4—9081no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—18161no—
254C2#2840921811yescharacteristic
subgroupG/H
centerC2#28409218
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
FittingB(4,𝔽2)×C227C2
socleC2×C227B(4,𝔽2)
hypercenterB(4,𝔽2)D227
hypocenterC227#2200
LCSG ▷ C2×D4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #28409771C2semidirect
2. #28409222C2semidirect
3. #28409200C2semidirect
4. #28408824C2semidirect
5. #28408821C2semidirect
6. #28408819C2semidirect
7. B(4,𝔽2)×C227C2semidirect
27. C2×D908D4semidirect
32. C22×Dic227D4semidirect
46. C22⋊Dic227D4semidirect
48. C4×Dic227D4semidirect
127. C227#2200semidirect
131. B(4,𝔽2)D227semidirect

Maximal subgroups

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

subgroupindex
#284097712
#284092222
#284092002
#284088242
#284088212
#284088192
B(4,𝔽2)×C2272
#2200227