Finite Group Databasebeta

← #44401744

Subgroups — #44401744

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#28409769C211yesmaximal, characteristic, complemented
2#28409223C211yesmaximal, characteristic, complemented
3#28409220C211yesmaximal, characteristic, complemented
4#28409199C211yesmaximal, characteristic, complemented
5#28408821C211yesmaximal, characteristic, complemented
6#28408818C211yesmaximal, characteristic, complemented
7#353×C227C211yesmaximal, characteristic, complemented
8#20300333C2211yescharacteristic
9#20300314—21no—
10#20300282—21no—
11#20300272—21no—
12#20300269—21no—
13#20300269C2211yescharacteristic
14#20300260—21no—
15#20300260—21no—
16#20300260—21no—
17#20300258—21no—
18#20300256—21no—
19#20300256C2211yescharacteristic
20#20300216—21no—
21#20300205C2211yescharacteristic
22#20300202—21no—
23#20300202—21no—
24#127×C227C2211yescharacteristic
25#99×C227C2211yescharacteristic
26#99×C227C2211yescharacteristic
27C23×D227D411yescharacteristic
28#16813884—21no—
29#16813883—41no—
30C2×D908—41no—
31C2×C22⋊D227—21no—
32C2×C22⋊D227—21no—
33C2×C22⋊D227—21no—
34C2×C22⋊D227—21no—
35C2×C22⋊D227—21no—
36D4×D227—41no—
37C22×Dic227—21no—
38C22×Dic227D411yescharacteristic, complemented
39#16813872—41no—
40#16813869—41no—
41#16813868—21no—
42C4⋊Dic227—41no—
43Dic227⋊C4—21no—
44Dic227⋊C4—21no—
45Dic227⋊C4—21no—
46C22⋊Dic227—21no—
47C22⋊Dic227—21no—
48C22⋊Dic227D411yescharacteristic
49C22⋊Dic227D411yescharacteristic, complemented
50(C2×D227)⋊C4—21no—
51(C2×D227)⋊C4—21no—
52(C2×D227)⋊C4—21no—
53(C2×D227)⋊C4—41no—
54(C2×D227)⋊C4—41no—
55C2×D4×C227—21no—
56C2×D4×C227—21no—
57C2×D4×C227D411yescharacteristic, complemented
58C2×D4×C227C2311yescharacteristic
59C22⋊C4×C227—21no—
60C22⋊C4×C227—21no—
61C42×C227D411yescharacteristic, complemented
62C22×D227—21no—
63C22×D227—21no—
64C22×D227—21no—
65C22×D227—41no—
66C22×D227—81no—
67Dic454—21no—
68D908—21no—
69D908—82no—
70C22⋊D227—41no—
71C22⋊D227—41no—
72C22⋊D227—41no—
73C22⋊D227—81no—
74C22⋊D227—81no—
75C22⋊D227—81no—
76C2×Dic227—21no—
77C2×Dic227—21no—
78C2×Dic227—21no—
79C2×Dic227—21no—
80C2×Dic227—21no—
81C2×Dic227—41no—
82C2×Dic227—81no—
83C4×D227—41no—
84C227⋊C8—41no—
85C23×C227—21no—
86C23×C227C2×D411yescharacteristic
87C23×C227C2×D411yescharacteristic
88D4×C227—41no—
89D4×C227—41no—
90D4×C227—81no—
91D4×C227—81no—
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—41no—
103C22×C227#12011yescharacteristic
104C2×D227—41no—
105C2×D227—41no—
106C2×D227—42no—
107C2×D227—41no—
108C2×D227—41no—
109C2×D227—81no—
110C2×D227—81no—
111C2×D227—81no—
112C4×C227—21no—
113C4×C227—41no—
114C4×C227—81no—
115C4×C227—81no—
116Dic227—41no—
117Dic227—41no—
118Dic227—41no—
119Dic227—41no—
120C2×C227—21no—
121C2×C227—41no—
122C2×C227—41no—
123C2×C227—81no—
124C2×C227B(4,𝔽2)11yescharacteristic
125D227—81no—
126D227—81no—
127C227#220011yescharacteristic, complemented, Sylow
128#2200—2271nomaximal, Sylow
129#560—2271no—
130#458—2271no—
131B(4,𝔽2)—2271no—
132#453—2271no—
133#353D22711yescharacteristic, complemented
134#352—2271no—
135AΣL(1,𝔽16)—2271no—
136H(5,𝔽2)—2271no—
137AGL(1,ℤ/8)—4541no—
138#127C2×D22711yescharacteristic
139#126—4541no—
140#123—4541no—
141#123—4541no—
142#123—2271no—
143#121—4541no—
144#121—4541no—
145#121—4541no—
146#120—4541no—
147#120—4541no—
148#120—2271no—
149#104—4541no—
150#100—2271no—
151#99—4541no—
152#99—4541no—
153#99C2×D22711yescharacteristic
154#99C2×D22711yescharacteristic
155C24—2271no—
156#41—9081no—
157#41—4541no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—9081no—
161C2×D4—9081no—
162C2×D4—4541no—
163C2×D4—4541no—
164C2×D4—4541no—
165C2×D4—4541no—
166C2×D4—4541no—
167C2×D4D90811yescharacteristic, complemented
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—9081no—
181C22⋊C4—9081no—
182C22⋊C4—4541no—
183C22⋊C4—4541no—
184C22⋊C4—4541no—
185C22⋊C4—4541no—
186C22⋊C4—4541no—
187C22⋊C4—2271no—
188C22⋊C4—2271no—
189C42D90811yescharacteristic, complemented
190C23—21no—
191C23—18161no—
192C23—9081no—
193C23—4541no—
194C23—4541no—
195C23—4541no—
196C23D4×D22711yescharacteristic
197C23D4×D22711yescharacteristic
198Q8—4541no—
199D4—41no—
200D4—41no—
201D4—81no—
202D4—81no—
203D4—18162no—
204D4—18161no—
205D4—18161no—
206D4—18161no—
207D4—9081no—
208D4—9081no—
209D4—9081no—
210D4—4541no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—18161no—
215C2×C4—9081no—
216C2×C4—9081no—
217C2×C4—4541no—
218C2×C4—4541no—
219C2×C4—4541no—
220C2×C4—4541no—
221C2×C4—4541no—
222C2×C4C2×D90811yescharacteristic
223C8—9081no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—18161no—
232C22—18161no—
233C22—18161no—
234C22—9081no—
235C22—9081no—
236C22—9082no—
237C22—9081no—
238C22—9081no—
239C22#2030025811yescharacteristic
240C4—21no—
241C4—41no—
242C4—81no—
243C4—81no—
244C4—9081no—
245C4—9081no—
246C4—9081no—
247C4—9081no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—18161no—
253C2—18161no—
254C2#2840922011yescharacteristic
subgroupG/H
centerC2#28409220
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
Fitting#353×C227C2
socleC2×C227B(4,𝔽2)
hypercenter#353D227
hypocenterC227#2200
LCSG ▷ C2×D4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #28409769C2semidirect
2. #28409223C2semidirect
3. #28409220C2semidirect
4. #28409199C2semidirect
5. #28408821C2semidirect
6. #28408818C2semidirect
7. #353×C227C2semidirect
38. C22×Dic227D4semidirect
49. C22⋊Dic227D4semidirect
57. C2×D4×C227D4semidirect
61. C42×C227D4semidirect
127. C227#2200semidirect
133. #353D227semidirect
167. C2×D4D908semidirect
189. C42D908semidirect

Maximal subgroups

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

subgroupindex
#284097692
#284092232
#284092202
#284091992
#284088212
#284088182
#353×C2272
#2200227