Finite Group Databasebeta

← #44401748

Subgroups — #44401748

Order 29056.

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

normal subgroups only

This group has 226 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#28409785C211yesmaximal, characteristic, complemented
2#28409223C211yesmaximal, characteristic
3#28409221C211yesmaximal, characteristic
4#28409212C211yesmaximal, characteristic
5#28408826C211yesmaximal, characteristic, complemented
6#28408823C211yesmaximal, characteristic, complemented
7#352×C227C211yesmaximal, characteristic, complemented
8#20300333C2211yescharacteristic
9#20300321—21no—
10#20300287—21no—
11#20300281—21no—
12#20300281—21no—
13#20300278—21no—
14#20300275C2211yescharacteristic
15#20300272—21no—
16#20300271—21no—
17#20300269C2211yescharacteristic
18#20300267—21no—
19#20300265—21no—
20#20300214—21no—
21#20300203C2211yescharacteristic
22#20300202—21no—
23#20300202—21no—
24#123×C227C2211yescharacteristic
25#100×C227C2211yescharacteristic
26#99×C227C2211yescharacteristic
27#16813884—21no—
28#16813883—41no—
29C2×Dic454—21no—
30C2×Dic454D411yescharacteristic
31C2×C22⋊D227—21no—
32C2×C22⋊D227—21no—
33D4×D227—41no—
34C22×Dic227D411yescharacteristic
35Dic908—41no—
36#16813873—41no—
37C4⋊Dic227—21no—
38C4⋊Dic227—21no—
39C4⋊Dic227—41no—
40Dic227⋊C4—21no—
41Dic227⋊C4—21no—
42Dic227⋊C4—21no—
43Dic227⋊C4—21no—
44Dic227⋊C4—21no—
45Dic227⋊C4—41no—
46C22⋊Dic227—21no—
47C22⋊Dic227—21no—
48C22⋊Dic227—21no—
49C22⋊Dic227—21no—
50C22⋊Dic227D411yescharacteristic
51(C2×D227)⋊C4—41no—
52(C2×D227)⋊C4—41no—
53C4×Dic227—21no—
54C4×Dic227D411yescharacteristic
55C2×D4×C227C2311yescharacteristic
56C22×C4×C227D411yescharacteristic
57M16×C227—21no—
58C4⋊C4×C227—21no—
59C22⋊C4×C227—21no—
60C22⋊C4×C227—21no—
61C22⋊C4×C227D411yescharacteristic
62C22×D227—21no—
63C22×D227—21no—
64Dic454—21no—
65Dic454—41no—
66Dic454—81no—
67D908—21no—
68C22⋊D227—41no—
69C22⋊D227—41no—
70C22⋊D227—41no—
71C2×Dic227—21no—
72C2×Dic227—21no—
73C2×Dic227—21no—
74C2×Dic227—21no—
75C2×Dic227—21no—
76C2×Dic227—21no—
77C2×Dic227—41no—
78C2×Dic227—41no—
79C4×D227—41no—
80C23×C227C2×D411yescharacteristic
81C23×C227C2×D411yescharacteristic
82D4×C227—41no—
83C2×C4×C227—21no—
84C2×C4×C227—21no—
85C2×C4×C227—41no—
86C2×C4×C227—41no—
87C2×C4×C227C2×D411yescharacteristic
88C8×C227—41no—
89C22×C227—21no—
90C22×C227—21no—
91C22×C227—41no—
92C22×C227—41no—
93C22×C227#12011yescharacteristic
94C2×D227—41no—
95C2×D227—81no—
96C2×D227—81no—
97C4×C227—21no—
98C4×C227—41no—
99C4×C227—41no—
100C4×C227—81no—
101Dic227—41no—
102Dic227—41no—
103Dic227—41no—
104Dic227—41no—
105Dic227—41no—
106Dic227—81no—
107C2×C227—21no—
108C2×C227—41no—
109C2×C227—41no—
110C2×C227B(4,𝔽2)11yescharacteristic
111D227—81no—
112C227#220111yescharacteristic, complemented, Sylow
113#2201—2271nomaximal, Sylow
114#563—2271no—
115#458—2271no—
116#458—2271no—
117#455—2271no—
118#354—2271no—
119#352—2271no—
120#352D22711yescharacteristic, complemented
121H(5,𝔽2)—2271no—
122#137—4541no—
123C4⋊Q8—4541no—
124#126—4541no—
125#126—4541no—
126#125—4541no—
127#124—2271no—
128#123—4541no—
129#123—4541no—
130#123—2271no—
131#123C2×D22711yescharacteristic
132C22⋊Q8—4541no—
133C22⋊Q8—4541no—
134#104—4541no—
135#100C2×D22711yescharacteristic
136#99—4541no—
137#99—4541no—
138#99—2271no—
139#99C2×D22711yescharacteristic
140#41—9081no—
141#41—4541no—
142C2×Q8—4541no—
143C2×Q8—2271no—
144C2×D4—9081no—
145C2×D4—4541no—
146C2×D4—4541no—
147C2×D4C22×D22711yescharacteristic
148C22×C4—2271no—
149C22×C4D90811yescharacteristic
150Q16—9081no—
151SD16—9081no—
152M16—21no—
153C4⋊C4—21no—
154C4⋊C4—9081no—
155C4⋊C4—9081no—
156C4⋊C4—4541no—
157C4⋊C4—4541no—
158C4⋊C4—4541no—
159C4⋊C4—4541no—
160C4⋊C4—4541no—
161C4⋊C4—4541no—
162C4⋊C4—4541no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—9081no—
166C22⋊C4—9081no—
167C22⋊C4—4541no—
168C22⋊C4—4541no—
169C22⋊C4—4541no—
170C22⋊C4—4541no—
171C22⋊C4—2271no—
172C22⋊C4D90811yescharacteristic
173C42—4541no—
174C42—2271no—
175C23—4541no—
176C23—4541no—
177C23D4×D22711yescharacteristic
178C23C2×D90811yescharacteristic
179Q8—18161no—
180Q8—9081no—
181Q8—4541no—
182D4—41no—
183D4—9081no—
184D4—9081no—
185D4—9081no—
186D4—4541no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—9081no—
192C2×C4—9081no—
193C2×C4—9081no—
194C2×C4—4541no—
195C2×C4—4541no—
196C2×C4—4541no—
197C2×C4—4541no—
198C2×C4—4541no—
199C2×C4—4541no—
200C2×C4D4×D22711yescharacteristic
201C8—41no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—18161no—
207C22—18161no—
208C22—9081no—
209C22#2030025811yescharacteristic
210C4—21no—
211C4—41no—
212C4—41no—
213C4—81no—
214C4—18161no—
215C4—9081no—
216C4—9081no—
217C4—9081no—
218C4—9081no—
219C4—9081no—
220C2—21no—
221C2—41no—
222C2—41no—
223C2—18161no—
224C2#2840922011yescharacteristic
subgroupG/H
centerC2#28409220
commutator, cosocleC2×D4×C227C23
FrattiniC2×D4C22×D227
Fitting#352×C227C2
socleC2×C227B(4,𝔽2)
prime radical#20300333C22
hypercenter#352D227
hypocenterC227#2201
LCSG ▷ C2×D4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
1. #28409785C2semidirect
5. #28408826C2semidirect
6. #28408823C2semidirect
7. #352×C227C2semidirect
112. C227#2201semidirect
120. #352D227semidirect

Maximal subgroups

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

subgroupindex
#284097852
#284092232
#284092212
#284092122
#284088262
#284088232
#352×C2272
#2201227