Finite Group Databasebeta

← #44408083

Subgroups — #44408083

Order 29056.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#20300286C211yesmaximal, characteristic, complemented
2C2×#20300268C211yesmaximal, characteristic
3C2×#20300264C211yesmaximal, characteristic
4C2×C42×D227C211yesmaximal, characteristic
5#28408938C211yesmaximal, characteristic, complemented
6#28408929C214yesmaximal, complemented
7#28408901C211yesmaximal, characteristic, complemented
8#28408891C214yesmaximal
9#391×C227C211yesmaximal, characteristic, complemented
10C22×Dic454C2211yescharacteristic
11C22×C4×D227C2211yescharacteristic
12C22×C4×D227C2212yes—
13#20300286—24no—
14#20300268—24no—
15#20300264—24no—
16C2×C4⋊Dic227C2211yescharacteristic
17C2×C4⋊Dic227C2212yes—
18C2×Dic227⋊C4C2214yes—
19C2×Dic227⋊C4C2214yes—
20C2×Dic227⋊C4C2212yes—
21C2×(C2×D227)⋊C4C2212yes—
22C2×(C2×D227)⋊C4C2212yes—
23C2×C4×Dic227C2212yes—
24C2×C4×Dic227C2211yescharacteristic
25C42×D227—24no—
26C227⋊#95C2214yes—
27C22×Q8×C227C2211yescharacteristic
28C2×C4⋊C4×C227C2211yescharacteristic
29C2×C42×C227C2211yescharacteristic
30#95×C227C2214yes—
31C23×D227C2311yescharacteristic
32C2×Dic454—22no—
33C2×Dic454—22no—
34C2×Dic454—22no—
35C22×Dic227C2312yes—
36C22×Dic227C2312yes—
37C22×Dic227C2311yescharacteristic
38C22×Dic227C2312yes—
39C2×C4×D227—24no—
40C2×C4×D227—24no—
41C2×C4×D227—22no—
42C2×C4×D227—22no—
43C2×C4×D227D414yes—
44C2×C4×D227Q818yescomplemented
45C4⋊Dic227—24no—
46C4⋊Dic227—22no—
47Dic227⋊C4—24no—
48Dic227⋊C4—28no—
49Dic227⋊C4—28no—
50(C2×D227)⋊C4—24no—
51(C2×D227)⋊C4—24no—
52C4×Dic227—24no—
53C4×Dic227D414yes—
54C2×Q8×C227—22no—
55C2×Q8×C227—22no—
56C2×Q8×C227—22no—
57C22×C4×C227C2312yes—
58C22×C4×C227C2312yes—
59C22×C4×C227C2312yes—
60C22×C4×C227C2311yescharacteristic
61C4⋊C4×C227—22no—
62C42×C227—22no—
63C22×D227—21no—
64C22×D227—21no—
65C22×D227—22no—
66C22×D227C2×D412yes—
67C22×D227C2×Q814yes—
68Dic454—44no—
69C2×Dic227—21no—
70C2×Dic227—21no—
71C2×Dic227—22no—
72C2×Dic227—22no—
73C2×Dic227—22no—
74C2×Dic227—24no—
75C2×Dic227—22no—
76C2×Dic227C2×D412yes—
77C2×Dic227C2×D414yes—
78C2×Dic227C2×Q814yes—
79C2×Dic227#4114yes—
80C4×D227—28no—
81C4×D227—216no—
82C23×C227C2411yescharacteristic
83Q8×C227—44no—
84C2×C4×C227—22no—
85C2×C4×C227—22no—
86C2×C4×C227—22no—
87C2×C4×C227—22no—
88C2×C4×C227—22no—
89C2×C4×C227—22no—
90C2×C4×C227—24no—
91C2×C4×C227C2×D412yes—
92C2×C4×C227C2×D412yes—
93C2×C4×C227C2×Q814yes—
94C2×C4×C227#4112yes—
95C22×C227C22×D411yescharacteristic
96C22×C227C22×D411yescharacteristic
97C22×C227C22×Q812yes—
98C22×C227C2×#4112yes—
99C22×C227C2×#4111yescharacteristic
100C2×D227—22no—
101C2×D227—24no—
102C2×D227—22no—
103C2×D227—24no—
104C2×D227C4⋊Q814yescomplemented
105C4×C227—24no—
106C4×C227—42no—
107C4×C227—42no—
108C4×C227#12114yes—
109Dic227—22no—
110Dic227—42no—
111Dic227—42no—
112Dic227C22⋊Q818yescomplemented
113C2×C227C2×#12111yescharacteristic
114C2×C227C2×C22⋊Q812yes—
115C2×C227C2×C4⋊Q811yescharacteristic
116C2×C227#53211yescharacteristic
117C2×C227#53312yes—
118D227—24no—
119C227#285711yescharacteristic, complemented, Sylow
120#2857—2271nomaximal, Sylow
121C2×C4⋊Q8—2271no—
122C2×C22⋊Q8—2271no—
123C2×C22⋊Q8—2271no—
124C22×C42—2271no—
125#391—2271no—
126#391D22711yescharacteristic, complemented
127#389—2274no—
128#384—2271no—
129#380—2274no—
130C22×Q8—2271no—
131C22×Q8C2×D22711yescharacteristic
132C23×C4—2271no—
133C23×C4—2272no—
134C4⋊Q8—4544no—
135C22⋊Q8—4544no—
136C22⋊Q8—4544no—
137C2×C4⋊C4—2271no—
138C2×C4⋊C4—2272no—
139C2×C4⋊C4—2272no—
140C2×C4⋊C4—2274no—
141C2×C4⋊C4—2274no—
142C2×C4⋊C4C2×D22711yescharacteristic
143C2×C22⋊C4—2272no—
144C2×C22⋊C4—2272no—
145C2×C42—4544no—
146C2×C42—2272no—
147C2×C42—2271no—
148C2×C42C2×D22711yescharacteristic
149#95—2274no—
150#95C2×D22714yes—
151C24—2271no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—4542no—
156C2×Q8—4542no—
157C2×Q8—4542no—
158C22×C4—4544no—
159C22×C4—4544no—
160C22×C4—4542no—
161C22×C4—4542no—
162C22×C4—2272no—
163C22×C4—2272no—
164C22×C4—2271no—
165C22×C4—2274no—
166C22×C4—2278no—
167C22×C4—2272no—
168C22×C4C22×D22712yes—
169C22×C4C22×D22712yes—
170C22×C4C22×D22712yes—
171C22×C4C22×D22711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—4548no—
174C4⋊C4—4544no—
175C4⋊C4—4548no—
176C4⋊C4—4544no—
177C4⋊C4—4542no—
178C22⋊C4—4544no—
179C22⋊C4—4544no—
180C42—22no—
181C42—4544no—
182C42—2274no—
183C23—4541no—
184C23—4542no—
185C23—4541no—
186C23—2272no—
187C23—2274no—
188C23C23×D22711yescharacteristic
189Q8—44no—
190Q8—9084no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—45416no—
199C2×C4—4542no—
200C2×C4—4542no—
201C2×C4—4542no—
202C2×C4—4544no—
203C2×C4—4542no—
204C2×C4—4541no—
205C2×C4—4541no—
206C2×C4—4548no—
207C2×C4—2272no—
208C2×C4—2274no—
209C2×C4—2274no—
210C2×C4—2274no—
211C2×C4D4×D22712yes—
212C2×C4D4×D22712yes—
213C2×C4Q8×D22714yes—
214C2×C4#1681388412yes—
215C22—4542no—
216C22—4544no—
217C22—4542no—
218C22—4544no—
219C22—2274no—
220C22C2×D4×D22711yescharacteristic
221C22C2×D4×D22711yescharacteristic
222C22C2×Q8×D22712yes—
223C22#41×D22712yes—
224C22C2×#1681388411yescharacteristic
225C4—24no—
226C4—42no—
227C4—42no—
228C4—9082no—
229C4—9082no—
230C4—4542no—
231C4—2278no—
232C4#2030026114yes—
233C2—4544no—
234C2C2×#2030026111yescharacteristic
235C2C22⋊Q8×D22712yes—
236C2C4⋊Q8×D22711yescharacteristic
237C2#2840958111yescharacteristic
238C2#2840958612yes—
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#391×C227C2
prime radicalC23×D227C23
hypercenter#391D227
hypocenterC227#2857
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#20300286C2semidirect
5. #28408938C2semidirect
6. #28408929C2semidirect
7. #28408901C2semidirect
9. #391×C227C2semidirect
44. C2×C4×D227Q8semidirect
104. C2×D227C4⋊Q8semidirect
112. Dic227C22⋊Q8semidirect
119. C227#2857semidirect
126. #391D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002862
C2×#203002682
C2×#203002642
C2×C42×D2272
#284089382
#284089292
#284089012
#284088912
#391×C2272
#2857227