Finite Group Databasebeta

← #44399708

Subgroups — #44399708

Order 29056.

536 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 97 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×#20300335C211yesmaximal, characteristic
2#28409049C211yesmaximal, characteristic, complemented
3#28409042C211yesmaximal, characteristic, complemented
4#28409040C211yesmaximal, characteristic, complemented
5#28409014C211yesmaximal, characteristic, complemented
6#28409005C211yesmaximal, characteristic, complemented
7#421×C227C211yesmaximal, characteristic, complemented
8#20300335—22no—
9#20300335—22no—
10#20300335—22no—
11#20300335—22no—
12C2×#16813884—21no—
13C2×#16813884C2211yescharacteristic
14C2×#16813883—22no—
15C2×#16813883C412yescomplemented
16C22×Dic454—21no—
17C22×Dic454C2211yescharacteristic
18C2×Q8×D227C412yescomplemented
19C2×#16813868C2211yescharacteristic
20#20300247C2211yescharacteristic
21#20300215—22no—
22#20300213—22no—
23#20300211—22no—
24#20300206—21no—
25#20300204—21no—
26#20300202—22no—
27C2×#41×C227C2211yescharacteristic
28C2×M16×C227C2211yescharacteristic
29#117×C227C2211yescharacteristic
30#104×C227—22no—
31#16813884—24no—
32#16813884—24no—
33#16813884—21no—
34#16813884—21no—
35#16813884—21no—
36#16813884—21no—
37#16813883—28no—
38#16813883—28no—
39#16813883—24no—
40#16813883—24no—
41C2×Dic454—24no—
42C2×Dic454—24no—
43C2×Dic454—22no—
44C2×Dic454—21no—
45C2×Dic454—21no—
46C2×Dic454—21no—
47C2×Dic454C2×C412yes—
48C2×Dic454D412yes—
49C2×Dic454D412yes—
50C2×Dic454D411yescharacteristic
51Q8×D227—24no—
52Q8×D227—24no—
53C2×D908D411yescharacteristic
54C2×C22⋊D227—22no—
55C2×C22⋊D227D412yes—
56C22×Dic227—22no—
57C22×Dic227—21no—
58C2×C4×D227—22no—
59C2×C4×D227C2×C412yes—
60#16813868—21no—
61#16813868—21no—
62C2×C227⋊C8—21no—
63C4⋊Dic227—21no—
64C22⋊Dic227—21no—
65C4×Dic227—21no—
66#41×C227—21no—
67#41×C227—21no—
68#41×C227D412yes—
69#41×C227D412yes—
70C2×Q8×C227C2×C411yescharacteristic
71C2×D4×C227—21no—
72C2×D4×C227C2×C411yescharacteristic
73C22×C4×C227—21no—
74C22×C4×C227C2311yescharacteristic
75M16×C227—21no—
76M16×C227—21no—
77C2×C8×C227—21no—
78C4⋊C4×C227—21no—
79C22⋊C4×C227—21no—
80C42×C227—21no—
81C22×D227C22⋊C412yes—
82Dic454—28no—
83Dic454—22no—
84Dic454—22no—
85Dic454—22no—
86Dic454—21no—
87Dic454—21no—
88Dic454C22⋊C414yes—
89D908—21no—
90D908—21no—
91C22⋊D227—28no—
92C22⋊D227—22no—
93C22⋊D227—22no—
94C2×Dic227—28no—
95C2×Dic227—22no—
96C2×Dic227—22no—
97C2×Dic227—22no—
98C2×Dic227—41no—
99C2×Dic227C22⋊C412yes—
100C4×D227—28no—
101C4×D227—22no—
102C4×D227—22no—
103C227⋊C8—41no—
104C23×C227—21no—
105C23×C227C2×D411yescharacteristic
106Q8×C227—21no—
107Q8×C227C22⋊C412yes—
108D4×C227—22no—
109D4×C227—22no—
110D4×C227—21no—
111D4×C227C22⋊C412yes—
112C2×C4×C227—21no—
113C2×C4×C227—22no—
114C2×C4×C227—22no—
115C2×C4×C227—41no—
116C2×C4×C227C22×C411yescharacteristic
117C2×C4×C227C2×D411yescharacteristic
118C2×C4×C227C2×D411yescharacteristic
119C2×C4×C227C2×D411yescharacteristic
120C2×C4×C227C2×D411yescharacteristic
121C2×C4×C227C2×D411yescharacteristic
122C8×C227—41no—
123C22×C227—22no—
124C22×C227—41no—
125C22×C227—42no—
126C22×C227C2×C22⋊C411yescharacteristic
127C22×C227#12011yescharacteristic
128C22×C227#12011yescharacteristic
129C2×D227—22no—
130C2×D227—42no—
131C4×C227—22no—
132C4×C227—81no—
133C4×C227C2×C22⋊C411yescharacteristic
134C4×C227C2×C22⋊C411yescharacteristic
135C4×C227#12011yescharacteristic
136C4×C227#12011yescharacteristic
137Dic227—24no—
138Dic227—22no—
139Dic227—81no—
140C2×C227—21no—
141C2×C227—21no—
142C2×C227—21no—
143C2×C227—42no—
144C2×C227#37911yescharacteristic
145D227—42no—
146C227#179611yescharacteristic, complemented, Sylow
147#1796—2271nomaximal, Sylow
148C2×2−1+4—2271no—
149#421—2271no—
150#421D22711yescharacteristic, complemented
151#419—2271no—
152#419—2271no—
153#413—2271no—
154#410—2271no—
1552−1+4—4542no—
1562−1+4—4542no—
1572−1+4—4542no—
1582−1+4—4542no—
159C2×#41—4541no—
160C2×#41—4542no—
161C2×#41—2272no—
162C2×#41—2271no—
163C2×#41C2×D22711yescharacteristic
164C22×Q8—4541no—
165C22×Q8—2272no—
166C22×Q8—2271no—
167C2×M16—2271no—
168C2×M16C2×D22711yescharacteristic
169#117—2271no—
170#117C2×D22711yescharacteristic
171#104—22no—
172#104—4542no—
173#103—4542no—
174#103—4542no—
175#101—4541no—
176#100—4541no—
177#99—4542no—
178#41—21no—
179#41—21no—
180#41—4544no—
181#41—4544no—
182#41—4548no—
183#41—4548no—
184#41—4544no—
185#41—4544no—
186#41—4541no—
187#41—4541no—
188#41—4541no—
189#41—4541no—
190#41C22⋊D22712yes—
191#41D90812yes—
192C2×Q8—4544no—
193C2×Q8—4544no—
194C2×Q8—4542no—
195C2×Q8—4544no—
196C2×Q8—4544no—
197C2×Q8—4541no—
198C2×Q8—4541no—
199C2×Q8—4541no—
200C2×Q8—2272no—
201C2×Q8—2272no—
202C2×Q8—2272no—
203C2×Q8—2271no—
204C2×Q8C4×D22711yescharacteristic
205C2×D4—21no—
206C2×D4—4542no—
207C2×D4—2272no—
208C2×D4—2271no—
209C2×D4C4×D22711yescharacteristic
210C22×C4—21no—
211C22×C4—4542no—
212C22×C4—4541no—
213C22×C4—4542no—
214C22×C4—2272no—
215C22×C4C22×D22711yescharacteristic
216M16—21no—
217M16—21no—
218M16—4541no—
219M16—4541no—
220C2×C8—21no—
221C2×C8—4541no—
222C4⋊C4—21no—
223C4⋊C4—4541no—
224C22⋊C4—21no—
225C22⋊C4—4541no—
226C42—21no—
227C42—4541no—
228C23—21no—
229C23—2272no—
230C23D4×D22711yescharacteristic
231Q8—21no—
232Q8—4548no—
233Q8—4542no—
234Q8—4542no—
235Q8—4541no—
236Q8—4542no—
237Q8—4541no—
238Q8—2274no—
239Q8(C2×D227)⋊C412yes—
240D4—22no—
241D4—22no—
242D4—21no—
243D4—4548no—
244D4—4542no—
245D4—4542no—
246D4—4541no—
247D4—4541no—
248D4(C2×D227)⋊C412yes—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—9081no—
254C2×C4—4548no—
255C2×C4—4548no—
256C2×C4—4542no—
257C2×C4—4542no—
258C2×C4—4542no—
259C2×C4—4542no—
260C2×C4—4542no—
261C2×C4—2272no—
262C2×C4C2×C4×D22711yescharacteristic
263C2×C4D4×D22711yescharacteristic
264C2×C4D4×D22711yescharacteristic
265C2×C4D4×D22711yescharacteristic
266C2×C4C2×C22⋊D22711yescharacteristic
267C2×C4C2×D90811yescharacteristic
268C8—41no—
269C8—9081no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—9082no—
274C22—4542no—
275C22C22⋊C4×D22711yescharacteristic
276C22#2030025611yescharacteristic
277C22#2030025811yescharacteristic
278C4—22no—
279C4—81no—
280C4—18161no—
281C4—4544no—
282C4—4542no—
283C4C22⋊C4×D22711yescharacteristic
284C4C2×(C2×D227)⋊C411yescharacteristic
285C4#2030025611yescharacteristic
286C4#2030025811yescharacteristic
287C2—21no—
288C2—21no—
289C2—21no—
290C2—42no—
291C2—9082no—
292C2#2840888711yescharacteristic
subgroupG/H
centerC2#28408887
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
Fitting#421×C227C2
socleC2×C227#379
prime radicalC2×#20300335C2
cosocleC22×C4×C227C23
hypercenter#421D227
hypocenterC227#1796
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #28409049C2semidirect
3. #28409042C2semidirect
4. #28409040C2semidirect
5. #28409014C2semidirect
6. #28409005C2semidirect
7. #421×C227C2semidirect
15. C2×#16813883C4semidirect
18. C2×Q8×D227C4semidirect
146. C227#1796semidirect
150. #421D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003352
#284090492
#284090422
#284090402
#284090142
#284090052
#421×C2272
#1796227