Finite Group Databasebeta

← #46846310

Subgroups — #46846310

Order 34688.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#22576136C211yesmaximal, characteristic
2#117×D271C211yesmaximal, characteristic
3#31774237C212yesmaximal, complemented
4#31774236C212yesmaximal, complemented
5#31774231C212yesmaximal, complemented
6#31774226C212yesmaximal, complemented
7#31773940C211yesmaximal, characteristic, complemented
8#31773939C211yesmaximal, characteristic, complemented
9#31773906C211yesmaximal, characteristic, complemented
10#31773860C211yesmaximal, characteristic
11#419×C271C211yesmaximal, characteristic, complemented
12C22×C4×D271C2211yescharacteristic
13C2×#16864638C2211yescharacteristic
14#22576136—21no—
15#22576136—21no—
16#22576136C2212yes—
17#22576136C2212yes—
18#22576119C2211yescharacteristic
19#22576118C2211yescharacteristic
20#22576118C2211yescharacteristic
21#22576117—21no—
22#22576116—21no—
23C2×C4⋊Dic271C2211yescharacteristic
24C2×Dic271⋊C4—21no—
25C4⋊C4×D271C2211yescharacteristic
26C4⋊C4×D271C2211yescharacteristic
27C2×(C2×D271)⋊C4—21no—
28C22⋊C4×D271—21no—
29C42×D271—21no—
30#22576094C2212yescomplemented
31#22576093C2212yescomplemented
32#22576092C2212yescomplemented
33#22576091C2212yescomplemented
34#22576084C2212yescomplemented
35#22576084C2212yescomplemented
36(C2×D271)⋊C8C2212yes—
37(C2×D271)⋊C8C2212yes—
38C22×Q8×C271C2211yescharacteristic
39C2×M16×C271C2211yescharacteristic
40#117×C271C2211yescharacteristic
41#103×C271C2212yescomplemented
42#103×C271C2212yescomplemented
43C23×D271D411yescharacteristic
44C22×Dic271—21no—
45C22×Dic271D411yescharacteristic
46C2×C4×D271—21no—
47C2×C4×D271—21no—
48C2×C4×D271—22no—
49C2×C4×D271—22no—
50C2×C4×D271D411yescharacteristic
51C2×C4×D271D411yescharacteristic
52C2×C4×D271C2311yescharacteristic
53C2×C4×D271C2311yescharacteristic
54#16864638—22no—
55C2×C271⋊C8C2312yes—
56C4⋊Dic271—21no—
57C4⋊Dic271C2312yes—
58C4⋊Dic271C2311yescharacteristic
59C4⋊Dic271C2311yescharacteristic
60Dic271⋊C4—22no—
61Dic271⋊C4—22no—
62Dic271⋊C4—21no—
63Dic271⋊C4—21no—
64C22⋊Dic271—21no—
65(C2×D271)⋊C4—22no—
66(C2×D271)⋊C4—22no—
67(C2×D271)⋊C4—21no—
68(C2×D271)⋊C4—21no—
69C4×Dic271—21no—
70C4×Dic271—21no—
71C4×Dic271—21no—
72C2×Q8×C271—21no—
73C2×Q8×C271—22no—
74C2×Q8×C271—22no—
75C2×Q8×C271C2312yes—
76C22×C4×C271—21no—
77C22×C4×C271C2311yescharacteristic
78M16×C271—22no—
79C2×C8×C271C2312yes—
80C4⋊C4×C271C2311yescharacteristic
81C4⋊C4×C271C2311yescharacteristic
82C22⋊C4×C271—21no—
83C42×C271—21no—
84C22×D271—22no—
85C22×D271—22no—
86C22×D271—41no—
87C22×D271C2×D411yescharacteristic
88C22×D271C2×D411yescharacteristic
89C2×Dic271—22no—
90C2×Dic271—21no—
91C2×Dic271—21no—
92C2×Dic271—22no—
93C2×Dic271—42no—
94C2×Dic271C2×D411yescharacteristic
95C2×Dic271C2×D411yescharacteristic
96C4×D271—22no—
97C4×D271—22no—
98C4×D271—21no—
99C4×D271—21no—
100C4×D271—21no—
101C4×D271—21no—
102C4×D271—42no—
103C4×D271—42no—
104C271⋊C8—22no—
105C23×C271C2×D411yescharacteristic
106Q8×C271—22no—
107Q8×C271—22no—
108Q8×C271—42no—
109C2×C4×C271—21no—
110C2×C4×C271—21no—
111C2×C4×C271—22no—
112C2×C4×C271—42no—
113C2×C4×C271C2×D411yescharacteristic
114C2×C4×C271#4112yes—
115C2×C4×C271#4112yes—
116C2×C4×C271C2411yescharacteristic
117C8×C271—22no—
118C22×C271—21no—
119C22×C271—21no—
120C22×C271#12312yes—
121C22×C271C22×D411yescharacteristic
122C2×D271—21no—
123C2×D271—21no—
124C2×D271—21no—
125C2×D271—21no—
126C2×D271—21no—
127C2×D271—44no—
128C2×D271#13712yes—
129C4×C271—42no—
130C4×C271—41no—
131C4×C271—41no—
132C4×C271#12312yes—
133C4×C271C2×#4111yescharacteristic
134C4×C271C2×#4111yescharacteristic
135Dic271—21no—
136Dic271—21no—
137Dic271—41no—
138Dic271—41no—
139Dic271—42no—
140C2×C271—22no—
141C2×C271C2×#12311yescharacteristic
142C2×C271C2×#13711yescharacteristic
143C2×C271#57511yescharacteristic
144D271—22no—
145D271—41no—
146C271#309511yescharacteristic, complemented, Sylow
147#3095—2711nomaximal, Sylow
148C2×C22⋊Q8—2711no—
149C2×#117—2711no—
150#485—2712no—
151#485—2712no—
152#484—2712no—
153#483—2712no—
154#428—2711no—
155#428—2711no—
156#419—2711no—
157#419D27111yescharacteristic, complemented
158#407—2711no—
159C22×Q8C2×D27111yescharacteristic
160C23×C4—2711no—
161C2×M16—2711no—
162C2×M16C2×D27111yescharacteristic
163C22⋊Q8—2712no—
164C22⋊Q8—2712no—
165C22⋊Q8—5421no—
166C22⋊Q8—5421no—
167#117—2711no—
168#117—2711no—
169#117—2711no—
170#117—5421no—
171#117—5421no—
172#117C2×D27111yescharacteristic
173C2×C4⋊C4—2711no—
174C2×C4⋊C4—2711no—
175C2×C4⋊C4—2711no—
176C2×C4⋊C4—5421no—
177C2×C22⋊C4—5421no—
178C2×C22⋊C4—5421no—
179C2×C42—5421no—
180#107—2712no—
181#107—2712no—
182#106—2712no—
183#106—2712no—
184#103—2712no—
185#103—2712no—
186#103C2×D27112yescomplemented
187#103C2×D27112yescomplemented
188C22⋊C8—2712no—
189C22⋊C8—2712no—
190C24—2711no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D27112yes—
195C22×C4—21no—
196C22×C4—2711no—
197C22×C4—2711no—
198C22×C4—2711no—
199C22×C4—2711no—
200C22×C4—2711no—
201C22×C4—5421no—
202C22×C4—5421no—
203C22×C4—5422no—
204C22×C4—5422no—
205C22×C4—5421no—
206C22×C4C22×D27111yescharacteristic
207M16—22no—
208M16—5422no—
209C2×C8—2712no—
210C2×C8C22×D27112yes—
211C4⋊C4—2711no—
212C4⋊C4—2711no—
213C4⋊C4—2712no—
214C4⋊C4—5421no—
215C4⋊C4—5421no—
216C4⋊C4—5422no—
217C4⋊C4—5422no—
218C4⋊C4—5421no—
219C4⋊C4C22×D27111yescharacteristic
220C4⋊C4C22×D27111yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—5421no—
223C22⋊C4—5421no—
224C22⋊C4—5422no—
225C22⋊C4—5422no—
226C22⋊C4—5421no—
227C42—21no—
228C42—5421no—
229C42—5421no—
230C42—5421no—
231C23—2711no—
232C23—2711no—
233C23—5422no—
234C23—5422no—
235C23—10841no—
236C23D4×D27111yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—2711no—
245C2×C4—2711no—
246C2×C4—5422no—
247C2×C4—5422no—
248C2×C4—5422no—
249C2×C4—5421no—
250C2×C4—5421no—
251C2×C4—5421no—
252C2×C4—5421no—
253C2×C4—5421no—
254C2×C4—5421no—
255C2×C4—5422no—
256C2×C4—10842no—
257C2×C4—10842no—
258C2×C4—10842no—
259C2×C4D4×D27111yescharacteristic
260C2×C4#1686465312yes—
261C2×C4#1686465412yes—
262C2×C4C23×D27111yescharacteristic
263C8—22no—
264C8—5422no—
265C22—21no—
266C22—21no—
267C22—2712no—
268C22—5421no—
269C22—5421no—
270C22—5421no—
271C22—5421no—
272C22—5421no—
273C22—10844no—
274C22#2257614212yes—
275C22C2×D4×D27111yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—5421no—
280C4—5421no—
281C4—10841no—
282C4—10841no—
283C4—10842no—
284C4#2257614212yes—
285C4C2×#1686465311yescharacteristic
286C4C2×#1686465411yescharacteristic
287C2—22no—
288C2—5422no—
289C2—10841no—
290C2C2×#2257614211yescharacteristic
291C2#137×D27111yescharacteristic
292C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#419×C271C2
socleC22×C271C22×D4
prime radicalC23×D271D4
hypercenter#419D271
hypocenterC271#3095
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #31774237C2semidirect
4. #31774236C2semidirect
5. #31774231C2semidirect
6. #31774226C2semidirect
7. #31773940C2semidirect
8. #31773939C2semidirect
9. #31773906C2semidirect
11. #419×C271C2semidirect
30. #22576094C22semidirect
31. #22576093C22semidirect
32. #22576092C22semidirect
33. #22576091C22semidirect
34. #22576084C22semidirect
35. #22576084C22semidirect
41. #103×C271C22semidirect
42. #103×C271C22semidirect
146. C271#3095semidirect
157. #419D271semidirect
186. #103C2×D271semidirect
187. #103C2×D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761362
#117×D2712
#317742372
#317742362
#317742312
#317742262
#317739402
#317739392
#317739062
#317738602
#419×C2712
#3095271