Finite Group Databasebeta

← #46838943

Subgroups — #46838943

Order 34688.

1104 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×D4×D271C211yesmaximal, characteristic
2C2×#22576118C211yesmaximal, characteristic, complemented
3C2×#22576117C211yesmaximal, characteristic, complemented
4#31773800C211yesmaximal, characteristic, complemented
5#31773799C211yesmaximal, characteristic, complemented
6#31773798C211yesmaximal, characteristic, complemented
7#31773797C214yesmaximal, complemented
8#31773752C214yesmaximal, complemented
9#390×C271C211yesmaximal, characteristic, complemented
10C24×D271C2212yes—
11C22×D1084C2211yescharacteristic
12C22×C22⋊D271C2212yes—
13C2×D4×D271—24no—
14C2×D4×D271—24no—
15C2×D4×D271C418yescomplemented
16C22×C4×D271C2211yescharacteristic
17#22576118—24no—
18#22576117—24no—
19C2×C4⋊Dic271C2211yescharacteristic, complemented
20C2×Dic271⋊C4C2212yescomplemented
21C2×C22⋊Dic271C2214yescomplemented
22C2×(C2×D271)⋊C4C2212yes—
23C2×(C2×D271)⋊C4C2212yes—
24C2×(C2×D271)⋊C4C2214yescomplemented
25C2×(C2×D271)⋊C4C2214yescomplemented
26C2×C4×Dic271C2212yescomplemented
27C2×C4×Dic271C2211yescharacteristic, complemented
28C22×D4×C271C2211yescharacteristic
29C2×C4⋊C4×C271C2211yescharacteristic, complemented
30C2×C22⋊C4×C271C2214yescomplemented
31C2×C42×C271C2211yescharacteristic, complemented
32C23×D271—24no—
33C23×D271—24no—
34C23×D271—44no—
35C23×D271—44no—
36C23×D271C2×C418yes—
37C23×D271C2312yes—
38C23×D271C2311yescharacteristic
39C2×D1084—22no—
40C2×D1084—22no—
41C2×D1084C2×C414yes—
42C2×C22⋊D271—24no—
43C2×C22⋊D271—24no—
44C2×C22⋊D271C2×C418yes—
45D4×D271—232no—
46C22×Dic271C2312yes—
47C22×Dic271C2312yes—
48C22×Dic271C2311yescharacteristic
49C2×C4×D271C2×C414yes—
50C2×C4×D271D414yes—
51C2×C4×D271D414yes—
52C4⋊Dic271—22no—
53Dic271⋊C4—24no—
54C22⋊Dic271—44no—
55(C2×D271)⋊C4—24no—
56(C2×D271)⋊C4—24no—
57(C2×D271)⋊C4—44no—
58(C2×D271)⋊C4—44no—
59C4×Dic271—24no—
60C4×Dic271—22no—
61C24×C271C2312yes—
62C2×D4×C271—22no—
63C2×D4×C271—22no—
64C2×D4×C271C2×C414yes—
65C22×C4×C271C2312yes—
66C22×C4×C271C2312yes—
67C22×C4×C271C2311yescharacteristic
68C4⋊C4×C271—22no—
69C22⋊C4×C271—44no—
70C42×C271—22no—
71C22×D271—22no—
72C22×D271—22no—
73C22×D271—22no—
74C22×D271—22no—
75C22×D271—24no—
76C22×D271—24no—
77C22×D271—216no—
78C22×D271—416no—
79C22×D271—48no—
80C22×D271—48no—
81C22×D271—416no—
82C22×D271C22×C414yes—
83C22×D271C22×C412yes—
84C22×D271C2×D412yes—
85C22×D271C2×D412yes—
86D1084—28no—
87C22⋊D271—216no—
88C2×Dic271—22no—
89C2×Dic271—22no—
90C2×Dic271—22no—
91C2×Dic271—24no—
92C2×Dic271—22no—
93C2×Dic271C22×C412yes—
94C2×Dic271C2×D412yes—
95C2×Dic271C2×D412yes—
96C4×D271C22⋊C4116yescomplemented
97C23×C271—22no—
98C23×C271—22no—
99C23×C271—24no—
100C23×C271—24no—
101C23×C271C22×C414yes—
102C23×C271C2411yescharacteristic
103D4×C271—28no—
104C2×C4×C271—22no—
105C2×C4×C271—22no—
106C2×C4×C271—22no—
107C2×C4×C271—24no—
108C2×C4×C271—22no—
109C2×C4×C271C22×C412yes—
110C2×C4×C271C2×D412yes—
111C2×C4×C271C2×D412yes—
112C22×C271—24no—
113C22×C271—28no—
114C22×C271—42no—
115C22×C271—42no—
116C22×C271—44no—
117C22×C271C23×C411yescharacteristic
118C22×C271C22×D411yescharacteristic
119C22×C271C22×D411yescharacteristic
120C22×C271H(5,𝔽2)12yes—
121C22×C271H(5,𝔽2)12yes—
122C2×D271—24no—
123C2×D271—22no—
124C2×D271—22no—
125C2×D271—24no—
126C2×D271—24no—
127C2×D271—28no—
128C2×D271—432no—
129C2×D271—42no—
130C2×D271—42no—
131C2×D271—44no—
132C2×D271C2×C22⋊C414yes—
133C4×C271—42no—
134C4×C271—42no—
135C4×C271C2×C22⋊C414yes—
136Dic271—42no—
137Dic271—42no—
138Dic271C2×C22⋊C414yes—
139C2×C271—44no—
140C2×C271C22×C22⋊C411yescharacteristic
141C2×C271#51812yes—
142C2×C271#53511yescharacteristic
143C2×C271#53511yescharacteristic
144C2×C271#53512yes—
145D271—24no—
146D271—44no—
147C271#230911yescharacteristic, complemented, Sylow
148#2309—2711nomaximal, Sylow
149C23×D4—2711no—
150C2×#117—2711no—
151C2×#117—2711no—
152#390—2711no—
153#390—2711no—
154#390—2714no—
155#390—2711no—
156#390D27111yescharacteristic, complemented
157#379—2714no—
158C25—2712no—
159C22×D4—2712no—
160C22×D4—2718no—
161C22×D4—2711no—
162C22×D4—5424no—
163C22×D4—5424no—
164C22×D4C2×D27111yescharacteristic
165C23×C4—2711no—
166#117—5424no—
167#117—5424no—
168C2×C4⋊C4—2712no—
169C2×C4⋊C4—2711no—
170C2×C4⋊C4C2×D27111yescharacteristic, complemented
171C2×C22⋊C4—2712no—
172C2×C22⋊C4—2712no—
173C2×C22⋊C4—2714no—
174C2×C22⋊C4—2714no—
175C2×C22⋊C4—2714no—
176C2×C22⋊C4C2×D27114yescomplemented
177C2×C42—2712no—
178C2×C42—2711no—
179C2×C42C2×D27111yescharacteristic, complemented
180C24—2711no—
181C24—2712no—
182C24—2718no—
183C24—5424no—
184C24—5424no—
185C24—10844no—
186C24—10844no—
187C24C22×D27112yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—2718no—
191C2×D4—2714no—
192C2×D4—54232no—
193C2×D4—5424no—
194C2×D4—5424no—
195C2×D4—5422no—
196C2×D4—5422no—
197C2×D4C4×D27114yes—
198C22×C4—2711no—
199C22×C4—2714no—
200C22×C4—2714no—
201C22×C4—2714no—
202C22×C4—2712no—
203C22×C4—2712no—
204C22×C4C22×D27112yes—
205C22×C4C22×D27112yes—
206C22×C4C22×D27111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—5424no—
209C4⋊C4—5422no—
210C22⋊C4—44no—
211C22⋊C4—5424no—
212C22⋊C4—5424no—
213C22⋊C4—10844no—
214C22⋊C4—10844no—
215C22⋊C4—10844no—
216C42—22no—
217C42—5424no—
218C42—5422no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—24no—
223C23—2712no—
224C23—2712no—
225C23—2712no—
226C23—2714no—
227C23—54216no—
228C23—5422no—
229C23—5422no—
230C23—5422no—
231C23—5422no—
232C23—5424no—
233C23—5424no—
234C23—108416no—
235C23—10848no—
236C23—10848no—
237C23—108416no—
238C23C2×C4×D27114yes—
239C23C23×D27111yescharacteristic
240D4—28no—
241D4—54216no—
242D4—5428no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—2712no—
249C2×C4—2712no—
250C2×C4—2712no—
251C2×C4—27116no—
252C2×C4—5422no—
253C2×C4—5422no—
254C2×C4—5422no—
255C2×C4—5424no—
256C2×C4—5422no—
257C2×C4C2×C4×D27112yes—
258C2×C4D4×D27112yes—
259C2×C4D4×D27112yes—
260C22—24no—
261C22—28no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—2714no—
266C22—5422no—
267C22—5424no—
268C22—5422no—
269C22—5424no—
270C22—5424no—
271C22—5428no—
272C22—108432no—
273C22—10842no—
274C22—10842no—
275C22—10844no—
276C22C22×C4×D27111yescharacteristic
277C22C2×D4×D27111yescharacteristic
278C22C2×D4×D27111yescharacteristic
279C22#2257620512yes—
280C22#2257620612yes—
281C4—42no—
282C4—42no—
283C4—2714no—
284C4—10842no—
285C4—10842no—
286C4C22⋊C4×D27114yes—
287C2—44no—
288C2—5424no—
289C2—10844no—
290C2C2×C22⋊C4×D27111yescharacteristic
291C2#3177436912yes—
292C2#3177446011yescharacteristic
293C2#3177446312yes—
294C2#3177446511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutatorC22×C271C23×C4
Fitting#390×C271C2
socle, cosocleC23×C271C24
prime radicalC22×D4×D271C2
hypercenter#390D271
hypocenterC271#2309
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#22576118C2semidirect
3. C2×#22576117C2semidirect
4. #31773800C2semidirect
5. #31773799C2semidirect
6. #31773798C2semidirect
7. #31773797C2semidirect
8. #31773752C2semidirect
9. #390×C271C2semidirect
15. C2×D4×D271C4semidirect
19. C2×C4⋊Dic271C22semidirect
20. C2×Dic271⋊C4C22semidirect
21. C2×C22⋊Dic271C22semidirect
24. C2×(C2×D271)⋊C4C22semidirect
25. C2×(C2×D271)⋊C4C22semidirect
26. C2×C4×Dic271C22semidirect
27. C2×C4×Dic271C22semidirect
29. C2×C4⋊C4×C271C22semidirect
30. C2×C22⋊C4×C271C22semidirect
31. C2×C42×C271C22semidirect
96. C4×D271C22⋊C4semidirect
147. C271#2309semidirect
156. #390D271semidirect
170. C2×C4⋊C4C2×D271semidirect
176. C2×C22⋊C4C2×D271semidirect
179. C2×C42C2×D271semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D2712
C2×#225761182
C2×#225761172
#317738002
#317737992
#317737982
#317737972
#317737522
#390×C2712
#2309271