Finite Group Databasebeta

← #46839721

Subgroups — #46839721

Order 34688.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 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
1C23×D1084C211yesmaximal, characteristic, complemented
2C2×#22576135C212yesmaximal, complemented
3C2×#22576130C212yesmaximal, complemented
4C2×C4×C22⋊D271C211yesmaximal, characteristic, complemented
5#31773815C212yesmaximal, complemented
6#31773807C212yesmaximal, complemented
7#31773798C212yesmaximal, complemented
8#31773752C212yesmaximal, complemented
9#380×C271C211yesmaximal, characteristic, complemented
10C24×D271C2212yes—
11C22×D1084—28no—
12C22×D1084—24no—
13C22×D1084C2212yes—
14C22×D1084C2212yes—
15C22×D1084C2214yescomplemented
16C22×C22⋊D271C2211yescharacteristic
17C22×C4×D271C2211yescharacteristic, complemented
18#22576135—28no—
19#22576130—28no—
20C4×C22⋊D271—24no—
21C2×Dic271⋊C4C2211yescharacteristic, complemented
22C2×C22⋊Dic271C2211yescharacteristic
23C2×(C2×D271)⋊C4C2214yescomplemented
24C2×(C2×D271)⋊C4C2214yescomplemented
25C2×(C2×D271)⋊C4C2211yescharacteristic, complemented
26C2×(C2×D271)⋊C4C2214yescomplemented
27C2×C4×Dic271C2211yescharacteristic, complemented
28C23×C4×C271C2211yescharacteristic
29C2×C4⋊C4×C271C2212yescomplemented
30C2×C22⋊C4×C271C2212yes—
31#95×C271C2212yescomplemented
32C23×D271—28no—
33C23×D271—24no—
34C23×D271—24no—
35C23×D271—24no—
36C23×D271—44no—
37C23×D271C2314yes—
38C23×D271C2311yescharacteristic
39C2×D1084—232no—
40C2×D1084—28no—
41C2×D1084—28no—
42C2×D1084—28no—
43C2×D1084—24no—
44C2×D1084—24no—
45C2×D1084—24no—
46C2×D1084—44no—
47C2×D1084D418yescomplemented
48C2×C22⋊D271—22no—
49C2×C22⋊D271—22no—
50C2×C22⋊D271D414yescomplemented
51C22×Dic271C2311yescharacteristic
52C22×Dic271C2311yescharacteristic
53C22×Dic271C2311yescharacteristic
54C2×C4×D271—24no—
55C2×C4×D271—22no—
56Dic271⋊C4—22no—
57C22⋊Dic271—22no—
58(C2×D271)⋊C4—28no—
59(C2×D271)⋊C4—28no—
60(C2×D271)⋊C4—22no—
61(C2×D271)⋊C4—44no—
62C4×Dic271—22no—
63C24×C271C2311yescharacteristic
64C22×C4×C271—24no—
65C22×C4×C271D414yescomplemented
66C22×C4×C271C2314yes—
67C22×C4×C271C2311yescharacteristic
68C22×C4×C271C2311yescharacteristic
69C4⋊C4×C271—24no—
70C22⋊C4×C271—24no—
71C22×D271—216no—
72C22×D271—21no—
73C22×D271—24no—
74C22×D271—22no—
75C22×D271—24no—
76C22×D271—24no—
77C22×D271—21no—
78C22×D271—24no—
79C22×D271—24no—
80C22×D271—22no—
81C22×D271—24no—
82C22×D271—416no—
83C22×D271—416no—
84C22×D271—48no—
85C22×D271—48no—
86C22×D271C2×D418yes—
87C22×D271C2×D412yescomplemented
88D1084—216no—
89D1084—216no—
90D1084—416no—
91C22⋊D271—28no—
92C2×Dic271—21no—
93C2×Dic271—21no—
94C2×Dic271—21no—
95C2×Dic271—22no—
96C2×Dic271—21no—
97C2×Dic271—22no—
98C2×Dic271C2×D412yescomplemented
99C4×D271—44no—
100C23×C271—22no—
101C23×C271—22no—
102C23×C271C2×D412yes—
103C23×C271C2×D412yes—
104C23×C271#4112yes—
105C23×C271C2411yescharacteristic
106C2×C4×C271—22no—
107C2×C4×C271—28no—
108C2×C4×C271—24no—
109C2×C4×C271—24no—
110C2×C4×C271—24no—
111C2×C4×C271C2×D414yes—
112C2×C4×C271C2×D412yes—
113C2×C4×C271C2×D412yes—
114C22×C271—22no—
115C22×C271—22no—
116C22×C271—24no—
117C22×C271—24no—
118C22×C271#12114yescomplemented
119C22×C271C22×D412yes—
120C22×C271C22×D411yescharacteristic
121C22×C271C22×D411yescharacteristic
122C22×C271C2×#4111yescharacteristic
123C22×C271H(5,𝔽2)12yes—
124C2×D271—28no—
125C2×D271—22no—
126C2×D271—24no—
127C2×D271—28no—
128C2×D271—28no—
129C2×D271—432no—
130C2×D271—41no—
131C2×D271—44no—
132C2×D271—41no—
133C2×D271—44no—
134C2×D271—44no—
135C2×D271—42no—
136C2×D271—44no—
137C4×C271—22no—
138C4×C271—44no—
139C4×C271#12014yescomplemented
140Dic271—22no—
141Dic271—41no—
142Dic271—41no—
143C2×C271—24no—
144C2×C271C2×#12011yescharacteristic
145C2×C271C2×#12111yescharacteristic
146C2×C271#53511yescharacteristic
147C2×C271D4212yes—
148C2×C271#54612yes—
149D271—42no—
150D271—48no—
151C271#240811yescharacteristic, complemented, Sylow
152#2408—2711nomaximal, Sylow
153C23×D4—2711no—
154C2×#121—2712no—
155C2×#120—2712no—
156C2×C4×D4—2711no—
157#394—2712no—
158#392—2712no—
159#390—2712no—
160#380D27111yescharacteristic, complemented
161#379—2712no—
162C25—2712no—
163C22×D4—2711no—
164C22×D4—2712no—
165C22×D4—2712no—
166C22×D4—2714no—
167C22×D4—5428no—
168C22×D4—5424no—
169C23×C4—2711no—
170C23×C4C2×D27111yescharacteristic
171#121—5428no—
172#120—5428no—
173C4×D4—5424no—
174C2×C4⋊C4—2711no—
175C2×C4⋊C4C2×D27112yescomplemented
176C2×C22⋊C4—2711no—
177C2×C22⋊C4—2711no—
178C2×C22⋊C4—2714no—
179C2×C22⋊C4—2714no—
180C2×C22⋊C4—2714no—
181C2×C22⋊C4C2×D27112yes—
182C2×C42—2711no—
183#95C2×D27112yescomplemented
184C24—2714no—
185C24—2711no—
186C24—5428no—
187C24—5424no—
188C24—5424no—
189C24—5424no—
190C24—10844no—
191C24C22×D27111yescharacteristic
192C2×D4—2714no—
193C2×D4—2718no—
194C2×D4—5428no—
195C2×D4—54232no—
196C2×D4—5428no—
197C2×D4—5428no—
198C2×D4—5422no—
199C2×D4—5422no—
200C2×D4—5424no—
201C2×D4—5424no—
202C2×D4—5424no—
203C2×D4—10844no—
204C22×C4—24no—
205C22×C4—2711no—
206C22×C4—2711no—
207C22×C4—2711no—
208C22×C4—5424no—
209C22×C4—5422no—
210C22×C4D108414yescomplemented
211C22×C4C22×D27114yes—
212C22×C4C22×D27111yescharacteristic
213C22×C4C22×D27111yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—5422no—
216C22⋊C4—24no—
217C22⋊C4—5428no—
218C22⋊C4—5428no—
219C22⋊C4—5422no—
220C22⋊C4—5422no—
221C22⋊C4—10844no—
222C42—5422no—
223C23—22no—
224C23—22no—
225C23—2718no—
226C23—2712no—
227C23—54216no—
228C23—5424no—
229C23—5422no—
230C23—5424no—
231C23—5421no—
232C23—5424no—
233C23—5421no—
234C23—5424no—
235C23—5424no—
236C23—5422no—
237C23—5424no—
238C23—108416no—
239C23—108416no—
240C23—10848no—
241C23—10848no—
242C23D4×D27112yes—
243C23C2×D108412yes—
244C23#1686465512yes—
245C23C23×D27111yescharacteristic
246D4—54216no—
247D4—5428no—
248D4—54216no—
249D4—108416no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—2712no—
256C2×C4—5421no—
257C2×C4—5421no—
258C2×C4—5422no—
259C2×C4—5421no—
260C2×C4—5421no—
261C2×C4—5422no—
262C2×C4—10844no—
263C2×C4D4×D27114yes—
264C2×C4C2×D108412yes—
265C2×C4C2×D108412yes—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—5424no—
271C22—5428no—
272C22—5428no—
273C22—5428no—
274C22—5422no—
275C22—108432no—
276C22—10844no—
277C22—10841no—
278C22—10844no—
279C22—10844no—
280C22—10842no—
281C22—10841no—
282C22—10844no—
283C22#2257613514yescomplemented
284C22C2×D4×D27111yescharacteristic
285C22C2×D4×D27112yes—
286C22C22×D108411yescharacteristic
287C22C2×#1686465511yescharacteristic
288C22#2257620612yes—
289C4—22no—
290C4—44no—
291C4—5422no—
292C4—10841no—
293C4—10841no—
294C4#2257613014yescomplemented
295C2—24no—
296C2—10848no—
297C2—10842no—
298C2C2×#2257613011yescharacteristic
299C2C2×#2257613511yescharacteristic
300C2#3177446611yescharacteristic
301C2#3177453112yes—
302C2#3177453812yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#380×C271C2
hypercenter#380D271
hypocenterC271#2408
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D1084C2semidirect
2. C2×#22576135C2semidirect
3. C2×#22576130C2semidirect
4. C2×C4×C22⋊D271C2semidirect
5. #31773815C2semidirect
6. #31773807C2semidirect
7. #31773798C2semidirect
8. #31773752C2semidirect
9. #380×C271C2semidirect
15. C22×D1084C22semidirect
17. C22×C4×D271C22semidirect
21. C2×Dic271⋊C4C22semidirect
23. C2×(C2×D271)⋊C4C22semidirect
24. C2×(C2×D271)⋊C4C22semidirect
25. C2×(C2×D271)⋊C4C22semidirect
26. C2×(C2×D271)⋊C4C22semidirect
27. C2×C4×Dic271C22semidirect
29. C2×C4⋊C4×C271C22semidirect
31. #95×C271C22semidirect
47. C2×D1084D4semidirect
50. C2×C22⋊D271D4semidirect
65. C22×C4×C271D4semidirect
87. C22×D271C2×D4semidirect
98. C2×Dic271C2×D4semidirect
118. C22×C271#121semidirect
139. C4×C271#120semidirect
151. C271#2408semidirect
160. #380D271semidirect
175. C2×C4⋊C4C2×D271semidirect
183. #95C2×D271semidirect
210. C22×C4D1084semidirect
283. C22#22576135semidirect
294. C4#22576130semidirect

Maximal subgroups

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

subgroupindex
C23×D10842
C2×#225761352
C2×#225761302
C2×C4×C22⋊D2712
#317738152
#317738072
#317737982
#317737522
#380×C2712
#2408271