Finite Group Databasebeta

← #46846964

Subgroups — #46846964

Order 34688.

364 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#31774431C211yesmaximal, characteristic
2#31774303C211yesmaximal, characteristic, complemented
3#31774289C211yesmaximal, characteristic, complemented
4#31774265C211yesmaximal, characteristic, complemented
5#31774259C211yesmaximal, characteristic, complemented
6#31774153C212yesmaximal, complemented
7#31774145C212yesmaximal
8C4×#16864644C212yesmaximal, complemented
9C4×#16864641C212yesmaximal, complemented
10C4×#16864638C211yesmaximal, characteristic, complemented
11#533×C271C211yesmaximal, characteristic, complemented
12C2×#16864644C2212yes—
13C2×#16864641C2212yes—
14C2×#16864638C2212yes—
15#22576154—21no—
16#22576152C2211yescharacteristic
17#22576148C2211yescharacteristic
18#22576143—21no—
19#22576139C2211yescharacteristic
20#22576134C2211yescharacteristic
21C4×Dic542C2211yescharacteristic
22C4×D1084C2211yescharacteristic
23C4×C22⋊D271—21no—
24#22576117—21no—
25#22576093C2212yes—
26#22576091C2212yes—
27#22576084C2212yes—
28#22576084C2212yes—
29#22576083C2212yes—
30#22576080C2212yes—
31#22576071C2211yescharacteristic
32#22576071C2211yescharacteristic
33C4×C271⋊C8C2212yes—
34C4⋊Q8×C271C2211yescharacteristic
35C4⋊Q8×C271C2211yescharacteristic
36#125×C271—21no—
37C22⋊Q8×C271—21no—
38C22⋊Q8×C271C2212yes—
39C4×Q8×C271—21no—
40C4×Q8×C271C2212yes—
41#117×C271—21no—
42C2×C42×C271C2211yescharacteristic
43C2×Dic542C2311yescharacteristic
44C2×D1084C2311yescharacteristic
45C2×C22⋊D271—21no—
46C2×C4×D271—21no—
47#16864644—24no—
48#16864641—24no—
49#16864638—24no—
50C2×C271⋊C8C2312yes—
51C2×C271⋊C8C2312yes—
52C4⋊Dic271C2311yescharacteristic
53C4⋊Dic271C2311yescharacteristic
54Dic271⋊C4—21no—
55Dic271⋊C4—21no—
56Dic271⋊C4—21no—
57Dic271⋊C4—21no—
58C22⋊Dic271—21no—
59(C2×D271)⋊C4—21no—
60(C2×D271)⋊C4—21no—
61(C2×D271)⋊C4—21no—
62(C2×D271)⋊C4—21no—
63C4×Dic271—21no—
64C2×Q8×C271—21no—
65C2×Q8×C271C2312yes—
66C22×C4×C271D411yescharacteristic
67C22×C4×C271D411yescharacteristic
68C22×C4×C271C2311yescharacteristic
69C4⋊C4×C271—21no—
70C4⋊C4×C271—21no—
71C4⋊C4×C271—21no—
72C4⋊C4×C271—22no—
73C4⋊C4×C271—22no—
74C4⋊C4×C271C2312yes—
75C22⋊C4×C271—22no—
76C42×C271—22no—
77C42×C271D411yescharacteristic
78C42×C271D411yescharacteristic
79C42×C271C2311yescharacteristic
80C42×C271C2311yescharacteristic
81C22×D271—21no—
82Dic542—22no—
83D1084—22no—
84C22⋊D271—42no—
85C2×Dic271—21no—
86C2×Dic271—21no—
87C2×Dic271—21no—
88C4×D271—42no—
89C271⋊C8—22no—
90C271⋊C8#4114yes—
91C23×C271C2×D411yescharacteristic
92Q8×C271—22no—
93Q8×C271—22no—
94Q8×C271—42no—
95C2×C4×C271—22no—
96C2×C4×C271—22no—
97C2×C4×C271—22no—
98C2×C4×C271—22no—
99C2×C4×C271—41no—
100C2×C4×C271C2×D411yescharacteristic
101C2×C4×C271C2×D411yescharacteristic
102C2×C4×C271C2×D411yescharacteristic
103C2×C4×C271C2×D411yescharacteristic
104C2×C4×C271C2×D411yescharacteristic
105C2×C4×C271C2411yescharacteristic
106C22×C271—21no—
107C22×C271—21no—
108C22×C271—21no—
109C22×C271C22×D411yescharacteristic
110C2×D271—41no—
111C2×D271—41no—
112C2×D271—41no—
113C4×C271—21no—
114C4×C271—21no—
115C4×C271—21no—
116C4×C271—21no—
117C4×C271—42no—
118C4×C271—42no—
119C4×C271#13712yes—
120C4×C271C2×#4112yes—
121Dic271—41no—
122Dic271—41no—
123Dic271—41no—
124C2×C271—41no—
125C2×C271#53211yescharacteristic
126C2×C271C2×#13711yescharacteristic
127C2×C271#57511yescharacteristic
128D271—81no—
129C271#317211yescharacteristic, complemented, Sylow
130#3172—2711nomaximal, Sylow
131#533D27111yescharacteristic, complemented
132#529—2711no—
133#501—2711no—
134#497—2711no—
135#491—2711no—
136#489—2711no—
137#470—2712no—
138#468—2712no—
139C4×Q16—2712no—
140C4×SD16—2712no—
141C4×M16—2711no—
142C2×Q16—2712no—
143C2×SD16—2712no—
144C2×M16—2712no—
145C4⋊Q8C2×D27111yescharacteristic
146C4⋊Q8C2×D27111yescharacteristic
147#126—5421no—
148#125—21no—
149#125—2711no—
150#124—2711no—
151#123—5421no—
152C22⋊Q8—21no—
153C22⋊Q8—2711no—
154C22⋊Q8C2×D27112yes—
155#121—2711no—
156C4×Q8—21no—
157C4×Q8—2711no—
158C4×Q8C2×D27112yes—
159C4×D4—2711no—
160C4×D4—5421no—
161#117—21no—
162#117—5421no—
163C2×C42C2×D27111yescharacteristic
164#107—2712no—
165#106—2712no—
166#103—2712no—
167#103—2712no—
168#103—2712no—
169#102—2712no—
170#97—2711no—
171#97—2711no—
172C4×C8—2712no—
173C2×Q8—21no—
174C2×Q8—2711no—
175C2×Q8C22×D27112yes—
176C2×D4—2711no—
177C2×D4—5421no—
178C22×C4—5421no—
179C22×C4C22⋊D27111yescharacteristic
180C22×C4C22⋊D27111yescharacteristic
181C22×C4C22×D27111yescharacteristic
182Q16—5424no—
183SD16—5424no—
184M16—5424no—
185C2×C8—2712no—
186C2×C8—2712no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—2711no—
193C4⋊C4—2711no—
194C4⋊C4—5421no—
195C4⋊C4—5421no—
196C4⋊C4—5421no—
197C4⋊C4—5421no—
198C4⋊C4C22×D27112yes—
199C22⋊C4—22no—
200C22⋊C4—5421no—
201C22⋊C4—5421no—
202C22⋊C4—5421no—
203C22⋊C4—5421no—
204C22⋊C4—5421no—
205C42—22no—
206C42—5421no—
207C42C22⋊D27111yescharacteristic
208C42C22⋊D27111yescharacteristic
209C42C22×D27111yescharacteristic
210C42C22×D27111yescharacteristic
211C23—5421no—
212C23C2×C22⋊D27111yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—5422no—
217D4—5422no—
218D4—10842no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—5421no—
225C2×C4—5421no—
226C2×C4—5421no—
227C2×C4—10842no—
228C2×C4C2×C22⋊D27111yescharacteristic
229C2×C4C2×C22⋊D27111yescharacteristic
230C2×C4C2×C22⋊D27111yescharacteristic
231C2×C4C2×C22⋊D27111yescharacteristic
232C2×C4C2×C22⋊D27111yescharacteristic
233C2×C4C23×D27111yescharacteristic
234C8—2714no—
235C8—5422no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—10841no—
240C22—10841no—
241C22—10841no—
242C22C22×C22⋊D27111yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—10841no—
250C4—10841no—
251C4—10841no—
252C4#2257619112yes—
253C4#41×D27112yes—
254C2—41no—
255C2—21681no—
256C2#3177444311yescharacteristic
257C2C2×#2257619111yescharacteristic
258C2#3177469511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#533×C271C2
socleC22×C271C22×D4
prime radical#22576134C22
hypercenter#533D271
hypocenterC271#3172
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #31774303C2semidirect
3. #31774289C2semidirect
4. #31774265C2semidirect
5. #31774259C2semidirect
6. #31774153C2semidirect
8. C4×#16864644C2semidirect
9. C4×#16864641C2semidirect
10. C4×#16864638C2semidirect
11. #533×C271C2semidirect
129. C271#3172semidirect
131. #533D271semidirect

Maximal subgroups

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

subgroupindex
#317744312
#317743032
#317742892
#317742652
#317742592
#317741532
#317741452
C4×#168646442
C4×#168646412
C4×#168646382
#533×C2712
#3172271