Finite Group Databasebeta

← #46834869

Subgroups — #46834869

Order 34688.

334 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 77 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#31774408C211yesmaximal, characteristic
2C2×C271⋊#95C211yesmaximal, characteristic, complemented
3#31773635C212yesmaximal, complemented
4#31773632C212yesmaximal, complemented
5#407×C271C211yesmaximal, characteristic, complemented
6C2×#16864654C411yescharacteristic, complemented
7C23×Dic271—21no—
8#22576143—21no—
9#22576139C411yescharacteristic, complemented
10#22576139C2211yescharacteristic
11#22576134C411yescharacteristic, complemented
12#22576134C2211yescharacteristic
13#22576129—21no—
14C4×C22⋊D271—21no—
15C4×C22⋊D271—21no—
16#22576119C411yescharacteristic, complemented
17C271⋊#95—21no—
18C271⋊#95C2212yes—
19C23×C4×C271C2211yescharacteristic
20C2×M16×C271—21no—
21C22⋊C8×C271—21no—
22C22⋊C8×C271C2212yes—
23#16864654—42no—
24C2×Dic542C2×C411yescharacteristic, complemented
25C2×D1084C2×C411yescharacteristic, complemented
26C2×C22⋊D271—21no—
27C2×C22⋊D271—21no—
28C2×C22⋊D271—21no—
29C22×Dic271—24no—
30C22×Dic271—22no—
31C22×Dic271—42no—
32C22×Dic271—42no—
33C2×C4×D271—21no—
34C4⋊Dic271C2×C411yescharacteristic, complemented
35C4⋊Dic271C2×C411yescharacteristic, complemented
36Dic271⋊C4—21no—
37Dic271⋊C4—21no—
38C22⋊Dic271—21no—
39C22⋊Dic271—21no—
40C22⋊Dic271—21no—
41(C2×D271)⋊C4—21no—
42(C2×D271)⋊C4—21no—
43C4×Dic271—21no—
44C24×C271D411yescharacteristic
45C22×C4×C271—42no—
46C22×C4×C271C2×C411yescharacteristic
47C22×C4×C271C2×C411yescharacteristic
48C22×C4×C271D411yescharacteristic
49C22×C4×C271D411yescharacteristic
50C22×C4×C271D411yescharacteristic
51C22×C4×C271C2311yescharacteristic
52M16×C271—42no—
53C2×C8×C271—22no—
54C22×D271—21no—
55Dic542—41no—
56D1084—41no—
57C22⋊D271—42no—
58C22⋊D271—42no—
59C22⋊D271—41no—
60C22⋊D271—41no—
61C2×Dic271—24no—
62C2×Dic271—21no—
63C2×Dic271—21no—
64C2×Dic271—21no—
65C2×Dic271—48no—
66C2×Dic271—42no—
67C2×Dic271—42no—
68C4×D271—42no—
69C23×C271—22no—
70C23×C271—22no—
71C23×C271—21no—
72C23×C271—21no—
73C23×C271C22⋊C411yescharacteristic
74C23×C271C22⋊C411yescharacteristic
75C23×C271C2×D411yescharacteristic
76C2×C4×C271—22no—
77C2×C4×C271—22no—
78C2×C4×C271—41no—
79C2×C4×C271—41no—
80C2×C4×C271—41no—
81C2×C4×C271—41no—
82C2×C4×C271C22⋊C411yescharacteristic
83C2×C4×C271C22⋊C411yescharacteristic
84C2×C4×C271C22×C411yescharacteristic
85C2×C4×C271C2×D411yescharacteristic
86C8×C271—42no—
87C22×C271—24no—
88C22×C271—24no—
89C22×C271—21no—
90C22×C271—21no—
91C22×C271—21no—
92C22×C271—21no—
93C22×C271—21no—
94C22×C271—21no—
95C22×C271#9912yes—
96C22×C271#10412yes—
97C22×C271C2×C22⋊C411yescharacteristic
98C22×C271AGL(1,ℤ/8)11yescharacteristic
99C22×C271#13711yescharacteristic
100C2×D271—41no—
101C2×D271—41no—
102C2×D271—41no—
103C4×C271—21no—
104C4×C271—21no—
105C4×C271—21no—
106C4×C271—21no—
107Dic271—44no—
108Dic271—41no—
109Dic271—41no—
110Dic271—41no—
111C2×C271—21no—
112C2×C271—22no—
113C2×C271—22no—
114C2×C271—21no—
115C2×C271C2×#9911yescharacteristic
116C2×C271#41711yescharacteristic
117C2×C271C2×#10411yescharacteristic
118D271—81no—
119C271#150411yescharacteristic, complemented, Sylow
120#1504—2711nomaximal, Sylow
121#525—2711no—
122#407D27111yescharacteristic, complemented
123C2×#95—2711no—
124#328—2712no—
125#327—2712no—
126C2×#41—2711no—
127C23×C4—5421no—
128C23×C4C2×D27111yescharacteristic
129C2×M16—21no—
130#123—5421no—
131C22⋊Q8—2711no—
132C22⋊Q8—2711no—
133#121—2711no—
134#121—2711no—
135#120—5421no—
136C4×D4—5421no—
137C4×D4—5421no—
138#117—2711no—
139C22⋊C8—21no—
140C22⋊C8C2×D27112yes—
141#95—2712no—
142#95—5421no—
143C24D108411yescharacteristic
144#41—10842no—
145C2×Q8—2711no—
146C2×D4—2711no—
147C2×D4—5421no—
148C2×D4—5421no—
149C2×D4—5421no—
150C22×C4—42no—
151C22×C4—5424no—
152C22×C4—5422no—
153C22×C4—5421no—
154C22×C4—10842no—
155C22×C4—10842no—
156C22×C4C4×D27111yescharacteristic
157C22×C4C4×D27111yescharacteristic
158C22×C4C22⋊D27111yescharacteristic
159C22×C4C22⋊D27111yescharacteristic
160C22×C4D108411yescharacteristic
161C22×C4C22×D27111yescharacteristic
162M16—42no—
163C2×C8—22no—
164C4⋊C4—2711no—
165C4⋊C4—2711no—
166C4⋊C4—5421no—
167C4⋊C4—5421no—
168C22⋊C4—5421no—
169C22⋊C4—5421no—
170C22⋊C4—5421no—
171C22⋊C4—5421no—
172C22⋊C4—5421no—
173C42—5421no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—5421no—
179C23(C2×D271)⋊C411yescharacteristic
180C23(C2×D271)⋊C411yescharacteristic
181C23C2×D108411yescharacteristic
182Q8—10841no—
183D4—10841no—
184D4—10841no—
185D4—10841no—
186D4—10842no—
187D4—10842no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—5424no—
195C2×C4—5421no—
196C2×C4—5421no—
197C2×C4—5421no—
198C2×C4—10842no—
199C2×C4—10848no—
200C2×C4—10842no—
201C2×C4—10842no—
202C2×C4(C2×D271)⋊C411yescharacteristic
203C2×C4(C2×D271)⋊C411yescharacteristic
204C2×C4C2×C4×D27111yescharacteristic
205C2×C4C2×C22⋊D27111yescharacteristic
206C8—42no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—10841no—
216C22—10841no—
217C22—10841no—
218C22#2257607412yes—
219C22#2257608612yes—
220C22C2×(C2×D271)⋊C411yescharacteristic
221C22#2257618811yescharacteristic
222C22#2257619311yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—10844no—
228C4—10841no—
229C4—10841no—
230C4—10841no—
231C2—21no—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—21681no—
236C2C2×#2257607411yescharacteristic
237C2#3177389911yescharacteristic
238C2C2×#2257608611yescharacteristic
subgroupG/H
centerC22C2×(C2×D271)⋊C4
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#407×C271C2
socleC22×C271C2×C22⋊C4
prime radical#31774408C2
cosocleC22×C4×C271C23
hypercenter#407D271
hypocenterC271#1504
LCSG ▷ C2×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×C271⋊#95C2semidirect
3. #31773635C2semidirect
4. #31773632C2semidirect
5. #407×C271C2semidirect
6. C2×#16864654C4semidirect
9. #22576139C4semidirect
11. #22576134C4semidirect
16. #22576119C4semidirect
24. C2×Dic542C2×C4semidirect
25. C2×D1084C2×C4semidirect
34. C4⋊Dic271C2×C4semidirect
35. C4⋊Dic271C2×C4semidirect
119. C271#1504semidirect
122. #407D271semidirect

Maximal subgroups

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

subgroupindex
#317744082
C2×C271⋊#952
#317736352
#317736322
#407×C2712
#1504271