Finite Group Databasebeta

← C2×#31774123 #46845921

Subgroups — C2×#31774123 #46845921

Order 34688.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×C8×D271C211yesmaximal, characteristic, complemented
2C2×#22576136C211yesmaximal, characteristic, complemented
3C2×#22576131C211yesmaximal, characteristic
4#31774123C218yesmaximal, direct product factor
5C2×#22576092C211yesmaximal, characteristic, complemented
6C2×#22576084C211yesmaximal, characteristic, complemented
7C2×#22576081C211yesmaximal, characteristic, complemented
8C22×SD16×C271C211yesmaximal, characteristic, complemented
9C22×C22⋊D271—21no—
10C22×C4×D271C2211yescharacteristic
11C22×C271⋊C8C2211yescharacteristic
12C2×C8×D271—24no—
13C2×C8×D271C2214yescomplemented
14#22576136—22no—
15#22576136C2214yescomplemented
16#22576131—22no—
17#22576131C2214yes—
18C2×C4⋊Dic271C2211yescharacteristic
19C2×C4⋊Dic271C2211yescharacteristic
20C2×Dic271⋊C4—21no—
21C2×C22⋊Dic271—21no—
22C2×(C2×D271)⋊C4—21no—
23#22576092C2214yescomplemented
24#22576084C2214yescomplemented
25#22576081C2214yescomplemented
26C22×Q8×C271C2211yescharacteristic
27C22×D4×C271C2211yescharacteristic
28C2×SD16×C271—24no—
29C2×SD16×C271C2214yescomplemented
30C22×C8×C271C2211yescharacteristic
31C23×D271D411yescharacteristic
32C2×C22⋊D271—24no—
33C2×C22⋊D271—44no—
34C22×Dic271—21no—
35C22×Dic271—21no—
36C22×Dic271D411yescharacteristic
37C2×C4×D271—22no—
38C2×C4×D271—22no—
39C2×C4×D271D412yes—
40C2×C4×D271C2312yes—
41C2×C271⋊C8—22no—
42C2×C271⋊C8C2312yes—
43C8×D271—28no—
44C4⋊Dic271—21no—
45C4⋊Dic271—21no—
46C4⋊Dic271C2312yes—
47C4⋊Dic271C2312yes—
48Dic271⋊C4—24no—
49C22⋊Dic271—24no—
50(C2×D271)⋊C4—24no—
51C24×C271—21no—
52C2×Q8×C271—21no—
53C2×Q8×C271—22no—
54C2×Q8×C271—22no—
55C2×Q8×C271C2312yes—
56C2×D4×C271—22no—
57C2×D4×C271—21no—
58C2×D4×C271—22no—
59C2×D4×C271C2312yes—
60C22×C4×C271—21no—
61C22×C4×C271C2311yescharacteristic
62SD16×C271—28no—
63C2×C8×C271D414yescomplemented
64C2×C8×C271C2312yes—
65C22×D271—22no—
66C22×D271—22no—
67C22×D271SD1614yescomplemented
68C22×D271C2×D412yes—
69C22⋊D271—48no—
70C2×Dic271—22no—
71C2×Dic271—22no—
72C2×Dic271—22no—
73C2×Dic271—42no—
74C2×Dic271—42no—
75C2×Dic271C2×D412yes—
76C4×D271—24no—
77C4×D271—22no—
78C4×D271—22no—
79C271⋊C8—22no—
80C23×C271—22no—
81C23×C271—42no—
82C23×C271—42no—
83C23×C271—42no—
84C23×C271C2×D411yescharacteristic
85Q8×C271—22no—
86Q8×C271—22no—
87Q8×C271—42no—
88D4×C271—22no—
89D4×C271—22no—
90D4×C271—42no—
91C2×C4×C271—22no—
92C2×C4×C271—42no—
93C2×C4×C271C2×D411yescharacteristic
94C2×C4×C271C2×D412yes—
95C2×C4×C271#4112yes—
96C2×C4×C271C2411yescharacteristic
97C8×C271C2×D414yescomplemented
98C22×C271—42no—
99C22×C271—42no—
100C22×C271—42no—
101C22×C271—84no—
102C22×C271#12112yes—
103C22×C271C2×SD1612yes—
104C22×C271#13512yes—
105C22×C271C22×D411yescharacteristic
106C2×D271—22no—
107C2×D271—22no—
108C2×D271—28no—
109C2×D271C2×SD1614yescomplemented
110C4×C271—42no—
111C4×C271#12112yes—
112C4×C271C22×D411yescharacteristic
113C4×C271C2×#4111yescharacteristic
114Dic271—22no—
115Dic271—42no—
116Dic271—42no—
117C2×C271—82no—
118C2×C271#46514yescomplemented
119C2×C271C2×#12111yescharacteristic
120C2×C271C22×SD1611yescharacteristic
121C2×C271C2×#13511yescharacteristic
122D271—24no—
123C271C2×#46511yescharacteristic, complemented, Sylow
124C2×#465—2711nomaximal, Sylow
125C22×SD16D27111yescharacteristic, complemented
126C23×C8—2711no—
127C2×C22⋊Q8—2711no—
128C2×#121—2711no—
129#465—2718no—
130C2×#106—2711no—
131C2×#103—2711no—
132C2×#102—2711no—
133C22×Q8C2×D27111yescharacteristic
134C22×D4—5421no—
135C22×D4C2×D27111yescharacteristic
136C23×C4—2711no—
137C2×SD16—24no—
138C2×SD16C2×D27114yescomplemented
139C22×C8—2711no—
140C22×C8—2714no—
141C22×C8—5424no—
142C22×C8C2×D27111yescharacteristic
143C22⋊Q8—2714no—
144C22⋊Q8—5422no—
145#121—2714no—
146#121—5422no—
147C2×C4⋊C4—2711no—
148C2×C4⋊C4—2711no—
149C2×C4⋊C4—5421no—
150C2×C22⋊C4—5421no—
151C2×C22⋊C4—5421no—
152#106—2714no—
153#103—2714no—
154#102—2714no—
155C24—21no—
156C24—2711no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D27112yes—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—5424no—
165C2×D4—10844no—
166C2×D4C22×D27112yes—
167C22×C4—21no—
168C22×C4—2711no—
169C22×C4—2712no—
170C22×C4—2712no—
171C22×C4—5422no—
172C22×C4—5422no—
173C22×C4—5421no—
174C22×C4—5421no—
175C22×C4C22×D27111yescharacteristic
176SD16—28no—
177C2×C8—2712no—
178C2×C8—5422no—
179C2×C8—5428no—
180C2×C8C22⋊D27114yescomplemented
181C2×C8C22×D27112yes—
182C4⋊C4—2712no—
183C4⋊C4—2712no—
184C4⋊C4—5424no—
185C4⋊C4—5421no—
186C4⋊C4—5421no—
187C22⋊C4—5424no—
188C22⋊C4—5424no—
189C23—22no—
190C23—42no—
191C23—42no—
192C23—42no—
193C23—2714no—
194C23—2712no—
195C23—5422no—
196C23—5422no—
197C23D4×D27111yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—22no—
202D4—22no—
203D4—42no—
204D4—10848no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—2712no—
208C2×C4—5422no—
209C2×C4—5424no—
210C2×C4—5422no—
211C2×C4—5422no—
212C2×C4—5422no—
213C2×C4—5422no—
214C2×C4—10842no—
215C2×C4—10842no—
216C2×C4D4×D27111yescharacteristic
217C2×C4C2×C22⋊D27112yes—
218C2×C4#1686465312yes—
219C2×C4C23×D27111yescharacteristic
220C8—5422no—
221C8C2×C22⋊D27114yescomplemented
222C22—42no—
223C22—42no—
224C22—42no—
225C22—84no—
226C22—2714no—
227C22—5428no—
228C22—5422no—
229C22—5422no—
230C22#2257613112yes—
231C22SD16×D27112yes—
232C22#2257618112yes—
233C22C2×D4×D27111yescharacteristic
234C4—42no—
235C4—5422no—
236C4—10842no—
237C4—10842no—
238C4#2257613112yes—
239C4C22×C22⋊D27111yescharacteristic
240C4C2×#1686465311yescharacteristic
241C2—82no—
242C2—5424no—
243C2#3177412314yesdirect product factor
244C2C2×#2257613111yescharacteristic
245C2C2×SD16×D27111yescharacteristic
246C2C2×#2257618111yescharacteristic
subgroupG/H
centerC23D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC22×SD16×C271C2
socleC23×C271C2×D4
prime radicalC2×#22576131C2
hypercenterC22×SD16D271
hypocenterC271C2×#465
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×C8×D271C2semidirect
2. C2×#22576136C2semidirect
4. #31774123C2direct (N × H)
5. C2×#22576092C2semidirect
6. C2×#22576084C2semidirect
7. C2×#22576081C2semidirect
8. C22×SD16×C271C2semidirect
13. C2×C8×D271C22semidirect
15. #22576136C22semidirect
23. #22576092C22semidirect
24. #22576084C22semidirect
25. #22576081C22semidirect
29. C2×SD16×C271C22semidirect
63. C2×C8×C271D4semidirect
67. C22×D271SD16semidirect
97. C8×C271C2×D4semidirect
109. C2×D271C2×SD16semidirect
118. C2×C271#465semidirect
123. C271C2×#465semidirect
125. C22×SD16D271semidirect
138. C2×SD16C2×D271semidirect
180. C2×C8C22⋊D271semidirect
221. C8C2×C22⋊D271semidirect
243. C2#31774123direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C8×D2712
C2×#225761362
C2×#225761312
#317741232
C2×#225760922
C2×#225760842
C2×#225760812
C22×SD16×C2712
C2×#465271