Finite Group Databasebeta

← C2×#23757504 #34109698

Subgroups — C2×#23757504 #34109698

Order 20096.

564 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 173 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×#14578941C211yesmaximal, characteristic, complemented
2C23×#11929434C211yesmaximal, characteristic, complemented
3C2×#19097745C211yesmaximal, characteristic
4#23757504C218yesmaximal, direct product factor
5C2×#19097702C211yesmaximal, characteristic, complemented
6C2×#19097695C211yesmaximal, characteristic, complemented
7C2×#19097691C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C157C211yesmaximal, characteristic, complemented
9C22×D628C2211yescharacteristic
10C22×C22⋊D157—21no—
11C2×#14578941—24no—
12C2×#14578941C2214yescomplemented
13C22×#11929434—24no—
14C22×#11929434C2211yescharacteristic
15C22×#11929434C2211yescharacteristic
16C22×#11929434C2214yescomplemented
17#19097745—22no—
18#19097745C2214yes—
19C2×C4⋊Dic157C2211yescharacteristic
20C2×#14578931—21no—
21#19097702C2214yescomplemented
22#19097695C2214yescomplemented
23#19097691C2214yescomplemented
24C22×Q8×C157C2211yescharacteristic
25C23×C4×C157C2211yescharacteristic
26C22⋊Q8×C157—22no—
27C22⋊Q8×C157C2214yescomplemented
28C2×C4⋊C4×C157—21no—
29C2×C4⋊C4×C157C2211yescharacteristic
30C2×C22⋊C4×C157—21no—
31C23×D157—21no—
32C2×D628—22no—
33C2×D628—21no—
34C2×D628—22no—
35C2×D628C2312yes—
36C2×C22⋊D157—24no—
37C2×C22⋊D157—44no—
38C22×Dic157—21no—
39#14578941—28no—
40C2×#11929434—28no—
41C2×#11929434—22no—
42C2×#11929434D414yescomplemented
43C2×#11929434C2312yes—
44C2×#11929434C2312yes—
45C4⋊Dic157—21no—
46C4⋊Dic157C2312yes—
47#14578931—24no—
48C24×C157D411yescharacteristic
49C2×Q8×C157—21no—
50C2×Q8×C157—22no—
51C2×Q8×C157—22no—
52C2×Q8×C157C2312yes—
53C22×C4×C157—22no—
54C22×C4×C157—22no—
55C22×C4×C157—21no—
56C22×C4×C157—21no—
57C22×C4×C157D411yescharacteristic
58C22×C4×C157D412yes—
59C22×C4×C157C2312yes—
60C22×C4×C157C2311yescharacteristic
61C4⋊C4×C157—24no—
62C4⋊C4×C157—21no—
63C4⋊C4×C157C2312yes—
64C22⋊C4×C157—24no—
65C22×D157—22no—
66C22×D157—42no—
67C22×D157—42no—
68C22×D157—42no—
69D628—22no—
70D628—22no—
71D628—42no—
72C22⋊D157—48no—
73C2×Dic157—22no—
74C2×Dic157—42no—
75#11929434—22no—
76#11929434C2×D414yescomplemented
77C23×C157—22no—
78C23×C157—22no—
79C23×C157SD1614yescomplemented
80C23×C157C2×D412yes—
81C23×C157C2×D411yescharacteristic
82Q8×C157—22no—
83Q8×C157—22no—
84Q8×C157—42no—
85C2×C4×C157—22no—
86C2×C4×C157—24no—
87C2×C4×C157—22no—
88C2×C4×C157—22no—
89C2×C4×C157—22no—
90C2×C4×C157—22no—
91C2×C4×C157—42no—
92C2×C4×C157—42no—
93C2×C4×C157C2×D412yes—
94C2×C4×C157C2×D412yes—
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157#4112yes—
97C2×C4×C157C2411yescharacteristic
98C22×C157—22no—
99C22×C157—22no—
100C22×C157—28no—
101C22×C157#12112yes—
102C22×C157C2×SD1612yes—
103C22×C157C2×SD1614yescomplemented
104C22×C157#13512yes—
105C22×C157C22×D411yescharacteristic
106C2×D157—42no—
107C2×D157—42no—
108C2×D157—42no—
109C2×D157—84no—
110C4×C157—22no—
111C4×C157—42no—
112C4×C157—42no—
113C4×C157#12112yes—
114C4×C157C22×D411yescharacteristic
115C4×C157C2×#4111yescharacteristic
116Dic157—42no—
117C2×C157—24no—
118C2×C157#46514yescomplemented
119C2×C157C2×#12111yescharacteristic
120C2×C157C22×SD1611yescharacteristic
121C2×C157C2×#13511yescharacteristic
122D157—82no—
123C157C2×#46511yescharacteristic, complemented, Sylow
124C2×#465—1571nomaximal, Sylow
125C22×SD16—1571no—
126C23×C8—1571no—
127C2×C22⋊Q8D15711yescharacteristic, complemented
128C2×#121—1571no—
129#465—1578no—
130C2×#106—1571no—
131C2×#103—1571no—
132C2×#102—1571no—
133C22×Q8C2×D15711yescharacteristic
134C22×D4—3141no—
135C22×D4—1571no—
136C23×C4C2×D15711yescharacteristic
137C2×SD16—3144no—
138C2×SD16—1574no—
139C22×C8—3144no—
140C22×C8—1571no—
141C22×C8—1574no—
142C22×C8—1571no—
143C22⋊Q8—22no—
144C22⋊Q8C2×D15714yescomplemented
145#121—3142no—
146#121—1574no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—1571no—
149C2×C4⋊C4C2×D15711yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—3141no—
152#106—1574no—
153#103—1574no—
154#102—1574no—
155C24—3141no—
156C24C22⋊D15711yescharacteristic
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D15712yes—
161C2×D4—3144no—
162C2×D4—3142no—
163C2×D4—3141no—
164C2×D4—3142no—
165C2×D4—6284no—
166C2×D4—1572no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—3141no—
172C22×C4C22⋊D15712yes—
173C22×C4C22⋊D15711yescharacteristic
174C22×C4C22×D15712yes—
175C22×C4C22×D15711yescharacteristic
176SD16—3148no—
177C2×C8—3142no—
178C2×C8—3148no—
179C2×C8—1572no—
180C2×C8—1574no—
181C2×C8—1572no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—3141no—
185C4⋊C4—1572no—
186C4⋊C4C22×D15712yes—
187C22⋊C4—24no—
188C22⋊C4—3144no—
189C23—22no—
190C23—22no—
191C23—3142no—
192C23—6282no—
193C23—6282no—
194C23—6282no—
195C23#1457894114yescomplemented
196C23C2×C22⋊D15712yes—
197C23C2×C22⋊D15711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—3142no—
202D4—3142no—
203D4—6288no—
204D4—6282no—
205C2×C4—22no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—3142no—
214C2×C4—6282no—
215C2×C4D4×D15712yes—
216C2×C4C2×C22⋊D15712yes—
217C2×C4C2×C22⋊D15711yescharacteristic
218C2×C4#1457895312yes—
219C2×C4C23×D15711yescharacteristic
220C8—3142no—
221C8—1574no—
222C22—28no—
223C22—22no—
224C22—22no—
225C22—6282no—
226C22—6282no—
227C22—6282no—
228C22—12564no—
229C22#1909774312yes—
230C22C2×#1457894112yes—
231C22C2×#1457894114yescomplemented
232C22#1909779112yes—
233C22C22×C22⋊D15711yescharacteristic
234C4—22no—
235C4—42no—
236C4—42no—
237C4—6282no—
238C4#1909774312yes—
239C4C2×D4×D15711yescharacteristic
240C4C2×#1457895311yescharacteristic
241C2—24no—
242C2—12562no—
243C2#2375750414yesdirect product factor
244C2C2×#1909774311yescharacteristic
245C2C22×#1457894111yescharacteristic
246C2C2×#1909779111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×C22⋊Q8×C157C2
socleC23×C157C2×D4
prime radicalC2×#19097745C2
hypercenterC2×C22⋊Q8D157
hypocenterC157C2×#465
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14578941C2semidirect
2. C23×#11929434C2semidirect
4. #23757504C2direct (N × H)
5. C2×#19097702C2semidirect
6. C2×#19097695C2semidirect
7. C2×#19097691C2semidirect
8. C2×C22⋊Q8×C157C2semidirect
12. C2×#14578941C22semidirect
16. C22×#11929434C22semidirect
21. #19097702C22semidirect
22. #19097695C22semidirect
23. #19097691C22semidirect
27. C22⋊Q8×C157C22semidirect
42. C2×#11929434D4semidirect
76. #11929434C2×D4semidirect
79. C23×C157SD16semidirect
103. C22×C157C2×SD16semidirect
118. C2×C157#465semidirect
123. C157C2×#465semidirect
127. C2×C22⋊Q8D157semidirect
144. C22⋊Q8C2×D157semidirect
195. C23#14578941semidirect
231. C22C2×#14578941semidirect
243. C2#23757504direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145789412
C23×#119294342
C2×#190977452
#237575042
C2×#190977022
C2×#190976952
C2×#190976912
C2×C22⋊Q8×C1572
C2×#465157