Finite Group Databasebeta

← C2×#23757450 #34109465

Subgroups — C2×#23757450 #34109465

Order 20096.

760 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19097816C211yesmaximal, characteristic
2C2×#19097802C212yesmaximal, complemented
3#23757450C218yesmaximal, direct product factor
4C2×#19097699C212yesmaximal, complemented
5C2×#19097688C211yesmaximal, characteristic, complemented
6C2×#124×C157C211yesmaximal, characteristic, complemented
7#19097816—22no—
8#19097816—24no—
9#19097816C2214yes—
10C2×#14578954C2212yes—
11C2×#14578953—22no—
12C22×C22⋊D157—22no—
13C2×D4×D157—22no—
14#19097802—24no—
15#19097802C2218yescomplemented
16C2×#14578944—22no—
17C2×#14578941—22no—
18C2×#14578938C2212yescomplemented
19#19097699C2218yescomplemented
20#19097688C2214yescomplemented
21C22×Q8×C157C2211yescharacteristic, complemented
22C22×D4×C157C2211yescharacteristic
23#124×C157—22no—
24#124×C157C2214yescomplemented
25C2×C22⋊C4×C157—22no—
26C2×C42×C157C2211yescharacteristic, complemented
27C23×D157—22no—
28#14578954—22no—
29#14578954—22no—
30#14578954D414yes—
31#14578954C2314yes—
32#14578953—28no—
33#14578953—28no—
34C2×Dic314D412yes—
35C2×D628D412yes—
36C2×C22⋊D157—24no—
37C2×C22⋊D157—28no—
38C2×C22⋊D157—22no—
39C2×C22⋊D157—28no—
40D4×D157—28no—
41D4×D157—28no—
42C22×Dic157—22no—
43C2×C4×D157—22no—
44#14578944—28no—
45#14578941—28no—
46#14578938—22no—
47#14578938C2314yescomplemented
48C2×#11929434—22no—
49C24×C157D412yes—
50C2×Q8×C157—21no—
51C2×Q8×C157—22no—
52C2×Q8×C157—22no—
53C2×Q8×C157C2312yescomplemented
54C2×D4×C157—22no—
55C2×D4×C157—22no—
56C2×D4×C157D412yes—
57C2×D4×C157C2312yes—
58C22×C4×C157—21no—
59C22×C4×C157—21no—
60C22×C4×C157C2311yescharacteristic
61C22⋊C4×C157—28no—
62C42×C157—21no—
63C42×C157C2312yescomplemented
64C22×D157—22no—
65C22×D157—28no—
66C22×D157—84no—
67Dic314—22no—
68Dic314C2×D414yes—
69D628—22no—
70D628C2×D414yes—
71C22⋊D157—216no—
72C22⋊D157—24no—
73C22⋊D157—24no—
74C2×Dic157—28no—
75C2×Dic157—22no—
76C4×D157—24no—
77C4×D157—24no—
78#11929434—24no—
79C23×C157—22no—
80C23×C157—22no—
81C23×C157—44no—
82C23×C157C2×D411yescharacteristic
83C23×C157C2×D414yes—
84Q8×C157—24no—
85Q8×C157—42no—
86D4×C157—24no—
87D4×C157—24no—
88C2×C4×C157—22no—
89C2×C4×C157—22no—
90C2×C4×C157—42no—
91C2×C4×C157—41no—
92C2×C4×C157—41no—
93C2×C4×C157C2×D412yes—
94C2×C4×C157C2×D412yes—
95C2×C4×C157C2×D411yescharacteristic
96C2×C4×C157C2411yescharacteristic
97C22×C157—22no—
98C22×C157—24no—
99C22×C157—44no—
100C22×C157—44no—
101C22×C157—82no—
102C22×C157#12011yescharacteristic
103C22×C157#12011yescharacteristic
104C22×C157C22×D411yescharacteristic
105C2×D157—24no—
106C2×D157—44no—
107C2×D157—88no—
108C4×C157—42no—
109C4×C157—42no—
110C4×C157#12012yes—
111C4×C157C22×D412yes—
112Dic157—24no—
113C2×C157—21no—
114C2×C157—21no—
115C2×C157—44no—
116C2×C157#45512yescomplemented
117C2×C157C2×#12011yescharacteristic
118D157—44no—
119C157C2×#45511yescharacteristic, complemented, Sylow
120C2×#455—1571nomaximal, Sylow
121C2×H(5,𝔽2)—1571no—
122C2×#137—1572no—
123C2×#124D15711yescharacteristic, complemented
124#455—1578no—
125C2×#104—1572no—
126C2×#100—1571no—
127H(5,𝔽2)—3144no—
128H(5,𝔽2)—3142no—
129H(5,𝔽2)—1574no—
130C2×#41—3142no—
131C2×#41—1572no—
132C22×Q8C2×D15711yescharacteristic, complemented
133C22×D4—3142no—
134C22×D4—3142no—
135C22×D4C2×D15711yescharacteristic
136#137—3144no—
137#137—1578no—
138C2×Q16—3142no—
139C2×SD16—3142no—
140C2×M16—1572no—
141#124—22no—
142#124C2×D15714yescomplemented
143C2×C22⋊C4—22no—
144C2×C42C2×D15711yescharacteristic, complemented
145#104—1578no—
146#100—1574no—
147C24—3142no—
148C24C22⋊D15712yes—
149#41—3148no—
150#41—3148no—
151#41—3142no—
152#41—3142no—
153#41—1574no—
154#41—1574no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—1572no—
159C2×Q8C22×D15712yescomplemented
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—3148no—
163C2×D4—3148no—
164C2×D4—3144no—
165C2×D4—3148no—
166C2×D4—3142no—
167C2×D4—3148no—
168C2×D4—1572no—
169C2×D4C22⋊D15712yes—
170C2×D4C22×D15712yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—3142no—
174C22×C4—3142no—
175C22×C4C22×D15711yescharacteristic
176Q16—3148no—
177SD16—3148no—
178M16—3142no—
179M16—1574no—
180C2×C8—3142no—
181C22⋊C4—28no—
182C42—21no—
183C42C22×D15712yescomplemented
184C23—22no—
185C23—22no—
186C23—44no—
187C23—3148no—
188C23—3142no—
189C23—12564no—
190C23C2×C22⋊D15711yescharacteristic
191C23C2×C22⋊D15714yes—
192Q8—24no—
193Q8—42no—
194Q8—3142no—
195Q8—1574no—
196D4—24no—
197D4—24no—
198D4—31416no—
199D4—3144no—
200D4—3144no—
201D4—3142no—
202D4—1574no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—3148no—
209C2×C4—3142no—
210C2×C4—3144no—
211C2×C4—3144no—
212C2×C4D4×D15712yes—
213C2×C4D4×D15712yes—
214C2×C4C2×C22⋊D15711yescharacteristic
215C2×C4C23×D15711yescharacteristic
216C8—3144no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—3144no—
223C22—6284no—
224C22—12568no—
225C22#1909773911yescharacteristic
226C22#1909773911yescharacteristic
227C22C22×C22⋊D15711yescharacteristic
228C4—42no—
229C4—42no—
230C4—3144no—
231C4#1909773912yes—
232C4C2×D4×D15712yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—6284no—
237C2#2375745012yesdirect product factor
238C2C2×#1909773911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#124×C157C2
socleC22×C157#120
prime radicalC2×#19097816C2
hypercenterC2×#124D157
hypocenterC157C2×#455
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#19097802C2semidirect
3. #23757450C2direct (N × H)
4. C2×#19097699C2semidirect
5. C2×#19097688C2semidirect
6. C2×#124×C157C2semidirect
15. #19097802C22semidirect
18. C2×#14578938C22semidirect
19. #19097699C22semidirect
20. #19097688C22semidirect
21. C22×Q8×C157C22semidirect
24. #124×C157C22semidirect
26. C2×C42×C157C22semidirect
47. #14578938C23semidirect
53. C2×Q8×C157C23semidirect
63. C42×C157C23semidirect
116. C2×C157#455semidirect
119. C157C2×#455semidirect
123. C2×#124D157semidirect
132. C22×Q8C2×D157semidirect
142. #124C2×D157semidirect
144. C2×C42C2×D157semidirect
159. C2×Q8C22×D157semidirect
183. C42C22×D157semidirect
237. C2#23757450direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190978162
C2×#190978022
#237574502
C2×#190976992
C2×#190976882
C2×#124×C1572
C2×#455157