Finite Group Databasebeta

← C2×#23757466 #34109546

Subgroups — C2×#23757466 #34109546

Order 20096.

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

normal subgroups only

This group has 648 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×#14578939C211yesmaximal, characteristic, complemented
2C2×#19097767C211yesmaximal, characteristic, complemented
3C2×D4×Dic157C211yesmaximal, characteristic, complemented
4#23757466C218yesmaximal, direct product factor
5C2×#19097700C211yesmaximal, characteristic, complemented
6C2×#19097693C211yesmaximal, characteristic, complemented
7C2×#19097692C211yesmaximal, characteristic, complemented
8C22×D8×C157C211yesmaximal, characteristic, complemented
9C22×D628C2211yescharacteristic
10C22×C22⋊D157—21no—
11C22×C22⋊D157—21no—
12C23×Dic157—21no—
13C2×#14578939—24no—
14C2×#14578939C2214yescomplemented
15C22×#11929434C2211yescharacteristic
16#19097767—22no—
17#19097767C2214yescomplemented
18D4×Dic157—22no—
19D4×Dic157C2214yescomplemented
20C2×C4⋊Dic157C2211yescharacteristic
21C2×C22⋊Dic157—21no—
22C2×C4×Dic157C2211yescharacteristic
23#19097700C2214yescomplemented
24#19097693C2214yescomplemented
25#19097692C2214yescomplemented
26C22×D4×C157C2211yescharacteristic
27C22×D4×C157C2211yescharacteristic
28C2×D8×C157—24no—
29C2×D8×C157C2214yescomplemented
30C22×C8×C157C2211yescharacteristic
31C23×D157—21no—
32C2×D628—22no—
33C2×D628—21no—
34C2×D628—22no—
35C2×D628C2312yes—
36C2×C22⋊D157—24no—
37C2×C22⋊D157—24no—
38C2×C22⋊D157—44no—
39C2×C22⋊D157—44no—
40C22×Dic157—24no—
41C22×Dic157—21no—
42C22×Dic157—44no—
43C22×Dic157D411yescharacteristic
44C22×Dic157D411yescharacteristic
45#14578939—28no—
46C2×#11929434—22no—
47C2×#11929434C2312yes—
48C4⋊Dic157—21no—
49C4⋊Dic157C2312yes—
50C22⋊Dic157—24no—
51C4×Dic157D412yes—
52C4×Dic157C2312yes—
53C24×C157—21no—
54C24×C157—21no—
55C2×D4×C157—21no—
56C2×D4×C157—22no—
57C2×D4×C157—21no—
58C2×D4×C157—22no—
59C2×D4×C157—22no—
60C2×D4×C157D414yescomplemented
61C2×D4×C157C2312yes—
62C2×D4×C157C2312yes—
63C22×C4×C157C2311yescharacteristic
64D8×C157—28no—
65C2×C8×C157—22no—
66C2×C8×C157C2312yes—
67C22×D157—22no—
68C22×D157—42no—
69C22×D157—42no—
70C22×D157—42no—
71D628—22no—
72D628—22no—
73D628—42no—
74C22⋊D157—48no—
75C22⋊D157—48no—
76C2×Dic157—22no—
77C2×Dic157—22no—
78C2×Dic157—42no—
79C2×Dic157—48no—
80C2×Dic157D814yescomplemented
81C2×Dic157C2×D412yes—
82C2×Dic157C2×D412yes—
83#11929434—22no—
84C23×C157—24no—
85C23×C157—22no—
86C23×C157—22no—
87C23×C157—42no—
88C23×C157—42no—
89C23×C157—42no—
90C23×C157—42no—
91C23×C157—42no—
92C23×C157C2×D411yescharacteristic
93D4×C157—24no—
94D4×C157—22no—
95D4×C157—22no—
96D4×C157—22no—
97D4×C157—42no—
98D4×C157C2×D414yescomplemented
99C2×C4×C157C2×D411yescharacteristic
100C2×C4×C157C2×D412yes—
101C2×C4×C157#4112yes—
102C2×C4×C157C2411yescharacteristic
103C8×C157—22no—
104C22×C157—24no—
105C22×C157—42no—
106C22×C157—42no—
107C22×C157—48no—
108C22×C157—42no—
109C22×C157—42no—
110C22×C157—42no—
111C22×C157—84no—
112C22×C157#12112yes—
113C22×C157C2×D812yes—
114C22×C157AGL(1,ℤ/8)12yes—
115C22×C157C22×D411yescharacteristic
116C2×D157—42no—
117C2×D157—42no—
118C2×D157—42no—
119C2×D157—84no—
120C4×C157#12112yes—
121C4×C157C22×D411yescharacteristic
122C4×C157C2×#4111yescharacteristic
123Dic157—22no—
124Dic157—42no—
125Dic157C2×D814yescomplemented
126C2×C157—44no—
127C2×C157—82no—
128C2×C157#45914yescomplemented
129C2×C157C2×#12111yescharacteristic
130C2×C157C22×D811yescharacteristic
131C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
132D157—82no—
133C157C2×#45911yescharacteristic, complemented, Sylow
134C2×#459—1571nomaximal, Sylow
135C22×D8—1571no—
136C22×D8D15711yescharacteristic, complemented
137C2×#127—1571no—
138C2×C4×D4—1571no—
139#459—1578no—
140C2×C4⋊C8—1571no—
141C2×#102—1571no—
142C2×#102—1571no—
143C22×D4—3141no—
144C22×D4—3141no—
145C22×D4—1571no—
146C22×D4C2×D15711yescharacteristic
147C22×D4C2×D15711yescharacteristic
148C23×C4—3141no—
149C2×D8—24no—
150C2×D8—3144no—
151C2×D8—1574no—
152C2×D8C2×D15714yescomplemented
153C22×C8—1571no—
154C22×C8C2×D15711yescharacteristic
155#127—3142no—
156#127—1574no—
157C4×D4—3142no—
158C4×D4—1574no—
159C2×C4⋊C4—1571no—
160C2×C22⋊C4—3141no—
161C2×C42—1571no—
162C4⋊C8—1574no—
163#102—1574no—
164#102—1574no—
165C24—21no—
166C24—21no—
167C24—3141no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—22no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—3144no—
174C2×D4—3144no—
175C2×D4—3141no—
176C2×D4—3142no—
177C2×D4—3142no—
178C2×D4—6284no—
179C2×D4—6284no—
180C2×D4—1572no—
181C2×D4C22⋊D15714yescomplemented
182C2×D4C22×D15712yes—
183C2×D4C22×D15712yes—
184C22×C4—3141no—
185C22×C4—3144no—
186C22×C4—6284no—
187C22×C4—1571no—
188C22×C4—1571no—
189C22×C4C22×D15711yescharacteristic
190D8—28no—
191D8—3148no—
192C2×C8—22no—
193C2×C8—3142no—
194C2×C8—1572no—
195C2×C8C22×D15712yes—
196C4⋊C4—3141no—
197C4⋊C4—1572no—
198C22⋊C4—3144no—
199C42—1572no—
200C42—1572no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—3142no—
210C23—6282no—
211C23—6282no—
212C23—6282no—
213C23D4×D15711yescharacteristic
214D4—24no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—42no—
219D4—3142no—
220D4—3142no—
221D4—6288no—
222D4—6288no—
223D4—6282no—
224D4C2×C22⋊D15714yescomplemented
225C2×C4—3142no—
226C2×C4—3142no—
227C2×C4—6282no—
228C2×C4—6288no—
229C2×C4—1572no—
230C2×C4—1574no—
231C2×C4—1572no—
232C2×C4D4×D15711yescharacteristic
233C2×C4C2×C22⋊D15712yes—
234C2×C4#1457895312yes—
235C2×C4C23×D15711yescharacteristic
236C8—22no—
237C8—3142no—
238C22—24no—
239C22—48no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—84no—
246C22—6282no—
247C22—6282no—
248C22—6282no—
249C22—12564no—
250C22#1909774312yes—
251C22D8×D15712yes—
252C22#1909779512yes—
253C22C2×D4×D15711yescharacteristic
254C4—3142no—
255C4—6282no—
256C4—1574no—
257C4#1909774312yes—
258C4C22×C22⋊D15711yescharacteristic
259C4C2×#1457895311yescharacteristic
260C2—44no—
261C2—82no—
262C2—12562no—
263C2#2375746614yesdirect product factor
264C2C2×#1909774311yescharacteristic
265C2C2×D8×D15711yescharacteristic
266C2C2×#1909779511yescharacteristic
subgroupG/H
centerC23D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC22×D8×C157C2
socleC23×C157C2×D4
hypercenterC22×D8D157
hypocenterC157C2×#459
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#14578939C2semidirect
2. C2×#19097767C2semidirect
3. C2×D4×Dic157C2semidirect
4. #23757466C2direct (N × H)
5. C2×#19097700C2semidirect
6. C2×#19097693C2semidirect
7. C2×#19097692C2semidirect
8. C22×D8×C157C2semidirect
14. C2×#14578939C22semidirect
17. #19097767C22semidirect
19. D4×Dic157C22semidirect
23. #19097700C22semidirect
24. #19097693C22semidirect
25. #19097692C22semidirect
29. C2×D8×C157C22semidirect
60. C2×D4×C157D4semidirect
80. C2×Dic157D8semidirect
98. D4×C157C2×D4semidirect
125. Dic157C2×D8semidirect
128. C2×C157#459semidirect
133. C157C2×#459semidirect
136. C22×D8D157semidirect
152. C2×D8C2×D157semidirect
181. C2×D4C22⋊D157semidirect
224. D4C2×C22⋊D157semidirect
263. C2#23757466direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145789392
C2×#190977672
C2×D4×Dic1572
#237574662
C2×#190977002
C2×#190976932
C2×#190976922
C22×D8×C1572
C2×#459157