Finite Group Databasebeta

← C2×#23757449 #34109464

Subgroups — C2×#23757449 #34109464

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×#19097817C211yesmaximal, characteristic
2C2×#19097802C211yesmaximal, characteristic, complemented
3C2×#19097760C211yesmaximal, characteristic, complemented
4#23757449C218yesmaximal, direct product factor
5C2×#19097698C211yesmaximal, characteristic, complemented
6C2×#19097697C211yesmaximal, characteristic, complemented
7C2×#19097687C211yesmaximal, characteristic, complemented
8C2×#137×C157C211yesmaximal, characteristic, complemented
9#19097817—22no—
10#19097817—22no—
11#19097817—22no—
12#19097817C2214yes—
13C2×#14578955—21no—
14C2×#14578954—21no—
15C2×#14578954C2211yescharacteristic
16C22×D628—21no—
17C22×D628C2211yescharacteristic
18C2×D4×D157—21no—
19C2×D4×D157—22no—
20#19097802—22no—
21#19097802C2214yescomplemented
22C2×#14578944—21no—
23C2×#14578941—21no—
24C2×#14578938C2211yescharacteristic, complemented
25#19097760—22no—
26#19097760C2214yescomplemented
27C2×#14578931—22no—
28C2×C4×Dic157C2211yescharacteristic, complemented
29#19097698C2214yescomplemented
30#19097697C2214yescomplemented
31#19097687C2214yescomplemented
32C2×#41×C157C2211yescharacteristic
33C22×Q8×C157C2211yescharacteristic, complemented
34#137×C157—22no—
35#137×C157C2214yescomplemented
36C2×Q16×C157—21no—
37C2×SD16×C157—21no—
38C2×M16×C157C2211yescharacteristic, complemented
39C23×D157—22no—
40C23×D157D412yes—
41#14578955—24no—
42#14578955—24no—
43#14578954—24no—
44#14578954—24no—
45#14578954—21no—
46#14578954—21no—
47#14578954D412yes—
48#14578954C2312yes—
49C2×Dic314D411yescharacteristic
50C2×D628—22no—
51C2×D628—24no—
52C2×D628—24no—
53C2×D628—21no—
54C2×D628—21no—
55C2×D628—21no—
56C2×D628—21no—
57C2×D628D412yes—
58C2×D628D411yescharacteristic
59C2×D628C2312yes—
60C2×C22⋊D157—22no—
61C2×C22⋊D157—21no—
62D4×D157—28no—
63D4×D157—24no—
64D4×D157—28no—
65D4×D157—24no—
66C22×Dic157—21no—
67C2×C4×D157—22no—
68C2×C4×D157—21no—
69#14578944—24no—
70#14578941—24no—
71#14578938—21no—
72#14578938C2312yescomplemented
73C2×#11929434—21no—
74#14578931—28no—
75C4×Dic157—21no—
76C4×Dic157C2312yescomplemented
77#41×C157—21no—
78#41×C157—21no—
79#41×C157D412yes—
80#41×C157C2312yes—
81C2×Q8×C157—21no—
82C2×Q8×C157—21no—
83C2×Q8×C157—21no—
84C2×Q8×C157—21no—
85C2×Q8×C157—21no—
86C2×Q8×C157D411yescharacteristic
87C2×Q8×C157C2312yescomplemented
88C2×D4×C157—21no—
89C2×D4×C157D411yescharacteristic
90C22×C4×C157—21no—
91C22×C4×C157—21no—
92C22×C4×C157C2311yescharacteristic
93Q16×C157—24no—
94SD16×C157—24no—
95M16×C157—21no—
96M16×C157C2312yescomplemented
97C2×C8×C157—21no—
98C22×D157—22no—
99C22×D157—21no—
100C22×D157—28no—
101C22×D157—22no—
102C22×D157—44no—
103C22×D157—84no—
104C22×D157C2×D414yes—
105Dic314—21no—
106Dic314C2×D412yes—
107D628—28no—
108D628—22no—
109D628—22no—
110D628—21no—
111D628—22no—
112D628—22no—
113D628C2×D412yes—
114C22⋊D157—28no—
115C22⋊D157—22no—
116C22⋊D157—22no—
117C2×Dic157—21no—
118C2×Dic157—22no—
119C2×Dic157—42no—
120C4×D157—28no—
121C4×D157—22no—
122C4×D157—22no—
123#11929434—22no—
124C23×C157—21no—
125C23×C157C2×D411yescharacteristic
126Q8×C157—21no—
127Q8×C157—22no—
128Q8×C157—22no—
129Q8×C157—41no—
130Q8×C157—41no—
131Q8×C157C2×D412yes—
132D4×C157—22no—
133D4×C157—22no—
134D4×C157—21no—
135D4×C157C2×D412yes—
136C2×C4×C157—21no—
137C2×C4×C157—22no—
138C2×C4×C157—22no—
139C2×C4×C157—22no—
140C2×C4×C157—41no—
141C2×C4×C157—41no—
142C2×C4×C157C2×D411yescharacteristic
143C2×C4×C157C2×D411yescharacteristic
144C2×C4×C157C2×D411yescharacteristic
145C2×C4×C157C2×D411yescharacteristic
146C2×C4×C157C2×D411yescharacteristic
147C2×C4×C157C2411yescharacteristic
148C8×C157—22no—
149C22×C157—22no—
150C22×C157—22no—
151C22×C157—42no—
152C22×C157#12011yescharacteristic
153C22×C157#12011yescharacteristic
154C22×C157C22×D411yescharacteristic
155C2×D157—24no—
156C2×D157—22no—
157C2×D157—44no—
158C2×D157—42no—
159C2×D157—44no—
160C2×D157—82no—
161C2×D157—88no—
162C4×C157—22no—
163C4×C157—42no—
164C4×C157#12011yescharacteristic
165C4×C157#12011yescharacteristic
166C4×C157C22×D411yescharacteristic
167C4×C157C22×D411yescharacteristic
168Dic157—22no—
169Dic157—42no—
170C2×C157—21no—
171C2×C157—21no—
172C2×C157—42no—
173C2×C157#45512yescomplemented
174C2×C157C2×#12011yescharacteristic
175D157—44no—
176D157—42no—
177C157C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—1571nomaximal, Sylow
179C2×H(5,𝔽2)—1571no—
180C2×#137—1571no—
181C2×#137D15711yescharacteristic, complemented
182C2×#124—1571no—
183#455—1578no—
184C2×#104—1571no—
185C2×#104—1571no—
186C2×#100—1571no—
187H(5,𝔽2)—3142no—
188H(5,𝔽2)—3142no—
189H(5,𝔽2)—3142no—
190H(5,𝔽2)—1574no—
191C2×#41—3141no—
192C2×#41—3141no—
193C2×#41—1571no—
194C2×#41C2×D15711yescharacteristic
195C22×Q8C2×D15711yescharacteristic, complemented
196C22×D4—3141no—
197C22×D4—3142no—
198C22×D4—3141no—
199C22×D4—1571no—
200#137—22no—
201#137—3142no—
202#137—1574no—
203#137C2×D15714yescomplemented
204C2×Q16—21no—
205C2×Q16—3141no—
206C2×SD16—21no—
207C2×SD16—3141no—
208C2×M16—1571no—
209C2×M16C2×D15711yescharacteristic, complemented
210#124—3142no—
211#124—1574no—
212C2×C22⋊C4—3142no—
213C2×C42—1571no—
214#104—1574no—
215#104—1574no—
216#100—1574no—
217C24—3142no—
218C24—1572no—
219#41—21no—
220#41—21no—
221#41—3144no—
222#41—3144no—
223#41—3144no—
224#41—3144no—
225#41—3141no—
226#41—3141no—
227#41—1572no—
228#41—1572no—
229#41C22⋊D15712yes—
230#41C22×D15712yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—1571no—
237C2×Q8C22⋊D15711yescharacteristic
238C2×Q8C22×D15712yescomplemented
239C2×D4—21no—
240C2×D4—3142no—
241C2×D4—3144no—
242C2×D4—3144no—
243C2×D4—3148no—
244C2×D4—3144no—
245C2×D4—3148no—
246C2×D4—3142no—
247C2×D4—3141no—
248C2×D4—3144no—
249C2×D4—3141no—
250C2×D4—3141no—
251C2×D4—3141no—
252C2×D4—3141no—
253C2×D4—1572no—
254C2×D4—1571no—
255C2×D4—1572no—
256C2×D4C22⋊D15711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3142no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4C22×D15711yescharacteristic
263Q16—24no—
264Q16—3144no—
265SD16—24no—
266SD16—3144no—
267M16—21no—
268M16—3141no—
269M16—1572no—
270M16C22×D15712yescomplemented
271C2×C8—21no—
272C2×C8—3141no—
273C22⋊C4—3148no—
274C42—3141no—
275C42—1572no—
276C23—21no—
277C23—3148no—
278C23—3142no—
279C23—3141no—
280C23—3142no—
281C23—6284no—
282C23—1574no—
283C23—12564no—
284C23D4×D15711yescharacteristic
285Q8—21no—
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—3141no—
291Q8—1572no—
292Q8C2×C22⋊D15712yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—3148no—
297D4—3148no—
298D4—3142no—
299D4—3142no—
300D4—3142no—
301D4—3141no—
302D4—3142no—
303D4—3142no—
304D4—3142no—
305D4—1572no—
306D4C2×C22⋊D15712yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—3148no—
314C2×C4—3141no—
315C2×C4—3142no—
316C2×C4—3142no—
317C2×C4—3142no—
318C2×C4—6282no—
319C2×C4D4×D15711yescharacteristic
320C2×C4D4×D15711yescharacteristic
321C2×C4D4×D15711yescharacteristic
322C2×C4C2×C22⋊D15711yescharacteristic
323C2×C4C2×C22⋊D15711yescharacteristic
324C2×C4C23×D15711yescharacteristic
325C8—22no—
326C8—3142no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—3144no—
331C22—3142no—
332C22—6284no—
333C22—6284no—
334C22—6282no—
335C22—12562no—
336C22—12568no—
337C22#1909773911yescharacteristic
338C22#1909773911yescharacteristic
339C22C2×D4×D15711yescharacteristic
340C4—22no—
341C4—42no—
342C4—3142no—
343C4—6282no—
344C4#1909773911yescharacteristic
345C4#1909773911yescharacteristic
346C4C2×D4×D15711yescharacteristic
347C4C22×C22⋊D15711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—6284no—
352C2—6282no—
353C2#2375744912yesdirect product factor
354C2C2×#1909773911yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#137×C157C2
socleC22×C157#120
prime radicalC2×#19097817C2
hypercenterC2×#137D157
hypocenterC157C2×#455
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#19097802C2semidirect
3. C2×#19097760C2semidirect
4. #23757449C2direct (N × H)
5. C2×#19097698C2semidirect
6. C2×#19097697C2semidirect
7. C2×#19097687C2semidirect
8. C2×#137×C157C2semidirect
21. #19097802C22semidirect
24. C2×#14578938C22semidirect
26. #19097760C22semidirect
28. C2×C4×Dic157C22semidirect
29. #19097698C22semidirect
30. #19097697C22semidirect
31. #19097687C22semidirect
33. C22×Q8×C157C22semidirect
35. #137×C157C22semidirect
38. C2×M16×C157C22semidirect
72. #14578938C23semidirect
76. C4×Dic157C23semidirect
87. C2×Q8×C157C23semidirect
96. M16×C157C23semidirect
173. C2×C157#455semidirect
177. C157C2×#455semidirect
181. C2×#137D157semidirect
195. C22×Q8C2×D157semidirect
203. #137C2×D157semidirect
209. C2×M16C2×D157semidirect
238. C2×Q8C22×D157semidirect
270. M16C22×D157semidirect
353. C2#23757449direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190978172
C2×#190978022
C2×#190977602
#237574492
C2×#190976982
C2×#190976972
C2×#190976872
C2×#137×C1572
C2×#455157