Finite Group Databasebeta

← C2×#23757452 #34109467

Subgroups — C2×#23757452 #34109467

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×#19097804C211yesmaximal, characteristic, complemented
3C2×#19097801C211yesmaximal, characteristic, complemented
4C2×#19097759C211yesmaximal, characteristic, complemented
5#23757452C218yesmaximal, direct product factor
6C2×#19097699C211yesmaximal, characteristic, complemented
7C2×#19097687C211yesmaximal, characteristic, complemented
8C2×#104×C157C211yesmaximal, characteristic, complemented
9#19097817—22no—
10#19097817—22no—
11#19097817—22no—
12#19097817C2214yes—
13C2×#14578955—21no—
14C2×#14578954—21no—
15C2×#14578954C2211yescharacteristic
16C22×Dic314C2211yescharacteristic, complemented
17C22×D628—21no—
18C22×D628C2211yescharacteristic
19C2×D4×D157—21no—
20C2×D4×D157—22no—
21#19097804—22no—
22#19097804C2214yescomplemented
23#19097801—22no—
24#19097801C2214yescomplemented
25C2×Dic628—21no—
26C2×#14578944—21no—
27C2×#14578943—21no—
28C2×#14578942—21no—
29C2×#14578938C2211yescharacteristic, complemented
30#19097759—22no—
31#19097759C2214yescomplemented
32C2×#14578931—22no—
33#19097699C2214yescomplemented
34#19097687C2214yescomplemented
35C2×#41×C157C2211yescharacteristic
36C2×M16×C157C2211yescharacteristic, complemented
37C2×C42×C157C2211yescharacteristic, complemented
38#104×C157C2214yescomplemented
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×Dic314—21no—
50C2×Dic314—21no—
51C2×Dic314—21no—
52C2×Dic314—21no—
53C2×Dic314—21no—
54C2×Dic314D411yescharacteristic
55C2×Dic314C2312yescomplemented
56C2×D628—22no—
57C2×D628—24no—
58C2×D628—24no—
59C2×D628—21no—
60C2×D628—21no—
61C2×D628—21no—
62C2×D628—21no—
63C2×D628D412yes—
64C2×D628D411yescharacteristic
65C2×D628C2312yes—
66C2×C22⋊D157—22no—
67C2×C22⋊D157—21no—
68D4×D157—28no—
69D4×D157—24no—
70D4×D157—28no—
71D4×D157—24no—
72C22×Dic157—21no—
73C2×C4×D157—22no—
74C2×C4×D157—21no—
75Dic628—24no—
76#14578944—24no—
77#14578943—24no—
78#14578942—24no—
79#14578938—21no—
80#14578938C2312yescomplemented
81C2×#11929434—21no—
82#14578931—28no—
83#41×C157—21no—
84#41×C157—21no—
85#41×C157D412yes—
86#41×C157C2312yes—
87C2×Q8×C157D411yescharacteristic
88C2×D4×C157—21no—
89C2×D4×C157D411yescharacteristic
90C22×C4×C157—21no—
91C22×C4×C157—21no—
92C22×C4×C157C2311yescharacteristic
93M16×C157—21no—
94M16×C157C2312yescomplemented
95C2×C8×C157—21no—
96C42×C157—21no—
97C42×C157C2312yescomplemented
98C22×D157—22no—
99C22×D157—21no—
100C22×D157—28no—
101C22×D157—22no—
102C22×D157—44no—
103C22×D157—84no—
104C22×D157C2×D414yes—
105Dic314—21no—
106Dic314—22no—
107Dic314—22no—
108Dic314—41no—
109Dic314—41no—
110Dic314C2×D412yes—
111D628—28no—
112D628—22no—
113D628—22no—
114D628—21no—
115D628—22no—
116D628—22no—
117D628C2×D412yes—
118C22⋊D157—28no—
119C22⋊D157—22no—
120C22⋊D157—22no—
121C2×Dic157—21no—
122C2×Dic157—22no—
123C2×Dic157—41no—
124C2×Dic157—41no—
125C4×D157—28no—
126C4×D157—22no—
127C4×D157—22no—
128#11929434—22no—
129C23×C157—21no—
130C23×C157C2×D411yescharacteristic
131Q8×C157—21no—
132Q8×C157C2×D412yes—
133D4×C157—22no—
134D4×C157—22no—
135D4×C157—21no—
136D4×C157C2×D412yes—
137C2×C4×C157—21no—
138C2×C4×C157—22no—
139C2×C4×C157—22no—
140C2×C4×C157—22no—
141C2×C4×C157—42no—
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×#137—1571no—
182C2×#124—1571no—
183#455—1578no—
184C2×#104—1571no—
185C2×#104D15711yescharacteristic, complemented
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×Q8—1571no—
196C22×D4—3141no—
197C22×D4—3142no—
198C22×D4—3141no—
199C22×D4—1571no—
200#137—3142no—
201#137—3142no—
202#137—1574no—
203#137—1574no—
204C2×Q16—3141no—
205C2×Q16—3141no—
206C2×SD16—3141no—
207C2×SD16—3141no—
208C2×M16—1571no—
209C2×M16C2×D15711yescharacteristic, complemented
210#124—3142no—
211#124—1574no—
212C2×C22⋊C4—3142no—
213C2×C42C2×D15711yescharacteristic, complemented
214#104—1574no—
215#104C2×D15714yescomplemented
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#41D62812yes—
230#41C22×D15712yes—
231C2×Q8—3141no—
232C2×Q8—3141no—
233C2×Q8—3141no—
234C2×Q8—3141no—
235C2×Q8—3141no—
236C2×Q8—1571no—
237C2×Q8—1572no—
238C2×Q8D62811yescharacteristic
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×D4D62811yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3142no—
260C22×C4—3141no—
261C22×C4—3141no—
262C22×C4C22×D15711yescharacteristic
263Q16—3144no—
264Q16—3144no—
265SD16—3144no—
266SD16—3144no—
267M16—21no—
268M16—3141no—
269M16—1572no—
270M16C22×D15712yescomplemented
271C2×C8—21no—
272C2×C8—3141no—
273C22⋊C4—3148no—
274C42—21no—
275C42C22×D15712yescomplemented
276C23—21no—
277C23—3148no—
278C23—3142no—
279C23—3141no—
280C23—3142no—
281C23—6284no—
282C23—1574no—
283C23—12564no—
284C23D4×D15711yescharacteristic
285Q8—21no—
286Q8—3141no—
287Q8—3142no—
288Q8—3142no—
289Q8—6281no—
290Q8—6281no—
291Q8—1572no—
292Q8C2×D62812yes—
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×D62812yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—3148no—
313C2×C4—3141no—
314C2×C4—3142no—
315C2×C4—3142no—
316C2×C4—3142no—
317C2×C4—6281no—
318C2×C4—6281no—
319C2×C4D4×D15711yescharacteristic
320C2×C4D4×D15711yescharacteristic
321C2×C4D4×D15711yescharacteristic
322C2×C4C2×D62811yescharacteristic
323C2×C4C2×D62811yescharacteristic
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#1909774111yescharacteristic
338C22#1909774111yescharacteristic
339C22C2×D4×D15711yescharacteristic
340C4—22no—
341C4—42no—
342C4—3142no—
343C4—6282no—
344C4#1909774111yescharacteristic
345C4#1909774111yescharacteristic
346C4C2×D4×D15711yescharacteristic
347C4C22×D62811yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—6284no—
352C2—6282no—
353C2#2375745212yesdirect product factor
354C2C2×#1909774111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#104×C157C2
socleC22×C157#120
prime radicalC2×#19097817C2
hypercenterC2×#104D157
hypocenterC157C2×#455
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#19097804C2semidirect
3. C2×#19097801C2semidirect
4. C2×#19097759C2semidirect
5. #23757452C2direct (N × H)
6. C2×#19097699C2semidirect
7. C2×#19097687C2semidirect
8. C2×#104×C157C2semidirect
16. C22×Dic314C22semidirect
22. #19097804C22semidirect
24. #19097801C22semidirect
29. C2×#14578938C22semidirect
31. #19097759C22semidirect
33. #19097699C22semidirect
34. #19097687C22semidirect
36. C2×M16×C157C22semidirect
37. C2×C42×C157C22semidirect
38. #104×C157C22semidirect
55. C2×Dic314C23semidirect
80. #14578938C23semidirect
94. M16×C157C23semidirect
97. C42×C157C23semidirect
173. C2×C157#455semidirect
177. C157C2×#455semidirect
185. C2×#104D157semidirect
209. C2×M16C2×D157semidirect
213. C2×C42C2×D157semidirect
215. #104C2×D157semidirect
270. M16C22×D157semidirect
275. C42C22×D157semidirect
353. C2#23757452direct (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×#190978042
C2×#190978012
C2×#190977592
#237574522
C2×#190976992
C2×#190976872
C2×#104×C1572
C2×#455157