Finite Group Databasebeta

← C2×#23757461 #34109521

Subgroups — C2×#23757461 #34109521

Order 20096.

1018 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1018 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, complemented
2C2×#19097741C212yesmaximal, complemented
3C2×#19097739C211yesmaximal, characteristic, complemented
4#23757461C218yesmaximal, direct product factor
5C2×#19097685C212yesmaximal, complemented
6C2×#99×C157C211yesmaximal, characteristic, complemented
7C24×D157C2211yescharacteristic, complemented
8#19097816—22no—
9#19097816—24no—
10#19097816C2214yescomplemented
11C2×#14578954—21no—
12C2×#14578953—22no—
13C22×D628—22no—
14C22×C22⋊D157—21no—
15C22×C22⋊D157—21no—
16C22×C22⋊D157C2212yes—
17C2×D4×D157—22no—
18#19097741—24no—
19#19097741C2218yescomplemented
20#19097739—22no—
21#19097739C2214yescomplemented
22C2×C22⋊Dic157C2211yescharacteristic, complemented
23C2×#14578931—21no—
24C2×#14578931—22no—
25#19097685C2218yescomplemented
26C22×D4×C157C2211yescharacteristic
27C2×C22⋊C4×C157C2212yescomplemented
28#99×C157C2214yescomplemented
29C23×D157—21no—
30C23×D157—21no—
31C23×D157—22no—
32C23×D157—21no—
33C23×D157—22no—
34C23×D157—44no—
35C23×D157D412yescomplemented
36C23×D157C2312yescomplemented
37#14578954—24no—
38#14578954—24no—
39#14578953—28no—
40#14578953—28no—
41C2×Dic314—21no—
42C2×D628—21no—
43C2×D628—28no—
44C2×D628—48no—
45C2×C22⋊D157—22no—
46C2×C22⋊D157—24no—
47C2×C22⋊D157—22no—
48C2×C22⋊D157—22no—
49C2×C22⋊D157—22no—
50C2×C22⋊D157—22no—
51C2×C22⋊D157—24no—
52C2×C22⋊D157—24no—
53C2×C22⋊D157—44no—
54C2×C22⋊D157D414yescomplemented
55C2×C22⋊D157C2314yes—
56D4×D157—28no—
57D4×D157—28no—
58C22×Dic157—21no—
59C22×Dic157—21no—
60C22×Dic157D412yescomplemented
61C2×C4×D157—22no—
62C22⋊Dic157—21no—
63C22⋊Dic157C2312yescomplemented
64#14578931—28no—
65#14578931—24no—
66C24×C157D411yescharacteristic, complemented
67C24×C157C2311yescharacteristic
68C2×D4×C157—22no—
69C2×D4×C157—22no—
70C2×D4×C157D412yescomplemented
71C2×D4×C157C2312yes—
72C22×C4×C157—22no—
73C22×C4×C157D411yescharacteristic, complemented
74C22⋊C4×C157—22no—
75C22⋊C4×C157C2314yescomplemented
76C22×D157—24no—
77C22×D157—28no—
78C22×D157—22no—
79C22×D157—22no—
80C22×D157—24no—
81C22×D157—24no—
82C22×D157—22no—
83C22×D157—44no—
84C22×D157—44no—
85C22×D157—48no—
86C22×D157—44no—
87C22×D157—42no—
88C22×D157—41no—
89C22×D157—42no—
90C22×D157—41no—
91C22×D157—44no—
92C22×D157—82no—
93C22×D157C2×D414yescomplemented
94Dic314—24no—
95D628—24no—
96D628—416no—
97C22⋊D157—28no—
98C22⋊D157—24no—
99C22⋊D157—24no—
100C22⋊D157—24no—
101C22⋊D157—24no—
102C22⋊D157—48no—
103C2×Dic157—24no—
104C2×Dic157—22no—
105C2×Dic157—22no—
106C2×Dic157—22no—
107C2×Dic157—42no—
108C2×Dic157C2×D414yescomplemented
109C4×D157—28no—
110C23×C157—21no—
111C23×C157—21no—
112C23×C157—42no—
113C23×C157—42no—
114C23×C157C2×D412yes—
115C23×C157C2×D412yes—
116C23×C157C2×D411yescharacteristic
117C23×C157C2×D411yescharacteristic
118C23×C157C2×D412yescomplemented
119C23×C157C2411yescharacteristic
120D4×C157—24no—
121D4×C157—24no—
122C2×C4×C157—21no—
123C2×C4×C157—21no—
124C2×C4×C157—24no—
125C2×C4×C157—44no—
126C2×C4×C157C2×D412yescomplemented
127C22×C157—24no—
128C22×C157—22no—
129C22×C157—22no—
130C22×C157—42no—
131C22×C157—41no—
132C22×C157—41no—
133C22×C157—42no—
134C22×C157—42no—
135C22×C157—81no—
136C22×C157#12011yescharacteristic
137C22×C157#12012yes—
138C22×C157#12011yescharacteristic
139C22×C157C22×D412yes—
140C22×C157C22×D411yescharacteristic
141C2×D157—24no—
142C2×D157—28no—
143C2×D157—24no—
144C2×D157—42no—
145C2×D157—44no—
146C2×D157—48no—
147C2×D157—44no—
148C2×D157—42no—
149C2×D157—44no—
150C2×D157—44no—
151C2×D157—44no—
152C2×D157—82no—
153C2×D157—84no—
154C4×C157—22no—
155C4×C157—44no—
156Dic157—24no—
157Dic157—42no—
158C2×C157—21no—
159C2×C157—22no—
160C2×C157—22no—
161C2×C157—21no—
162C2×C157—42no—
163C2×C157B(4,𝔽2)12yescomplemented
164C2×C157C2×#12011yescharacteristic
165D157—44no—
166D157—44no—
167C157C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
168C2×B(4,𝔽2)—1571nomaximal, Sylow
169C2×H(5,𝔽2)—1571no—
170C2×#120—1571no—
171C2×#120—1572no—
172B(4,𝔽2)—1578no—
173C2×#99—1572no—
174C2×#99D15711yescharacteristic, complemented
175C25—1571no—
176H(5,𝔽2)—3144no—
177H(5,𝔽2)—3142no—
178H(5,𝔽2)—1574no—
179C2×#41—3142no—
180C2×#41—3141no—
181C22×D4—3142no—
182C22×D4—3142no—
183C22×D4—3141no—
184C22×D4—3141no—
185C22×D4—1572no—
186C22×D4C2×D15711yescharacteristic
187#120—3144no—
188#120—3142no—
189#120—1574no—
190#120—1578no—
191C2×C22⋊C4—3142no—
192C2×C22⋊C4—3141no—
193C2×C22⋊C4—1571no—
194C2×C22⋊C4C2×D15712yescomplemented
195#99—1578no—
196#99C2×D15714yescomplemented
197C24—3141no—
198C24—3142no—
199C24—3141no—
200C24—3142no—
201C24—3141no—
202C24—6284no—
203C24—1572no—
204C24—1572no—
205C24D62811yescharacteristic, complemented
206C24C22×D15711yescharacteristic
207#41—3144no—
208#41—3148no—
209#41—3148no—
210#41—3144no—
211C2×Q8—3141no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—3141no—
215C2×D4—3148no—
216C2×D4—3148no—
217C2×D4—3148no—
218C2×D4—3142no—
219C2×D4—3144no—
220C2×D4—3142no—
221C2×D4—3142no—
222C2×D4—3142no—
223C2×D4—3142no—
224C2×D4—3144no—
225C2×D4—3144no—
226C2×D4—6288no—
227C2×D4—6284no—
228C2×D4—1574no—
229C2×D4—1574no—
230C2×D4D62812yescomplemented
231C2×D4C22×D15712yes—
232C22×C4—22no—
233C22×C4—3142no—
234C22×C4—3141no—
235C22×C4—3141no—
236C22×C4—1572no—
237C22×C4D62811yescharacteristic, complemented
238C22⋊C4—22no—
239C22⋊C4—3148no—
240C22⋊C4—3141no—
241C22⋊C4—3144no—
242C22⋊C4—1572no—
243C22⋊C4C22×D15714yescomplemented
244C23—21no—
245C23—21no—
246C23—42no—
247C23—42no—
248C23—3148no—
249C23—3144no—
250C23—3144no—
251C23—3142no—
252C23—3144no—
253C23—3142no—
254C23—3142no—
255C23—6284no—
256C23—6282no—
257C23—6284no—
258C23—6284no—
259C23—6288no—
260C23—6284no—
261C23—6281no—
262C23—6282no—
263C23—6281no—
264C23—1574no—
265C23—12562no—
266C23D4×D15712yes—
267C23D4×D15712yes—
268C23C2×D62811yescharacteristic
269C23C2×D62811yescharacteristic
270C23C2×D62812yescomplemented
271C23C23×D15711yescharacteristic
272Q8—3144no—
273D4—24no—
274D4—24no—
275D4—3144no—
276D4—3148no—
277D4—3144no—
278D4—3144no—
279D4—3144no—
280D4—3144no—
281D4—62816no—
282D4—6288no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—44no—
287C2×C4—3148no—
288C2×C4—3144no—
289C2×C4—3142no—
290C2×C4—3142no—
291C2×C4—3142no—
292C2×C4—6282no—
293C2×C4—1574no—
294C2×C4C2×D62812yescomplemented
295C22—24no—
296C22—22no—
297C22—22no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—81no—
304C22—3148no—
305C22—3144no—
306C22—3144no—
307C22—6284no—
308C22—6284no—
309C22—6282no—
310C22—6284no—
311C22—6288no—
312C22—6284no—
313C22—6284no—
314C22—6282no—
315C22—12562no—
316C22—12564no—
317C22#1909774111yescharacteristic
318C22#1909774112yes—
319C22#1909774111yescharacteristic
320C22C2×D4×D15712yes—
321C22C22×D62811yescharacteristic
322C4—22no—
323C4—44no—
324C4—3144no—
325C4—6282no—
326C2—22no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—42no—
331C2—6284no—
332C2—6284no—
333C2#2375746112yesdirect product factor
334C2C2×#1909774111yescharacteristic
subgroupG/H
centerC22#19097741
commutator, cosocleC23×C157C24
FrattiniC23C23×D157
FittingC2×#99×C157C2
socleC22×C157#120
hypercenterC2×#99D157
hypocenterC157C2×B(4,𝔽2)
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#19097816C2semidirect
2. C2×#19097741C2semidirect
3. C2×#19097739C2semidirect
4. #23757461C2direct (N × H)
5. C2×#19097685C2semidirect
6. C2×#99×C157C2semidirect
7. C24×D157C22semidirect
10. #19097816C22semidirect
19. #19097741C22semidirect
21. #19097739C22semidirect
22. C2×C22⋊Dic157C22semidirect
25. #19097685C22semidirect
27. C2×C22⋊C4×C157C22semidirect
28. #99×C157C22semidirect
35. C23×D157D4semidirect
36. C23×D157C23semidirect
54. C2×C22⋊D157D4semidirect
60. C22×Dic157D4semidirect
63. C22⋊Dic157C23semidirect
66. C24×C157D4semidirect
70. C2×D4×C157D4semidirect
73. C22×C4×C157D4semidirect
75. C22⋊C4×C157C23semidirect
93. C22×D157C2×D4semidirect
108. C2×Dic157C2×D4semidirect
118. C23×C157C2×D4semidirect
126. C2×C4×C157C2×D4semidirect
163. C2×C157B(4,𝔽2)semidirect
167. C157C2×B(4,𝔽2)semidirect
174. C2×#99D157semidirect
194. C2×C22⋊C4C2×D157semidirect
196. #99C2×D157semidirect
205. C24D628semidirect
230. C2×D4D628semidirect
237. C22×C4D628semidirect
243. C22⋊C4C22×D157semidirect
270. C23C2×D628semidirect
294. C2×C4C2×D628semidirect
333. C2#23757461direct (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×#190977412
C2×#190977392
#237574612
C2×#190976852
C2×#99×C1572
C2×B(4,𝔽2)157