Finite Group Databasebeta

← C2×#20265363 #27180515

Subgroups — C2×#20265363 #27180515

Order 12928.

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×#15692300C211yesmaximal, characteristic, complemented
2C2×#15692224C212yesmaximal, complemented
3#20265363C218yesmaximal, direct product factor
4C2×#15692170C211yesmaximal, characteristic, complemented
5C2×#15692169C212yesmaximal, complemented
6C2×#120×C101C211yesmaximal, characteristic, complemented
7#15692300—24no—
8#15692300—22no—
9#15692300C2214yescomplemented
10C2×#13108074—21no—
11C2×#13108073—22no—
12C22×C22⋊D101—22no—
13C22×C22⋊D101—21no—
14C22×C22⋊D101C2212yes—
15C2×D4×D101—22no—
16#15692224—24no—
17#15692224C2218yescomplemented
18C2×C22⋊Dic101—22no—
19C2×C22⋊Dic101C2212yescomplemented
20#15692170C2214yescomplemented
21#15692169C2218yescomplemented
22C25×C101C2211yescharacteristic, complemented
23C22×D4×C101—21no—
24C22×D4×C101C2211yescharacteristic
25#120×C101—22no—
26#120×C101C2214yescomplemented
27C2×C22⋊C4×C101—21no—
28C2×C22⋊C4×C101C2211yescharacteristic, complemented
29C23×D101—21no—
30C23×D101D412yescomplemented
31#13108074—24no—
32#13108074—24no—
33#13108073—28no—
34#13108073—28no—
35C2×Dic202—21no—
36C2×D404—21no—
37C2×C22⋊D101—22no—
38C2×C22⋊D101—24no—
39C2×C22⋊D101—22no—
40C2×C22⋊D101—22no—
41C2×C22⋊D101—22no—
42C2×C22⋊D101—22no—
43C2×C22⋊D101—24no—
44C2×C22⋊D101—28no—
45C2×C22⋊D101—48no—
46C2×C22⋊D101D414yescomplemented
47C2×C22⋊D101C2314yes—
48D4×D101—28no—
49D4×D101—28no—
50C22×Dic101—21no—
51C22×Dic101—22no—
52C22×Dic101D412yescomplemented
53C2×C4×D101—22no—
54C22⋊Dic101—22no—
55C22⋊Dic101—28no—
56C22⋊Dic101C2314yescomplemented
57C24×C101—22no—
58C24×C101—21no—
59C24×C101—21no—
60C24×C101—22no—
61C24×C101—44no—
62C24×C101D411yescharacteristic, complemented
63C24×C101C2311yescharacteristic
64C24×C101C2312yescomplemented
65C2×D4×C101—22no—
66C2×D4×C101—22no—
67C2×D4×C101—24no—
68C2×D4×C101—44no—
69C2×D4×C101D412yescomplemented
70C2×D4×C101C2312yes—
71C22×C4×C101—21no—
72C22×C4×C101D411yescharacteristic, complemented
73C22⋊C4×C101—21no—
74C22⋊C4×C101—24no—
75C22⋊C4×C101C2312yescomplemented
76C23×C101—24no—
77C23×C101—28no—
78C23×C101—21no—
79C23×C101—22no—
80C23×C101—24no—
81C23×C101—21no—
82C23×C101—42no—
83C23×C101—44no—
84C23×C101—44no—
85C23×C101—42no—
86C23×C101—48no—
87C23×C101—44no—
88C23×C101—42no—
89C23×C101—42no—
90C23×C101—41no—
91C23×C101—41no—
92C23×C101C2×D412yes—
93C23×C101C2×D412yes—
94C23×C101C2×D411yescharacteristic
95C23×C101C2×D411yescharacteristic
96C23×C101C2×D412yescomplemented
97C23×C101C2411yescharacteristic
98C22×D101—22no—
99C22×D101—24no—
100C22×D101—22no—
101C22×D101—44no—
102C22×D101—82no—
103C22×D101C2×D414yescomplemented
104D4×C101—24no—
105D4×C101—24no—
106D4×C101—48no—
107C2×C4×C101—21no—
108C2×C4×C101—21no—
109C2×C4×C101—22no—
110C2×C4×C101—42no—
111C2×C4×C101C2×D412yescomplemented
112C22⋊D101—28no—
113C22⋊D101—24no—
114C22⋊D101—24no—
115C22⋊D101—24no—
116C22⋊D101—24no—
117C22⋊D101—416no—
118C2×Dic101—24no—
119C2×Dic101—22no—
120C2×Dic101—22no—
121C2×Dic101—24no—
122C2×Dic101—44no—
123C2×Dic101C2×D414yescomplemented
124D404—24no—
125C4×D101—28no—
126Dic202—24no—
127C22×C101—24no—
128C22×C101—22no—
129C22×C101—28no—
130C22×C101—24no—
131C22×C101—22no—
132C22×C101—42no—
133C22×C101—41no—
134C22×C101—41no—
135C22×C101—42no—
136C22×C101—44no—
137C22×C101—42no—
138C22×C101—48no—
139C22×C101—44no—
140C22×C101—42no—
141C22×C101—42no—
142C22×C101—44no—
143C22×C101—81no—
144C22×C101#12012yes—
145C22×C101#12011yescharacteristic
146C22×C101#12011yescharacteristic
147C22×C101C22×D412yes—
148C22×C101C22×D411yescharacteristic
149C2×D101—24no—
150C2×D101—44no—
151C2×D101—44no—
152C2×D101—82no—
153C2×D101—84no—
154C4×C101—22no—
155C4×C101—42no—
156Dic101—24no—
157Dic101—44no—
158C2×C101—22no—
159C2×C101—21no—
160C2×C101—22no—
161C2×C101—21no—
162C2×C101—42no—
163C2×C101—44no—
164C2×C101B(4,𝔽2)12yescomplemented
165C2×C101C2×#12011yescharacteristic
166D101—44no—
167C2×B(4,𝔽2)—1011nomaximal, Sylow
168C101C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
169C2×H(5,𝔽2)—1011no—
170C2×#120—1012no—
171C2×#120D10111yescharacteristic, complemented
172B(4,𝔽2)—1018no—
173C2×#99—1011no—
174C2×#99—1012no—
175C25C2×D10111yescharacteristic, complemented
176H(5,𝔽2)—1014no—
177H(5,𝔽2)—2024no—
178H(5,𝔽2)—2022no—
179C2×#41—2022no—
180C2×#41—2021no—
181C22×D4—21no—
182C22×D4—1012no—
183C22×D4—2022no—
184C22×D4—2022no—
185C22×D4—2021no—
186C22×D4C2×D10111yescharacteristic
187#120—22no—
188#120—1018no—
189#120—2024no—
190#120C2×D10114yescomplemented
191C2×C22⋊C4—21no—
192C2×C22⋊C4—1012no—
193C2×C22⋊C4—2022no—
194C2×C22⋊C4C2×D10111yescharacteristic, complemented
195#99—1014no—
196#99—1018no—
197C24—22no—
198C24—21no—
199C24—22no—
200C24—21no—
201C24—44no—
202C24—1012no—
203C24—2021no—
204C24C22⋊D10111yescharacteristic, complemented
205C24C22×D10111yescharacteristic
206C24C22×D10112yescomplemented
207#41—2024no—
208#41—2028no—
209#41—2028no—
210#41—2024no—
211C2×Q8—2021no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—44no—
216C2×D4—1014no—
217C2×D4—1014no—
218C2×D4—4048no—
219C2×D4—2021no—
220C2×D4—2028no—
221C2×D4—2028no—
222C2×D4—2028no—
223C2×D4—2022no—
224C2×D4—2024no—
225C2×D4—2022no—
226C2×D4—2022no—
227C2×D4—2022no—
228C2×D4—2022no—
229C2×D4—2024no—
230C2×D4C22⋊D10112yescomplemented
231C2×D4C22×D10112yes—
232C22×C4—21no—
233C22×C4—1012no—
234C22×C4—2022no—
235C22×C4—2022no—
236C22×C4—2021no—
237C22×C4C22⋊D10111yescharacteristic, complemented
238C22⋊C4—21no—
239C22⋊C4—24no—
240C22⋊C4—1014no—
241C22⋊C4—2028no—
242C22⋊C4—2022no—
243C22⋊C4C22×D10112yescomplemented
244C23—24no—
245C23—28no—
246C23—24no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—42no—
251C23—44no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23—48no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—8082no—
261C23—1014no—
262C23—4044no—
263C23—2024no—
264C23—2022no—
265C23—2022no—
266C23D4×D10112yes—
267C23D4×D10112yes—
268C23C2×C22⋊D10111yescharacteristic
269C23C2×C22⋊D10111yescharacteristic
270C23C2×C22⋊D10112yescomplemented
271C23C23×D10111yescharacteristic
272Q8—2024no—
273D4—24no—
274D4—24no—
275D4—48no—
276D4—40416no—
277D4—2024no—
278D4—2028no—
279D4—2024no—
280D4—2024no—
281D4—2024no—
282D4—2024no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—1014no—
288C2×C4—4044no—
289C2×C4—2028no—
290C2×C4—2024no—
291C2×C4—2024no—
292C2×C4—2022no—
293C2×C4—2022no—
294C2×C4C2×C22⋊D10112yescomplemented
295C22—22no—
296C22—24no—
297C22—24no—
298C22—28no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—44no—
306C22—44no—
307C22—48no—
308C22—44no—
309C22—42no—
310C22—42no—
311C22—81no—
312C22—8082no—
313C22—8084no—
314C22—4044no—
315C22—4044no—
316C22—2024no—
317C22#1569222311yescharacteristic
318C22#1569222312yes—
319C22#1569222311yescharacteristic
320C22C2×D4×D10112yes—
321C22C22×C22⋊D10111yescharacteristic
322C4—22no—
323C4—42no—
324C4—4044no—
325C4—2024no—
326C2—21no—
327C2—22no—
328C2—21no—
329C2—22no—
330C2—44no—
331C2—42no—
332C2—4044no—
333C2#2026536312yesdirect product factor
334C2C2×#1569222311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC23×C101C24
FrattiniC23C23×D101
FittingC2×#120×C101C2
socleC22×C101#120
hypercenterC2×#120D101
hypocenterC101C2×B(4,𝔽2)
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#15692300C2semidirect
2. C2×#15692224C2semidirect
3. #20265363C2direct (N × H)
4. C2×#15692170C2semidirect
5. C2×#15692169C2semidirect
6. C2×#120×C101C2semidirect
9. #15692300C22semidirect
17. #15692224C22semidirect
19. C2×C22⋊Dic101C22semidirect
20. #15692170C22semidirect
21. #15692169C22semidirect
22. C25×C101C22semidirect
26. #120×C101C22semidirect
28. C2×C22⋊C4×C101C22semidirect
30. C23×D101D4semidirect
46. C2×C22⋊D101D4semidirect
52. C22×Dic101D4semidirect
56. C22⋊Dic101C23semidirect
62. C24×C101D4semidirect
64. C24×C101C23semidirect
69. C2×D4×C101D4semidirect
72. C22×C4×C101D4semidirect
75. C22⋊C4×C101C23semidirect
96. C23×C101C2×D4semidirect
103. C22×D101C2×D4semidirect
111. C2×C4×C101C2×D4semidirect
123. C2×Dic101C2×D4semidirect
164. C2×C101B(4,𝔽2)semidirect
168. C101C2×B(4,𝔽2)semidirect
171. C2×#120D101semidirect
175. C25C2×D101semidirect
190. #120C2×D101semidirect
194. C2×C22⋊C4C2×D101semidirect
204. C24C22⋊D101semidirect
206. C24C22×D101semidirect
230. C2×D4C22⋊D101semidirect
237. C22×C4C22⋊D101semidirect
243. C22⋊C4C22×D101semidirect
270. C23C2×C22⋊D101semidirect
294. C2×C4C2×C22⋊D101semidirect
333. C2#20265363direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923002
C2×#156922242
#202653632
C2×#156921702
C2×#156921692
C2×#120×C1012
C2×B(4,𝔽2)101