Finite Group Databasebeta

← C2×#22550055 #30648061

Subgroups — C2×#22550055 #30648061

Order 16256.

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×#16861255C211yesmaximal, characteristic
2C2×#16861242C211yesmaximal, characteristic, complemented
3C2×#16861239C211yesmaximal, characteristic, complemented
4C2×#16861197C211yesmaximal, characteristic, complemented
5#22550055C218yesmaximal, direct product factor
6C2×#16861137C211yesmaximal, characteristic, complemented
7C2×#16861125C211yesmaximal, characteristic, complemented
8C2×#104×C127C211yesmaximal, characteristic, complemented
9#16861255—22no—
10#16861255—22no—
11#16861255—22no—
12#16861255C2214yes—
13C2×#14523249—21no—
14C2×#14523248—21no—
15C2×#14523248C2211yescharacteristic
16C22×Dic254C2211yescharacteristic, complemented
17C22×D508—21no—
18C22×D508C2211yescharacteristic
19C2×D4×D127—22no—
20C2×D4×D127—21no—
21#16861242—22no—
22#16861242C2214yescomplemented
23#16861239—22no—
24#16861239C2214yescomplemented
25C2×Dic508—21no—
26C2×#14523238—21no—
27C2×#14523237—21no—
28C2×#14523236—21no—
29C2×#14523232C2211yescharacteristic, complemented
30#16861197—22no—
31#16861197C2214yescomplemented
32C2×(C2×D127)⋊C4—22no—
33#16861137C2214yescomplemented
34#16861125C2214yescomplemented
35C2×#41×C127C2211yescharacteristic
36C2×M16×C127C2211yescharacteristic, complemented
37C2×C42×C127C2211yescharacteristic, complemented
38#104×C127C2214yescomplemented
39C23×D127—22no—
40C23×D127D412yes—
41#14523249—24no—
42#14523249—24no—
43#14523248—24no—
44#14523248—24no—
45#14523248—21no—
46#14523248—21no—
47#14523248D412yes—
48#14523248C2312yes—
49C2×Dic254—21no—
50C2×Dic254—21no—
51C2×Dic254—21no—
52C2×Dic254—21no—
53C2×Dic254—21no—
54C2×Dic254D411yescharacteristic
55C2×Dic254C2312yescomplemented
56C2×D508—22no—
57C2×D508—24no—
58C2×D508—24no—
59C2×D508—21no—
60C2×D508—21no—
61C2×D508—21no—
62C2×D508—21no—
63C2×D508D412yes—
64C2×D508D411yescharacteristic
65C2×D508C2312yes—
66C2×C22⋊D127—22no—
67C2×C22⋊D127—21no—
68D4×D127—28no—
69D4×D127—24no—
70D4×D127—28no—
71D4×D127—24no—
72C22×Dic127—21no—
73C2×C4×D127—22no—
74C2×C4×D127—21no—
75Dic508—24no—
76#14523238—24no—
77#14523237—24no—
78#14523236—24no—
79#14523232—21no—
80#14523232C2312yescomplemented
81C2×C127⋊C8—21no—
82(C2×D127)⋊C4—28no—
83#41×C127—21no—
84#41×C127—21no—
85#41×C127D412yes—
86#41×C127C2312yes—
87C2×Q8×C127D411yescharacteristic
88C2×D4×C127—21no—
89C2×D4×C127D411yescharacteristic
90C22×C4×C127—21no—
91C22×C4×C127—21no—
92C22×C4×C127C2311yescharacteristic
93M16×C127—21no—
94M16×C127C2312yescomplemented
95C2×C8×C127—21no—
96C42×C127—21no—
97C42×C127C2312yescomplemented
98C22×D127—22no—
99C22×D127—21no—
100C22×D127—28no—
101C22×D127—22no—
102C22×D127—44no—
103C22×D127—84no—
104C22×D127C2×D414yes—
105Dic254—21no—
106Dic254—22no—
107Dic254—22no—
108Dic254—41no—
109Dic254—41no—
110Dic254C2×D412yes—
111D508—28no—
112D508—22no—
113D508—22no—
114D508—21no—
115D508—22no—
116D508—22no—
117D508C2×D412yes—
118C22⋊D127—28no—
119C22⋊D127—22no—
120C22⋊D127—22no—
121C2×Dic127—21no—
122C2×Dic127—22no—
123C2×Dic127—41no—
124C2×Dic127—41no—
125C4×D127—28no—
126C4×D127—22no—
127C4×D127—22no—
128C127⋊C8—22no—
129C23×C127—21no—
130C23×C127C2×D411yescharacteristic
131Q8×C127—21no—
132Q8×C127C2×D412yes—
133D4×C127—22no—
134D4×C127—22no—
135D4×C127—21no—
136D4×C127C2×D412yes—
137C2×C4×C127—21no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—42no—
142C2×C4×C127C2×D411yescharacteristic
143C2×C4×C127C2×D411yescharacteristic
144C2×C4×C127C2×D411yescharacteristic
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2411yescharacteristic
148C8×C127—22no—
149C22×C127—22no—
150C22×C127—22no—
151C22×C127—42no—
152C22×C127#12011yescharacteristic
153C22×C127#12011yescharacteristic
154C22×C127C22×D411yescharacteristic
155C2×D127—24no—
156C2×D127—22no—
157C2×D127—44no—
158C2×D127—42no—
159C2×D127—44no—
160C2×D127—82no—
161C2×D127—88no—
162C4×C127—22no—
163C4×C127—42no—
164C4×C127#12011yescharacteristic
165C4×C127#12011yescharacteristic
166C4×C127C22×D411yescharacteristic
167C4×C127C22×D411yescharacteristic
168Dic127—22no—
169Dic127—42no—
170C2×C127—21no—
171C2×C127—21no—
172C2×C127—42no—
173C2×C127#45512yescomplemented
174C2×C127C2×#12011yescharacteristic
175D127—44no—
176D127—42no—
177C2×#455—1271nomaximal, Sylow
178C127C2×#45511yescharacteristic, complemented, Sylow
179C2×H(5,𝔽2)—1271no—
180C2×#137—1271no—
181C2×#137—1271no—
182C2×#124—1271no—
183#455—1278no—
184C2×#104—1271no—
185C2×#104D12711yescharacteristic, complemented
186C2×#100—1271no—
187H(5,𝔽2)—1274no—
188H(5,𝔽2)—2542no—
189H(5,𝔽2)—2542no—
190H(5,𝔽2)—2542no—
191C2×#41—1271no—
192C2×#41—2541no—
193C2×#41—2541no—
194C2×#41C2×D12711yescharacteristic
195C22×Q8—1271no—
196C22×D4—1271no—
197C22×D4—2541no—
198C22×D4—2542no—
199C22×D4—2541no—
200#137—1274no—
201#137—1274no—
202#137—2542no—
203#137—2542no—
204C2×Q16—2541no—
205C2×Q16—2541no—
206C2×SD16—2541no—
207C2×SD16—2541no—
208C2×M16—1271no—
209C2×M16C2×D12711yescharacteristic, complemented
210#124—1274no—
211#124—2542no—
212C2×C22⋊C4—2542no—
213C2×C42C2×D12711yescharacteristic, complemented
214#104—1274no—
215#104C2×D12714yescomplemented
216#100—1274no—
217C24—1272no—
218C24—2542no—
219#41—21no—
220#41—21no—
221#41—1272no—
222#41—1272no—
223#41—2544no—
224#41—2544no—
225#41—2544no—
226#41—2544no—
227#41—2541no—
228#41—2541no—
229#41D50812yes—
230#41C22×D12712yes—
231C2×Q8—1271no—
232C2×Q8—1272no—
233C2×Q8—2541no—
234C2×Q8—2541no—
235C2×Q8—2541no—
236C2×Q8—2541no—
237C2×Q8—2541no—
238C2×Q8D50811yescharacteristic
239C2×D4—21no—
240C2×D4—1271no—
241C2×D4—1272no—
242C2×D4—1272no—
243C2×D4—2544no—
244C2×D4—2542no—
245C2×D4—2544no—
246C2×D4—2548no—
247C2×D4—2548no—
248C2×D4—2544no—
249C2×D4—2542no—
250C2×D4—2541no—
251C2×D4—2544no—
252C2×D4—2541no—
253C2×D4—2541no—
254C2×D4—2541no—
255C2×D4—2541no—
256C2×D4D50811yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—2542no—
260C22×C4—2541no—
261C22×C4—2541no—
262C22×C4C22×D12711yescharacteristic
263Q16—2544no—
264Q16—2544no—
265SD16—2544no—
266SD16—2544no—
267M16—21no—
268M16—1272no—
269M16—2541no—
270M16C22×D12712yescomplemented
271C2×C8—21no—
272C2×C8—2541no—
273C22⋊C4—2548no—
274C42—21no—
275C42C22×D12712yescomplemented
276C23—21no—
277C23—1274no—
278C23—10164no—
279C23—5084no—
280C23—2548no—
281C23—2542no—
282C23—2541no—
283C23—2542no—
284C23D4×D12711yescharacteristic
285Q8—21no—
286Q8—1272no—
287Q8—5081no—
288Q8—5081no—
289Q8—2542no—
290Q8—2542no—
291Q8—2541no—
292Q8C2×D50812yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—1272no—
297D4—2548no—
298D4—2548no—
299D4—2542no—
300D4—2542no—
301D4—2542no—
302D4—2542no—
303D4—2541no—
304D4—2542no—
305D4—2542no—
306D4C2×D50812yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—5081no—
313C2×C4—5081no—
314C2×C4—2548no—
315C2×C4—2542no—
316C2×C4—2541no—
317C2×C4—2542no—
318C2×C4—2542no—
319C2×C4D4×D12711yescharacteristic
320C2×C4D4×D12711yescharacteristic
321C2×C4D4×D12711yescharacteristic
322C2×C4C2×D50811yescharacteristic
323C2×C4C2×D50811yescharacteristic
324C2×C4C23×D12711yescharacteristic
325C8—22no—
326C8—2542no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—10162no—
331C22—10168no—
332C22—5084no—
333C22—5082no—
334C22—5084no—
335C22—2542no—
336C22—2544no—
337C22#1686117911yescharacteristic
338C22#1686117911yescharacteristic
339C22C2×D4×D12711yescharacteristic
340C4—22no—
341C4—42no—
342C4—5082no—
343C4—2542no—
344C4#1686117911yescharacteristic
345C4#1686117911yescharacteristic
346C4C2×D4×D12711yescharacteristic
347C4C22×D50811yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—5082no—
352C2—5084no—
353C2#2255005512yesdirect product factor
354C2C2×#1686117911yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#104×C127C2
socleC22×C127#120
prime radicalC2×#16861255C2
hypercenterC2×#104D127
hypocenterC127C2×#455
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16861242C2semidirect
3. C2×#16861239C2semidirect
4. C2×#16861197C2semidirect
5. #22550055C2direct (N × H)
6. C2×#16861137C2semidirect
7. C2×#16861125C2semidirect
8. C2×#104×C127C2semidirect
16. C22×Dic254C22semidirect
22. #16861242C22semidirect
24. #16861239C22semidirect
29. C2×#14523232C22semidirect
31. #16861197C22semidirect
33. #16861137C22semidirect
34. #16861125C22semidirect
36. C2×M16×C127C22semidirect
37. C2×C42×C127C22semidirect
38. #104×C127C22semidirect
55. C2×Dic254C23semidirect
80. #14523232C23semidirect
94. M16×C127C23semidirect
97. C42×C127C23semidirect
173. C2×C127#455semidirect
178. C127C2×#455semidirect
185. C2×#104D127semidirect
209. C2×M16C2×D127semidirect
213. C2×C42C2×D127semidirect
215. #104C2×D127semidirect
270. M16C22×D127semidirect
275. C42C22×D127semidirect
353. C2#22550055direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612422
C2×#168612392
C2×#168611972
#225500552
C2×#168611372
C2×#168611252
C2×#104×C1272
C2×#455127