Finite Group Databasebeta

← C2×#22550045 #30648055

Subgroups — C2×#22550045 #30648055

Order 16256.

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

normal subgroups only

This group has 844 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, complemented
2C2×#16861237C211yesmaximal, characteristic, complemented
3C2×#16861234C211yesmaximal, characteristic, complemented
4C2×#16861206C211yesmaximal, characteristic, complemented
5#22550045C218yesmaximal, 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#16861255C2214yescomplemented
13C2×#14523249—21no—
14C2×#14523248—21no—
15C2×#14523248C2211yescharacteristic
16C22×D508—21no—
17C22×D508—22no—
18C22×D508C2211yescharacteristic
19C22×D508C2211yescharacteristic, complemented
20C2×D4×D127—22no—
21C2×D4×D127—21no—
22#16861237—22no—
23#16861237C2214yescomplemented
24#16861234—22no—
25#16861234C2214yescomplemented
26C2×#14523237—21no—
27C2×#14523235—21no—
28C2×D1016—21no—
29C2×#14523233—21no—
30C2×#14523232C2211yescharacteristic, complemented
31#16861206—22no—
32#16861206C2214yescomplemented
33#16861137C2214yescomplemented
34#16861125C2214yescomplemented
35C2×#41×C127C2211yescharacteristic
36C2×M16×C127C2211yescharacteristic, complemented
37C2×C42×C127C2211yescharacteristic, complemented
38#104×C127C2214yescomplemented
39C23×D127—22no—
40C23×D127—21no—
41C23×D127D412yes—
42#14523249—24no—
43#14523249—24no—
44#14523248—24no—
45#14523248—24no—
46#14523248—21no—
47#14523248—21no—
48#14523248D412yescomplemented
49#14523248C2312yes—
50C2×Dic254D411yescharacteristic, complemented
51C2×D508—22no—
52C2×D508—24no—
53C2×D508—24no—
54C2×D508—28no—
55C2×D508—21no—
56C2×D508—21no—
57C2×D508—21no—
58C2×D508—21no—
59C2×D508—21no—
60C2×D508—21no—
61C2×D508—21no—
62C2×D508—21no—
63C2×D508—21no—
64C2×D508—48no—
65C2×D508D412yes—
66C2×D508D411yescharacteristic, complemented
67C2×D508C2312yes—
68C2×D508C2312yescomplemented
69C2×C22⋊D127—22no—
70C2×C22⋊D127—21no—
71D4×D127—28no—
72D4×D127—24no—
73D4×D127—28no—
74D4×D127—24no—
75C2×C4×D127—22no—
76C2×C4×D127—21no—
77#14523237—24no—
78#14523235—24no—
79D1016—24no—
80#14523233—24no—
81#14523232—21no—
82#14523232C2312yescomplemented
83C2×C127⋊C8—21no—
84#41×C127—21no—
85#41×C127—21no—
86#41×C127D412yescomplemented
87#41×C127C2312yes—
88C2×Q8×C127D411yescharacteristic, complemented
89C2×D4×C127—21no—
90C2×D4×C127D411yescharacteristic, complemented
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127C2311yescharacteristic
94M16×C127—21no—
95M16×C127C2312yescomplemented
96C2×C8×C127—21no—
97C42×C127—21no—
98C42×C127C2312yescomplemented
99C22×D127—22no—
100C22×D127—21no—
101C22×D127—28no—
102C22×D127—22no—
103C22×D127—22no—
104C22×D127—44no—
105C22×D127—44no—
106C22×D127—41no—
107C22×D127—41no—
108C22×D127—84no—
109C22×D127C2×D414yes—
110Dic254—21no—
111Dic254C2×D412yescomplemented
112D508—28no—
113D508—22no—
114D508—22no—
115D508—21no—
116D508—22no—
117D508—22no—
118D508—22no—
119D508—22no—
120D508—416no—
121D508—41no—
122D508—41no—
123D508C2×D412yescomplemented
124C22⋊D127—28no—
125C22⋊D127—22no—
126C22⋊D127—22no—
127C2×Dic127—21no—
128C4×D127—28no—
129C4×D127—22no—
130C4×D127—22no—
131C127⋊C8—22no—
132C23×C127—21no—
133C23×C127C2×D411yescharacteristic
134Q8×C127—21no—
135Q8×C127C2×D412yescomplemented
136D4×C127—22no—
137D4×C127—22no—
138D4×C127—21no—
139D4×C127C2×D412yescomplemented
140C2×C4×C127—24no—
141C2×C4×C127—21no—
142C2×C4×C127—22no—
143C2×C4×C127—22no—
144C2×C4×C127—22no—
145C2×C4×C127C2×D411yescharacteristic
146C2×C4×C127C2×D411yescharacteristic
147C2×C4×C127C2×D411yescharacteristic
148C2×C4×C127C2×D411yescharacteristic
149C2×C4×C127C2×D411yescharacteristic
150C2×C4×C127C2411yescharacteristic
151C8×C127—22no—
152C22×C127—22no—
153C22×C127—22no—
154C22×C127—42no—
155C22×C127#12011yescharacteristic
156C22×C127#12011yescharacteristic
157C22×C127C22×D411yescharacteristic
158C2×D127—24no—
159C2×D127—22no—
160C2×D127—44no—
161C2×D127—42no—
162C2×D127—44no—
163C2×D127—44no—
164C2×D127—42no—
165C2×D127—82no—
166C2×D127—88no—
167C2×D127—82no—
168C2×D127—82no—
169C4×C127—24no—
170C4×C127—22no—
171C4×C127#12011yescharacteristic
172C4×C127#12011yescharacteristic
173C4×C127C22×D411yescharacteristic
174C4×C127C22×D411yescharacteristic
175Dic127—22no—
176C2×C127—21no—
177C2×C127—21no—
178C2×C127—42no—
179C2×C127#45312yescomplemented
180C2×C127C2×#12011yescharacteristic
181D127—44no—
182D127—42no—
183D127—82no—
184C2×#453—1271nomaximal, Sylow
185C127C2×#45311yescharacteristic, complemented, Sylow
186C2×H(5,𝔽2)—1271no—
187C2×AGL(1,ℤ/8)—1271no—
188C2×AGL(1,ℤ/8)—1271no—
189C2×#127—1271no—
190#453—1278no—
191C2×#104—1271no—
192C2×#104D12711yescharacteristic, complemented
193C2×#100—1271no—
194H(5,𝔽2)—1274no—
195H(5,𝔽2)—2542no—
196H(5,𝔽2)—2542no—
197H(5,𝔽2)—2542no—
198C2×#41—1271no—
199C2×#41—2541no—
200C2×#41—2541no—
201C2×#41C2×D12711yescharacteristic
202C22×D4—1271no—
203C22×D4—1271no—
204C22×D4—2541no—
205C22×D4—2542no—
206C22×D4—2542no—
207C22×D4—2541no—
208AGL(1,ℤ/8)—1274no—
209AGL(1,ℤ/8)—1274no—
210AGL(1,ℤ/8)—2542no—
211AGL(1,ℤ/8)—2542no—
212C2×SD16—2541no—
213C2×SD16—2541no—
214C2×D8—2541no—
215C2×D8—2541no—
216C2×M16—1271no—
217C2×M16C2×D12711yescharacteristic, complemented
218#127—1274no—
219#127—2542no—
220C2×C42C2×D12711yescharacteristic, complemented
221#104—1274no—
222#104C2×D12714yescomplemented
223#100—1274no—
224C24—1272no—
225C24—2542no—
226C24—2541no—
227#41—21no—
228#41—21no—
229#41—1272no—
230#41—1272no—
231#41—2544no—
232#41—2544no—
233#41—2544no—
234#41—2544no—
235#41—2541no—
236#41—2541no—
237#41D50812yescomplemented
238#41C22×D12712yes—
239C2×Q8—1271no—
240C2×Q8D50811yescharacteristic, complemented
241C2×D4—21no—
242C2×D4—1271no—
243C2×D4—1272no—
244C2×D4—1272no—
245C2×D4—1272no—
246C2×D4—5088no—
247C2×D4—2544no—
248C2×D4—2548no—
249C2×D4—2542no—
250C2×D4—2544no—
251C2×D4—2548no—
252C2×D4—2548no—
253C2×D4—2544no—
254C2×D4—2542no—
255C2×D4—2541no—
256C2×D4—2544no—
257C2×D4—2541no—
258C2×D4—2541no—
259C2×D4—2541no—
260C2×D4—2541no—
261C2×D4—2541no—
262C2×D4—2541no—
263C2×D4—2541no—
264C2×D4—2541no—
265C2×D4—2541no—
266C2×D4D50811yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—2542no—
270C22×C4—2541no—
271C22×C4C22×D12711yescharacteristic
272SD16—2544no—
273SD16—2544no—
274D8—2544no—
275D8—2544no—
276M16—21no—
277M16—1272no—
278M16—2541no—
279M16C22×D12712yescomplemented
280C2×C8—21no—
281C2×C8—2541no—
282C42—21no—
283C42C22×D12712yescomplemented
284C23—21no—
285C23—1274no—
286C23—10164no—
287C23—5084no—
288C23—5084no—
289C23—5081no—
290C23—5081no—
291C23—2548no—
292C23—2542no—
293C23—2542no—
294C23—2541no—
295C23—2542no—
296C23D4×D12711yescharacteristic
297Q8—21no—
298Q8—1272no—
299Q8—2541no—
300Q8C2×D50812yescomplemented
301D4—22no—
302D4—22no—
303D4—21no—
304D4—1272no—
305D4—50816no—
306D4—5081no—
307D4—5081no—
308D4—2548no—
309D4—2548no—
310D4—2542no—
311D4—2542no—
312D4—2542no—
313D4—2542no—
314D4—2542no—
315D4—2542no—
316D4—2541no—
317D4—2542no—
318D4—2542no—
319D4C2×D50812yescomplemented
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—2548no—
326C2×C4—2541no—
327C2×C4—2542no—
328C2×C4—2542no—
329C2×C4D4×D12711yescharacteristic
330C2×C4D4×D12711yescharacteristic
331C2×C4D4×D12711yescharacteristic
332C2×C4C2×D50811yescharacteristic
333C2×C4C2×D50811yescharacteristic
334C2×C4C23×D12711yescharacteristic
335C8—22no—
336C8—2542no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—10162no—
341C22—10162no—
342C22—10168no—
343C22—10162no—
344C22—5084no—
345C22—5084no—
346C22—5082no—
347C22—5082no—
348C22—5084no—
349C22—2542no—
350C22—2544no—
351C22#1686117911yescharacteristic
352C22#1686117911yescharacteristic
353C22C2×D4×D12711yescharacteristic
354C4—24no—
355C4—22no—
356C4—2542no—
357C4#1686117911yescharacteristic
358C4#1686117911yescharacteristic
359C4C2×D4×D12711yescharacteristic
360C4C22×D50811yescharacteristic
361C2—21no—
362C2—21no—
363C2—42no—
364C2—10162no—
365C2—5082no—
366C2—5084no—
367C2#2255004512yesdirect product factor
368C2C2×#1686117911yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#104×C127C2
socleC22×C127#120
hypercenterC2×#104D127
hypocenterC127C2×#453
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
1. C2×#16861255C2semidirect
2. C2×#16861237C2semidirect
3. C2×#16861234C2semidirect
4. C2×#16861206C2semidirect
5. #22550045C2direct (N × H)
6. C2×#16861137C2semidirect
7. C2×#16861125C2semidirect
8. C2×#104×C127C2semidirect
12. #16861255C22semidirect
19. C22×D508C22semidirect
23. #16861237C22semidirect
25. #16861234C22semidirect
30. C2×#14523232C22semidirect
32. #16861206C22semidirect
33. #16861137C22semidirect
34. #16861125C22semidirect
36. C2×M16×C127C22semidirect
37. C2×C42×C127C22semidirect
38. #104×C127C22semidirect
48. #14523248D4semidirect
50. C2×Dic254D4semidirect
66. C2×D508D4semidirect
68. C2×D508C23semidirect
82. #14523232C23semidirect
86. #41×C127D4semidirect
88. C2×Q8×C127D4semidirect
90. C2×D4×C127D4semidirect
95. M16×C127C23semidirect
98. C42×C127C23semidirect
111. Dic254C2×D4semidirect
123. D508C2×D4semidirect
135. Q8×C127C2×D4semidirect
139. D4×C127C2×D4semidirect
179. C2×C127#453semidirect
185. C127C2×#453semidirect
192. C2×#104D127semidirect
217. C2×M16C2×D127semidirect
220. C2×C42C2×D127semidirect
222. #104C2×D127semidirect
237. #41D508semidirect
240. C2×Q8D508semidirect
266. C2×D4D508semidirect
279. M16C22×D127semidirect
283. C42C22×D127semidirect
300. Q8C2×D508semidirect
319. D4C2×D508semidirect
367. C2#22550045direct (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×#168612372
C2×#168612342
C2×#168612062
#225500452
C2×#168611372
C2×#168611252
C2×#104×C1272
C2×#453127