Finite Group Databasebeta

← #30638697

Subgroups — #30638697

Order 16256.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861254C211yesmaximal, characteristic
2#22549894C211yesmaximal, characteristic, complemented
3#22549892C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5#22549858C211yesmaximal, characteristic, complemented
6#22549849C211yesmaximal, characteristic, complemented
7#418×C127C211yesmaximal, characteristic, complemented
8#16861254—22no—
9#16861254—22no—
10#16861254—22no—
11#16861254—22no—
12C2×#14523248C2211yescharacteristic
13C2×#14523248C2211yescharacteristic
14C2×#14523247—21no—
15C2×#14523247C412yescomplemented
16C22×C22⋊D127—22no—
17C2×D4×D127—21no—
18C2×D4×D127C412yescomplemented
19C2×#14523232C2211yescharacteristic, complemented
20#16861168C2211yescharacteristic, complemented
21#16861137—22no—
22#16861135—22no—
23#16861130—22no—
24#16861127—21no—
25#16861125—21no—
26#16861123—22no—
27C22×D4×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic, complemented
29#117×C127C2211yescharacteristic, complemented
30#102×C127—22no—
31C23×D127—22no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36#14523248—21no—
37#14523248—21no—
38#14523248D412yes—
39#14523248D412yes—
40#14523247—24no—
41#14523247—24no—
42#14523247—24no—
43#14523247—24no—
44C2×Dic254C2×C411yescharacteristic
45C2×Dic254D411yescharacteristic
46C2×D508C2×C411yescharacteristic
47C2×D508D411yescharacteristic
48C2×C22⋊D127—28no—
49C2×C22⋊D127—28no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—22no—
52C2×C22⋊D127—21no—
53C2×C22⋊D127D412yes—
54D4×D127—24no—
55D4×D127—24no—
56D4×D127—24no—
57D4×D127—24no—
58C22×Dic127—22no—
59C2×C4×D127—21no—
60C2×C4×D127C2×C412yes—
61#14523232—21no—
62#14523232—21no—
63C2×C127⋊C8—21no—
64C4⋊Dic127—21no—
65C22⋊Dic127—21no—
66C4×Dic127—21no—
67C24×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127C2×C412yes—
72C2×D4×C127D412yes—
73C2×D4×C127D412yes—
74C22×C4×C127C2311yescharacteristic
75M16×C127—21no—
76M16×C127—21no—
77C2×C8×C127—21no—
78C4⋊C4×C127—21no—
79C22⋊C4×C127—21no—
80C42×C127—21no—
81C22×D127—21no—
82C22×D127—28no—
83C22×D127—84no—
84C22×D127C22⋊C412yes—
85Dic254—21no—
86Dic254—21no—
87Dic254—21no—
88Dic254C22⋊C412yes—
89D508—21no—
90D508—21no—
91D508—21no—
92D508C22⋊C412yes—
93C22⋊D127—28no—
94C22⋊D127—28no—
95C22⋊D127—22no—
96C22⋊D127—22no—
97C22⋊D127—22no—
98C22⋊D127—22no—
99C2×Dic127—28no—
100C2×Dic127—21no—
101C2×Dic127—41no—
102C2×Dic127C22⋊C412yes—
103C4×D127—22no—
104C4×D127—22no—
105C4×D127—22no—
106C4×D127—22no—
107C127⋊C8—41no—
108C23×C127—22no—
109C23×C127—22no—
110C23×C127—22no—
111C23×C127—82no—
112C23×C127C2×D411yescharacteristic
113D4×C127—22no—
114D4×C127—22no—
115D4×C127—22no—
116D4×C127C22⋊C414yes—
117C2×C4×C127—41no—
118C2×C4×C127C22×C411yescharacteristic
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D411yescharacteristic
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127C2×D411yescharacteristic
124C8×C127—41no—
125C22×C127—24no—
126C22×C127—41no—
127C22×C127—44no—
128C22×C127—82no—
129C22×C127—82no—
130C22×C127C2×C22⋊C411yescharacteristic
131C22×C127#12011yescharacteristic
132C22×C127#12011yescharacteristic
133C2×D127—22no—
134C2×D127—22no—
135C2×D127—42no—
136C2×D127—42no—
137C2×D127—88no—
138C4×C127—81no—
139C4×C127C2×C22⋊C411yescharacteristic
140C4×C127C2×C22⋊C411yescharacteristic
141C4×C127#12011yescharacteristic
142C4×C127#12011yescharacteristic
143Dic127—22no—
144Dic127—22no—
145Dic127—81no—
146C2×C127—21no—
147C2×C127—21no—
148C2×C127—21no—
149C2×C127—44no—
150C2×C127#37911yescharacteristic
151D127—42no—
152D127—42no—
153#1795—1271nomaximal, Sylow
154C127#179511yescharacteristic, complemented, Sylow
155C2×H(5,𝔽2)—1271no—
156#421—1271no—
157#421—1271no—
158#418—1271no—
159#418D12711yescharacteristic, complemented
160#413—1271no—
161#410—1271no—
162H(5,𝔽2)—2542no—
163H(5,𝔽2)—2542no—
164H(5,𝔽2)—2542no—
165H(5,𝔽2)—2542no—
166C2×#41—1271no—
167C2×#41—1272no—
168C2×#41—1271no—
169C2×#41—2541no—
170C22×D4—1272no—
171C22×D4—2541no—
172C22×D4—2542no—
173C22×D4C2×D12711yescharacteristic
174C2×M16—1271no—
175C2×M16C2×D12711yescharacteristic, complemented
176#117—1271no—
177#117C2×D12711yescharacteristic, complemented
178#104—2542no—
179#104—2542no—
180#102—22no—
181#102—2542no—
182#101—2541no—
183#100—2541no—
184#99—2542no—
185C24—21no—
186C24—2542no—
187#41—1272no—
188#41—1272no—
189#41—2544no—
190#41—2544no—
191#41—2544no—
192#41—2544no—
193#41—2541no—
194#41—2541no—
195#41—2541no—
196#41—2541no—
197#41—2541no—
198#41—2541no—
199C2×Q8—1271no—
200C2×Q8—1271no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—1272no—
205C2×D4—1271no—
206C2×D4—1271no—
207C2×D4—2542no—
208C2×D4—2544no—
209C2×D4—2544no—
210C2×D4—2544no—
211C2×D4—2542no—
212C2×D4—2548no—
213C2×D4—2548no—
214C2×D4—2541no—
215C2×D4—2544no—
216C2×D4C4×D12712yes—
217C2×D4C22⋊D12712yes—
218C2×D4D50812yes—
219C22×C4—1272no—
220C22×C4—2542no—
221C22×C4—2541no—
222C22×C4C22×D12711yescharacteristic
223M16—21no—
224M16—21no—
225M16—2541no—
226M16—2541no—
227C2×C8—21no—
228C2×C8—2541no—
229C4⋊C4—21no—
230C4⋊C4—2541no—
231C22⋊C4—21no—
232C22⋊C4—2541no—
233C42—21no—
234C42—2541no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—82no—
239C23—1272no—
240C23—10164no—
241C23—2548no—
242C23—2541no—
243C23D4×D12711yescharacteristic
244Q8—1272no—
245Q8—2541no—
246Q8—2541no—
247Q8—2541no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—1272no—
252D4—2548no—
253D4—2548no—
254D4—2542no—
255D4—2542no—
256D4—2542no—
257D4—2542no—
258D4—2541no—
259D4—2541no—
260D4—2541no—
261D4(C2×D127)⋊C414yes—
262C2×C4—41no—
263C2×C4—1272no—
264C2×C4—5081no—
265C2×C4—2548no—
266C2×C4—2541no—
267C2×C4—2542no—
268C2×C4—2542no—
269C2×C4—2542no—
270C2×C4—2542no—
271C2×C4C2×C4×D12711yescharacteristic
272C2×C4D4×D12711yescharacteristic
273C2×C4D4×D12711yescharacteristic
274C2×C4D4×D12711yescharacteristic
275C2×C4C2×C22⋊D12711yescharacteristic
276C2×C4C2×D50811yescharacteristic
277C8—41no—
278C8—5081no—
279C22—24no—
280C22—41no—
281C22—44no—
282C22—82no—
283C22—82no—
284C22—10168no—
285C22—5082no—
286C22—5082no—
287C22—2542no—
288C22—2542no—
289C22C22⋊C4×D12711yescharacteristic
290C22#1686117711yescharacteristic
291C22#1686117911yescharacteristic
292C4—81no—
293C4—10161no—
294C4—2542no—
295C4—2542no—
296C4C22⋊C4×D12711yescharacteristic
297C4C2×(C2×D127)⋊C411yescharacteristic
298C4#1686117711yescharacteristic
299C4#1686117911yescharacteristic
300C2—21no—
301C2—21no—
302C2—21no—
303C2—44no—
304C2—5082no—
305C2—5082no—
306C2#2254973111yescharacteristic
subgroupG/H
centerC2#22549731
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#418×C127C2
socleC2×C127#379
prime radicalC2×#16861254C2
cosocleC22×C4×C127C23
hypercenter#418D127
hypocenterC127#1795
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #22549894C2semidirect
3. #22549892C2semidirect
4. #22549881C2semidirect
5. #22549858C2semidirect
6. #22549849C2semidirect
7. #418×C127C2semidirect
15. C2×#14523247C4semidirect
18. C2×D4×D127C4semidirect
19. C2×#14523232C22semidirect
20. #16861168C22semidirect
28. C2×M16×C127C22semidirect
29. #117×C127C22semidirect
154. C127#1795semidirect
159. #418D127semidirect
175. C2×M16C2×D127semidirect
177. #117C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
#225498942
#225498922
#225498812
#225498582
#225498492
#418×C1272
#1795127