Finite Group Databasebeta

← #30638692

Subgroups — #30638692

Order 16256.

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

normal subgroups only

This group has 548 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
1#22550712C211yesmaximal, characteristic
2#22549892C211yesmaximal, characteristic, complemented
3#22549872C211yesmaximal, characteristic, complemented
4#22549870C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549844C211yesmaximal, characteristic, complemented
7#421×C127C211yesmaximal, characteristic, complemented
8#16861257—21no—
9#16861256—21no—
10#16861256C412yescomplemented
11#16861255—21no—
12#16861255C412yescomplemented
13#16861254—22no—
14#16861254—21no—
15C2×#14523248—22no—
16C2×#14523248C2211yescharacteristic
17C2×#14523248C2211yescharacteristic
18#41×D127—22no—
19#41×D127—22no—
20C22×C127⋊C8C2211yescharacteristic, complemented
21#16861168C2211yescharacteristic, complemented
22#16861135—22no—
23#16861133—21no—
24#16861132—21no—
25#16861130—21no—
26#16861129—21no—
27#16861123—22no—
28(C2×D127)⋊C8—22no—
29C2×#41×C127C2211yescharacteristic
30C2×M16×C127C2211yescharacteristic, complemented
31#117×C127C2211yescharacteristic, complemented
32#104×C127—22no—
33#14523249—22no—
34#14523249—22no—
35#14523248—24no—
36#14523248—24no—
37#14523248—22no—
38#14523248—22no—
39#14523248—21no—
40#14523248—21no—
41#14523248—21no—
42#14523248—21no—
43#14523248—21no—
44#14523248—21no—
45#14523248C2×C412yes—
46#14523248D412yes—
47#14523247—24no—
48#14523247—22no—
49#14523247—24no—
50#14523247—22no—
51C2×Dic254—22no—
52C2×Dic254C2×C411yescharacteristic
53C2×Dic254D411yescharacteristic
54Q8×D127—22no—
55Q8×D127—22no—
56C2×D508—22no—
57C2×D508C2×C411yescharacteristic
58C2×D508D411yescharacteristic
59C2×C22⋊D127—24no—
60C2×C22⋊D127—21no—
61C2×C22⋊D127D412yes—
62D4×D127—24no—
63D4×D127—22no—
64D4×D127—24no—
65D4×D127—22no—
66C2×C4×D127—24no—
67C2×C4×D127—21no—
68C2×C4×D127D412yes—
69C2×C127⋊C8—21no—
70C2×C127⋊C8—21no—
71C2×C127⋊C8—21no—
72C4⋊Dic127—21no—
73C22⋊Dic127—21no—
74C4×Dic127—21no—
75#41×C127—21no—
76#41×C127—21no—
77#41×C127C2×C412yes—
78#41×C127D412yes—
79C2×Q8×C127D411yescharacteristic
80C2×D4×C127—21no—
81C2×D4×C127D411yescharacteristic
82C22×C4×C127—21no—
83C22×C4×C127C2311yescharacteristic
84M16×C127—21no—
85M16×C127—21no—
86C2×C8×C127—21no—
87C4⋊C4×C127—21no—
88C22⋊C4×C127—21no—
89C42×C127—21no—
90C22×D127—24no—
91C22×D127—21no—
92C22×D127C22⋊C412yes—
93Dic254—24no—
94Dic254—21no—
95Dic254—21no—
96Dic254—21no—
97Dic254C22⋊C412yes—
98D508—24no—
99D508—21no—
100D508—21no—
101D508—21no—
102D508C22⋊C412yes—
103C22⋊D127—24no—
104C22⋊D127—24no—
105C22⋊D127—22no—
106C22⋊D127—22no—
107C22⋊D127—22no—
108C22⋊D127—22no—
109C2×Dic127—24no—
110C2×Dic127—21no—
111C2×Dic127—41no—
112C2×Dic127C22⋊C412yes—
113C4×D127—24no—
114C4×D127—24no—
115C4×D127—22no—
116C4×D127—22no—
117C4×D127—22no—
118C4×D127—22no—
119C127⋊C8—41no—
120C23×C127—21no—
121C23×C127C2×D411yescharacteristic
122Q8×C127—21no—
123Q8×C127C22⋊C412yes—
124D4×C127—22no—
125D4×C127—22no—
126D4×C127—21no—
127D4×C127C22⋊C412yes—
128C2×C4×C127—21no—
129C2×C4×C127—22no—
130C2×C4×C127—22no—
131C2×C4×C127—41no—
132C2×C4×C127C22×C411yescharacteristic
133C2×C4×C127C2×D411yescharacteristic
134C2×C4×C127C2×D411yescharacteristic
135C2×C4×C127C2×D411yescharacteristic
136C2×C4×C127C2×D411yescharacteristic
137C2×C4×C127C2×D411yescharacteristic
138C8×C127—41no—
139C22×C127—22no—
140C22×C127—41no—
141C22×C127—81no—
142C22×C127C2×C22⋊C411yescharacteristic
143C22×C127#12011yescharacteristic
144C22×C127#12011yescharacteristic
145C2×D127—22no—
146C2×D127—22no—
147C2×D127—42no—
148C2×D127—81no—
149C2×D127—84no—
150C4×C127—22no—
151C4×C127—81no—
152C4×C127C2×C22⋊C411yescharacteristic
153C4×C127C2×C22⋊C411yescharacteristic
154C4×C127#12011yescharacteristic
155C4×C127#12011yescharacteristic
156Dic127—22no—
157Dic127—22no—
158Dic127—81no—
159C2×C127—21no—
160C2×C127—21no—
161C2×C127—21no—
162C2×C127—42no—
163C2×C127#37911yescharacteristic
164D127—42no—
165D127—42no—
166#1794—1271nomaximal, Sylow
167C127#179411yescharacteristic, complemented, Sylow
168#585—1271no—
169#421—1271no—
170#421D12711yescharacteristic, complemented
171#416—1271no—
172#416—1271no—
173#410—1271no—
174#408—1271no—
1752−1+4—1272no—
1762−1+4—2541no—
1772−1+4—2541no—
178H(5,𝔽2)—1272no—
179H(5,𝔽2)—2542no—
180H(5,𝔽2)—2541no—
181H(5,𝔽2)—2541no—
182C2×#41—1271no—
183C2×#41—1271no—
184C2×#41—2542no—
185C2×#41—2542no—
186C2×#41—2542no—
187C2×#41C2×D12711yescharacteristic
188C2×M16C2×D12711yescharacteristic, complemented
189C22×C8—1271no—
190#117—1271no—
191#117C2×D12711yescharacteristic, complemented
192#104—22no—
193#104—2542no—
194#103—2541no—
195#103—2541no—
196#102—2541no—
197#102—2541no—
198#99—2542no—
199C22⋊C8—2542no—
200#41—21no—
201#41—21no—
202#41—1272no—
203#41—1272no—
204#41—2542no—
205#41—2542no—
206#41—2544no—
207#41—2544no—
208#41—2544no—
209#41—2544no—
210#41—2542no—
211#41—2542no—
212#41—2542no—
213#41—2542no—
214#41—2541no—
215#41—2541no—
216#41—2541no—
217#41—2541no—
218#41—2541no—
219#41—2541no—
220#41C4×D12712yes—
221#41C22⋊D12712yes—
222C2×Q8—1271no—
223C2×Q8—1271no—
224C2×Q8—2542no—
225C2×Q8—2542no—
226C2×Q8—2542no—
227C2×Q8D50811yescharacteristic
228C2×D4—21no—
229C2×D4—1272no—
230C2×D4—1271no—
231C2×D4—1271no—
232C2×D4—2542no—
233C2×D4—2544no—
234C2×D4—2544no—
235C2×D4—2542no—
236C2×D4—2544no—
237C2×D4—2541no—
238C2×D4—2542no—
239C2×D4D50811yescharacteristic
240C22×C4—21no—
241C22×C4—1272no—
242C22×C4—2544no—
243C22×C4—2541no—
244C22×C4C22×D12711yescharacteristic
245M16—21no—
246M16—21no—
247C2×C8—21no—
248C2×C8—2541no—
249C2×C8—2541no—
250C2×C8—2541no—
251C4⋊C4—21no—
252C4⋊C4—2541no—
253C22⋊C4—21no—
254C22⋊C4—2541no—
255C42—21no—
256C42—2541no—
257C23—21no—
258C23—1272no—
259C23—2544no—
260C23—2541no—
261C23D4×D12711yescharacteristic
262Q8—21no—
263Q8—1272no—
264Q8—2544no—
265Q8—2541no—
266Q8—2541no—
267Q8—2541no—
268Q8(C2×D127)⋊C412yes—
269D4—22no—
270D4—22no—
271D4—21no—
272D4—1272no—
273D4—2544no—
274D4—2544no—
275D4—2544no—
276D4—2542no—
277D4—2542no—
278D4—2542no—
279D4—2542no—
280D4—2541no—
281D4—2541no—
282D4—2541no—
283D4(C2×D127)⋊C412yes—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—1272no—
289C2×C4—5081no—
290C2×C4—2544no—
291C2×C4—2544no—
292C2×C4—2544no—
293C2×C4—2541no—
294C2×C4—2542no—
295C2×C4—2542no—
296C2×C4—2542no—
297C2×C4—2542no—
298C2×C4C2×C4×D12711yescharacteristic
299C2×C4D4×D12711yescharacteristic
300C2×C4D4×D12711yescharacteristic
301C2×C4D4×D12711yescharacteristic
302C2×C4C2×C22⋊D12711yescharacteristic
303C2×C4C2×D50811yescharacteristic
304C8—41no—
305C8—5081no—
306C22—22no—
307C22—41no—
308C22—81no—
309C22—10161no—
310C22—10164no—
311C22—5082no—
312C22—2542no—
313C22—2542no—
314C22C22⋊C4×D12711yescharacteristic
315C22#1686117711yescharacteristic
316C22#1686117911yescharacteristic
317C4—22no—
318C4—81no—
319C4—10161no—
320C4—2542no—
321C4—2542no—
322C4C22⋊C4×D12711yescharacteristic
323C4C2×(C2×D127)⋊C411yescharacteristic
324C4#1686117711yescharacteristic
325C4#1686117911yescharacteristic
326C2—21no—
327C2—21no—
328C2—21no—
329C2—42no—
330C2—5082no—
331C2—5082no—
332C2#2254973111yescharacteristic
subgroupG/H
centerC4#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#421×C127C2
socleC2×C127#379
prime radical#22550712C2
cosocleC22×C4×C127C23
hypercenter#421D127
hypocenterC127#1794
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #22549892C2semidirect
3. #22549872C2semidirect
4. #22549870C2semidirect
5. #22549849C2semidirect
6. #22549844C2semidirect
7. #421×C127C2semidirect
10. #16861256C4semidirect
12. #16861255C4semidirect
20. C22×C127⋊C8C22semidirect
21. #16861168C22semidirect
30. C2×M16×C127C22semidirect
31. #117×C127C22semidirect
167. C127#1794semidirect
170. #421D127semidirect
188. C2×M16C2×D127semidirect
191. #117C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
#225498922
#225498722
#225498702
#225498492
#225498442
#421×C1272
#1794127