Finite Group Databasebeta

← #30639795

Subgroups — #30639795

Order 16256.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#22550674C211yesmaximal, characteristic, complemented
2#22550387C211yesmaximal, characteristic, complemented
3#22549894C211yesmaximal, characteristic, complemented
4#22549893C211yesmaximal, characteristic, complemented
5#22549840C211yesmaximal, characteristic, complemented
6#22549632C211yesmaximal, characteristic, complemented
7#525×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239—21no—
11#16861235—21no—
12#16861234—21no—
13#16861229—22no—
14C2×#14523232C2211yescharacteristic
15C2×#14523232C2211yescharacteristic
16#16861192—21no—
17#16861188—21no—
18#16861183—21no—
19#16861178—21no—
20C4×C22⋊D127—22no—
21#16861168C2211yescharacteristic
22#16861137—22no—
23#16861136—22no—
24C22⋊(C127⋊C8)—22no—
25C2×#41×C127C2211yescharacteristic
26C23×C4×C127C2211yescharacteristic
27#123×C127—21no—
28C22⋊Q8×C127—21no—
29#121×C127—21no—
30#120×C127—21no—
31C4×D4×C127—22no—
32#117×C127C2211yescharacteristic
33#14523248—21no—
34#14523248—21no—
35#14523248—21no—
36#14523248—21no—
37C2×Dic254D411yescharacteristic, complemented
38C2×D508D411yescharacteristic, complemented
39C2×C22⋊D127—22no—
40C2×C22⋊D127—21no—
41C2×C4×D127—21no—
42#14523238—22no—
43#14523236—22no—
44#14523235—22no—
45#14523233—22no—
46#14523232—21no—
47#14523232—21no—
48#14523232—21no—
49#14523232—21no—
50C2×C127⋊C8—21no—
51C2×C127⋊C8D412yescomplemented
52C4⋊Dic127—21no—
53Dic127⋊C4—22no—
54C22⋊Dic127—22no—
55C22⋊Dic127—21no—
56(C2×D127)⋊C4—22no—
57C4×Dic127D412yescomplemented
58C24×C127D411yescharacteristic
59#41×C127—21no—
60#41×C127—21no—
61#41×C127—21no—
62#41×C127—21no—
63C2×Q8×C127D411yescharacteristic, complemented
64C2×D4×C127—22no—
65C2×D4×C127—21no—
66C2×D4×C127D411yescharacteristic, complemented
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127D411yescharacteristic
75C22×C4×C127C2311yescharacteristic
76C4⋊C4×C127—22no—
77C4⋊C4×C127—21no—
78C22⋊C4×C127—22no—
79C22⋊C4×C127—22no—
80C22⋊C4×C127—21no—
81C42×C127D412yescomplemented
82C22×D127—21no—
83Dic254—21no—
84Dic254—21no—
85D508—21no—
86D508—21no—
87C22⋊D127—44no—
88C22⋊D127—41no—
89C22⋊D127—41no—
90C2×Dic127—22no—
91C2×Dic127—21no—
92C4×D127—41no—
93C4×D127—41no—
94C127⋊C8—22no—
95C127⋊C8—41no—
96C23×C127—21no—
97C23×C127—21no—
98C23×C127—21no—
99C23×C127—21no—
100C23×C127—42no—
101C23×C127C2×D411yescharacteristic
102Q8×C127—21no—
103Q8×C127—21no—
104D4×C127—21no—
105D4×C127—21no—
106D4×C127—44no—
107D4×C127—41no—
108D4×C127—41no—
109C2×C4×C127—22no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D411yescharacteristic
122C2×C4×C127C2×D411yescharacteristic
123C2×C4×C127C2×D411yescharacteristic
124C2×C4×C127C2×D411yescharacteristic
125C2×C4×C127#4111yescharacteristic
126C22×C127—22no—
127C22×C127—22no—
128C22×C127—22no—
129C22×C127—41no—
130C22×C127—41no—
131C22×C127—41no—
132C22×C127—41no—
133C22×C127—42no—
134C22×C127—41no—
135C22×C127—41no—
136C22×C127#12011yescharacteristic
137C22×C127#12011yescharacteristic
138C22×C127#12111yescharacteristic
139C2×D127—42no—
140C2×D127—41no—
141C4×C127—42no—
142C4×C127—41no—
143C4×C127—41no—
144C4×C127#12011yescharacteristic
145C4×C127#12111yescharacteristic
146C4×C127#12111yescharacteristic
147C4×C127#12711yescharacteristic
148Dic127—42no—
149Dic127—41no—
150C2×C127—21no—
151C2×C127—21no—
152C2×C127—21no—
153C2×C127—42no—
154C2×C127—81no—
155C2×C127#39211yescharacteristic
156D127—81no—
157#2010—1271nomaximal, Sylow
158C127#201011yescharacteristic, complemented, Sylow
159#575—1271no—
160#525—1271no—
161#525D12711yescharacteristic, complemented
162#421—1271no—
163#421—1271no—
164#407—1271no—
165H(3,ℤ/4)—1271no—
166C2×#41—1271no—
167C2×#41C2×D12711yescharacteristic
168C23×C4C2×D12711yescharacteristic
169#137—2541no—
170#137—2541no—
171AGL(1,ℤ/8)—2541no—
172AGL(1,ℤ/8)—2541no—
173#135—2542no—
174C2×M16—1271no—
175C2×M16—1271no—
176#123—21no—
177#123—2541no—
178C22⋊Q8—21no—
179C22⋊Q8—2541no—
180#121—21no—
181#121—2541no—
182#120—21no—
183#120—2541no—
184C4×D4—22no—
185C4×D4—2542no—
186#117—1271no—
187#117C2×D12711yescharacteristic
188#104—2542no—
189#104—2542no—
190C22⋊C8—2542no—
191C24C22⋊D12711yescharacteristic
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196#41—2541no—
197#41—2541no—
198#41—2541no—
199#41—2541no—
200C2×Q8—1271no—
201C2×Q8C22⋊D12711yescharacteristic, complemented
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—1271no—
205C2×D4—2542no—
206C2×D4—2541no—
207C2×D4C22⋊D12711yescharacteristic, complemented
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2541no—
216C22×C4C22⋊D12711yescharacteristic
217C22×C4C22×D12711yescharacteristic
218Q16—2542no—
219SD16—2542no—
220SD16—2542no—
221D8—2542no—
222M16—2541no—
223M16—2541no—
224M16—2541no—
225M16—2541no—
226C2×C8—1272no—
227C2×C8—2541no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—2542no—
231C4⋊C4—2541no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—21no—
235C22⋊C4—2542no—
236C22⋊C4—2542no—
237C22⋊C4—2541no—
238C42—1272no—
239C42C22⋊D12712yescomplemented
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—42no—
245C23—2541no—
246C23C2×C22⋊D12711yescharacteristic
247Q8—21no—
248Q8—21no—
249Q8—2541no—
250Q8—2541no—
251D4—21no—
252D4—21no—
253D4—44no—
254D4—41no—
255D4—41no—
256D4—5084no—
257D4—5081no—
258D4—5081no—
259D4—2541no—
260D4—2541no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—5081no—
273C2×C4—5081no—
274C2×C4—2542no—
275C2×C4—2541no—
276C2×C4D4×D12711yescharacteristic
277C2×C4D4×D12711yescharacteristic
278C2×C4D4×D12711yescharacteristic
279C2×C4C2×C22⋊D12711yescharacteristic
280C2×C4C2×C22⋊D12711yescharacteristic
281C2×C4#1452324711yescharacteristic
282C8—5081no—
283C8—2542no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—5082no—
295C22—5081no—
296C22#1686117711yescharacteristic
297C22#1686117811yescharacteristic
298C22#1686118111yescharacteristic
299C4—42no—
300C4—41no—
301C4—41no—
302C4—5082no—
303C4—5081no—
304C4#1686117711yescharacteristic
305C4#1686118011yescharacteristic
306C4#1686118111yescharacteristic
307C4#1686120511yescharacteristic
308C2—21no—
309C2—21no—
310C2—21no—
311C2—42no—
312C2—81no—
313C2—10161no—
314C2#2254978411yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#525×C127C2
socleC2×C127#392
hypercenter#525D127
hypocenterC127#2010
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. #22550387C2semidirect
3. #22549894C2semidirect
4. #22549893C2semidirect
5. #22549840C2semidirect
6. #22549632C2semidirect
7. #525×C127C2semidirect
37. C2×Dic254D4semidirect
38. C2×D508D4semidirect
51. C2×C127⋊C8D4semidirect
57. C4×Dic127D4semidirect
63. C2×Q8×C127D4semidirect
66. C2×D4×C127D4semidirect
81. C42×C127D4semidirect
158. C127#2010semidirect
161. #525D127semidirect
201. C2×Q8C22⋊D127semidirect
207. C2×D4C22⋊D127semidirect
239. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
#225503872
#225498942
#225498932
#225498402
#225496322
#525×C1272
#2010127