Finite Group Databasebeta

← #30639783

Subgroups — #30639783

Order 16256.

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

normal subgroups only

This group has 380 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
2C2×#16861235C211yesmaximal, characteristic, complemented
3#22549893C211yesmaximal, characteristic, complemented
4#22549881C211yesmaximal, characteristic, complemented
5#22549849C211yesmaximal, characteristic, complemented
6#22549646C211yesmaximal, characteristic, complemented
7#535×C127C211yesmaximal, characteristic, complemented
8C2×#14523248C2211yescharacteristic
9#16861240—21no—
10#16861239—21no—
11#16861235—22no—
12#16861235—22no—
13#16861235—21no—
14#16861234—21no—
15#16861229—22no—
16C2×#14523236—21no—
17C2×#14523233—21no—
18C2×#14523232C2211yescharacteristic
19C2×#14523232C2211yescharacteristic
20#16861168C2211yescharacteristic
21#16861136—22no—
22#16861130—22no—
23#16861123—22no—
24C2×#41×C127C2211yescharacteristic
25C22×D4×C127C2211yescharacteristic
26#127×C127—21no—
27#124×C127—21no—
28#121×C127—22no—
29#120×C127—22no—
30#117×C127C2211yescharacteristic
31#14523248—21no—
32#14523248—21no—
33#14523248—21no—
34#14523248—21no—
35C2×Dic254D411yescharacteristic, complemented
36C2×D508D411yescharacteristic, complemented
37C2×C22⋊D127—21no—
38C2×C4×D127—21no—
39#14523238—22no—
40#14523236—24no—
41#14523236—22no—
42#14523235—22no—
43#14523233—24no—
44#14523233—22no—
45#14523232—21no—
46#14523232—21no—
47#14523232—21no—
48#14523232D412yescomplemented
49C2×C127⋊C8—21no—
50C2×C127⋊C8D412yescomplemented
51C4⋊Dic127—21no—
52C22⋊Dic127—21no—
53C4×Dic127—21no—
54C24×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57#41×C127—21no—
58#41×C127—21no—
59C2×Q8×C127D411yescharacteristic
60C2×D4×C127—21no—
61C2×D4×C127—24no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127—21no—
67C2×D4×C127D412yes—
68C2×D4×C127D411yescharacteristic
69C22×C4×C127—21no—
70C22×C4×C127C2311yescharacteristic
71C4⋊C4×C127—21no—
72C22⋊C4×C127—24no—
73C22⋊C4×C127D412yes—
74C42×C127—21no—
75C22×D127—21no—
76Dic254—21no—
77Dic254—21no—
78D508—21no—
79D508—21no—
80C22⋊D127—41no—
81C22⋊D127—41no—
82C2×Dic127—21no—
83C2×Dic127—41no—
84C4×D127—41no—
85C4×D127—41no—
86C127⋊C8—22no—
87C127⋊C8—22no—
88C23×C127—22no—
89C23×C127—21no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127—82no—
93C23×C127C2×D411yescharacteristic
94Q8×C127—21no—
95Q8×C127—21no—
96D4×C127—22no—
97D4×C127—21no—
98D4×C127—22no—
99D4×C127—21no—
100D4×C127—41no—
101D4×C127—41no—
102D4×C127—41no—
103D4×C127—41no—
104D4×C127—84no—
105C2×C4×C127—22no—
106C2×C4×C127—21no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127C2×D411yescharacteristic
110C2×C4×C127C2×D411yescharacteristic
111C2×C4×C127C2×D411yescharacteristic
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D411yescharacteristic
114C2×C4×C127#4111yescharacteristic
115C22×C127—41no—
116C22×C127—42no—
117C22×C127—44no—
118C22×C127—41no—
119C22×C127—42no—
120C22×C127—82no—
121C22×C127—82no—
122C22×C127#12011yescharacteristic
123C22×C127#12111yescharacteristic
124C22×C127#12711yescharacteristic
125C2×D127—41no—
126C2×D127—81no—
127C4×C127—42no—
128C4×C127—41no—
129C4×C127#12011yescharacteristic
130C4×C127#12011yescharacteristic
131C4×C127#12111yescharacteristic
132C4×C127#12111yescharacteristic
133Dic127—41no—
134Dic127—81no—
135C2×C127—21no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—81no—
139C2×C127—82no—
140C2×C127#39211yescharacteristic
141D127—81no—
142#2008—1271nomaximal, Sylow
143C127#200811yescharacteristic, complemented, Sylow
144#575—1271no—
145C2×AGL(1,ℤ/8)—1271no—
146#535D12711yescharacteristic, complemented
147#421—1271no—
148#418—1271no—
149#410—1271no—
150#344—1271no—
151C2×#41—1271no—
152C2×#41C2×D12711yescharacteristic
153C22×D4C2×D12711yescharacteristic
154#137—2541no—
155#137—2541no—
156AGL(1,ℤ/8)—2542no—
157AGL(1,ℤ/8)—2541no—
158AGL(1,ℤ/8)—2542no—
159AGL(1,ℤ/8)—2541no—
160#135—2542no—
161C2×SD16—2541no—
162C2×D8—2541no—
163C2×M16—1271no—
164C2×M16—1271no—
165#127—21no—
166#124—21no—
167#121—22no—
168#120—22no—
169#117—1271no—
170#117C2×D12711yescharacteristic
171#104—2542no—
172#102—2542no—
173#99—2542no—
174C24—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—2541no—
180#41—2541no—
181#41—2541no—
182#41—2541no—
183C2×Q8—1271no—
184C2×Q8C22⋊D12711yescharacteristic
185C2×D4—21no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—1271no—
193C2×D4—2541no—
194C2×D4C22⋊D12712yes—
195C2×D4C22⋊D12711yescharacteristic
196C22×C4—21no—
197C22×C4—2541no—
198C22×C4C22×D12711yescharacteristic
199Q16—2542no—
200SD16—2544no—
201SD16—2542no—
202SD16—2542no—
203D8—2544no—
204D8—2542no—
205M16—1272no—
206M16—2541no—
207M16—2541no—
208M16—2541no—
209C2×C8—1272no—
210C2×C8—2541no—
211C4⋊C4—21no—
212C4⋊C4—2541no—
213C22⋊C4—24no—
214C22⋊C4—2541no—
215C22⋊C4C22⋊D12712yes—
216C42—21no—
217C42—2541no—
218C23—22no—
219C23—21no—
220C23—41no—
221C23—41no—
222C23—82no—
223C23—2541no—
224C23C2×C22⋊D12711yescharacteristic
225Q8—21no—
226Q8—21no—
227Q8—2541no—
228Q8—2541no—
229D4—22no—
230D4—21no—
231D4—22no—
232D4—21no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—84no—
238D4—5081no—
239D4—5081no—
240D4—2541no—
241D4—2541no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—5081no—
247C2×C4—5081no—
248C2×C4—5081no—
249C2×C4—2541no—
250C2×C4D4×D12711yescharacteristic
251C2×C4D4×D12711yescharacteristic
252C2×C4D4×D12711yescharacteristic
253C2×C4C2×C22⋊D12711yescharacteristic
254C2×C4C2×C22⋊D12711yescharacteristic
255C2×C4#1452324711yescharacteristic
256C8—2542no—
257C8—2542no—
258C22—41no—
259C22—42no—
260C22—44no—
261C22—41no—
262C22—42no—
263C22—82no—
264C22—82no—
265C22—10161no—
266C22—5081no—
267C22#1686117811yescharacteristic
268C22#1686118011yescharacteristic
269C22#1686120511yescharacteristic
270C4—42no—
271C4—41no—
272C4—10161no—
273C4—5081no—
274C4#1686117711yescharacteristic
275C4#1686117711yescharacteristic
276C4#1686118111yescharacteristic
277C4#1686118111yescharacteristic
278C2—21no—
279C2—21no—
280C2—21no—
281C2—82no—
282C2—81no—
283C2—10161no—
284C2#2254978411yescharacteristic
subgroupG/H
centerC2#22549784
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#535×C127C2
socleC2×C127#392
hypercenter#535D127
hypocenterC127#2008
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22550674C2semidirect
2. C2×#16861235C2semidirect
3. #22549893C2semidirect
4. #22549881C2semidirect
5. #22549849C2semidirect
6. #22549646C2semidirect
7. #535×C127C2semidirect
35. C2×Dic254D4semidirect
36. C2×D508D4semidirect
48. #14523232D4semidirect
50. C2×C127⋊C8D4semidirect
143. C127#2008semidirect
146. #535D127semidirect

Maximal subgroups

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

subgroupindex
#225506742
C2×#168612352
#225498932
#225498812
#225498492
#225496462
#535×C1272
#2008127