Finite Group Databasebeta

← #458×D127 #30648116

Subgroups — #458×D127 #30648116

Order 16256.

794 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 135 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
1H(5,𝔽2)×D127C211yesmaximal, characteristic
2#123×D127C213yesmaximal, complemented
3#22550067C213yesmaximal, complemented
4#22550066C211yesmaximal, characteristic, complemented
5#22550065C213yesmaximal, complemented
6#99×D127C213yesmaximal, complemented
7#458×C127C211yesmaximal, characteristic, complemented
8#16861255—23no—
9#16861254—23no—
10#16861254C2213yes—
11#41×D127—23no—
12C2×D4×D127—23no—
13C2×D4×D127C2213yes—
14C22×C4×D127C2211yescharacteristic, complemented
15#16861194—23no—
16#16861193C2213yescomplemented
17#16861192C2213yescomplemented
18#16861191—23no—
19#16861190C2213yescomplemented
20C4⋊C4×D127—23no—
21C22⋊C4×D127—23no—
22C22⋊C4×D127C2213yescomplemented
23#16861124C2213yescomplemented
24#16861123C2216yescomplemented
25H(5,𝔽2)×C127C2211yescharacteristic
26#123×C127C2213yescomplemented
27#99×C127C2213yescomplemented
28C23×D127—21no—
29C23×D127D413yes—
30C23×D127C2311yescharacteristic
31#14523249—23no—
32#14523248—26no—
33#14523248—23no—
34#14523247—23no—
35#14523247—26no—
36Q8×D127—21no—
37C2×D508—23no—
38C2×D508D413yes—
39C2×C22⋊D127—23no—
40C2×C22⋊D127—23no—
41C2×C22⋊D127D413yes—
42C2×C22⋊D127C2313yes—
43D4×D127—23no—
44D4×D127—26no—
45D4×D127—26no—
46D4×D127—26no—
47D4×D127—26no—
48C22×Dic127C2311yescharacteristic, complemented
49C2×C4×D127—23no—
50C2×C4×D127—23no—
51C2×C4×D127—23no—
52C2×C4×D127—23no—
53C2×C4×D127D413yes—
54C4⋊Dic127—23no—
55Dic127⋊C4—23no—
56Dic127⋊C4—23no—
57C22⋊Dic127—23no—
58C22⋊Dic127C2313yescomplemented
59(C2×D127)⋊C4—23no—
60(C2×D127)⋊C4—23no—
61(C2×D127)⋊C4—23no—
62#41×C127—23no—
63C2×D4×C127—23no—
64C2×D4×C127C2313yes—
65C22×C4×C127C2311yescharacteristic, complemented
66C4⋊C4×C127—23no—
67C22⋊C4×C127—23no—
68C22⋊C4×C127C2313yescomplemented
69C22×D127—23no—
70C22×D127—23no—
71C22×D127—23no—
72C22×D127—42no—
73C22×D127—46no—
74C22×D127—82no—
75C22×D127C2×D413yes—
76C22×D127C2×D413yes—
77C22×D127C2×D413yes—
78Dic254—23no—
79D508—23no—
80D508—26no—
81C22⋊D127—26no—
82C22⋊D127—26no—
83C22⋊D127—26no—
84C2×Dic127—23no—
85C2×Dic127—23no—
86C2×Dic127—23no—
87C2×Dic127C2×D413yes—
88C4×D127—26no—
89C4×D127—23no—
90C4×D127—23no—
91C4×D127—46no—
92C4×D127—43no—
93C4×D127—41no—
94C23×C127—21no—
95C23×C127C2×D413yes—
96C23×C127C2411yescharacteristic
97Q8×C127—21no—
98D4×C127—23no—
99D4×C127—26no—
100C2×C4×C127—23no—
101C2×C4×C127—23no—
102C2×C4×C127—23no—
103C2×C4×C127C2×D413yes—
104C22×C127—23no—
105C22×C127—41no—
106C22×C127—43no—
107C22×C127—81no—
108C22×C127C22×D413yes—
109C2×D127—26no—
110C2×D127—23no—
111C2×D127—43no—
112C2×D127—46no—
113C2×D127—43no—
114C2×D127—83no—
115C2×D127—83no—
116C2×D127#12013yes—
117C2×D127#12011yescharacteristic
118C4×C127—23no—
119C4×C127—43no—
120C4×C127—41no—
121Dic127—23no—
122Dic127—43no—
123Dic127—41no—
124C2×C127—23no—
125C2×C127—43no—
126C2×C127C2×#12011yescharacteristic
127D127—23no—
128D127—43no—
129D127#45812yesdirect product factor
130C2×#458—1271nomaximal, Sylow
131C127C2×#45811yescharacteristic, complemented, Sylow
132C2×H(5,𝔽2)—1271no—
133C2×#123—1273no—
134#458—1271no—
135#458—1273no—
136#458—1273no—
137#458D12711yescharacteristic, direct product factor
138C2×#99—1273no—
139H(5,𝔽2)—1273no—
140H(5,𝔽2)—2543no—
141H(5,𝔽2)—2543no—
142H(5,𝔽2)C2×D12711yescharacteristic
143C2×#41—2543no—
144C22×D4—1273no—
145C22×D4—2543no—
146C23×C4—1271no—
147#123—1273no—
148#123—1273no—
149#123—1273no—
150#123—2543no—
151#123—2543no—
152#123C2×D12713yescomplemented
153C2×C4⋊C4—2543no—
154C2×C22⋊C4—1273no—
155C2×C22⋊C4—2543no—
156#99—1273no—
157#99—1276no—
158#99C2×D12713yescomplemented
159C24—1271no—
160C24—1273no—
161C24—2541no—
162#41—23no—
163#41—2546no—
164#41—2546no—
165#41—2543no—
166#41—2543no—
167#41—2543no—
168C2×Q8—2541no—
169C2×D4—23no—
170C2×D4—1273no—
171C2×D4—1273no—
172C2×D4—1273no—
173C2×D4—2543no—
174C2×D4—2546no—
175C2×D4—2546no—
176C2×D4—2546no—
177C2×D4—2543no—
178C2×D4—2546no—
179C2×D4—2543no—
180C2×D4—2543no—
181C2×D4C22×D12713yes—
182C22×C4—1273no—
183C22×C4—1271no—
184C22×C4—2543no—
185C22×C4—2543no—
186C22×C4—2543no—
187C22×C4—2543no—
188C22×C4C22×D12711yescharacteristic, complemented
189C4⋊C4—23no—
190C4⋊C4—2543no—
191C4⋊C4—2543no—
192C4⋊C4—2543no—
193C22⋊C4—23no—
194C22⋊C4—1273no—
195C22⋊C4—2543no—
196C22⋊C4—2543no—
197C22⋊C4—2543no—
198C22⋊C4—2543no—
199C22⋊C4C22×D12713yescomplemented
200C23—21no—
201C23—1273no—
202C23—1273no—
203C23—1273no—
204C23—10162no—
205C23—5086no—
206C23—5082no—
207C23—2543no—
208C23—2543no—
209C23—2543no—
210C23D4×D12713yes—
211C23C23×D12711yescharacteristic
212Q8—21no—
213Q8—2543no—
214D4—23no—
215D4—26no—
216D4—2546no—
217D4—2543no—
218D4—2546no—
219D4—2546no—
220D4—2546no—
221C2×C4—23no—
222C2×C4—23no—
223C2×C4—23no—
224C2×C4—1273no—
225C2×C4—5081no—
226C2×C4—5083no—
227C2×C4—5086no—
228C2×C4—2543no—
229C2×C4—2543no—
230C2×C4—2546no—
231C2×C4—2543no—
232C2×C4—2543no—
233C2×C4—2543no—
234C2×C4D4×D12713yes—
235C22—23no—
236C22—41no—
237C22—43no—
238C22—81no—
239C22—1271no—
240C22—1273no—
241C22—10163no—
242C22—10163no—
243C22—5083no—
244C22—5083no—
245C22—5086no—
246C22—2546no—
247C22—2543no—
248C22C2×D4×D12713yes—
249C4—23no—
250C4—43no—
251C4—41no—
252C4—5083no—
253C4—5081no—
254C4—2543no—
255C2—23no—
256C2—43no—
257C2—1272no—
258C2—5083no—
259C2—2543no—
260C2#120×D12711yescharacteristic
subgroupG/H
centerC2#120×D127
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
Fitting#458×C127C2
socleC2×C127C2×#120
prime radicalH(5,𝔽2)×D127C2
hypercenter#458D127
hypocenterC127C2×#458
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ #458

Complements

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

NH ≅ G/Ntype
2. #123×D127C2semidirect
3. #22550067C2semidirect
4. #22550066C2semidirect
5. #22550065C2semidirect
6. #99×D127C2semidirect
7. #458×C127C2semidirect
14. C22×C4×D127C22semidirect
16. #16861193C22semidirect
17. #16861192C22semidirect
19. #16861190C22semidirect
22. C22⋊C4×D127C22semidirect
23. #16861124C22semidirect
24. #16861123C22semidirect
26. #123×C127C22semidirect
27. #99×C127C22semidirect
48. C22×Dic127C23semidirect
58. C22⋊Dic127C23semidirect
65. C22×C4×C127C23semidirect
68. C22⋊C4×C127C23semidirect
129. D127#458direct (N × H)
131. C127C2×#458semidirect
137. #458D127direct (N × H)
152. #123C2×D127semidirect
158. #99C2×D127semidirect
188. C22×C4C22×D127semidirect
199. C22⋊C4C22×D127semidirect

Maximal subgroups

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

subgroupindex
H(5,𝔽2)×D1272
#123×D1272
#225500672
#225500662
#225500652
#99×D1272
#458×C1272
C2×#458127