Finite Group Databasebeta

← #30650507

Subgroups — #30650507

Order 16256.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550691C212yesmaximal, complemented
2#22550686C212yesmaximal, complemented
3#22550271C211yesmaximal, characteristic, complemented
4#22550150C212yesmaximal, complemented
5#22550137C212yesmaximal, complemented
6#22550060C212yesmaximal, complemented
7#22550053C211yesmaximal, characteristic, complemented
8#22549995C212yesmaximal, complemented
9#454×C127C211yesmaximal, characteristic, complemented
10#16861256C2212yes—
11#16861254C2211yescharacteristic
12#16861242C2212yes—
13#16861240—22no—
14#16861240C2212yes—
15#16861239—22no—
16#16861239C2212yescomplemented
17#16861238—22no—
18#16861233—22no—
19#16861229—22no—
20#16861229C2212yes—
21#16861228—22no—
22C2×#14523238—22no—
23C2×#14523238C2212yescomplemented
24C2×#14523236C2212yes—
25SD16×D127—22no—
26#16861217C2212yes—
27#16861216C2212yescomplemented
28#16861209C2211yescharacteristic, complemented
29#16861144C2212yescomplemented
30#16861137C2212yes—
31#16861127C2212yescomplemented
32#16861126C2211yescharacteristic
33#16861120C2211yescharacteristic, complemented
342−1+4×C127C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C127C2212yes—
36#124×C127C2211yescharacteristic
37#104×C127C2212yescomplemented
38#101×C127C2211yescharacteristic, complemented
39#14523248—22no—
40#14523248C2312yes—
41#14523247—22no—
42#14523247—22no—
43#14523247—42no—
44C2×Dic254—22no—
45C2×Dic254C2312yes—
46Q8×D127—22no—
47C2×C22⋊D127—41no—
48D4×D127—22no—
49Dic508—22no—
50#14523238—22no—
51#14523238—22no—
52#14523238—22no—
53#14523238—22no—
54#14523238D412yes—
55#14523237—22no—
56#14523236—22no—
57#14523236—22no—
58#14523236—22no—
59#14523236D412yes—
60#14523235—22no—
61#14523235D412yes—
62#14523233D412yes—
63#14523232—22no—
64#14523232C2312yes—
65#14523231—22no—
66C2×C127⋊C8—22no—
67C2×C127⋊C8C2312yes—
68C8×D127—22no—
69C4⋊Dic127—41no—
70#41×C127—22no—
71#41×C127—41no—
72#41×C127C2312yes—
73C2×Q8×C127—22no—
74C2×Q8×C127C2311yescharacteristic
75C2×D4×C127C2311yescharacteristic
76SD16×C127—22no—
77D8×C127—22no—
78M16×C127C2312yes—
79C22⋊C4×C127—41no—
80C42×C127C2311yescharacteristic
81C22×D127—41no—
82Dic254—22no—
83Dic254—22no—
84Dic254—42no—
85Dic254C2×D412yes—
86D508C2×D412yes—
87C22⋊D127—22no—
88C22⋊D127—42no—
89C22⋊D127—42no—
90C2×Dic127—22no—
91C2×Dic127—41no—
92C2×Dic127—42no—
93C4×D127—22no—
94C4×D127—42no—
95C127⋊C8—22no—
96C127⋊C8C2×D412yes—
97C127⋊C8C2×D412yes—
98C23×C127—21no—
99Q8×C127—22no—
100Q8×C127—41no—
101Q8×C127C2×D412yes—
102D4×C127—22no—
103D4×C127—22no—
104D4×C127—41no—
105D4×C127C2×D412yes—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—42no—
110C2×C4×C127C2411yescharacteristic
111C8×C127—22no—
112C22×C127—22no—
113C22×C127—41no—
114C22×C127—81no—
115C22×C127H(5,𝔽2)11yescharacteristic
116C2×D127—22no—
117C2×D127—161no—
118C4×C127—22no—
119C4×C127—41no—
120C4×C127—41no—
121C4×C127C22×D412yes—
122Dic127—22no—
123Dic127—42no—
124C2×C127—21no—
125C2×C127—42no—
126C2×C127—81no—
127C2×C127D4211yescharacteristic
128D127—42no—
129#3289—1271nomaximal, Sylow
130C127#328911yescharacteristic, complemented, Sylow
131#578—1272no—
132#577—1272no—
133#497—1271no—
134#473—1272no—
135#471—1272no—
136#456—1272no—
137#455—1271no—
138#454D12711yescharacteristic, complemented
139#444—1272no—
1402−1+4—1272no—
1412−1+4C2×D12711yescharacteristic, complemented
142H(5,𝔽2)—1271no—
143#137—1272no—
144#137—1272no—
145#137—1272no—
146#137—2542no—
147#137—2542no—
148#137—2542no—
149AGL(1,ℤ/8)—2542no—
150AGL(1,ℤ/8)C2×D12712yes—
151#135—1272no—
152#135—2542no—
153#135—2542no—
154C2×Q16—1272no—
155C2×Q16—2542no—
156C2×SD16—1272no—
157C2×SD16—2542no—
158#131—1272no—
159#131—1272no—
160C4⋊Q8—1271no—
161#124C2×D12711yescharacteristic
162#108—1272no—
163#104—1272no—
164#104C2×D12712yescomplemented
165#101—1272no—
166#101C2×D12711yescharacteristic, complemented
167#100—1271no—
168#97—1271no—
169#41—22no—
170#41—41no—
171#41—1272no—
172#41—5082no—
173#41—2542no—
174#41—2542no—
175#41—2542no—
176#41C22×D12712yes—
177C2×Q8—22no—
178C2×Q8—1272no—
179C2×Q8—2542no—
180C2×Q8—2542no—
181C2×Q8C22×D12711yescharacteristic
182C2×D4—5081no—
183C2×D4—2542no—
184C2×D4C22×D12711yescharacteristic
185Q16—1272no—
186Q16—2542no—
187Q16—2542no—
188Q16—2542no—
189Q16—2542no—
190Q16—2542no—
191SD16—22no—
192SD16—1272no—
193SD16—1272no—
194SD16—2542no—
195SD16—2542no—
196SD16—2542no—
197SD16—2542no—
198SD16—2542no—
199D8—22no—
200D8—1272no—
201M16—1272no—
202M16—2542no—
203M16—2542no—
204M16C22×D12712yes—
205C2×C8—1272no—
206C2×C8—2542no—
207C2×C8—2542no—
208C4⋊C4—5081no—
209C22⋊C4—41no—
210C42C22×D12711yescharacteristic
211C23—21no—
212C23—5081no—
213Q8—22no—
214Q8—41no—
215Q8—1272no—
216Q8—5082no—
217Q8—2542no—
218Q8—2542no—
219Q8D4×D12712yes—
220D4—22no—
221D4—22no—
222D4—41no—
223D4—1272no—
224D4—5082no—
225D4—5082no—
226D4—2542no—
227D4D4×D12712yes—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—5082no—
233C2×C4—5082no—
234C2×C4—5081no—
235C2×C4—2542no—
236C2×C4—2542no—
237C2×C4C23×D12711yescharacteristic
238C8—22no—
239C8—1272no—
240C8—1272no—
241C8—2542no—
242C22—22no—
243C22—41no—
244C22—81no—
245C22—20321no—
246C22—2542no—
247C22#1686125411yescharacteristic
248C4—22no—
249C4—41no—
250C4—41no—
251C4—5082no—
252C4—2542no—
253C4C2×D4×D12712yes—
254C2—21no—
255C2—42no—
256C2—81no—
257C2—5082no—
258C2#2255050811yescharacteristic
subgroupG/H
centerC2#22550508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#454×C127C2
socleC2×C127D42
hypercenter#454D127
hypocenterC127#3289
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #22550691C2semidirect
2. #22550686C2semidirect
3. #22550271C2semidirect
4. #22550150C2semidirect
5. #22550137C2semidirect
6. #22550060C2semidirect
7. #22550053C2semidirect
8. #22549995C2semidirect
9. #454×C127C2semidirect
16. #16861239C22semidirect
23. C2×#14523238C22semidirect
27. #16861216C22semidirect
28. #16861209C22semidirect
29. #16861144C22semidirect
31. #16861127C22semidirect
33. #16861120C22semidirect
34. 2−1+4×C127C22semidirect
37. #104×C127C22semidirect
38. #101×C127C22semidirect
130. C127#3289semidirect
138. #454D127semidirect
141. 2−1+4C2×D127semidirect
164. #104C2×D127semidirect
166. #101C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506912
#225506862
#225502712
#225501502
#225501372
#225500602
#225500532
#225499952
#454×C1272
#3289127