Finite Group Databasebeta

← #497×D127 #30649166

Subgroups — #497×D127 #30649166

Order 16256.

564 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×Q16×D127C212yesmaximal, complemented
2C2×SD16×D127C212yesmaximal
3C4⋊Q8×D127C211yesmaximal, characteristic, complemented
4#124×D127C211yesmaximal, characteristic
5#22550271C211yesmaximal, characteristic, complemented
6#22550270C211yesmaximal, characteristic, complemented
7#22550269C212yesmaximal, complemented
8#22550268C211yesmaximal, characteristic, complemented
9#22550267C212yesmaximal, complemented
10#97×D127C211yesmaximal, characteristic, complemented
11#497×C127C211yesmaximal, characteristic, complemented
12C2×Q8×D127C2211yescharacteristic
13C2×Q8×D127C2212yes—
14C2×D4×D127C2211yescharacteristic
15C2×Dic508C2212yescomplemented
16C2×#14523238C2212yescomplemented
17C2×#14523238C2212yescomplemented
18Q16×D127—28no—
19C2×#14523237C2212yes—
20C2×#14523236C2212yes—
21C2×#14523235C2212yes—
22SD16×D127—28no—
23C2×C8×D127C2212yes—
24#16861209C2211yescharacteristic, complemented
25#16861208—22no—
26#16861207C2212yescomplemented
27#16861198C2211yescharacteristic
28#16861197C2211yescharacteristic
29#16861196C2211yescharacteristic
30#16861195—22no—
31C4⋊C4×D127—22no—
32C22⋊C4×D127—22no—
33C42×D127C2211yescharacteristic
34#16861120C2211yescharacteristic, complemented
35#16861119C2212yescomplemented
36C2×Q16×C127C2212yescomplemented
37C2×SD16×C127C2212yes—
38C4⋊Q8×C127C2211yescharacteristic, complemented
39#124×C127C2211yescharacteristic
40#97×C127C2211yescharacteristic, complemented
41C23×D127—21no—
42C2×Dic254—21no—
43C2×Dic254—22no—
44C2×Dic254C2311yescharacteristic
45C2×Dic254C2312yes—
46Q8×D127—24no—
47Q8×D127—24no—
48Q8×D127—24no—
49C2×D508C2311yescharacteristic
50C2×C22⋊D127—21no—
51D4×D127—24no—
52C2×C4×D127—21no—
53C2×C4×D127—22no—
54C2×C4×D127D412yes—
55C2×C4×D127C2311yescharacteristic
56Dic508—24no—
57#14523238—24no—
58#14523238—24no—
59#14523237—24no—
60#14523236—24no—
61#14523235—24no—
62C2×C127⋊C8C2312yes—
63C8×D127D418yes—
64C4⋊Dic127—22no—
65Dic127⋊C4—24no—
66C22⋊Dic127—22no—
67(C2×D127)⋊C4—22no—
68(C2×D127)⋊C4—22no—
69C4×Dic127D412yes—
70C4×Dic127C2311yescharacteristic
71C2×Q8×C127C2311yescharacteristic
72C2×Q8×C127C2312yes—
73C2×D4×C127C2311yescharacteristic
74Q16×C127—24no—
75SD16×C127—24no—
76C2×C8×C127C2312yes—
77C4⋊C4×C127—22no—
78C22⋊C4×C127—22no—
79C42×C127C2311yescharacteristic
80C22×D127—21no—
81C22×D127—42no—
82C22×D127—44no—
83C22×D127C2×D411yescharacteristic
84Dic254—22no—
85Dic254—22no—
86Dic254—22no—
87Dic254—44no—
88Dic254—42no—
89D508—22no—
90C22⋊D127—42no—
91C2×Dic127—21no—
92C2×Dic127—22no—
93C2×Dic127C2×D412yes—
94C2×Dic127C2×D411yescharacteristic
95C4×D127—24no—
96C4×D127—42no—
97C4×D127—44no—
98C4×D127C2×D414yes—
99C127⋊C8C2×D414yes—
100C23×C127—21no—
101Q8×C127—22no—
102Q8×C127—22no—
103Q8×C127—22no—
104D4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2411yescharacteristic
109C8×C127C2×D414yes—
110C22×C127—41no—
111C22×C127—42no—
112C22×C127C22×D411yescharacteristic
113C2×D127—41no—
114C2×D127—42no—
115C2×D127—84no—
116C2×D127#12712yes—
117C2×D127#13714yes—
118C4×C127—22no—
119C4×C127—41no—
120C4×C127—42no—
121C4×C127C22×D412yes—
122Dic127—22no—
123Dic127—41no—
124Dic127—42no—
125Dic127#12712yes—
126C2×C127—81no—
127C2×C127C2×#12711yescharacteristic
128C2×C127C2×#13712yes—
129D127—81no—
130D127#49714yesdirect product factor
131C2×#497—1271nomaximal, Sylow
132C127C2×#49711yescharacteristic, complemented, Sylow
133C22×Q16—1272no—
134C22×SD16—1272no—
135C2×C4⋊Q8—1271no—
136C2×#124—1271no—
137#497—1272no—
138#497—1272no—
139#497—1271no—
140#497—1271no—
141#497—1271no—
142#497D12711yescharacteristic, direct product factor
143C2×#97—1271no—
144C22×Q8—1271no—
145C22×Q8—1272no—
146C22×D4—1271no—
147C2×Q16—1272no—
148C2×Q16—1272no—
149C2×Q16—1272no—
150C2×Q16—2548no—
151C2×Q16C2×D12712yescomplemented
152C2×SD16—1272no—
153C2×SD16—1272no—
154C2×SD16—1272no—
155C2×SD16—2548no—
156C2×SD16C2×D12712yes—
157C22×C8—1272no—
158C4⋊Q8—1272no—
159C4⋊Q8—1271no—
160C4⋊Q8—2542no—
161C4⋊Q8C2×D12711yescharacteristic, complemented
162#124—1271no—
163#124—1271no—
164#124—1271no—
165#124—2542no—
166#124C2×D12711yescharacteristic
167C2×C4⋊C4—2542no—
168C2×C22⋊C4—2542no—
169C2×C42—1271no—
170#97—1272no—
171#97—1271no—
172#97C2×D12711yescharacteristic, complemented
173C24—2541no—
174C2×Q8—1271no—
175C2×Q8—1272no—
176C2×Q8—2541no—
177C2×Q8—2542no—
178C2×Q8—2544no—
179C2×Q8—2544no—
180C2×Q8—2544no—
181C2×Q8C22×D12711yescharacteristic
182C2×Q8C22×D12712yes—
183C2×D4—1271no—
184C2×D4—2544no—
185C2×D4—2541no—
186C2×D4C22×D12711yescharacteristic
187C22×C4—1271no—
188C22×C4—1272no—
189C22×C4—2541no—
190C22×C4—2542no—
191Q16—24no—
192Q16—2544no—
193Q16—2544no—
194Q16—2544no—
195SD16—24no—
196SD16—2544no—
197SD16—2544no—
198SD16—2544no—
199C2×C8—1272no—
200C2×C8—1278no—
201C2×C8C22×D12712yes—
202C4⋊C4—22no—
203C4⋊C4—2544no—
204C4⋊C4—2542no—
205C22⋊C4—22no—
206C22⋊C4—2542no—
207C22⋊C4—2542no—
208C22⋊C4—2542no—
209C42—1272no—
210C42—1271no—
211C42C22×D12711yescharacteristic
212C23—21no—
213C23—1271no—
214C23—5082no—
215C23—5084no—
216C23—2541no—
217Q8—22no—
218Q8—22no—
219Q8—22no—
220Q8—5082no—
221Q8—5084no—
222Q8—2542no—
223Q8—2542no—
224Q8—2542no—
225D4—22no—
226D4—5082no—
227D4—2542no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—1271no—
231C2×C4—1274no—
232C2×C4—1272no—
233C2×C4—5082no—
234C2×C4—5084no—
235C2×C4—2544no—
236C2×C4—2541no—
237C2×C4—2542no—
238C2×C4D4×D12712yes—
239C2×C4C23×D12711yescharacteristic
240C8—1274no—
241C8D4×D12714yes—
242C22—42no—
243C22—41no—
244C22—1274no—
245C22—1272no—
246C22—10164no—
247C22—5082no—
248C22—5081no—
249C22C2×D4×D12711yescharacteristic
250C4—22no—
251C4—41no—
252C4—42no—
253C4—1272no—
254C4—5081no—
255C4—5082no—
256C4—2542no—
257C4C2×D4×D12712yes—
258C2—81no—
259C2—1274no—
260C2—10161no—
261C2#127×D12711yescharacteristic
262C2#137×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#497×C127C2
socleC22×C127C22×D4
prime radicalC2×D4×D127C22
hypercenter#497D127
hypocenterC127C2×#497
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D127C2semidirect
3. C4⋊Q8×D127C2semidirect
5. #22550271C2semidirect
6. #22550270C2semidirect
7. #22550269C2semidirect
8. #22550268C2semidirect
9. #22550267C2semidirect
10. #97×D127C2semidirect
11. #497×C127C2semidirect
15. C2×Dic508C22semidirect
16. C2×#14523238C22semidirect
17. C2×#14523238C22semidirect
24. #16861209C22semidirect
26. #16861207C22semidirect
34. #16861120C22semidirect
35. #16861119C22semidirect
36. C2×Q16×C127C22semidirect
38. C4⋊Q8×C127C22semidirect
40. #97×C127C22semidirect
130. D127#497direct (N × H)
132. C127C2×#497semidirect
142. #497D127direct (N × H)
151. C2×Q16C2×D127semidirect
161. C4⋊Q8C2×D127semidirect
172. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1272
C2×SD16×D1272
C4⋊Q8×D1272
#124×D1272
#225502712
#225502702
#225502692
#225502682
#225502672
#97×D1272
#497×C1272
C2×#497127