Finite Group Databasebeta

← #30648968

Subgroups — #30648968

Order 16256.

418 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 121 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#22550350C211yesmaximal, characteristic, complemented
2#22550197C212yesmaximal, complemented
3#22550173C212yesmaximal, complemented
4#22550068C212yesmaximal, complemented
5#22550009C212yesmaximal, complemented
6#22549986C212yesmaximal, complemented
7C4×#14523233C212yesmaximal, complemented
8#22549937C211yesmaximal, characteristic, complemented
9#540×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×#14523233C2212yes—
12C2×#14523233C2212yes—
13C4×D508C2212yes—
14C4×D508C2212yes—
15#16861171—22no—
16#16861168C2211yescharacteristic
17C22⋊C4×D127—22no—
18#16861142C2212yescomplemented
19#16861140C2212yescomplemented
20C4⋊(C127⋊C8)C2212yescomplemented
21#16861130C2212yescomplemented
22#16861130C2212yescomplemented
23#16861129C2212yes—
24#16861129C2212yes—
25C22⋊(C127⋊C8)C2212yes—
26#16861120C2211yescharacteristic, complemented
27C4×C127⋊C8C2211yescharacteristic, complemented
28#127×C127C2211yescharacteristic, complemented
29#125×C127C2211yescharacteristic, complemented
30#123×C127—22no—
31#121×C127—22no—
32#121×C127C2212yes—
33C4×D4×C127C2212yescomplemented
34#117×C127C2211yescharacteristic
35C23×D127—21no—
36C2×D508—24no—
37C2×D508—21no—
38C2×D508C2312yes—
39C2×C4×D127—22no—
40C2×C4×D127—22no—
41#14523233—24no—
42#14523233—24no—
43C2×C127⋊C8C2312yes—
44C2×C127⋊C8C2312yes—
45C4⋊Dic127C2312yes—
46C22⋊Dic127—21no—
47(C2×D127)⋊C4—22no—
48(C2×D127)⋊C4—22no—
49C4×Dic127—21no—
50C2×D4×C127—22no—
51C2×D4×C127—22no—
52C2×D4×C127C2312yes—
53C22×C4×C127—22no—
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C4⋊C4×C127D412yes—
57C4⋊C4×C127C2312yes—
58C22⋊C4×C127—22no—
59C22⋊C4×C127—22no—
60C22⋊C4×C127D412yes—
61C42×C127C2311yescharacteristic
62C42×C127C2311yescharacteristic
63C22×D127—22no—
64C22×D127—24no—
65C22×D127—82no—
66D508—24no—
67D508—22no—
68D508#4114yes—
69C2×Dic127—22no—
70C4×D127—82no—
71C4×D127—82no—
72C127⋊C8—22no—
73C127⋊C8—22no—
74C23×C127—22no—
75C23×C127C2×D411yescharacteristic
76D4×C127—22no—
77D4×C127—22no—
78D4×C127—82no—
79D4×C127—82no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—82no—
83C2×C4×C127C2×D412yes—
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2411yescharacteristic
87C22×C127—21no—
88C22×C127—22no—
89C22×C127—42no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127C22×D411yescharacteristic
93C2×D127—24no—
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—84no—
97C4×C127—21no—
98C4×C127—22no—
99C4×C127—22no—
100C4×C127—42no—
101C4×C127C2×#4112yes—
102Dic127—42no—
103C2×C127—41no—
104C2×C127—82no—
105C2×C127#52511yescharacteristic
106C2×C127#57612yes—
107D127—44no—
108#3122—1271nomaximal, Sylow
109C127#312211yescharacteristic, complemented, Sylow
110#540D12711yescharacteristic, complemented
111#518—1271no—
112#482—1272no—
113#478—1272no—
114#459—1272no—
115#447—1272no—
116#442—1272no—
117C4×D8—1272no—
118#433—1271no—
119C22×D4—1271no—
120C2×D8—1272no—
121C2×D8—1272no—
122#127C2×D12711yescharacteristic, complemented
123#125C2×D12711yescharacteristic, complemented
124#123—22no—
125#121—22no—
126#121C2×D12712yes—
127C4×D4—1272no—
128C4×D4—1272no—
129C4×D4—2542no—
130C4×D4C2×D12712yescomplemented
131#117—1271no—
132#117C2×D12711yescharacteristic
133C2×C22⋊C4—2542no—
134#107—1272no—
135#106—1272no—
136C4⋊C8—1272no—
137#102—1272no—
138#102—1272no—
139#102—1272no—
140#102—1272no—
141C22⋊C8—1272no—
142#97—1271no—
143C4×C8—1271no—
144C24—2541no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—1272no—
148C2×D4—2544no—
149C2×D4—2541no—
150C2×D4C22×D12712yes—
151C22×C4—22no—
152C22×C4—2542no—
153C22×C4—2542no—
154C22×C4C22×D12711yescharacteristic
155D8—2544no—
156D8—2544no—
157C2×C8—1272no—
158C2×C8—1272no—
159C4⋊C4—22no—
160C4⋊C4—1272no—
161C4⋊C4C22⋊D12712yes—
162C4⋊C4C22×D12712yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—2541no—
166C22⋊C4—2542no—
167C22⋊C4—2542no—
168C22⋊C4C22⋊D12712yes—
169C42—2541no—
170C42C22×D12711yescharacteristic
171C42C22×D12711yescharacteristic
172C23—22no—
173C23—10162no—
174C23—2544no—
175C23—2542no—
176C23C2×C22⋊D12711yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—82no—
181D4—1274no—
182D4—2544no—
183D4—2542no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—10162no—
188C2×C4—10162no—
189C2×C4—2542no—
190C2×C4C2×C22⋊D12711yescharacteristic
191C2×C4C2×C22⋊D12712yes—
192C2×C4C2×C22⋊D12712yes—
193C2×C4C23×D12711yescharacteristic
194C8—2542no—
195C8—2542no—
196C22—21no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—10164no—
202C22—5082no—
203C22—5082no—
204C22—2544no—
205C22C22×C22⋊D12711yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—5082no—
211C4#41×D12712yes—
212C2—41no—
213C2—82no—
214C2—5084no—
215C2#2255038311yescharacteristic
216C2#2255067912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#540×C127C2
socleC22×C127C22×D4
hypercenter#540D127
hypocenterC127#3122
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #22550350C2semidirect
2. #22550197C2semidirect
3. #22550173C2semidirect
4. #22550068C2semidirect
5. #22550009C2semidirect
6. #22549986C2semidirect
7. C4×#14523233C2semidirect
8. #22549937C2semidirect
9. #540×C127C2semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. C4⋊(C127⋊C8)C22semidirect
21. #16861130C22semidirect
22. #16861130C22semidirect
26. #16861120C22semidirect
27. C4×C127⋊C8C22semidirect
28. #127×C127C22semidirect
29. #125×C127C22semidirect
33. C4×D4×C127C22semidirect
109. C127#3122semidirect
110. #540D127semidirect
122. #127C2×D127semidirect
123. #125C2×D127semidirect
130. C4×D4C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225503502
#225501972
#225501732
#225500682
#225500092
#225499862
C4×#145232332
#225499372
#540×C1272
#3122127