Finite Group Databasebeta

← #30649118

Subgroups — #30649118

Order 16256.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550429C211yesmaximal, characteristic, complemented
2C2×#16861198C211yesmaximal, characteristic
3#22550237C214yesmaximal, complemented
4#22550232C214yesmaximal, complemented
5#22549885C211yesmaximal, characteristic, complemented
6#22549882C211yesmaximal, characteristic
7#22549880C211yesmaximal, characteristic
8M16×Dic127C211yesmaximal, characteristic, complemented
9#419×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic
11C2×#14523232C2211yescharacteristic
12#16861208C2212yescomplemented
13#16861200C2212yescomplemented
14#16861198—22no—
15#16861198C2212yes—
16#16861198C2212yes—
17#16861186—22no—
18C4×Dic254—22no—
19#16861168C2211yescharacteristic
20C2×(C2×D127)⋊C4—22no—
21C2×C4×Dic127C2211yescharacteristic
22#16861133C2214yescomplemented
23#16861131C2214yes—
24#16861129C2214yes—
25#16861119C2212yescomplemented
26C8×Dic127C2212yescomplemented
27C22×Q8×C127C2211yescharacteristic
28C2×M16×C127C2211yescharacteristic
29#117×C127C2211yescharacteristic
30#103×C127C2214yescomplemented
31C23×D127—21no—
32C2×Dic254—22no—
33C2×D508—21no—
34C2×D508—22no—
35C2×D508—22no—
36C2×D508C2312yes—
37C22×Dic127D412yes—
38#14523232—22no—
39C2×C127⋊C8C2312yes—
40C4⋊Dic127—22no—
41C4⋊Dic127C2312yes—
42Dic127⋊C4—22no—
43Dic127⋊C4—22no—
44C22⋊Dic127—21no—
45(C2×D127)⋊C4—24no—
46(C2×D127)⋊C4—24no—
47C4×Dic127—21no—
48C4×Dic127D412yes—
49C4×Dic127C2311yescharacteristic
50C4×Dic127C2311yescharacteristic
51C2×Q8×C127—21no—
52C2×Q8×C127—22no—
53C2×Q8×C127—22no—
54C2×Q8×C127C2312yes—
55C22×C4×C127—21no—
56C22×C4×C127C2311yescharacteristic
57M16×C127—22no—
58C2×C8×C127C2312yes—
59C4⋊C4×C127C2312yes—
60C22⋊C4×C127—21no—
61C42×C127—21no—
62C22×D127—22no—
63C22×D127—44no—
64C22×D127—42no—
65Dic254—44no—
66D508—22no—
67D508—22no—
68D508—42no—
69C2×Dic127—22no—
70C2×Dic127—42no—
71C2×Dic127C2×D412yes—
72C2×Dic127C2×D412yes—
73C127⋊C8—22no—
74C23×C127C2×D411yescharacteristic
75Q8×C127—22no—
76Q8×C127—22no—
77Q8×C127—42no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—42no—
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127#4112yes—
83C2×C4×C127#4112yes—
84C2×C4×C127C2411yescharacteristic
85C8×C127—22no—
86C22×C127—22no—
87C22×C127#12412yes—
88C22×C127C22×D411yescharacteristic
89C2×D127—42no—
90C2×D127—44no—
91C2×D127—84no—
92C4×C127—42no—
93C4×C127—42no—
94C4×C127#12412yes—
95C4×C127C2×#4111yescharacteristic
96C4×C127C2×#4111yescharacteristic
97Dic127—22no—
98Dic127—22no—
99Dic127—42no—
100C2×C127—22no—
101C2×C127C2×#12411yescharacteristic
102C2×C127#57512yes—
103D127—82no—
104#3140—1271nomaximal, Sylow
105C127#314011yescharacteristic, complemented, Sylow
106#533—1271no—
107C2×#124—1271no—
108#489—1274no—
109#488—1274no—
110#419—1271no—
111#419D12711yescharacteristic, complemented
112#418—1271no—
113#418—1271no—
114C4×M16—1271no—
115C22×Q8C2×D12711yescharacteristic
116C22×D4—1271no—
117C2×M16—1271no—
118C2×M16C2×D12711yescharacteristic
119C4⋊Q8—1272no—
120#125—1272no—
121#124—1272no—
122#124—1272no—
123#124—2542no—
124C22⋊Q8—2542no—
125C4×Q8—2542no—
126#117—1271no—
127#117C2×D12711yescharacteristic
128C2×C22⋊C4—2542no—
129C2×C42—1271no—
130#103—1274no—
131#103C2×D12714yescomplemented
132#102—1274no—
133#102—1274no—
134#97—1272no—
135C4×C8—1272no—
136C24—2541no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—2542no—
141C2×Q8C22×D12712yes—
142C2×D4—1272no—
143C2×D4—2541no—
144C2×D4—2542no—
145C2×D4—2542no—
146C22×C4—21no—
147C22×C4—1272no—
148C22×C4C22×D12711yescharacteristic
149M16—22no—
150M16—2542no—
151C2×C8—1272no—
152C2×C8C22×D12712yes—
153C4⋊C4—1272no—
154C4⋊C4—2542no—
155C4⋊C4—2542no—
156C4⋊C4—2542no—
157C4⋊C4C22×D12712yes—
158C22⋊C4—21no—
159C22⋊C4—2544no—
160C22⋊C4—2544no—
161C22⋊C4—2541no—
162C42—21no—
163C42—1272no—
164C42—1271no—
165C42—1271no—
166C42—2541no—
167C23—5084no—
168C23—5082no—
169C23—2542no—
170C23D4×D12711yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—42no—
174Q8—5084no—
175D4—5082no—
176D4—2542no—
177D4—2542no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—1272no—
182C2×C4—1272no—
183C2×C4—5082no—
184C2×C4—2542no—
185C2×C4D4×D12711yescharacteristic
186C2×C4#1452324712yes—
187C2×C4#1452324812yes—
188C2×C4C23×D12711yescharacteristic
189C8—22no—
190C8—2542no—
191C22—22no—
192C22—10164no—
193C22—5082no—
194C22—5084no—
195C22#1686119512yes—
196C22C2×D4×D12711yescharacteristic
197C4—42no—
198C4—42no—
199C4—5082no—
200C4—2542no—
201C4—2542no—
202C4#1686119512yes—
203C4C2×#1452324711yescharacteristic
204C4C2×#1452324811yescharacteristic
205C2—22no—
206C2—10162no—
207C2C2×#1686119511yescharacteristic
208C2#2255067512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#419D127
hypocenterC127#3140
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #22550429C2semidirect
3. #22550237C2semidirect
4. #22550232C2semidirect
5. #22549885C2semidirect
8. M16×Dic127C2semidirect
9. #419×C127C2semidirect
12. #16861208C22semidirect
13. #16861200C22semidirect
22. #16861133C22semidirect
25. #16861119C22semidirect
26. C8×Dic127C22semidirect
30. #103×C127C22semidirect
105. C127#3140semidirect
111. #419D127semidirect
131. #103C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504292
C2×#168611982
#225502372
#225502322
#225498852
#225498822
#225498802
M16×Dic1272
#419×C1272
#3140127