Finite Group Databasebeta

← #30637650

Subgroups — #30637650

Order 16256.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550438C211yesmaximal, characteristic, complemented
2#22550012C212yesmaximal, complemented
3#22549785C211yesmaximal, characteristic, complemented
4#22549611C212yesmaximal, complemented
5#323×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic, complemented
7C22×C22⋊D127—21no—
8C22×C22⋊D127—21no—
9C2×D1016—22no—
10#16861193—22no—
11#16861183—21no—
12#16861183C2212yes—
13#16861179—22no—
14C2×(C2×D127)⋊C4—21no—
15#16861131—22no—
16C127⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yescomplemented
19C23×D127—21no—
20C23×D127—21no—
21C2×D508—21no—
22C2×D508—22no—
23C2×D508—42no—
24C2×D508D412yescomplemented
25C2×C22⋊D127—22no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127—42no—
28C2×C22⋊D127—42no—
29C2×C22⋊D127—42no—
30C2×C22⋊D127—42no—
31C2×C22⋊D127—42no—
32C22×Dic127—21no—
33C22×Dic127—21no—
34D1016—44no—
35C4⋊Dic127—21no—
36C4⋊Dic127D412yescomplemented
37(C2×D127)⋊C4—22no—
38(C2×D127)⋊C4—22no—
39(C2×D127)⋊C4—42no—
40C24×C127D411yescharacteristic
41C22×C4×C127D411yescharacteristic
42C22×C4×C127C2311yescharacteristic
43C2×C8×C127—22no—
44C22⋊C4×C127—22no—
45C22×D127—22no—
46C22×D127—22no—
47C22×D127—41no—
48C22×D127—41no—
49C22×D127—42no—
50C22×D127—41no—
51C22×D127—41no—
52C22×D127—41no—
53C22×D127—41no—
54C22×D127—81no—
55C22×D127—81no—
56D508—24no—
57D508—42no—
58D508—42no—
59D508—81no—
60C22⋊D127—44no—
61C22⋊D127—48no—
62C22⋊D127—42no—
63C22⋊D127—42no—
64C2×Dic127—22no—
65C2×Dic127—22no—
66C2×Dic127—41no—
67C2×Dic127—42no—
68C2×Dic127—41no—
69C23×C127—21no—
70C23×C127—21no—
71C23×C127—41no—
72C23×C127—41no—
73C23×C127D812yescomplemented
74C23×C127C2×D411yescharacteristic
75C2×C4×C127—21no—
76C2×C4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127C2×D412yes—
79C8×C127—42no—
80C22×C127—21no—
81C22×C127—21no—
82C22×C127—22no—
83C22×C127—42no—
84C22×C127—41no—
85C22×C127—41no—
86C22×C127—81no—
87C22×C127#12011yescharacteristic
88C22×C127C2×D811yescharacteristic
89C22×C127AGL(1,ℤ/8)11yescharacteristic
90C2×D127—42no—
91C2×D127—42no—
92C2×D127—42no—
93C2×D127—42no—
94C2×D127—42no—
95C2×D127—42no—
96C2×D127—81no—
97C2×D127—81no—
98C2×D127—84no—
99C2×D127—81no—
100C2×D127—81no—
101C4×C127—22no—
102C4×C127—41no—
103Dic127—42no—
104Dic127—42no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—42no—
108C2×C127#44711yescharacteristic
109C2×C127#45311yescharacteristic
110C2×C127B(4,𝔽2)11yescharacteristic
111D127—82no—
112D127—82no—
113#1595—1271nomaximal, Sylow
114C127#159511yescharacteristic, complemented, Sylow
115#534—1271no—
116#447—1272no—
117#392—1271no—
118#327—1272no—
119#323D12711yescharacteristic, complemented
120C22×D4—1271no—
121C22×D4—2541no—
122C22×D4—2541no—
123C2×D8—2542no—
124#123—2542no—
125#121—1272no—
126#121—2541no—
127#120—2542no—
128C2×C22⋊C4—2541no—
129C2×C22⋊C4C2×D12711yescharacteristic
130#102—2542no—
131C22⋊C8C2×D12712yescomplemented
132#95—1271no—
133C24—2541no—
134C24—2541no—
135C24D50811yescharacteristic
136C2×D4—1272no—
137C2×D4—5082no—
138C2×D4—5082no—
139C2×D4—5082no—
140C2×D4—5082no—
141C2×D4—5082no—
142C2×D4—5082no—
143C2×D4—2541no—
144C2×D4—2542no—
145C2×D4—2542no—
146C2×D4—2542no—
147C22×C4—2541no—
148C22×C4—2541no—
149C22×C4D50811yescharacteristic
150C22×C4C22×D12711yescharacteristic
151D8—5084no—
152C2×C8—22no—
153C4⋊C4—1272no—
154C4⋊C4—2541no—
155C22⋊C4—22no—
156C22⋊C4—5082no—
157C22⋊C4—2542no—
158C22⋊C4—2542no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—10161no—
164C23—10161no—
165C23—5081no—
166C23—5081no—
167C23—5081no—
168C23—5082no—
169C23—5081no—
170C23—5081no—
171C23—5081no—
172C23—2542no—
173C23—2542no—
174C23D101612yescomplemented
175C23C2×D50811yescharacteristic
176D4—10161no—
177D4—5082no—
178D4—5082no—
179D4—5084no—
180D4—5088no—
181D4—5082no—
182D4—5082no—
183D4—2544no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—5081no—
188C2×C4—5082no—
189C2×C4—5081no—
190C2×C4—2542no—
191C2×C4—2542no—
192C2×C4D4×D12712yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—10161no—
202C22—10161no—
203C22—10164no—
204C22—10161no—
205C22—10161no—
206C22—5082no—
207C22—5082no—
208C22—5082no—
209C22—5082no—
210C22—5082no—
211C22—5082no—
212C22#1686117911yescharacteristic
213C22C2×D101611yescharacteristic
214C22#1686123711yescharacteristic
215C4—22no—
216C4—41no—
217C4—5082no—
218C4—5082no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—10162no—
223C2—10162no—
224C2#2255001211yescharacteristic
225C2#2255004611yescharacteristic
226C2#2255006411yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#323×C127C2
socleC22×C127#120
hypercenter#323D127
hypocenterC127#1595
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #22550438C2semidirect
2. #22550012C2semidirect
3. #22549785C2semidirect
4. #22549611C2semidirect
5. #323×C127C2semidirect
6. C22×D508C22semidirect
16. C127⋊#95C22semidirect
18. C22⋊C8×C127C22semidirect
24. C2×D508D4semidirect
36. C4⋊Dic127D4semidirect
73. C23×C127D8semidirect
114. C127#1595semidirect
119. #323D127semidirect
131. C22⋊C8C2×D127semidirect
174. C23D1016semidirect

Maximal subgroups

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

subgroupindex
#225504382
#225500122
#225497852
#225496112
#323×C1272
#1595127