Finite Group Databasebeta

← #30637681

Subgroups — #30637681

Order 16256.

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

normal subgroups only

This group has 308 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
2#22550015C212yesmaximal, complemented
3#22549609C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic, complemented
5#393×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127—21no—
7C2×#14523235—22no—
8#16861193—22no—
9#16861183—21no—
10#16861183C2212yes—
11#16861179—22no—
12#16861133—22no—
13C22⋊(C127⋊C8)C2212yescomplemented
14C22×Q8×C127C2211yescharacteristic, complemented
15C2×C22⋊C4×C127—21no—
16C2×C22⋊C4×C127C2211yescharacteristic
17#95×C127—21no—
18#95×C127C2211yescharacteristic, complemented
19C23×D127—21no—
20C2×D508—21no—
21C2×D508D412yes—
22C2×C22⋊D127—22no—
23C2×C22⋊D127—22no—
24C2×C22⋊D127—42no—
25C2×C22⋊D127—42no—
26C22×Dic127—21no—
27#14523235—44no—
28C2×C127⋊C8—22no—
29C4⋊Dic127—21no—
30C4⋊Dic127D412yes—
31(C2×D127)⋊C4—22no—
32(C2×D127)⋊C4—22no—
33C24×C127D411yescharacteristic
34C2×Q8×C127—22no—
35C2×Q8×C127—42no—
36C22×C4×C127—21no—
37C22×C4×C127—21no—
38C22×C4×C127D411yescharacteristic
39C22×C4×C127C2311yescharacteristic
40C22⋊C4×C127—22no—
41C22⋊C4×C127—42no—
42C22×D127—22no—
43C22×D127—42no—
44C22×D127—81no—
45C22×D127—81no—
46D508—42no—
47D508—81no—
48C22⋊D127—44no—
49C22⋊D127—42no—
50C22⋊D127—42no—
51C2×Dic127—22no—
52C2×Dic127—42no—
53C127⋊C8—42no—
54C23×C127—21no—
55C23×C127—21no—
56C23×C127—41no—
57C23×C127—41no—
58C23×C127SD1612yescomplemented
59C23×C127C2×D411yescharacteristic
60Q8×C127—24no—
61Q8×C127—42no—
62C2×C4×C127—22no—
63C2×C4×C127—21no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D412yes—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—22no—
75C22×C127—42no—
76C22×C127—41no—
77C22×C127—41no—
78C22×C127—81no—
79C22×C127#12011yescharacteristic
80C22×C127C2×SD1611yescharacteristic
81C22×C127AGL(1,ℤ/8)11yescharacteristic
82C2×D127—42no—
83C2×D127—42no—
84C2×D127—42no—
85C2×D127—84no—
86C4×C127—22no—
87C4×C127—41no—
88C4×C127—42no—
89C4×C127—81no—
90Dic127—42no—
91C2×C127—21no—
92C2×C127—21no—
93C2×C127—42no—
94C2×C127#45011yescharacteristic
95C2×C127#45511yescharacteristic
96C2×C127B(4,𝔽2)11yescharacteristic
97D127—82no—
98#1601—1271nomaximal, Sylow
99C127#160111yescharacteristic, complemented, Sylow
100#534—1271no—
101#448—1272no—
102#393D12711yescharacteristic, complemented
103#327—1272no—
104#323—1271no—
105C22×Q8C2×D12711yescharacteristic, complemented
106C22×D4—2541no—
107C2×SD16—2542no—
108#123—2542no—
109#121—1272no—
110#121—2541no—
111#120—2542no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C2×D12711yescharacteristic
114#103—2542no—
115C22⋊C8—1272no—
116#95—21no—
117#95C2×D12711yescharacteristic, complemented
118C24—2541no—
119C24C22⋊D12711yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—1272no—
123C2×D4—5082no—
124C2×D4—5082no—
125C2×D4—2541no—
126C2×D4—2542no—
127C2×D4—2542no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—2541no—
131C22×C4C22⋊D12711yescharacteristic
132C22×C4C22×D12711yescharacteristic
133SD16—5084no—
134C2×C8—2542no—
135C4⋊C4—1272no—
136C4⋊C4—2541no—
137C22⋊C4—22no—
138C22⋊C4—42no—
139C22⋊C4—2542no—
140C22⋊C4—2542no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—10161no—
146C23—10161no—
147C23—5082no—
148C23—2542no—
149C23#1452323512yescomplemented
150C23C2×C22⋊D12711yescharacteristic
151Q8—24no—
152Q8—42no—
153D4—10161no—
154D4—5082no—
155D4—5084no—
156D4—5082no—
157D4—5082no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—5082no—
168C2×C4—2542no—
169C2×C4D4×D12712yes—
170C8—5082no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—10164no—
179C22—5082no—
180C22—5082no—
181C22—5082no—
182C22#1686117711yescharacteristic
183C22C2×#1452323511yescharacteristic
184C22#1686123411yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—81no—
189C4—5082no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—10162no—
194C2#2255002611yescharacteristic
195C2#2255005311yescharacteristic
196C2#2255006211yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#393×C127C2
socleC22×C127#120
prime radical#22550438C2
hypercenter#393D127
hypocenterC127#1601
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #22550015C2semidirect
3. #22549609C2semidirect
4. #22549600C2semidirect
5. #393×C127C2semidirect
13. C22⋊(C127⋊C8)C22semidirect
14. C22×Q8×C127C22semidirect
18. #95×C127C22semidirect
58. C23×C127SD16semidirect
99. C127#1601semidirect
102. #393D127semidirect
105. C22×Q8C2×D127semidirect
117. #95C2×D127semidirect
149. C23#14523235semidirect

Maximal subgroups

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

subgroupindex
#225504382
#225500152
#225496092
#225496002
#393×C1272
#1601127