Finite Group Databasebeta

← #30637749

Subgroups — #30637749

Order 16256.

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

normal subgroups only

This group has 288 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#22550194C212yesmaximal, complemented
3#22549609C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic, complemented
5#399×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127—21no—
7#16861193—22no—
8#16861183—21no—
9#16861183C2212yes—
10#16861179—22no—
11#16861140—22no—
12#16861129—22no—
13C22⋊(C127⋊C8)C2212yescomplemented
14C2×C4⋊C4×C127—21no—
15C2×C4⋊C4×C127C2211yescharacteristic, complemented
16C2×C22⋊C4×C127—21no—
17C2×C22⋊C4×C127C2211yescharacteristic
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—
27C2×C127⋊C8—22no—
28C4⋊Dic127—21no—
29C4⋊Dic127D412yes—
30(C2×D127)⋊C4—22no—
31(C2×D127)⋊C4—22no—
32C24×C127D411yescharacteristic
33C22×C4×C127—21no—
34C22×C4×C127—21no—
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C4⋊C4×C127—22no—
38C4⋊C4×C127—42no—
39C22⋊C4×C127—22no—
40C22⋊C4×C127—42no—
41C22×D127—22no—
42C22×D127—42no—
43C22×D127—81no—
44C22×D127—81no—
45D508—42no—
46D508—81no—
47C22⋊D127—44no—
48C22⋊D127—42no—
49C22⋊D127—42no—
50C2×Dic127—22no—
51C2×Dic127—42no—
52C127⋊C8—42no—
53C23×C127—21no—
54C23×C127—21no—
55C23×C127—41no—
56C23×C127—41no—
57C23×C127SD1612yes—
58C23×C127C2×D411yescharacteristic
59C2×C4×C127—22no—
60C2×C4×C127—21no—
61C2×C4×C127—41no—
62C2×C4×C127—41no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127C2×D412yes—
69C22×C127—21no—
70C22×C127—21no—
71C22×C127—22no—
72C22×C127—42no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127—81no—
76C22×C127#12011yescharacteristic
77C22×C127C2×SD1611yescharacteristic
78C22×C127AGL(1,ℤ/8)11yescharacteristic
79C2×D127—42no—
80C2×D127—42no—
81C2×D127—42no—
82C2×D127—84no—
83C4×C127—22no—
84C4×C127—42no—
85C4×C127—41no—
86C4×C127—81no—
87Dic127—42no—
88C2×C127—21no—
89C2×C127—21no—
90C2×C127—42no—
91C2×C127#45011yescharacteristic
92C2×C127#45311yescharacteristic
93C2×C127#45811yescharacteristic
94D127—82no—
95#1613—1271nomaximal, Sylow
96C127#161311yescharacteristic, complemented, Sylow
97#534—1271no—
98#481—1272no—
99#399D12711yescharacteristic, complemented
100#327—1272no—
101#323—1271no—
102C22×D4—2541no—
103#123—2542no—
104#121—1272no—
105#121—2541no—
106#120—2542no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2×D12711yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C2×D12711yescharacteristic
111#106—2542no—
112#102—2542no—
113C22⋊C8—1272no—
114#95C2×D12711yescharacteristic, complemented
115C24—2541no—
116C24C22⋊D12711yescharacteristic
117C2×D4—1272no—
118C2×D4—5082no—
119C2×D4—5082no—
120C2×D4—2541no—
121C2×D4—2542no—
122C2×D4—2542no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—2541no—
126C22×C4C22⋊D12711yescharacteristic
127C22×C4C22×D12711yescharacteristic
128C2×C8—2542no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—1272no—
132C4⋊C4—2541no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C22⋊C4—2542no—
136C22⋊C4—2542no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—10161no—
142C23—10161no—
143C23—5082no—
144C23—2542no—
145C23#1452323512yes—
146C23C2×C22⋊D12711yescharacteristic
147D4—10161no—
148D4—5082no—
149D4—5084no—
150D4—5082no—
151D4—5082no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—5082no—
162C2×C4—2542no—
163C2×C4D4×D12712yes—
164C8—5082no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—10164no—
173C22—5082no—
174C22—5082no—
175C22—5082no—
176C22#1686117711yescharacteristic
177C22C2×#1452323511yescharacteristic
178C22#1686123411yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—5082no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—10162no—
188C2#2255002611yescharacteristic
189C2#2255004311yescharacteristic
190C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#399×C127C2
socleC22×C127#120
prime radical#22550438C2
hypercenter#399D127
hypocenterC127#1613
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #22550194C2semidirect
3. #22549609C2semidirect
4. #22549600C2semidirect
5. #399×C127C2semidirect
13. C22⋊(C127⋊C8)C22semidirect
15. C2×C4⋊C4×C127C22semidirect
18. #95×C127C22semidirect
96. C127#1613semidirect
99. #399D127semidirect
108. C2×C4⋊C4C2×D127semidirect
114. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504382
#225501942
#225496092
#225496002
#399×C1272
#1613127