Finite Group Databasebeta

← #30637715

Subgroups — #30637715

Order 16256.

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

normal subgroups only

This group has 280 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#22550189C212yesmaximal, complemented
3#22549609C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic, complemented
5#397×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127—21no—
7#16861193—22no—
8#16861183—21no—
9#16861183C2212yes—
10#16861179—22no—
11#16861142—22no—
12#16861129—22no—
13C22⋊(C127⋊C8)C2212yescomplemented
14C2×C4⋊C4×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—
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—
38C22⋊C4×C127—22no—
39C22⋊C4×C127—42no—
40C22×D127—22no—
41C22×D127—42no—
42C22×D127—81no—
43C22×D127—81no—
44D508—42no—
45D508—81no—
46C22⋊D127—44no—
47C22⋊D127—42no—
48C22⋊D127—42no—
49C2×Dic127—22no—
50C2×Dic127—42no—
51C127⋊C8—42no—
52C23×C127—21no—
53C23×C127—21no—
54C23×C127—41no—
55C23×C127—41no—
56C23×C127D812yes—
57C23×C127C2×D411yescharacteristic
58C2×C4×C127—21no—
59C2×C4×C127—41no—
60C2×C4×C127—41no—
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×C127C2×D412yes—
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—22no—
71C22×C127—42no—
72C22×C127—41no—
73C22×C127—41no—
74C22×C127—81no—
75C22×C127#12011yescharacteristic
76C22×C127C2×D811yescharacteristic
77C22×C127AGL(1,ℤ/8)11yescharacteristic
78C2×D127—42no—
79C2×D127—42no—
80C2×D127—42no—
81C2×D127—84no—
82C4×C127—22no—
83C4×C127—41no—
84C4×C127—81no—
85C4×C127—81no—
86Dic127—42no—
87C2×C127—21no—
88C2×C127—21no—
89C2×C127—42no—
90C2×C127#44711yescharacteristic
91C2×C127#45511yescharacteristic
92C2×C127#45811yescharacteristic
93D127—82no—
94#1607—1271nomaximal, Sylow
95C127#160711yescharacteristic, complemented, Sylow
96#534—1271no—
97#480—1272no—
98#397D12711yescharacteristic, complemented
99#327—1272no—
100#323—1271no—
101C22×D4—2541no—
102#123—2542no—
103#121—1272no—
104#121—2541no—
105#120—2542no—
106C2×C4⋊C4C2×D12711yescharacteristic, complemented
107C2×C22⋊C4—21no—
108C2×C22⋊C4C2×D12711yescharacteristic
109#107—2542no—
110#102—2542no—
111C22⋊C8—1272no—
112#95—21no—
113#95C2×D12711yescharacteristic, complemented
114C24—2541no—
115C24C22⋊D12711yescharacteristic
116C2×D4—1272no—
117C2×D4—5082no—
118C2×D4—5082no—
119C2×D4—2541no—
120C2×D4—2542no—
121C2×D4—2542no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—2541no—
125C22×C4C22⋊D12711yescharacteristic
126C22×C4C22×D12711yescharacteristic
127C2×C8—2542no—
128C4⋊C4—22no—
129C4⋊C4—1272no—
130C4⋊C4—2541no—
131C22⋊C4—22no—
132C22⋊C4—42no—
133C22⋊C4—2542no—
134C22⋊C4—2542no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—10161no—
140C23—10161no—
141C23—5082no—
142C23—2542no—
143C23#1452323312yes—
144C23C2×C22⋊D12711yescharacteristic
145D4—10161no—
146D4—5082no—
147D4—5084no—
148D4—5082no—
149D4—5082no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—5082no—
160C2×C4—2542no—
161C2×C4D4×D12712yes—
162C8—5082no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—10164no—
171C22—5082no—
172C22—5082no—
173C22—5082no—
174C22#1686117711yescharacteristic
175C22C2×#1452323311yescharacteristic
176C22#1686123411yescharacteristic
177C4—22no—
178C4—41no—
179C4—81no—
180C4—81no—
181C4—5082no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—10162no—
186C2#2255000911yescharacteristic
187C2#2255005311yescharacteristic
188C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#397×C127C2
socleC22×C127#120
prime radical#22550438C2
hypercenter#397D127
hypocenterC127#1607
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #22550189C2semidirect
3. #22549609C2semidirect
4. #22549600C2semidirect
5. #397×C127C2semidirect
13. C22⋊(C127⋊C8)C22semidirect
14. C2×C4⋊C4×C127C22semidirect
18. #95×C127C22semidirect
95. C127#1607semidirect
98. #397D127semidirect
106. C2×C4⋊C4C2×D127semidirect
113. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504382
#225501892
#225496092
#225496002
#397×C1272
#1607127