Finite Group Databasebeta

← #30638814

Subgroups — #30638814

Order 16256.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C22×C4⋊Dic127C211yesmaximal, characteristic
2C2×C22⋊(C127⋊C8)C211yesmaximal, characteristic
3#22549638C214yesmaximal, complemented
4#380×C127C211yesmaximal, characteristic
5C23×Dic127—21no—
6C22×C127⋊C8C2212yes—
7C2×C4⋊Dic127—21no—
8C2×C4⋊Dic127—22no—
9C2×C4⋊Dic127C414yescomplemented
10C2×C4⋊Dic127C2212yes—
11C22⋊(C127⋊C8)—22no—
12C23×C4×C127C2211yescharacteristic
13C2×C4⋊C4×C127C2212yes—
14C2×C22⋊C4×C127—21no—
15#95×C127—21no—
16C22×Dic127—24no—
17C22×Dic127—22no—
18C22×Dic127—42no—
19C22×Dic127—42no—
20C2×C127⋊C8—22no—
21C2×C127⋊C8—24no—
22C4⋊Dic127—24no—
23C4⋊Dic127—22no—
24C4⋊Dic127C2×C414yes—
25C24×C127D411yescharacteristic
26C22×C4×C127—24no—
27C22×C4×C127—22no—
28C22×C4×C127C2×C412yes—
29C22×C4×C127D411yescharacteristic
30C22×C4×C127D412yes—
31C22×C4×C127C2311yescharacteristic
32C4⋊C4×C127—24no—
33C22⋊C4×C127—42no—
34C2×Dic127—24no—
35C2×Dic127—48no—
36C2×Dic127—42no—
37C2×Dic127—42no—
38C127⋊C8—42no—
39C23×C127—22no—
40C23×C127—22no—
41C23×C127C22⋊C412yes—
42C23×C127D812yes—
43C23×C127SD1612yes—
44C23×C127C2×D411yescharacteristic
45C2×C4×C127—22no—
46C2×C4×C127—24no—
47C2×C4×C127—24no—
48C2×C4×C127—42no—
49C2×C4×C127—42no—
50C2×C4×C127—42no—
51C2×C4×C127C22⋊C412yes—
52C2×C4×C127C22×C411yescharacteristic
53C2×C4×C127C2×D411yescharacteristic
54C2×C4×C127#4114yes—
55C22×C127—22no—
56C22×C127—22no—
57C22×C127—24no—
58C22×C127—24no—
59C22×C127#10214yes—
60C22×C127C2×C22⋊C411yescharacteristic
61C22×C127#12312yes—
62C22×C127C2×D811yescharacteristic
63C22×C127C2×SD1611yescharacteristic
64C22×C127#13712yes—
65C4×C127—22no—
66C4×C127—82no—
67C4×C127#11712yes—
68C4×C127#12312yes—
69Dic127—44no—
70C2×C127—24no—
71C2×C127#38111yescharacteristic
72C2×C127C2×#10211yescharacteristic
73C2×C127#41911yescharacteristic
74C2×C127#48012yes—
75C2×C127#48312yes—
76#1823—1271nomaximal, Sylow
77C127#182311yescharacteristic, complemented, Sylow
78C22×C4⋊C4—1271no—
79C2×C22⋊C8—1271no—
80#380D12711yescharacteristic
81#340—1274no—
82C23×C4—2541no—
83C23×C4C2×D12711yescharacteristic
84C22×C8—1272no—
85C2×C4⋊C4—1274no—
86C2×C4⋊C4—1272no—
87C2×C4⋊C4—2541no—
88C2×C4⋊C4—2542no—
89C2×C4⋊C4C2×D12712yes—
90C2×C22⋊C4—21no—
91C22⋊C8—2542no—
92#95—21no—
93C24C22⋊D12711yescharacteristic
94C22×C4—24no—
95C22×C4—22no—
96C22×C4—5082no—
97C22×C4—5082no—
98C22×C4—2544no—
99C22×C4—2542no—
100C22×C4C4×D12712yes—
101C22×C4C22⋊D12711yescharacteristic
102C22×C4D50812yes—
103C22×C4C22×D12711yescharacteristic
104C2×C8—2542no—
105C2×C8—2544no—
106C4⋊C4—24no—
107C4⋊C4—1274no—
108C4⋊C4—2544no—
109C4⋊C4—2542no—
110C22⋊C4—42no—
111C23—22no—
112C23—22no—
113C23(C2×D127)⋊C412yes—
114C23#1452323312yes—
115C23#1452323512yes—
116C23C2×C22⋊D12711yescharacteristic
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—24no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4—42no—
123C2×C4—5082no—
124C2×C4—5082no—
125C2×C4—5088no—
126C2×C4—2544no—
127C2×C4(C2×D127)⋊C412yes—
128C2×C4C2×C4×D12711yescharacteristic
129C2×C4C2×D50811yescharacteristic
130C2×C4#1452324714yes—
131C8—5082no—
132C22—22no—
133C22—24no—
134C22—24no—
135C22—22no—
136C22#1686112914yes—
137C22C2×(C2×D127)⋊C411yescharacteristic
138C22#1686119012yes—
139C22C2×#1452323311yescharacteristic
140C22C2×#1452323511yescharacteristic
141C22#1686123912yes—
142C4—22no—
143C4—82no—
144C4—5084no—
145C4#1686116512yes—
146C4#1686119312yes—
147C2—24no—
148C2#2254973711yescharacteristic
149C2C2×#1686112911yescharacteristic
150C2#2254988411yescharacteristic
151C2#2255018712yes—
152C2#2255020412yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#380×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
cosocleC22×C4×C127C23
hypercenter#380D127
hypocenterC127#1823
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #22549638C2semidirect
9. C2×C4⋊Dic127C4semidirect
77. C127#1823semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1272
C2×C22⋊(C127⋊C8)2
#225496382
#380×C1272
#1823127