Finite Group Databasebeta

← #30638757

Subgroups — #30638757

Order 16256.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C23×C127⋊C8C211yesmaximal, characteristic
2C2×#16861142C212yesmaximal, complemented
3#22549736C211yesmaximal, characteristic
4#22549638C212yesmaximal, complemented
5#380×C127C211yesmaximal, characteristic
6C22×C127⋊C8—24no—
7C22×C127⋊C8C414yescomplemented
8C22×C127⋊C8C2211yescharacteristic
9C22×C127⋊C8C2211yescharacteristic
10C2×C4⋊Dic127C2212yes—
11C2×C22⋊Dic127—21no—
12#16861142—24no—
13C127⋊#95—21no—
14C23×C4×C127C2211yescharacteristic
15C2×C4⋊C4×C127C2212yes—
16C2×C22⋊C4×C127—21no—
17#95×C127—21no—
18C22×Dic127—22no—
19C2×C127⋊C8—28no—
20C2×C127⋊C8—22no—
21C2×C127⋊C8C2×C412yes—
22C2×C127⋊C8C2×C412yes—
23C2×C127⋊C8D414yes—
24C4⋊Dic127—22no—
25C4⋊Dic127—22no—
26C22⋊Dic127—42no—
27C24×C127D411yescharacteristic
28C22×C4×C127—22no—
29C22×C4×C127—22no—
30C22×C4×C127—22no—
31C22×C4×C127C2×C412yes—
32C22×C4×C127D411yescharacteristic
33C22×C4×C127Q812yes—
34C22×C4×C127C2311yescharacteristic
35C4⋊C4×C127—22no—
36C4⋊C4×C127—22no—
37C22⋊C4×C127—42no—
38C2×Dic127—42no—
39C2×Dic127—42no—
40C2×Dic127—42no—
41C127⋊C8—22no—
42C127⋊C8C22⋊C414yescomplemented
43C23×C127—22no—
44C23×C127—22no—
45C23×C127C4⋊C412yes—
46C23×C127D812yes—
47C23×C127Q1612yes—
48C23×C127C2×D411yescharacteristic
49C2×C4×C127—22no—
50C2×C4×C127—22no—
51C2×C4×C127—22no—
52C2×C4×C127—24no—
53C2×C4×C127—42no—
54C2×C4×C127—42no—
55C2×C4×C127—42no—
56C2×C4×C127C4⋊C412yes—
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D412yes—
59C2×C4×C127C2×Q811yescharacteristic
60C2×C4×C127#4112yes—
61C22×C127—22no—
62C22×C127—22no—
63C22×C127—24no—
64C22×C127—24no—
65C22×C127#10714yescomplemented
66C22×C127C2×C4⋊C411yescharacteristic
67C22×C127#12112yes—
68C22×C127C2×D811yescharacteristic
69C22×C127C2×Q1611yescharacteristic
70C22×C127#13512yes—
71C4×C127—22no—
72C4×C127—82no—
73C4×C127C2×C22⋊C411yescharacteristic
74C4×C127#11711yescharacteristic
75C4×C127C22⋊Q812yes—
76Dic127—82no—
77C2×C127—24no—
78C2×C127#38011yescharacteristic
79C2×C127C2×#10711yescharacteristic
80C2×C127#42711yescharacteristic
81C2×C127#46612yes—
82C2×C127#46712yes—
83#1808—1271nomaximal, Sylow
84C127#180811yescharacteristic, complemented, Sylow
85C23×C8—1271no—
86C2×#107—1272no—
87#380—1271no—
88#380D12711yescharacteristic
89#340—1272no—
90C23×C4C2×D12711yescharacteristic
91C22×C8—1271no—
92C22×C8—1274no—
93C22×C8—1271no—
94C22×C8—2544no—
95C2×C4⋊C4—1272no—
96C2×C4⋊C4C2×D12712yes—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—2541no—
99#107—2544no—
100#95—21no—
101#95—2541no—
102C24C22⋊D12711yescharacteristic
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—2542no—
107C22×C4C4×D12712yes—
108C22×C4C22⋊D12711yescharacteristic
109C22×C4Dic25412yes—
110C22×C4C22×D12711yescharacteristic
111C2×C8—1272no—
112C2×C8—1274no—
113C2×C8—1272no—
114C2×C8—2548no—
115C2×C8—2542no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—2542no—
119C4⋊C4—2542no—
120C22⋊C4—42no—
121C22⋊C4—5082no—
122C23—22no—
123C23—22no—
124C23Dic127⋊C412yes—
125C23#1452323312yes—
126C23#1452323812yes—
127C23C2×C22⋊D12711yescharacteristic
128C2×C4—22no—
129C2×C4—24no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—5082no—
136C2×C4—5082no—
137C2×C4—5082no—
138C2×C4Dic127⋊C412yes—
139C2×C4C2×C4×D12711yescharacteristic
140C2×C4D4×D12712yes—
141C2×C4C2×Dic25411yescharacteristic
142C2×C4#1452324712yes—
143C8—1274no—
144C8—2542no—
145C22—24no—
146C22—24no—
147C22—22no—
148C22—22no—
149C22#1686114214yescomplemented
150C22C2×Dic127⋊C411yescharacteristic
151C22#1686118112yes—
152C22C2×#1452323311yescharacteristic
153C22C2×#1452323811yescharacteristic
154C22#1686122912yes—
155C4—22no—
156C4—82no—
157C4—10162no—
158C4C22⋊C4×D12711yescharacteristic
159C4#1686116511yescharacteristic
160C4#1686118612yes—
161C2—24no—
162C2#2254973411yescharacteristic
163C2C2×#1686114211yescharacteristic
164C2#2254991211yescharacteristic
165C2#2255011012yes—
166C2#2255011512yes—
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#1808
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
2. C2×#16861142C2semidirect
4. #22549638C2semidirect
7. C22×C127⋊C8C4semidirect
42. C127⋊C8C22⋊C4semidirect
65. C22×C127#107semidirect
84. C127#1808semidirect
149. C22#16861142semidirect

Maximal subgroups

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

subgroupindex
C23×C127⋊C82
C2×#168611422
#225497362
#225496382
#380×C1272
#1808127