Finite Group Databasebeta

← #30639133

Subgroups — #30639133

Order 16256.

344 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#16861198C211yesmaximal, characteristic
2#22549880C212yesmaximal
3C2×#16861128C211yesmaximal, characteristic, complemented
4#22549635C212yesmaximal, complemented
5#519×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C2×#14523232C2212yes—
8#16861198—22no—
9#16861198C414yescomplemented
10C2×(C2×D127)⋊C4—22no—
11C2×C4×Dic127C2211yescharacteristic
12#16861129—24no—
13#16861128—22no—
14C22×Q8×C127C2211yescharacteristic
15C4×Q8×C127—24no—
16#117×C127—22no—
17#117×C127C2212yes—
18C23×D127—21no—
19C2×D508—22no—
20C2×D508—21no—
21C2×D508—22no—
22C2×D508C2×C412yes—
23C22×Dic127—21no—
24#14523232—22no—
25#14523232—22no—
26C2×C127⋊C8—22no—
27(C2×D127)⋊C4—28no—
28C4×Dic127—21no—
29C4×Dic127C2×C412yes—
30C2×Q8×C127—22no—
31C2×Q8×C127—22no—
32C2×Q8×C127C2×C412yes—
33C2×Q8×C127D412yes—
34C22×C4×C127D412yes—
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—24no—
37C4⋊C4×C127D414yes—
38C22⋊C4×C127—22no—
39C42×C127—24no—
40C42×C127—22no—
41C22×D127—22no—
42C22×D127—44no—
43C22×D127—41no—
44C22×D127—41no—
45D508—24no—
46D508—42no—
47C2×Dic127—22no—
48C2×Dic127—42no—
49C127⋊C8—42no—
50C23×C127C2×D411yescharacteristic
51Q8×C127—24no—
52Q8×C127—24no—
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—24no—
56C2×C4×C127C22⋊C414yes—
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D412yes—
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127#4112yes—
61C22×C127—22no—
62C22×C127C2×C22⋊C411yescharacteristic
63C22×C127#12011yescharacteristic
64C22×C127#12311yescharacteristic
65C2×D127—44no—
66C2×D127—42no—
67C2×D127—82no—
68C2×D127—82no—
69C4×C127—24no—
70C4×C127—44no—
71C4×C127C4×D412yes—
72C4×C127#12112yes—
73Dic127—42no—
74C2×C127—21no—
75C2×C127—21no—
76C2×C127#38611yescharacteristic
77C2×C127#45412yes—
78D127—82no—
79#1888—1271nomaximal, Sylow
80C127#188811yescharacteristic, complemented, Sylow
81C2×#124—1271no—
82#519D12711yescharacteristic, complemented
83#418—1272no—
84C2×#101—1271no—
85#339—1272no—
86C22×Q8C2×D12711yescharacteristic
87C22×D4—1271no—
88C2×M16—1272no—
89#124—1274no—
90#124—2542no—
91C4×Q8—24no—
92#117—22no—
93#117C2×D12712yes—
94C2×C22⋊C4—2542no—
95C2×C42—1271no—
96#102—2544no—
97#101—2542no—
98C24—2541no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8C4×D12712yes—
102C2×Q8D50812yes—
103C2×D4—1272no—
104C2×D4—2542no—
105C2×D4—2541no—
106C2×D4—2542no—
107C22×C4—2541no—
108C22×C4C22⋊D12712yes—
109C22×C4C22×D12711yescharacteristic
110M16—2542no—
111M16—2542no—
112C2×C8—2542no—
113C4⋊C4—24no—
114C4⋊C4C22⋊D12714yes—
115C22⋊C4—22no—
116C22⋊C4—2548no—
117C42—24no—
118C42—22no—
119C42—1272no—
120C42—2541no—
121C23—5084no—
122C23—5081no—
123C23—5081no—
124C23—2542no—
125C23C2×C22⋊D12711yescharacteristic
126Q8—24no—
127Q8—24no—
128D4—5082no—
129D4—2544no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—24no—
133C2×C4—5082no—
134C2×C4—2542no—
135C2×C4(C2×D127)⋊C414yes—
136C2×C4C2×C4×D12711yescharacteristic
137C2×C4C2×C22⋊D12712yes—
138C2×C4C2×D50811yescharacteristic
139C2×C4#1452324812yes—
140C8—5082no—
141C22—22no—
142C22—10162no—
143C22—10162no—
144C22—5084no—
145C22—5082no—
146C22C2×(C2×D127)⋊C411yescharacteristic
147C22#1686117811yescharacteristic
148C22#1686119211yescharacteristic
149C4—24no—
150C4—44no—
151C4—5082no—
152C4C4×C22⋊D12712yes—
153C4#1686118312yes—
154C2—21no—
155C2—21no—
156C2—10162no—
157C2#2254975511yescharacteristic
158C2#2255004912yes—
subgroupG/H
centerC22#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#519×C127C2
socleC22×C127#120
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#519D127
hypocenterC127#1888
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
3. C2×#16861128C2semidirect
4. #22549635C2semidirect
5. #519×C127C2semidirect
9. #16861198C4semidirect
80. C127#1888semidirect
82. #519D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611982
#225498802
C2×#168611282
#225496352
#519×C1272
#1888127