Finite Group Databasebeta

← #30636859

Subgroups — #30636859

Order 16256.

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

normal subgroups only

This group has 162 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#22549860C211yesmaximal, characteristic
2#22549850C211yesmaximal, characteristic
3#408×C127C211yesmaximal, characteristic
4C2×#14523232—21no—
5#16861168—21no—
6#16861128C412yescomplemented
7#16861126—21no—
8#16861124—21no—
9#16861124C412yes—
10C2×#41×C127C2211yescharacteristic
11C2×M16×C127C411yescharacteristic, complemented
12C22×C8×C127C411yescharacteristic, complemented
13C22⋊C8×C127—21no—
14C22⋊C8×C127—21no—
15#14523232—22no—
16#14523232—41no—
17C2×C127⋊C8—41no—
18C4⋊Dic127—41no—
19C22⋊Dic127—22no—
20C4×Dic127—41no—
21#41×C127—41no—
22#41×C127—41no—
23C2×Q8×C127C2×C411yescharacteristic
24C2×D4×C127C2×C411yescharacteristic
25C2×D4×C127D411yescharacteristic
26C2×D4×C127Q811yescharacteristic
27C22×C4×C127C2×C411yescharacteristic
28C22×C4×C127D411yescharacteristic
29C22×C4×C127D411yescharacteristic
30M16×C127—41no—
31C2×C8×C127—21no—
32C2×C8×C127—21no—
33C2×C8×C127—41no—
34C2×Dic127—42no—
35C127⋊C8—42no—
36C23×C127C22⋊C411yescharacteristic
37C23×C127C4⋊C411yescharacteristic
38C23×C127C4⋊C411yescharacteristic
39Q8×C127—41no—
40D4×C127—41no—
41D4×C127—41no—
42D4×C127—41no—
43C2×C4×C127—21no—
44C2×C4×C127—21no—
45C2×C4×C127—21no—
46C2×C4×C127—21no—
47C2×C4×C127—21no—
48C2×C4×C127—21no—
49C2×C4×C127C4211yescharacteristic
50C2×C4×C127C22⋊C411yescharacteristic
51C2×C4×C127C22⋊C411yescharacteristic
52C2×C4×C127C4⋊C411yescharacteristic
53C8×C127—41no—
54C8×C127—41no—
55C22×C127—21no—
56C22×C127—21no—
57C22×C127—21no—
58C22×C127—41no—
59C22×C127—41no—
60C22×C127—41no—
61C22×C127#9511yescharacteristic
62C4×C127—21no—
63C4×C127—21no—
64C4×C127—21no—
65C4×C127#9911yescharacteristic
66C4×C127#9911yescharacteristic
67Dic127—82no—
68C2×C127—21no—
69C2×C127—41no—
70C2×C127—41no—
71C2×C127—41no—
72C2×C127#34211yescharacteristic
73#1395—1271nomaximal, Sylow
74C127#139511yescharacteristic, complemented, Sylow
75#413—1271no—
76#410—1271no—
77#408D12711yescharacteristic
78C2×#41C2×D12711yescharacteristic
79C2×M16—2541no—
80C2×M16Dic12711yescharacteristic, complemented
81C22×C8Dic12711yescharacteristic, complemented
82#117—2541no—
83#101—1272no—
84#100—2541no—
85#99—1272no—
86#99—2541no—
87C22⋊C8—21no—
88C22⋊C8—21no—
89#41—41no—
90#41—41no—
91C2×Q8C4×D12711yescharacteristic
92C2×D4C4×D12711yescharacteristic
93C2×D4D50811yescharacteristic
94C2×D4Dic25411yescharacteristic
95C22×C4C2×Dic12711yescharacteristic
96C22×C4C22⋊D12711yescharacteristic
97C22×C4C22⋊D12711yescharacteristic
98M16—41no—
99M16—5081no—
100M16—2542no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—41no—
104C2×C8—5081no—
105C4⋊C4—5081no—
106C22⋊C4—2542no—
107C42—5081no—
108C23(C2×D127)⋊C411yescharacteristic
109C23Dic127⋊C411yescharacteristic
110C23C4⋊Dic12711yescharacteristic
111Q8—41no—
112D4—41no—
113D4—41no—
114D4—41no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—5082no—
122C2×C4C4×Dic12711yescharacteristic
123C2×C4(C2×D127)⋊C411yescharacteristic
124C2×C4C22⋊Dic12711yescharacteristic
125C2×C4Dic127⋊C411yescharacteristic
126C8—41no—
127C8—41no—
128C8—5082no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22—41no—
133C22—41no—
134C22—41no—
135C22C127⋊#9511yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—10162no—
140C4#1686112411yescharacteristic
141C4#1686112411yescharacteristic
142C2—21no—
143C2—41no—
144C2—41no—
145C2—41no—
146C2#2254964311yescharacteristic
subgroupG/H
centerC4#16861124
commutatorC2×C4×C127C42
FrattiniC2×#41C2×D127
Fitting#408×C127C2
socleC2×C127#342
prime radical, cosocleC2×#41×C127C22
hypercenter#408D127
hypocenterC127#1395
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
6. #16861128C4semidirect
11. C2×M16×C127C4semidirect
12. C22×C8×C127C4semidirect
74. C127#1395semidirect
80. C2×M16Dic127semidirect
81. C22×C8Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498602
#225498502
#408×C1272
#1395127