Finite Group Databasebeta

← #30636850

Subgroups — #30636850

Order 16256.

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

normal subgroups only

This group has 166 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#22549850C211yesmaximal, characteristic
2#22549843C211yesmaximal, characteristic
3#410×C127C211yesmaximal, characteristic
4C2×#14523232C411yescharacteristic, complemented
5C22×C127⋊C8C411yescharacteristic, complemented
6#16861168—21no—
7#16861124—21no—
8#16861124C412yescomplemented
9C22⋊(C127⋊C8)—21no—
10C22⋊(C127⋊C8)—21no—
11C2×#41×C127C2211yescharacteristic
12#117×C127—21no—
13#99×C127—21no—
14#99×C127C412yescomplemented
15#14523232—41no—
16C2×C127⋊C8—21no—
17C2×C127⋊C8—21no—
18C2×C127⋊C8—41no—
19C4⋊Dic127—41no—
20C22⋊Dic127—22no—
21C4×Dic127—41no—
22#41×C127—41no—
23#41×C127—41no—
24C2×Q8×C127Q811yescharacteristic
25C2×D4×C127C2×C411yescharacteristic
26C2×D4×C127C2×C411yescharacteristic
27C2×D4×C127D411yescharacteristic
28C22×C4×C127C2×C411yescharacteristic
29C22×C4×C127D411yescharacteristic
30C22×C4×C127D411yescharacteristic
31C4⋊C4×C127—41no—
32C22⋊C4×C127—22no—
33C42×C127—41no—
34C2×Dic127—42no—
35C127⋊C8—41no—
36C127⋊C8—41no—
37C23×C127C4211yescharacteristic
38C23×C127C22⋊C411yescharacteristic
39C23×C127C22⋊C411yescharacteristic
40Q8×C127—41no—
41D4×C127—41no—
42D4×C127—41no—
43D4×C127—41no—
44C2×C4×C127—21no—
45C2×C4×C127—21no—
46C2×C4×C127—21no—
47C2×C4×C127—21no—
48C2×C4×C127—21no—
49C2×C4×C127—21no—
50C2×C4×C127—42no—
51C2×C4×C127C22⋊C411yescharacteristic
52C2×C4×C127C4⋊C411yescharacteristic
53C2×C4×C127C4⋊C411yescharacteristic
54C2×C4×C127C4⋊C411yescharacteristic
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—82no—
66C4×C127#9911yescharacteristic
67C4×C127#9911yescharacteristic
68Dic127—82no—
69C2×C127—21no—
70C2×C127—41no—
71C2×C127—41no—
72C2×C127—41no—
73C2×C127#34211yescharacteristic
74#1391—1271nomaximal, Sylow
75C127#139111yescharacteristic, complemented, Sylow
76#410—1271no—
77#410D12711yescharacteristic
78#408—1271no—
79C2×#41C2×D12711yescharacteristic
80C2×M16—1271no—
81C22×C8—1271no—
82#117—21no—
83#117—2541no—
84#99—21no—
85#99—1272no—
86#99—2541no—
87#99Dic12712yescomplemented
88C22⋊C8—2541no—
89C22⋊C8—2541no—
90#41—41no—
91#41—41no—
92C2×Q8Dic25411yescharacteristic
93C2×D4C4×D12711yescharacteristic
94C2×D4C2×Dic12711yescharacteristic
95C2×D4C22⋊D12711yescharacteristic
96C22×C4C4×D12711yescharacteristic
97C22×C4C22⋊D12711yescharacteristic
98C22×C4D50811yescharacteristic
99M16—5081no—
100C2×C8—5081no—
101C2×C8—2541no—
102C2×C8—2541no—
103C4⋊C4—41no—
104C4⋊C4—5081no—
105C22⋊C4—22no—
106C22⋊C4—2542no—
107C42—41no—
108C42—5081no—
109C23C4×Dic12711yescharacteristic
110C23(C2×D127)⋊C411yescharacteristic
111C23C22⋊Dic12711yescharacteristic
112Q8—41no—
113D4—41no—
114D4—41no—
115D4—41no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—42no—
123C2×C4—5082no—
124C2×C4(C2×D127)⋊C411yescharacteristic
125C2×C4Dic127⋊C411yescharacteristic
126C2×C4Dic127⋊C411yescharacteristic
127C2×C4C4⋊Dic12711yescharacteristic
128C8—5081no—
129C8—5081no—
130C22—21no—
131C22—21no—
132C22—21no—
133C22—41no—
134C22—41no—
135C22—41no—
136C22C127⋊#9511yescharacteristic
137C4—21no—
138C4—21no—
139C4—21no—
140C4—82no—
141C4—10162no—
142C4#1686112311yescharacteristic
143C4#1686112311yescharacteristic
144C2—21no—
145C2—41no—
146C2—41no—
147C2—41no—
148C2#2254964211yescharacteristic
subgroupG/H
centerC4#16861123
commutatorC23×C127C42
FrattiniC2×#41C2×D127
Fitting#410×C127C2
socleC2×C127#342
prime radical, cosocleC2×#41×C127C22
hypercenter#410D127
hypocenterC127#1391
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
4. C2×#14523232C4semidirect
5. C22×C127⋊C8C4semidirect
8. #16861124C4semidirect
14. #99×C127C4semidirect
75. C127#1391semidirect
87. #99Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498502
#225498432
#410×C1272
#1391127