Finite Group Databasebeta

← #30638408

Subgroups — #30638408

Order 16256.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550279C211yesmaximal, characteristic
2#22550274C211yesmaximal, characteristic
3#22550255C211yesmaximal, characteristic
4#22550228C211yesmaximal, characteristic
5#22550226C211yesmaximal, characteristic
6#22549597C211yesmaximal, characteristic
7#500×C127C211yesmaximal, characteristic
8C2×#14523238—22no—
9#16861209C2211yescharacteristic
10#16861209C2211yescharacteristic
11#16861142—22no—
12#16861141—22no—
13#16861133—22no—
14#16861132—22no—
15C4×C127⋊C8C2211yescharacteristic
16C4×C127⋊C8C2211yescharacteristic
17C4⋊Q8×C127C2211yescharacteristic
18C4⋊Q8×C127C2211yescharacteristic
19#107×C127—22no—
20C4×C8×C127C2211yescharacteristic
21C2×Dic254—21no—
22C2×Dic254—21no—
23#14523238—44no—
24C2×C127⋊C8D412yes—
25C2×C127⋊C8D412yes—
26C4⋊Dic127—21no—
27C4⋊Dic127—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C2×Q8×C127—21no—
31C2×Q8×C127—21no—
32C2×C8×C127D412yes—
33C4⋊C4×C127—21no—
34C4⋊C4×C127—21no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—21no—
37C42×C127C2311yescharacteristic
38Dic254—42no—
39Dic254—42no—
40C2×Dic127—41no—
41C2×Dic127—41no—
42C127⋊C8—22no—
43C127⋊C8—22no—
44Q8×C127—42no—
45Q8×C127—42no—
46C2×C4×C127—41no—
47C2×C4×C127—41no—
48C2×C4×C127C2×D411yescharacteristic
49C2×C4×C127C2×D411yescharacteristic
50C2×C4×C127C2×D411yescharacteristic
51C8×C127Q1614yescomplemented
52C22×C127#12711yescharacteristic
53C4×C127—81no—
54C4×C127—81no—
55C4×C127C2×Q1612yes—
56C4×C127AGL(1,ℤ/8)12yes—
57C4×C127#13712yes—
58Dic127—81no—
59Dic127—81no—
60C2×C127#49411yescharacteristic
61C2×C127#49611yescharacteristic
62C2×C127#49711yescharacteristic
63#1723—1271nomaximal, Sylow
64C127#172311yescharacteristic, complemented, Sylow
65#500—1271no—
66#500D12711yescharacteristic
67#498—1271no—
68#494—1271no—
69#487—1271no—
70#487—1271no—
71#322—1271no—
72C2×Q16—2542no—
73C4⋊Q8—1271no—
74C4⋊Q8—1271no—
75C4⋊Q8C2×D12711yescharacteristic
76C4⋊Q8C2×D12711yescharacteristic
77#107—22no—
78#107—2542no—
79#106—2542no—
80#103—2542no—
81#103—2542no—
82C4×C8—1271no—
83C4×C8—1271no—
84C4×C8C2×D12711yescharacteristic
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—2541no—
88C2×Q8—2541no—
89Q16—5084no—
90C2×C8—1272no—
91C2×C8—1272no—
92C2×C8C22⋊D12712yes—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—2541no—
98C4⋊C4—2541no—
99C4⋊C4—2541no—
100C4⋊C4—2541no—
101C42C22×D12711yescharacteristic
102Q8—42no—
103Q8—42no—
104Q8—5082no—
105Q8—5082no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—5081no—
109C2×C4—5081no—
110C2×C4D4×D12711yescharacteristic
111C2×C4D4×D12711yescharacteristic
112C2×C4C2×C22⋊D12711yescharacteristic
113C8—2542no—
114C8—2542no—
115C8#1452323814yescomplemented
116C22#1686120511yescharacteristic
117C4—81no—
118C4—81no—
119C4—10161no—
120C4—10161no—
121C4C2×#1452323812yes—
122C4#1686123312yes—
123C4#1686124112yes—
124C2#2255025511yescharacteristic
125C2#2255026411yescharacteristic
126C2#2255026911yescharacteristic
subgroupG/H
centerC22#16861205
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#500×C127C2
socle, prime radicalC22×C127#127
hypercenter#500D127
hypocenterC127#1723
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
51. C8×C127Q16semidirect
64. C127#1723semidirect
115. C8#14523238semidirect

Maximal subgroups

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

subgroupindex
#225502792
#225502742
#225502552
#225502282
#225502262
#225495972
#500×C1272
#1723127