Finite Group Databasebeta

← #30638179

Subgroups — #30638179

Order 16256.

160 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550226C211yesmaximal, characteristic, complemented
2#22550166C211yesmaximal, characteristic
3D4×C127⋊C8C211yesmaximal, characteristic
4#22549628C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549622C211yesmaximal, characteristic, complemented
7#474×C127C211yesmaximal, characteristic
8C22×C127⋊C8—21no—
9#16861209C2211yescharacteristic
10#16861140—21no—
11C4⋊(C127⋊C8)C2211yescharacteristic
12C4⋊(C127⋊C8)C2211yescharacteristic
13#16861133—21no—
14#16861133—21no—
15#16861130—21no—
16C22⋊(C127⋊C8)—21no—
17C4×C127⋊C8C2211yescharacteristic
18C4⋊Q8×C127C2211yescharacteristic
19C4×D4×C127C2211yescharacteristic
20#107×C127—21no—
21C4⋊C8×C127C2211yescharacteristic
22#102×C127—21no—
23C2×Dic254—21no—
24C2×C127⋊C8—21no—
25C2×C127⋊C8—21no—
26C2×C127⋊C8—42no—
27C2×C127⋊C8D411yescharacteristic
28C2×C127⋊C8D411yescharacteristic
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C2×Q8×C127—21no—
32C2×D4×C127D411yescharacteristic
33C22×C4×C127—21no—
34C2×C8×C127—21no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127D411yescharacteristic
38C22⋊C4×C127—21no—
39C42×C127C2311yescharacteristic
40Dic254—81no—
41C2×Dic127—41no—
42C127⋊C8—21no—
43C127⋊C8—21no—
44C127⋊C8—41no—
45C127⋊C8—41no—
46C23×C127—21no—
47Q8×C127—41no—
48Q8×C127—41no—
49D4×C127—21no—
50D4×C127Q1612yes—
51C2×C4×C127—21no—
52C2×C4×C127—41no—
53C2×C4×C127—41no—
54C2×C4×C127—41no—
55C2×C4×C127C2×D411yescharacteristic
56C2×C4×C127C2×D411yescharacteristic
57C2×C4×C127#4111yescharacteristic
58C8×C127—41no—
59C22×C127—22no—
60C22×C127—41no—
61C22×C127—41no—
62C22×C127#12111yescharacteristic
63C4×C127—21no—
64C4×C127—41no—
65C4×C127—81no—
66C4×C127C2×Q1611yescharacteristic
67C4×C127#13511yescharacteristic
68C4×C127#13511yescharacteristic
69C4×C127AGL(1,ℤ/8)11yescharacteristic
70Dic127—81no—
71C2×C127—42no—
72C2×C127#46311yescharacteristic
73C2×C127#46711yescharacteristic
74C2×C127#47111yescharacteristic
75#1683—1271nomaximal, Sylow
76C127#168311yescharacteristic, complemented, Sylow
77#487—1271no—
78#476—1271no—
79#474D12711yescharacteristic
80C8×D4—1271no—
81#335—1271no—
82#333—1271no—
83#332—1271no—
84C22×C8—2541no—
85C4⋊Q8—1271no—
86C4⋊Q8C2×D12711yescharacteristic
87C4×D4C2×D12711yescharacteristic
88#107—21no—
89#106—2541no—
90C4⋊C8—1271no—
91C4⋊C8—1271no—
92C4⋊C8C2×D12711yescharacteristic
93#103—2541no—
94#103—2541no—
95#102—21no—
96#102—2541no—
97C22⋊C8—2541no—
98C4×C8—1271no—
99C2×Q8—21no—
100C2×Q8—2541no—
101C2×D4C22⋊D12711yescharacteristic
102C22×C4—21no—
103C2×C8—21no—
104C2×C8—1271no—
105C2×C8—1271no—
106C2×C8—5082no—
107C2×C8—2541no—
108C2×C8—2541no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—2541no—
112C4⋊C4—2541no—
113C4⋊C4C22⋊D12711yescharacteristic
114C22⋊C4—21no—
115C42C22×D12711yescharacteristic
116C23—21no—
117Q8—41no—
118Q8—41no—
119Q8—10161no—
120D4—21no—
121D4#1452323812yes—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—5081no—
127C2×C4D4×D12711yescharacteristic
128C2×C4C2×C22⋊D12711yescharacteristic
129C2×C4#1452324711yescharacteristic
130C8—41no—
131C8—5081no—
132C8—5081no—
133C8—2541no—
134C8—2541no—
135C22—22no—
136C22—41no—
137C22—41no—
138C22#1686118111yescharacteristic
139C4—21no—
140C4—41no—
141C4—81no—
142C4—10161no—
143C4C2×#1452323811yescharacteristic
144C4#1686122811yescharacteristic
145C4#1686122911yescharacteristic
146C4#1686123311yescharacteristic
147C2—42no—
148C2#2255008911yescharacteristic
149C2#2255011511yescharacteristic
150C2#2255014111yescharacteristic
subgroupG/H
centerC22#16861181
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#474×C127C2
socleC22×C127#121
prime radicalC2×D4×C127D4
hypercenter#474D127
hypocenterC127#1683
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #22550226C2semidirect
4. #22549628C2semidirect
5. #22549624C2semidirect
6. #22549622C2semidirect
76. C127#1683semidirect

Maximal subgroups

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

subgroupindex
#225502262
#225501662
D4×C127⋊C82
#225496282
#225496242
#225496222
#474×C1272
#1683127