Finite Group Databasebeta

← #30638774

Subgroups — #30638774

Order 16256.

260 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 89 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×#16861216C211yesmaximal, characteristic
2#22549912C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic
4#22549632C212yesmaximal, complemented
5#410×C127C211yesmaximal, characteristic
6#16861216—22no—
7#16861216—22no—
8C2×#14523232C412yescomplemented
9C2×#14523232C2211yescharacteristic
10C22×C127⋊C8C412yescomplemented
11C22×C127⋊C8C2211yescharacteristic
12#16861168C2212yes—
13#16861142—22no—
14#16861140—22no—
15#16861124—22no—
16C2×#41×C127C2211yescharacteristic
17#117×C127C2212yes—
18#99×C127—22no—
19#14523232—24no—
20#14523232—22no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—22no—
23C2×C127⋊C8C2×C412yes—
24C2×C127⋊C8C2×C411yescharacteristic
25C2×C127⋊C8C2×C411yescharacteristic
26C2×C127⋊C8D412yes—
27C2×C127⋊C8D412yes—
28C4⋊Dic127—22no—
29C22⋊Dic127—22no—
30C4×Dic127—22no—
31#41×C127—22no—
32#41×C127—22no—
33C2×Q8×C127Q811yescharacteristic
34C2×D4×C127D412yes—
35C2×D4×C127Q811yescharacteristic
36C22×C4×C127C2×C412yes—
37C22×C4×C127C2311yescharacteristic
38C4⋊C4×C127—22no—
39C22⋊C4×C127—22no—
40C42×C127—22no—
41C2×Dic127—42no—
42C127⋊C8—22no—
43C127⋊C8C22⋊C411yescharacteristic
44C127⋊C8C22⋊C411yescharacteristic
45C127⋊C8C22⋊C412yes—
46C23×C127C4⋊C412yes—
47C23×C127C2×D411yescharacteristic
48Q8×C127—22no—
49D4×C127—24no—
50D4×C127—22no—
51C2×C4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—42no—
54C2×C4×C127C4⋊C412yes—
55C2×C4×C127C22×C411yescharacteristic
56C2×C4×C127C2×D412yes—
57C2×C4×C127C2×Q811yescharacteristic
58C2×C4×C127#4112yes—
59C22×C127—22no—
60C22×C127—41no—
61C22×C127—42no—
62C22×C127C2×C4⋊C411yescharacteristic
63C22×C127#12112yes—
64C4×C127—22no—
65C4×C127—82no—
66C4×C127C2×C22⋊C411yescharacteristic
67C4×C127#11711yescharacteristic
68C4×C127C22⋊Q812yes—
69Dic127—82no—
70C2×C127—22no—
71C2×C127—21no—
72C2×C127—42no—
73C2×C127#38011yescharacteristic
74#1812—1271nomaximal, Sylow
75C127#181211yescharacteristic, complemented, Sylow
76C2×#131—1271no—
77#427—1272no—
78#410—1271no—
79#410D12711yescharacteristic
80H(3,ℤ/4)—1272no—
81C2×#41C2×D12711yescharacteristic
82#131—2542no—
83#131—2542no—
84C2×M16—1272no—
85C2×M16—1271no—
86C22×C8—1271no—
87C22×C8—1272no—
88#117—1272no—
89#117C2×D12712yes—
90#107—2542no—
91#106—2542no—
92#99—22no—
93#99—2542no—
94#41—22no—
95#41—22no—
96C2×Q8Dic25411yescharacteristic
97C2×D4C22⋊D12712yes—
98C2×D4Dic25411yescharacteristic
99C22×C4C4×D12712yes—
100C22×C4C22×D12711yescharacteristic
101M16—2544no—
102M16—2542no—
103C2×C8—1271no—
104C2×C8—1271no—
105C2×C8—1272no—
106C2×C8—1272no—
107C2×C8—1272no—
108C2×C8—2542no—
109C2×C8—2542no—
110C4⋊C4—22no—
111C4⋊C4—2542no—
112C22⋊C4—22no—
113C22⋊C4—2542no—
114C42—22no—
115C42—2542no—
116C23Dic127⋊C412yes—
117C23C2×C22⋊D12711yescharacteristic
118Q8—22no—
119D4—24no—
120D4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—42no—
124C2×C4—5082no—
125C2×C4Dic127⋊C412yes—
126C2×C4C2×C4×D12711yescharacteristic
127C2×C4D4×D12712yes—
128C2×C4C2×Dic25411yescharacteristic
129C2×C4#1452324712yes—
130C8—1271no—
131C8—1271no—
132C8—1272no—
133C8—2542no—
134C22—22no—
135C22—41no—
136C22—42no—
137C22C2×Dic127⋊C411yescharacteristic
138C22#1686118112yes—
139C4—22no—
140C4—82no—
141C4—10162no—
142C4C22⋊C4×D12711yescharacteristic
143C4#1686116511yescharacteristic
144C4#1686118612yes—
145C2—22no—
146C2—21no—
147C2—42no—
148C2#2254973411yescharacteristic
subgroupG/H
centerC4#16861165
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#380
prime radicalC2×#41×C127C22
cosocleC22×C4×C127C23
hypercenter#410D127
hypocenterC127#1812
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22549912C2semidirect
4. #22549632C2semidirect
8. C2×#14523232C4semidirect
10. C22×C127⋊C8C4semidirect
75. C127#1812semidirect

Maximal subgroups

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

subgroupindex
C2×#168612162
#225499122
#225498502
#225496322
#410×C1272
#1812127