Finite Group Databasebeta

← #30637737

Subgroups — #30637737

Order 16256.

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

normal subgroups only

This group has 210 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#22550507C211yesmaximal, characteristic
2#22550206C212yesmaximal
3#22549612C212yesmaximal
4#22549602C211yesmaximal, characteristic
5#398×C127C211yesmaximal, characteristic
6#16861208—22no—
7#16861200—22no—
8#16861188—21no—
9#16861188C2212yes—
10#16861168—21no—
11#16861140—22no—
12#16861132—22no—
13C22⋊(C127⋊C8)C2212yes—
14C2×C4⋊C4×C127—21no—
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C4⋊C4×C127C2211yescharacteristic
17#95×C127—21no—
18#95×C127C2211yescharacteristic
19C2×Dic254—21no—
20C2×Dic254D412yes—
21C2×C127⋊C8—22no—
22C4⋊Dic127—21no—
23C4⋊Dic127—21no—
24C4⋊Dic127—21no—
25C4⋊Dic127D412yes—
26Dic127⋊C4—22no—
27Dic127⋊C4—22no—
28C22⋊Dic127—22no—
29C4×Dic127—22no—
30C22×C4×C127—21no—
31C22×C4×C127—21no—
32C22×C4×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127C2311yescharacteristic
35C4⋊C4×C127—22no—
36C4⋊C4×C127—22no—
37C4⋊C4×C127—42no—
38Dic254—42no—
39Dic254—81no—
40C2×Dic127—22no—
41C2×Dic127—22no—
42C127⋊C8—42no—
43C23×C127C2×D411yescharacteristic
44C2×C4×C127—21no—
45C2×C4×C127—21no—
46C2×C4×C127—22no—
47C2×C4×C127—21no—
48C2×C4×C127—41no—
49C2×C4×C127—41no—
50C2×C4×C127—41no—
51C2×C4×C127—41no—
52C2×C4×C127—41no—
53C2×C4×C127—41no—
54C2×C4×C127—41no—
55C2×C4×C127SD1612yes—
56C2×C4×C127C2×D412yes—
57C22×C127—21no—
58C22×C127—21no—
59C22×C127#12011yescharacteristic
60C22×C127C2×SD1611yescharacteristic
61C22×C127#13711yescharacteristic
62C4×C127—22no—
63C4×C127—22no—
64C4×C127—42no—
65C4×C127—41no—
66C4×C127—81no—
67Dic127—42no—
68Dic127—42no—
69C2×C127—21no—
70C2×C127—21no—
71C2×C127#44811yescharacteristic
72C2×C127#45611yescharacteristic
73C2×C127#45811yescharacteristic
74#1611—1271nomaximal, Sylow
75C127#161111yescharacteristic, complemented, Sylow
76#544—1271no—
77#483—1272no—
78#398D12711yescharacteristic
79#328—1272no—
80#324—1271no—
81C4⋊Q8—2542no—
82#125—2542no—
83C22⋊Q8—1272no—
84C22⋊Q8—2541no—
85#117—2541no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4C2×D12711yescharacteristic
88C2×C4⋊C4C2×D12711yescharacteristic
89#106—2542no—
90#103—2542no—
91C22⋊C8—1272no—
92#95—21no—
93#95C2×D12711yescharacteristic
94C2×Q8—1272no—
95C2×Q8—2541no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4C22⋊D12711yescharacteristic
99C22×C4C22⋊D12711yescharacteristic
100C22×C4C22×D12711yescharacteristic
101C2×C8—2542no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—42no—
105C4⋊C4—1272no—
106C4⋊C4—2542no—
107C4⋊C4—2541no—
108C4⋊C4—2541no—
109C4⋊C4—2541no—
110C4⋊C4—2542no—
111C22⋊C4—2542no—
112C42—2542no—
113C23C2×C22⋊D12711yescharacteristic
114Q8—10161no—
115Q8—5082no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—2542no—
128C2×C4—2542no—
129C2×C4#1452323612yes—
130C2×C4D4×D12712yes—
131C8—5082no—
132C22—21no—
133C22—21no—
134C22#1686117711yescharacteristic
135C22C2×#1452323611yescharacteristic
136C22#1686123911yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—81no—
142C4—5082no—
143C4—5082no—
144C2—21no—
145C2—21no—
146C2#2255001411yescharacteristic
147C2#2255005811yescharacteristic
148C2#2255006511yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#398×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#398D127
hypocenterC127#1611
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
75. C127#1611semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225502062
#225496122
#225496022
#398×C1272
#1611127