Finite Group Databasebeta

← #30637774

Subgroups — #30637774

Order 16256.

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

normal subgroups only

This group has 202 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#22550212C212yesmaximal
3#22549815C211yesmaximal, characteristic
4#22549614C212yesmaximal
5#324×C127C211yesmaximal, characteristic
6#16861208—22no—
7#16861200—22no—
8#16861188—21no—
9#16861188C2212yes—
10#16861168—21no—
11C2×C4⋊Dic127C2211yescharacteristic
12#16861143—22no—
13#16861134—22no—
14C127⋊#95—21no—
15C127⋊#95—21no—
16C127⋊#95C2211yescharacteristic
17C2×C4⋊C4×C127C2211yescharacteristic
18C22⋊C8×C127C2212yes—
19C2×Dic254—21no—
20C2×Dic254D412yes—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C4⋊Dic127—21no—
24C4⋊Dic127—21no—
25C4⋊Dic127—21no—
26C4⋊Dic127—22no—
27C4⋊Dic127D412yes—
28Dic127⋊C4—22no—
29Dic127⋊C4—22no—
30C22⋊Dic127—22no—
31C4×Dic127—22no—
32C22×C4×C127D411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127C2311yescharacteristic
35C2×C8×C127—22no—
36C4⋊C4×C127—22no—
37Dic254—42no—
38Dic254—81no—
39C2×Dic127—22no—
40C2×Dic127—22no—
41C2×Dic127—41no—
42C2×Dic127—41no—
43C2×Dic127—42no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C23×C127C2×D411yescharacteristic
47C2×C4×C127—21no—
48C2×C4×C127—21no—
49C2×C4×C127—21no—
50C2×C4×C127—41no—
51C2×C4×C127—41no—
52C2×C4×C127Q1612yes—
53C2×C4×C127C2×D412yes—
54C8×C127—42no—
55C22×C127—21no—
56C22×C127—21no—
57C22×C127#12011yescharacteristic
58C22×C127C2×Q1611yescharacteristic
59C22×C127#13711yescharacteristic
60C4×C127—22no—
61C4×C127—22no—
62C4×C127—41no—
63Dic127—42no—
64Dic127—42no—
65Dic127—81no—
66Dic127—81no—
67C2×C127—21no—
68C2×C127—21no—
69C2×C127#45111yescharacteristic
70C2×C127#45411yescharacteristic
71C2×C127#45811yescharacteristic
72#1617—1271nomaximal, Sylow
73C127#161711yescharacteristic, complemented, Sylow
74#544—1271no—
75#484—1272no—
76#400—1271no—
77#328—1272no—
78#324D12711yescharacteristic
79C4⋊Q8—2542no—
80#125—2542no—
81C22⋊Q8—1272no—
82C22⋊Q8—2541no—
83#117—2541no—
84C2×C4⋊C4—1271no—
85C2×C4⋊C4C2×D12711yescharacteristic
86#107—2542no—
87#103—2542no—
88C22⋊C8C2×D12712yes—
89#95—1271no—
90#95—2541no—
91#95—2541no—
92C2×Q8—1272no—
93C2×Q8—2541no—
94C22×C4—2541no—
95C22×C4—2541no—
96C22×C4D50811yescharacteristic
97C22×C4D50811yescharacteristic
98C22×C4C22×D12711yescharacteristic
99C2×C8—22no—
100C4⋊C4—22no—
101C4⋊C4—1272no—
102C4⋊C4—2542no—
103C4⋊C4—2541no—
104C4⋊C4—2541no—
105C4⋊C4—2541no—
106C4⋊C4—2542no—
107C4⋊C4—2542no—
108C22⋊C4—2542no—
109C42—2542no—
110C23C2×D50811yescharacteristic
111Q8—10161no—
112Q8—5082no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—5081no—
119C2×C4—5081no—
120C2×C4—5082no—
121C2×C4—5081no—
122C2×C4—5081no—
123C2×C4—2542no—
124C2×C4—2542no—
125C2×C4Dic50812yes—
126C2×C4D4×D12712yes—
127C8—42no—
128C22—21no—
129C22—21no—
130C22#1686117911yescharacteristic
131C22C2×Dic50811yescharacteristic
132C22#1686124211yescharacteristic
133C4—22no—
134C4—22no—
135C4—41no—
136C4—10161no—
137C4—10161no—
138C4—5082no—
139C4—5082no—
140C2—21no—
141C2—21no—
142C2#2255003411yescharacteristic
143C2#2255005111yescharacteristic
144C2#2255006711yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#324×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#324D127
hypocenterC127#1617
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
73. C127#1617semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225502122
#225498152
#225496142
#324×C1272
#1617127