Finite Group Databasebeta

← #30638836

Subgroups — #30638836

Order 16256.

296 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 81 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#22550351C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic
3#22549643C212yesmaximal, complemented
4#22549642C212yesmaximal, complemented
5C2×#99×C127C211yesmaximal, characteristic
6D4×Dic127—22no—
7D4×Dic127—22no—
8#16861168C412yescomplemented
9C2×C22⋊Dic127C412yescomplemented
10C2×C22⋊Dic127C2212yes—
11C2×C22⋊Dic127C2211yescharacteristic
12C2×C22⋊Dic127C2211yescharacteristic
13#16861124—22no—
14C22×D4×C127C2211yescharacteristic
15C2×C22⋊C4×C127C2212yes—
16#99×C127—22no—
17C22×Dic127—22no—
18C22×Dic127—21no—
19C22×Dic127—21no—
20C22×Dic127—22no—
21C4⋊Dic127—22no—
22C22⋊Dic127—22no—
23C22⋊Dic127—21no—
24C22⋊Dic127—21no—
25C22⋊Dic127—22no—
26C22⋊Dic127—22no—
27C22⋊Dic127C2×C412yes—
28C22⋊Dic127C2×C412yes—
29C4×Dic127—22no—
30C24×C127C2×C411yescharacteristic
31C24×C127C2311yescharacteristic
32C2×D4×C127—22no—
33C2×D4×C127—22no—
34C2×D4×C127D412yes—
35C2×D4×C127D412yes—
36C22×C4×C127—22no—
37C22×C4×C127C2×C411yescharacteristic
38C22⋊C4×C127—22no—
39C22⋊C4×C127—22no—
40C2×Dic127—22no—
41C2×Dic127—22no—
42C2×Dic127—42no—
43C2×Dic127—81no—
44C2×Dic127—81no—
45C2×Dic127—82no—
46C2×Dic127—82no—
47C23×C127—21no—
48C23×C127—21no—
49C23×C127—81no—
50C23×C127—81no—
51C23×C127C22⋊C412yes—
52C23×C127C22×C411yescharacteristic
53C23×C127C2×D411yescharacteristic
54C23×C127C2×D411yescharacteristic
55C23×C127#4112yes—
56C23×C127#4112yes—
57D4×C127—24no—
58D4×C127—24no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127—42no—
62C2×C4×C127—82no—
63C2×C4×C127C22⋊C412yes—
64C22×C127—22no—
65C22×C127—42no—
66C22×C127—41no—
67C22×C127—41no—
68C22×C127—41no—
69C22×C127—42no—
70C22×C127—42no—
71C22×C127—81no—
72C22×C127—81no—
73C22×C127C2×C22⋊C411yescharacteristic
74C22×C127#11711yescharacteristic
75C22×C127#11711yescharacteristic
76C22×C127#12312yes—
77C22×C127#12312yes—
78C4×C127—22no—
79C4×C127—82no—
80Dic127—42no—
81Dic127—42no—
82Dic127—82no—
83C2×C127—21no—
84C2×C127—22no—
85C2×C127—22no—
86C2×C127—21no—
87C2×C127—21no—
88C2×C127—42no—
89C2×C127#38111yescharacteristic
90#1828—1271nomaximal, Sylow
91C127#182811yescharacteristic, complemented, Sylow
92#518—1271no—
93C2×#99—1271no—
94C2×#99D12711yescharacteristic
95#342—1272no—
96#342—1272no—
97C22×D4C2×D12711yescharacteristic
98C4×D4—2542no—
99C4×D4—2542no—
100#117—1272no—
101C2×C22⋊C4—1272no—
102C2×C22⋊C4—1271no—
103C2×C22⋊C4—1271no—
104C2×C22⋊C4—1272no—
105C2×C22⋊C4C2×D12712yes—
106#99—22no—
107#99—2542no—
108C24C4×D12711yescharacteristic
109C24C22×D12711yescharacteristic
110C2×D4—22no—
111C2×D4—22no—
112C2×D4C22⋊D12712yes—
113C2×D4D50812yes—
114C22×C4—22no—
115C22×C4—2542no—
116C22×C4—2542no—
117C22×C4—2541no—
118C22×C4—2541no—
119C22×C4C4×D12711yescharacteristic
120C4⋊C4—2542no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—1272no—
124C22⋊C4—1272no—
125C22⋊C4—2542no—
126C22⋊C4—2542no—
127C22⋊C4—2542no—
128C22⋊C4—2541no—
129C22⋊C4—2541no—
130C42—2542no—
131C23—21no—
132C23—21no—
133C23—81no—
134C23—81no—
135C23(C2×D127)⋊C412yes—
136C23C2×C4×D12711yescharacteristic
137C23C2×C22⋊D12711yescharacteristic
138C23C2×D50811yescharacteristic
139C23#1452324712yes—
140C23#1452324712yes—
141D4—24no—
142D4—24no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—82no—
147C2×C4—10162no—
148C2×C4—10161no—
149C2×C4—10161no—
150C2×C4—10162no—
151C2×C4—5082no—
152C2×C4—2542no—
153C2×C4—2542no—
154C2×C4(C2×D127)⋊C412yes—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—42no—
160C22—41no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22C2×(C2×D127)⋊C411yescharacteristic
165C22#1686116511yescharacteristic
166C22#1686116511yescharacteristic
167C22#1686119012yes—
168C22#1686119312yes—
169C4—22no—
170C4—82no—
171C4—10162no—
172C4—5082no—
173C4—5082no—
174C2—22no—
175C2—22no—
176C2—21no—
177C2—21no—
178C2—21no—
179C2—42no—
180C2#2254973711yescharacteristic
subgroupG/H
centerC2#22549737
commutatorC23×C127C22×C4
FrattiniC24C22×D127
FittingC2×#99×C127C2
socleC2×C127#381
prime radicalC22×D4×C127C22
cosocleC24×C127C23
hypercenterC2×#99D127
hypocenterC127#1828
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
3. #22549643C2semidirect
4. #22549642C2semidirect
8. #16861168C4semidirect
9. C2×C22⋊Dic127C4semidirect
91. C127#1828semidirect

Maximal subgroups

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

subgroupindex
#225503512
C2×#168611242
#225496432
#225496422
C2×#99×C1272
#1828127