Finite Group Databasebeta

← #30638942

Subgroups — #30638942

Order 16256.

324 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 97 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
2#22549918C212yesmaximal, complemented
3#22549779C211yesmaximal, characteristic
4#22549637C212yesmaximal, complemented
5C2×#100×C127C211yesmaximal, characteristic
6D4×Dic127—22no—
7D4×Dic127—22no—
8#16861168C2212yes—
9C2×C4⋊Dic127C2211yescharacteristic
10C2×C22⋊Dic127—22no—
11C2×C22⋊Dic127—22no—
12C2×C4×Dic127C2211yescharacteristic
13#16861143—22no—
14#16861141—22no—
15C22×D4×C127C2211yescharacteristic
16C2×M16×C127C2212yes—
17#100×C127C414yescomplemented
18C22×Dic127—22no—
19C22×Dic127—22no—
20C22×Dic127—21no—
21C22×Dic127—21no—
22C4⋊Dic127—22no—
23C4⋊Dic127D412yes—
24C4⋊Dic127D412yes—
25C22⋊Dic127—24no—
26C22⋊Dic127—22no—
27C22⋊Dic127—44no—
28C4×Dic127—22no—
29C4×Dic127—22no—
30C24×C127D412yes—
31C2×D4×C127—22no—
32C2×D4×C127—22no—
33C2×D4×C127C2×C412yes—
34C2×D4×C127Q812yes—
35C22×C4×C127C2311yescharacteristic
36M16×C127—22no—
37M16×C127C2×C414yescomplemented
38C2×C8×C127—22no—
39C2×Dic127—22no—
40C2×Dic127—22no—
41C2×Dic127—22no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C2×Dic127—42no—
45C2×Dic127—41no—
46C2×Dic127—82no—
47C2×Dic127—82no—
48C23×C127—22no—
49C23×C127—22no—
50C23×C127—42no—
51C23×C127—42no—
52C23×C127C4⋊C414yes—
53C23×C127C2×D411yescharacteristic
54D4×C127—24no—
55D4×C127—24no—
56C2×C4×C127C22×C411yescharacteristic
57C2×C4×C127C2×D412yes—
58C2×C4×C127C2×Q811yescharacteristic
59C2×C4×C127#4112yes—
60C8×C127—24no—
61C22×C127—21no—
62C22×C127—21no—
63C22×C127—24no—
64C22×C127—44no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127—82no—
68C22×C127C2×C4⋊C411yescharacteristic
69C22×C127#12011yescharacteristic
70C22×C127#12311yescharacteristic
71C4×C127C4×D412yes—
72C4×C127C22⋊Q812yes—
73Dic127—42no—
74Dic127—42no—
75Dic127—42no—
76Dic127—81no—
77C2×C127—21no—
78C2×C127—21no—
79C2×C127—44no—
80C2×C127#38511yescharacteristic
81C2×C127#45311yescharacteristic
82C2×C127#45511yescharacteristic
83#1856—1271nomaximal, Sylow
84C127#185611yescharacteristic, complemented, Sylow
85#518—1271no—
86#428—1272no—
87C2×#100D12711yescharacteristic
88#390—1271no—
89#339—1272no—
90C22×D4C2×D12711yescharacteristic
91C2×M16C2×D12712yes—
92C4×D4—2542no—
93C4×D4—2542no—
94#117—1272no—
95C2×C4⋊C4—1271no—
96C2×C22⋊C4—2542no—
97C2×C22⋊C4—2542no—
98C2×C42—1271no—
99#107—2542no—
100#106—2542no—
101#100Dic12714yescomplemented
102C24D50812yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4C2×Dic12712yes—
106C2×D4Dic25412yes—
107C22×C4—2542no—
108C22×C4—2542no—
109C22×C4—2541no—
110C22×C4—2541no—
111C22×C4C22×D12711yescharacteristic
112M16—22no—
113M16C2×Dic12714yescomplemented
114C2×C8—22no—
115C4⋊C4—1272no—
116C4⋊C4—1272no—
117C4⋊C4—2542no—
118C22⋊C4—5084no—
119C22⋊C4—2544no—
120C22⋊C4—2542no—
121C42—2542no—
122C42—2542no—
123C23—22no—
124C23—22no—
125C23—42no—
126C23—42no—
127C23C4⋊Dic12714yes—
128C23C2×D50811yescharacteristic
129D4—24no—
130D4—24no—
131C2×C4—10162no—
132C2×C4—10162no—
133C2×C4—5082no—
134C2×C4—5081no—
135C2×C4—5081no—
136C2×C4—5081no—
137C2×C4—2542no—
138C2×C4—2542no—
139C2×C4—2542no—
140C2×C4C22×Dic12711yescharacteristic
141C2×C4D4×D12712yes—
142C2×C4C2×Dic25411yescharacteristic
143C2×C4#1452324712yes—
144C8—24no—
145C22—21no—
146C22—21no—
147C22—24no—
148C22—44no—
149C22—42no—
150C22—42no—
151C22—82no—
152C22C2×C4⋊Dic12711yescharacteristic
153C22#1686117911yescharacteristic
154C22#1686119311yescharacteristic
155C4—10161no—
156C4—5082no—
157C4—5082no—
158C4—5082no—
159C4D4×Dic12712yes—
160C4#1686118612yes—
161C2—21no—
162C2—21no—
163C2—44no—
164C2#2254975111yescharacteristic
165C2#2255004611yescharacteristic
166C2#2255005611yescharacteristic
subgroupG/H
centerC22#16861179
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127#120
prime radicalC22×D4×C127C22
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1856
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #22549918C2semidirect
4. #22549637C2semidirect
17. #100×C127C4semidirect
37. M16×C127C2×C4semidirect
84. C127#1856semidirect
101. #100Dic127semidirect
113. M16C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
#225503512
#225499182
#225497792
#225496372
C2×#100×C1272
#1856127