Finite Group Databasebeta

← #30640742

Subgroups — #30640742

Order 16256.

226 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 41 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#22550626C211yesmaximal, characteristic, complemented
2#22550066C212yesmaximal
3#22549671C211yesmaximal, characteristic, complemented
4#22549667C212yesmaximal, complemented
5#455×C127C211yesmaximal, characteristic
6#16861207—21no—
7#16861202—22no—
8#16861201—21no—
9#16861190—22no—
10#16861190C2212yes—
11#16861186—22no—
12#16861124—22no—
13#16861124C2212yes—
14H(5,𝔽2)×C127C2211yescharacteristic
15#137×C127—21no—
16#124×C127C2211yescharacteristic
17#104×C127—21no—
18#100×C127C2211yescharacteristic
19C2×Dic254—21no—
20C22×Dic127D412yes—
21C4⋊Dic127—21no—
22C4⋊Dic127—21no—
23Dic127⋊C4—22no—
24Dic127⋊C4—22no—
25Dic127⋊C4—22no—
26Dic127⋊C4—42no—
27C22⋊Dic127—22no—
28C22⋊Dic127—22no—
29C22⋊Dic127—42no—
30C22⋊Dic127D412yes—
31C4×Dic127—21no—
32#41×C127—21no—
33#41×C127—41no—
34C2×Q8×C127D411yescharacteristic
35C2×D4×C127—22no—
36C2×D4×C127—41no—
37C2×D4×C127C2311yescharacteristic
38Q16×C127—41no—
39SD16×C127—41no—
40M16×C127—21no—
41C22⋊C4×C127—22no—
42C42×C127D411yescharacteristic
43Dic254—81no—
44C2×Dic127—22no—
45C2×Dic127—22no—
46C2×Dic127—42no—
47C2×Dic127—42no—
48C23×C127—22no—
49C23×C127C2×D412yes—
50Q8×C127—21no—
51Q8×C127—41no—
52D4×C127—21no—
53D4×C127—41no—
54D4×C127—42no—
55D4×C127—41no—
56C2×C4×C127—22no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—41no—
60C2×C4×C127C2×D411yescharacteristic
61C8×C127—41no—
62C22×C127—22no—
63C22×C127—42no—
64C22×C127—41no—
65C22×C127—82no—
66C22×C127#12011yescharacteristic
67C4×C127—21no—
68C4×C127—41no—
69C4×C127—41no—
70C4×C127—41no—
71Dic127—42no—
72Dic127—42no—
73Dic127—82no—
74C2×C127—21no—
75C2×C127—42no—
76C2×C127—81no—
77C2×C127B(4,𝔽2)11yescharacteristic
78#2201—1271nomaximal, Sylow
79C127#220111yescharacteristic, complemented, Sylow
80#563—1271no—
81#458—1272no—
82#455D12711yescharacteristic
83#354—1271no—
84#352—1272no—
85H(5,𝔽2)C2×D12711yescharacteristic
86#137—21no—
87C4⋊Q8—2541no—
88#126—2542no—
89#125—2541no—
90#124C2×D12711yescharacteristic
91#123—1272no—
92#123—2542no—
93C22⋊Q8—2542no—
94#104—21no—
95#100C2×D12711yescharacteristic
96#99—1272no—
97#99—2542no—
98#41—21no—
99#41—41no—
100C2×Q8—2541no—
101C2×Q8C22⋊D12711yescharacteristic
102C2×D4—22no—
103C2×D4—41no—
104C2×D4C22×D12711yescharacteristic
105C22×C4—1272no—
106Q16—41no—
107SD16—41no—
108M16—21no—
109C4⋊C4—5082no—
110C4⋊C4—2542no—
111C4⋊C4—2542no—
112C4⋊C4—2541no—
113C4⋊C4—2541no—
114C4⋊C4—2542no—
115C22⋊C4—22no—
116C22⋊C4—1272no—
117C22⋊C4—5082no—
118C22⋊C4—2542no—
119C22⋊C4—2542no—
120C42—2541no—
121C42C22⋊D12711yescharacteristic
122C23—22no—
123C23D4×D12712yes—
124Q8—21no—
125Q8—41no—
126Q8—10161no—
127D4—21no—
128D4—41no—
129D4—42no—
130D4—41no—
131C2×C4—22no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—5082no—
136C2×C4—5082no—
137C2×C4—2542no—
138C2×C4—2542no—
139C2×C4C2×C22⋊D12711yescharacteristic
140C8—41no—
141C22—22no—
142C22—42no—
143C22—41no—
144C22—82no—
145C22#1686117711yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—41no—
150C4—10162no—
151C4—5082no—
152C4—5082no—
153C2—21no—
154C2—42no—
155C2—81no—
156C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#455×C127C2
socleC2×C127B(4,𝔽2)
prime radicalH(5,𝔽2)×C127C22
hypercenter#455D127
hypocenterC127#2201
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #455

Complements

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

NH ≅ G/Ntype
1. #22550626C2semidirect
3. #22549671C2semidirect
4. #22549667C2semidirect
79. C127#2201semidirect

Maximal subgroups

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

subgroupindex
#225506262
#225500662
#225496712
#225496672
#455×C1272
#2201127