Finite Group Databasebeta

← #30640745

Subgroups — #30640745

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#22550067C212yesmaximal
3#22550053C211yesmaximal, characteristic
4#22549665C212yesmaximal, complemented
5#354×C127C211yesmaximal, characteristic, complemented
6#16861254C2211yescharacteristic
7#16861240—21no—
8#16861207—21no—
9#16861202—22no—
10#16861201—21no—
11#16861193—22no—
12#16861190C2212yes—
13#16861186—22no—
14#16861137—21no—
15#16861126C2211yescharacteristic
16#16861123—22no—
17#124×C127C2211yescharacteristic
18#99×C127C2212yes—
19#14523248—21no—
20#14523247—41no—
21C2×Dic254—21no—
22C2×C22⋊D127—22no—
23D4×D127—41no—
24C22×Dic127D412yes—
25#14523238—41no—
26#14523235—41no—
27#14523232—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—42no—
31Dic127⋊C4—22no—
32Dic127⋊C4—22no—
33Dic127⋊C4—22no—
34C22⋊Dic127—22no—
35C22⋊Dic127D412yes—
36(C2×D127)⋊C4—42no—
37C4×Dic127—21no—
38C2×Q8×C127D411yescharacteristic
39C2×D4×C127C2311yescharacteristic
40C22⋊C4×C127—22no—
41C22⋊C4×C127—22no—
42C42×C127D411yescharacteristic
43C22×D127—22no—
44Dic254—21no—
45Dic254—81no—
46D508—21no—
47C22⋊D127—42no—
48C22⋊D127—41no—
49C2×Dic127—22no—
50C2×Dic127—22no—
51C2×Dic127—22no—
52C2×Dic127—42no—
53C4×D127—41no—
54C127⋊C8—41no—
55C23×C127C2×D412yes—
56Q8×C127—41no—
57D4×C127—41no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—42no—
61C2×C4×C127C2×D411yescharacteristic
62C22×C127—22no—
63C22×C127—42no—
64C22×C127#12011yescharacteristic
65C2×D127—41no—
66C2×D127—82no—
67C4×C127—21no—
68C4×C127—41no—
69C4×C127—41no—
70C4×C127—82no—
71Dic127—42no—
72Dic127—41no—
73Dic127—42no—
74C2×C127—21no—
75C2×C127—42no—
76C2×C127B(4,𝔽2)11yescharacteristic
77D127—81no—
78#2201—1271nomaximal, Sylow
79C127#220111yescharacteristic, complemented, Sylow
80#563—1271no—
81#458—1272no—
82#455—1271no—
83#354D12711yescharacteristic, complemented
84#352—1272no—
85H(5,𝔽2)—1271no—
86#137—2541no—
87C4⋊Q8—2541no—
88#126—2542no—
89#125—2541no—
90#124C2×D12711yescharacteristic
91#123—1272no—
92#123—2542no—
93C22⋊Q8—2542no—
94#104—2541no—
95#100—1271no—
96#99—2542no—
97#99C2×D12712yes—
98#41—5081no—
99#41—2541no—
100C2×Q8—2541no—
101C2×Q8D50811yescharacteristic
102C2×D4—5081no—
103C2×D4—2542no—
104C2×D4C22×D12711yescharacteristic
105C22×C4—1272no—
106Q16—5081no—
107SD16—5081no—
108M16—2541no—
109C4⋊C4—5082no—
110C4⋊C4—2542no—
111C4⋊C4—2542no—
112C4⋊C4—2541no—
113C4⋊C4—2541no—
114C4⋊C4—2542no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—1272no—
118C22⋊C4—5082no—
119C22⋊C4—2542no—
120C42—2541no—
121C42D50811yescharacteristic
122C23—2542no—
123C23D4×D12712yes—
124Q8—41no—
125Q8—10161no—
126Q8—2541no—
127D4—41no—
128D4—5082no—
129D4—5081no—
130D4—2541no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—42no—
134C2×C4—5081no—
135C2×C4—5082no—
136C2×C4—2542no—
137C2×C4—2542no—
138C2×C4—2542no—
139C2×C4C2×D50811yescharacteristic
140C8—5081no—
141C22—22no—
142C22—42no—
143C22—10162no—
144C22—5081no—
145C22#1686117911yescharacteristic
146C4—21no—
147C4—41no—
148C4—41no—
149C4—82no—
150C4—5082no—
151C4—5081no—
152C4—5082no—
153C2—21no—
154C2—42no—
155C2—10161no—
156C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#354×C127C2
socleC2×C127B(4,𝔽2)
prime radical#16861254C22
hypercenter#354D127
hypocenterC127#2201
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
1. #22550626C2semidirect
4. #22549665C2semidirect
5. #354×C127C2semidirect
79. C127#2201semidirect
83. #354D127semidirect

Maximal subgroups

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

subgroupindex
#225506262
#225500672
#225500532
#225496652
#354×C1272
#2201127