Finite Group Databasebeta

← #30640743

Subgroups — #30640743

Order 16256.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550065C212yesmaximal
2#22550053C211yesmaximal, characteristic
3#22549671C211yesmaximal, characteristic, complemented
4#22549666C212yesmaximal, complemented
5#563×C127C211yesmaximal, characteristic, complemented
6#16861254C2211yescharacteristic
7#16861240—21no—
8#16861192—22no—
9#16861137—21no—
10#16861126C2211yescharacteristic
11#16861124C2212yes—
12#16861123—22no—
13C4⋊Q8×C127—21no—
14#126×C127—22no—
15#125×C127—21no—
16#124×C127C2211yescharacteristic
17#123×C127C2212yes—
18C22⋊Q8×C127—22no—
19#14523248—21no—
20#14523247—41no—
21C2×C22⋊D127—22no—
22D4×D127—41no—
23#14523238—41no—
24#14523235—41no—
25#14523232—21no—
26Dic127⋊C4—42no—
27C22⋊Dic127—22no—
28(C2×D127)⋊C4—42no—
29C2×Q8×C127—21no—
30C2×Q8×C127D411yescharacteristic
31C2×D4×C127C2311yescharacteristic
32C22×C4×C127D412yes—
33C4⋊C4×C127—21no—
34C4⋊C4×C127—22no—
35C4⋊C4×C127—22no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C22⋊C4×C127—22no—
40C22⋊C4×C127D412yes—
41C42×C127—21no—
42C42×C127D411yescharacteristic
43C22×D127—22no—
44Dic254—21no—
45D508—21no—
46C22⋊D127—42no—
47C22⋊D127—41no—
48C2×Dic127—22no—
49C2×Dic127—42no—
50C4×D127—41no—
51C127⋊C8—41no—
52C23×C127C2×D412yes—
53Q8×C127—41no—
54Q8×C127—81no—
55D4×C127—41no—
56C2×C4×C127—21no—
57C2×C4×C127—22no—
58C2×C4×C127—21no—
59C2×C4×C127—22no—
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—42no—
69C4×C127—41no—
70C4×C127—42no—
71C4×C127—41no—
72Dic127—41no—
73Dic127—82no—
74C2×C127—21no—
75C2×C127—42no—
76C2×C127B(4,𝔽2)11yescharacteristic
77D127—81no—
78#2201—1271nomaximal, Sylow
79C127#220111yescharacteristic, complemented, Sylow
80#563D12711yescharacteristic, complemented
81#458—1272no—
82#455—1271no—
83#354—1271no—
84#352—1272no—
85H(5,𝔽2)—1271no—
86#137—2541no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D12711yescharacteristic
91#123—2542no—
92#123C2×D12712yes—
93C22⋊Q8—22no—
94#104—2541no—
95#100—1271no—
96#99—1272no—
97#99—2542no—
98#41—5081no—
99#41—2541no—
100C2×Q8—21no—
101C2×Q8C22⋊D12711yescharacteristic
102C2×D4—5081no—
103C2×D4—2542no—
104C2×D4C22×D12711yescharacteristic
105C22×C4C22⋊D12712yes—
106Q16—5081no—
107SD16—5081no—
108M16—2541no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—5082no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—5082no—
118C22⋊C4—2542no—
119C22⋊C4C22⋊D12712yes—
120C42—21no—
121C42C22⋊D12711yescharacteristic
122C23—2542no—
123C23C2×C22⋊D12712yes—
124Q8—41no—
125Q8—81no—
126Q8—2541no—
127D4—41no—
128D4—5082no—
129D4—5081no—
130D4—2541no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—5081no—
137C2×C4—5082no—
138C2×C4—2542no—
139C2×C4C2×C22⋊D12711yescharacteristic
140C8—5081no—
141C22—22no—
142C22—42no—
143C22—10162no—
144C22—5081no—
145C22#1686117811yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—10162no—
152C4—5081no—
153C2—21no—
154C2—42no—
155C2—10161no—
156C2#2255006311yescharacteristic
subgroupG/H
centerC2#22550063
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#563×C127C2
socleC2×C127B(4,𝔽2)
prime radical#16861254C22
hypercenter#563D127
hypocenterC127#2201
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #22549671C2semidirect
4. #22549666C2semidirect
5. #563×C127C2semidirect
79. C127#2201semidirect
80. #563D127semidirect

Maximal subgroups

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

subgroupindex
#225500652
#225500532
#225496712
#225496662
#563×C1272
#2201127