Finite Group Databasebeta

← #30640728

Subgroups — #30640728

Order 16256.

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

normal subgroups only

This group has 246 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#22550612C211yesmaximal, characteristic, complemented
2#22550067C212yesmaximal, complemented
3#22550043C211yesmaximal, characteristic
4#22549665C212yesmaximal, complemented
5#353×C127C211yesmaximal, characteristic, complemented
6#16861254C2211yescharacteristic
7#16861235—21no—
8#16861196—22no—
9#16861193—22no—
10#16861190—21no—
11#16861190C2212yes—
12C4×Dic254—21no—
13D4×Dic127—22no—
14#16861137—21no—
15#16861126C2211yescharacteristic
16#16861123—22no—
17#127×C127C2211yescharacteristic
18#99×C127C2212yes—
19#14523248—21no—
20#14523247—41no—
21C2×Dic254—21no—
22C2×C22⋊D127—22no—
23D4×D127—41no—
24C22×Dic127—21no—
25C22×Dic127D412yes—
26#14523236—41no—
27#14523233—41no—
28#14523232—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—42no—
31Dic127⋊C4—22no—
32C22⋊Dic127—21no—
33C22⋊Dic127—22no—
34C22⋊Dic127—22no—
35C22⋊Dic127D412yes—
36(C2×D127)⋊C4—42no—
37C4×Dic127—22no—
38C2×D4×C127—22no—
39C2×D4×C127D411yescharacteristic
40C2×D4×C127C2311yescharacteristic
41C22⋊C4×C127—22no—
42C42×C127D411yescharacteristic
43C22×D127—22no—
44Dic254—21no—
45Dic254—42no—
46D508—21no—
47C22⋊D127—42no—
48C22⋊D127—41no—
49C2×Dic127—22no—
50C2×Dic127—22no—
51C2×Dic127—22no—
52C2×Dic127—42no—
53C2×Dic127—81no—
54C4×D127—41no—
55C127⋊C8—41no—
56C23×C127—21no—
57C23×C127C2×D412yes—
58D4×C127—41no—
59D4×C127—41no—
60D4×C127—44no—
61C2×C4×C127—21no—
62C2×C4×C127—42no—
63C2×C4×C127C2×D411yescharacteristic
64C22×C127—22no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127—41no—
68C22×C127#12011yescharacteristic
69C2×D127—41no—
70C2×D127—82no—
71C4×C127—22no—
72C4×C127—21no—
73C4×C127—82no—
74Dic127—42no—
75Dic127—41no—
76Dic127—42no—
77C2×C127—21no—
78C2×C127—42no—
79C2×C127—81no—
80C2×C127B(4,𝔽2)11yescharacteristic
81D127—81no—
82#2198—1271nomaximal, Sylow
83C127#219811yescharacteristic, complemented, Sylow
84#559—1271no—
85#458—1272no—
86#453—1271no—
87#353D12711yescharacteristic, complemented
88#352—1272no—
89H(5,𝔽2)—1271no—
90AGL(1,ℤ/8)—2541no—
91#127C2×D12711yescharacteristic
92#124—2542no—
93#123—1272no—
94#123—2542no—
95#123—2541no—
96C4×Q8—2541no—
97C4×D4—2542no—
98#104—2541no—
99#100—1271no—
100#99—2542no—
101#99C2×D12712yes—
102#41—5081no—
103#41—2541no—
104C2×Q8—2541no—
105C2×D4—22no—
106C2×D4—5081no—
107C2×D4—2542no—
108C2×D4D50811yescharacteristic
109C2×D4C22×D12711yescharacteristic
110C22×C4—1272no—
111C22×C4—2541no—
112SD16—5081no—
113D8—5081no—
114M16—2541no—
115C4⋊C4—5082no—
116C4⋊C4—2541no—
117C4⋊C4—2542no—
118C22⋊C4—22no—
119C22⋊C4—1272no—
120C22⋊C4—5082no—
121C22⋊C4—2542no—
122C22⋊C4—2541no—
123C22⋊C4—2542no—
124C42—2542no—
125C42D50811yescharacteristic
126C23—21no—
127C23—2542no—
128C23D4×D12712yes—
129Q8—5082no—
130Q8—2541no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—5082no—
135D4—5081no—
136D4—2541no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—10161no—
140C2×C4—5081no—
141C2×C4—5082no—
142C2×C4—2542no—
143C2×C4—2542no—
144C2×C4—2542no—
145C2×C4C2×D50811yescharacteristic
146C8—5081no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—10162no—
152C22—5081no—
153C22#1686117911yescharacteristic
154C4—22no—
155C4—21no—
156C4—82no—
157C4—5082no—
158C4—5082no—
159C4—5081no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—10161no—
164C2#2255006411yescharacteristic
subgroupG/H
centerC2#22550064
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#353×C127C2
socleC2×C127B(4,𝔽2)
prime radical#22550043C2
hypercenter#353D127
hypocenterC127#2198
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #22550612C2semidirect
2. #22550067C2semidirect
4. #22549665C2semidirect
5. #353×C127C2semidirect
83. C127#2198semidirect
87. #353D127semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225500672
#225500432
#225496652
#353×C1272
#2198127