Finite Group Databasebeta

← #30640725

Subgroups — #30640725

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#22550066C212yesmaximal, complemented
3#22549669C211yesmaximal, characteristic, complemented
4#22549667C212yesmaximal, complemented
5#453×C127C211yesmaximal, characteristic
6#16861196—22no—
7#16861190—22no—
8#16861190—21no—
9#16861190C2212yes—
10C4×Dic254—21no—
11D4×Dic127—22no—
12#16861124—22no—
13#16861124C2212yes—
14H(5,𝔽2)×C127C2211yescharacteristic
15AGL(1,ℤ/8)×C127—21no—
16#127×C127C2211yescharacteristic
17#104×C127—21no—
18#100×C127C2211yescharacteristic
19C2×Dic254—21no—
20C22×Dic127—21no—
21C22×Dic127D412yes—
22C4⋊Dic127—21no—
23Dic127⋊C4—22no—
24Dic127⋊C4—42no—
25C22⋊Dic127—21no—
26C22⋊Dic127—22no—
27C22⋊Dic127—22no—
28C22⋊Dic127—22no—
29C22⋊Dic127—42no—
30C22⋊Dic127D412yes—
31C4×Dic127—22no—
32#41×C127—21no—
33#41×C127—41no—
34C2×D4×C127—22no—
35C2×D4×C127—22no—
36C2×D4×C127—41no—
37C2×D4×C127D411yescharacteristic
38C2×D4×C127C2311yescharacteristic
39SD16×C127—41no—
40D8×C127—41no—
41M16×C127—21no—
42C42×C127D411yescharacteristic
43Dic254—42no—
44C2×Dic127—22no—
45C2×Dic127—22no—
46C2×Dic127—42no—
47C2×Dic127—42no—
48C2×Dic127—81no—
49C23×C127—22no—
50C23×C127—21no—
51C23×C127C2×D412yes—
52Q8×C127—21no—
53D4×C127—21no—
54D4×C127—41no—
55D4×C127—41no—
56D4×C127—44no—
57D4×C127—42no—
58D4×C127—41no—
59C2×C4×C127—21no—
60C2×C4×C127—22no—
61C2×C4×C127—41no—
62C2×C4×C127C2×D411yescharacteristic
63C8×C127—41no—
64C22×C127—22no—
65C22×C127—42no—
66C22×C127—42no—
67C22×C127—41no—
68C22×C127—41no—
69C22×C127—82no—
70C22×C127#12011yescharacteristic
71C4×C127—22no—
72C4×C127—21no—
73C4×C127—41no—
74Dic127—42no—
75Dic127—42no—
76Dic127—82no—
77C2×C127—21no—
78C2×C127—42no—
79C2×C127—81no—
80C2×C127—81no—
81C2×C127B(4,𝔽2)11yescharacteristic
82#2198—1271nomaximal, Sylow
83C127#219811yescharacteristic, complemented, Sylow
84#559—1271no—
85#458—1272no—
86#453D12711yescharacteristic
87#353—1271no—
88#352—1272no—
89H(5,𝔽2)C2×D12711yescharacteristic
90AGL(1,ℤ/8)—21no—
91#127C2×D12711yescharacteristic
92#124—2542no—
93#123—1272no—
94#123—2542no—
95#123—2541no—
96C4×Q8—2541no—
97C4×D4—2542no—
98#104—21no—
99#100C2×D12711yescharacteristic
100#99—1272no—
101#99—2542no—
102#41—21no—
103#41—41no—
104C2×Q8—2541no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—41no—
108C2×D4C22⋊D12711yescharacteristic
109C2×D4C22×D12711yescharacteristic
110C22×C4—1272no—
111C22×C4—2541no—
112SD16—41no—
113D8—41no—
114M16—21no—
115C4⋊C4—5082no—
116C4⋊C4—2541no—
117C4⋊C4—2542no—
118C22⋊C4—1272no—
119C22⋊C4—5082no—
120C22⋊C4—2542no—
121C22⋊C4—2542no—
122C22⋊C4—2541no—
123C22⋊C4—2542no—
124C42—2542no—
125C42C22⋊D12711yescharacteristic
126C23—22no—
127C23—21no—
128C23D4×D12712yes—
129Q8—21no—
130Q8—5082no—
131D4—21no—
132D4—41no—
133D4—41no—
134D4—44no—
135D4—42no—
136D4—41no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—10161no—
141C2×C4—5082no—
142C2×C4—5082no—
143C2×C4—2542no—
144C2×C4—2542no—
145C2×C4C2×C22⋊D12711yescharacteristic
146C8—41no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—82no—
153C22#1686117711yescharacteristic
154C4—22no—
155C4—21no—
156C4—41no—
157C4—10162no—
158C4—5082no—
159C4—5082no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—81no—
164C2#2255006211yescharacteristic
subgroupG/H
centerC2#22550062
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting, prime radical#453×C127C2
socleC2×C127B(4,𝔽2)
hypercenter#453D127
hypocenterC127#2198
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #22550612C2semidirect
2. #22550066C2semidirect
3. #22549669C2semidirect
4. #22549667C2semidirect
83. C127#2198semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225500662
#225496692
#225496672
#453×C1272
#2198127