Finite Group Databasebeta

← #30637352

Subgroups — #30637352

Order 16256.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D127C211yesmaximal, characteristic
2#22549931C212yesmaximal, complemented
3#22549616C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#329×C127C211yesmaximal, characteristic, complemented
6C2×Q8×D127C411yescharacteristic
7C2×D4×D127C411yescharacteristic
8#16861198C411yescharacteristic
9#16861197C2211yescharacteristic
10#16861196C411yescharacteristic
11#16861195—22no—
12C22⋊C4×D127—22no—
13C42×D127C2211yescharacteristic
14Dic127⋊C8—22no—
15(C2×D127)⋊C8—22no—
16#16861120C2212yescomplemented
17#124×C127C2211yescharacteristic
18#97×C127C2212yescomplemented
19C23×D127—21no—
20C2×Dic254—21no—
21C2×Dic254C2×C411yescharacteristic
22Q8×D127—42no—
23C2×D508C2×C411yescharacteristic
24C2×C22⋊D127—21no—
25D4×D127—42no—
26C2×C4×D127—21no—
27C2×C4×D127C2×C411yescharacteristic
28C2×C4×D127D412yes—
29C2×C127⋊C8—22no—
30C22⋊Dic127—22no—
31(C2×D127)⋊C4—22no—
32(C2×D127)⋊C4—22no—
33C4×Dic127C2×C411yescharacteristic
34C4×Dic127D412yes—
35C2×Q8×C127C2×C411yescharacteristic
36C2×D4×C127C2×C411yescharacteristic
37C2×C8×C127—22no—
38C22⋊C4×C127—22no—
39C42×C127C2311yescharacteristic
40C22×D127—21no—
41C22×D127—44no—
42C22×D127—42no—
43C22×D127C22⋊C411yescharacteristic
44Dic254—42no—
45Dic254—41no—
46D508—41no—
47C22⋊D127—42no—
48C2×Dic127—21no—
49C2×Dic127C22⋊C412yes—
50C2×Dic127C22⋊C411yescharacteristic
51C4×D127—22no—
52C4×D127—42no—
53C4×D127—42no—
54C127⋊C8—42no—
55C23×C127—21no—
56Q8×C127—41no—
57D4×C127—41no—
58C2×C4×C127—21no—
59C2×C4×C127C22×C411yescharacteristic
60C2×C4×C127C2×D412yes—
61C8×C127—42no—
62C22×C127—41no—
63C22×C127—42no—
64C22×C127C2×C22⋊C411yescharacteristic
65C2×D127—22no—
66C2×D127—41no—
67C2×D127—42no—
68C2×D127—84no—
69C2×D127#10012yes—
70C4×C127—21no—
71C4×C127—22no—
72C4×C127—41no—
73Dic127—21no—
74Dic127—22no—
75Dic127—41no—
76C2×C127—81no—
77C2×C127C2×#10011yescharacteristic
78C2×C127#42112yes—
79D127—22no—
80D127—81no—
81#1533—1271nomaximal, Sylow
82C127#153311yescharacteristic, complemented, Sylow
83C2×#124—1271no—
84#432—1272no—
85#329—1271no—
86#329—1272no—
87#329D12711yescharacteristic, complemented
88C22×Q8—1271no—
89C22×D4—1271no—
90#124—1271no—
91#124—1271no—
92#124—1271no—
93#124—2542no—
94#124C2×D12711yescharacteristic
95C2×C22⋊C4—2542no—
96C2×C42—1271no—
97C4⋊C8—2542no—
98C22⋊C8—2542no—
99#97—1272no—
100#97C2×D12712yescomplemented
101C24—2541no—
102C2×Q8—1271no—
103C2×Q8—5082no—
104C2×Q8—2541no—
105C2×Q8C4×D12711yescharacteristic
106C2×D4—1271no—
107C2×D4—5082no—
108C2×D4—2541no—
109C2×D4C4×D12711yescharacteristic
110C22×C4—1272no—
111C22×C4—1271no—
112C22×C4—2541no—
113C2×C8—22no—
114C2×C8—2542no—
115C22⋊C4—22no—
116C22⋊C4—2542no—
117C22⋊C4—2542no—
118C22⋊C4—2542no—
119C42—1271no—
120C42—1272no—
121C42C22×D12711yescharacteristic
122C23—21no—
123C23—1271no—
124C23—5082no—
125C23—5084no—
126C23—2541no—
127Q8—41no—
128Q8—5081no—
129Q8—5082no—
130D4—41no—
131D4—5081no—
132D4—5082no—
133C2×C4—21no—
134C2×C4—1272no—
135C2×C4—1271no—
136C2×C4—5082no—
137C2×C4—5082no—
138C2×C4—2542no—
139C2×C4—2541no—
140C2×C4C2×C4×D12711yescharacteristic
141C2×C4D4×D12712yes—
142C8—42no—
143C8—5082no—
144C22—41no—
145C22—42no—
146C22—1272no—
147C22—10164no—
148C22—5081no—
149C22—5082no—
150C22—2542no—
151C22C22⋊C4×D12711yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—5081no—
156C4—2541no—
157C4—2542no—
158C2—81no—
159C2—10161no—
160C2—2542no—
161C2#100×D12711yescharacteristic
162C2#2254989112yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×D4×D127C4
cosocleC42×C127C23
hypercenter#329D127
hypocenterC127#1533
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22549931C2semidirect
3. #22549616C2semidirect
4. #22549615C2semidirect
5. #329×C127C2semidirect
16. #16861120C22semidirect
18. #97×C127C22semidirect
82. C127#1533semidirect
87. #329D127semidirect
100. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#124×D1272
#225499312
#225496162
#225496152
#329×C1272
#1533127