Finite Group Databasebeta

← #30640392

Subgroups — #30640392

Order 16256.

258 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550424C211yesmaximal, characteristic
2#22549848C212yesmaximal, complemented
3#22549670C212yesmaximal, complemented
4#22549669C211yesmaximal, characteristic, complemented
5#353×C127C211yesmaximal, characteristic, complemented
6C2×#14523247C411yescharacteristic, complemented
7C2×#14523247C2211yescharacteristic
8#16861209C411yescharacteristic, complemented
9#16861196C411yescharacteristic, complemented
10#16861196C2211yescharacteristic
11#16861180—22no—
12D4×Dic127—22no—
13#16861165—22no—
14C42×D127C411yescharacteristic, complemented
15#16861124C2212yes—
16#16861123—22no—
17#127×C127C2211yescharacteristic
18#99×C127C2212yes—
19#14523247—41no—
20#14523247—41no—
21#14523247—41no—
22#14523247—41no—
23C2×Dic254C2×C411yescharacteristic
24C2×Dic254C2×C411yescharacteristic
25C2×C22⋊D127—21no—
26C2×C22⋊D127D412yes—
27C22×Dic127—21no—
28C22×Dic127D412yes—
29C2×C4×D127—21no—
30C2×C4×D127C2×C411yescharacteristic
31C4⋊Dic127—22no—
32Dic127⋊C4—42no—
33C22⋊Dic127—22no—
34C22⋊Dic127—22no—
35C22⋊Dic127—22no—
36C4×Dic127—21no—
37C4×Dic127—42no—
38C4×Dic127C2×C411yescharacteristic
39C2×D4×C127—22no—
40C2×D4×C127C2×C411yescharacteristic
41C2×D4×C127C2311yescharacteristic
42C22⋊C4×C127—22no—
43C42×C127C2×C411yescharacteristic
44C22×D127C22⋊C411yescharacteristic
45Dic254—41no—
46Dic254—41no—
47C22⋊D127—42no—
48C22⋊D127—42no—
49C2×Dic127—22no—
50C2×Dic127—22no—
51C2×Dic127—21no—
52C2×Dic127—21no—
53C2×Dic127—41no—
54C2×Dic127—41no—
55C2×Dic127—42no—
56C2×Dic127C22⋊C412yes—
57C2×Dic127C22⋊C411yescharacteristic
58C4×D127—21no—
59C4×D127—21no—
60C4×D127—42no—
61C23×C127—21no—
62C23×C127C2×D412yes—
63D4×C127—41no—
64D4×C127—41no—
65D4×C127—44no—
66C2×C4×C127—21no—
67C2×C4×C127—42no—
68C2×C4×C127C22×C411yescharacteristic
69C22×C127—22no—
70C22×C127—42no—
71C22×C127—42no—
72C22×C127—41no—
73C22×C127C2×C22⋊C411yescharacteristic
74C2×D127—21no—
75C2×D127—41no—
76C4×C127—22no—
77C4×C127—21no—
78C4×C127—82no—
79Dic127—22no—
80Dic127—41no—
81Dic127—41no—
82Dic127—82no—
83Dic127#9911yescharacteristic
84Dic127#9911yescharacteristic
85C2×C127—21no—
86C2×C127—42no—
87C2×C127—81no—
88C2×C127C2×#9911yescharacteristic
89D127—41no—
90#2126—1271nomaximal, Sylow
91C127#212611yescharacteristic, complemented, Sylow
92#532—1271no—
93#410—1272no—
94#354—1272no—
95#353—1271no—
96#353D12711yescharacteristic, complemented
97C2×#41—1271no—
98C2×#41—1271no—
99C4⋊Q8—1271no—
100#127C2×D12711yescharacteristic
101#124—1271no—
102#124—1271no—
103#121—2542no—
104C4×D4—2542no—
105#117—2542no—
106C2×C42—1271no—
107#99—1272no—
108#99—2542no—
109#99C2×D12712yes—
110#41—5081no—
111#41—5081no—
112#41—5081no—
113#41—5081no—
114C2×Q8—1271no—
115C2×Q8—1271no—
116C2×D4—22no—
117C2×D4—1272no—
118C2×D4—2541no—
119C2×D4C4×D12711yescharacteristic
120C2×D4C22×D12711yescharacteristic
121C22×C4—1272no—
122C22×C4—1271no—
123C22×C4—2541no—
124C22×C4—2541no—
125C4⋊C4—5082no—
126C4⋊C4—2542no—
127C22⋊C4—22no—
128C22⋊C4—2542no—
129C22⋊C4—2542no—
130C22⋊C4—2542no—
131C42—1271no—
132C42—5082no—
133C42—2541no—
134C42C4×D12711yescharacteristic
135C23—21no—
136C23—1271no—
137C23D4×D12712yes—
138Q8—5081no—
139Q8—5081no—
140D4—41no—
141D4—41no—
142D4—44no—
143D4—5082no—
144D4—5082no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—1271no—
148C2×C4—1272no—
149C2×C4—5082no—
150C2×C4—5081no—
151C2×C4—5081no—
152C2×C4—5082no—
153C2×C4—2541no—
154C2×C4—2541no—
155C2×C4—2542no—
156C2×C4—2542no—
157C2×C4—2541no—
158C2×C4—2541no—
159C2×C4C2×C4×D12711yescharacteristic
160C22—22no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—5081no—
165C22—2541no—
166C22C22⋊C4×D12711yescharacteristic
167C4—22no—
168C4—21no—
169C4—82no—
170C4—1271no—
171C4—1271no—
172C4—10162no—
173C4—5081no—
174C4—5081no—
175C4—2542no—
176C2—21no—
177C2—42no—
178C2—81no—
179C2—5081no—
180C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#353×C127C2
socleC2×C127C2×#99
prime radical#22550424C2
cosocleC2×D4×C127C23
hypercenter#353D127
hypocenterC127#2126
LCSG ▷ C2×C4×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
2. #22549848C2semidirect
3. #22549670C2semidirect
4. #22549669C2semidirect
5. #353×C127C2semidirect
6. C2×#14523247C4semidirect
8. #16861209C4semidirect
9. #16861196C4semidirect
14. C42×D127C4semidirect
91. C127#2126semidirect
96. #353D127semidirect

Maximal subgroups

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

subgroupindex
#225504242
#225498482
#225496702
#225496692
#353×C1272
#2126127