Finite Group Databasebeta

← #30640403

Subgroups — #30640403

Order 16256.

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

normal subgroups only

This group has 238 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#22550446C211yesmaximal, characteristic
2#22549848C212yesmaximal
3#22549671C211yesmaximal, characteristic, complemented
4#22549670C212yesmaximal, complemented
5#354×C127C211yesmaximal, characteristic, complemented
6C2×#14523247C2211yescharacteristic
7C2×Q8×D127C411yescharacteristic, complemented
8#16861209C411yescharacteristic, complemented
9#16861207C411yescharacteristic, complemented
10#16861196C2211yescharacteristic
11#16861191—22no—
12#16861186—22no—
13#16861166C411yescharacteristic, complemented
14#16861165—22no—
15#16861124C2212yes—
16#16861123—22no—
17#124×C127C2211yescharacteristic
18#99×C127C2212yes—
19#14523247—42no—
20C2×Dic254—21no—
21C2×Dic254C2×C411yescharacteristic
22C2×Dic254C2×C411yescharacteristic
23Q8×D127—42no—
24C2×C22⋊D127D412yes—
25C22×Dic127D412yes—
26C2×C4×D127—21no—
27C2×C4×D127C2×C411yescharacteristic
28C4⋊Dic127—22no—
29Dic127⋊C4—21no—
30Dic127⋊C4—22no—
31Dic127⋊C4—42no—
32C22⋊Dic127—22no—
33C22⋊Dic127—22no—
34(C2×D127)⋊C4—21no—
35C4×Dic127—42no—
36C4×Dic127C2×C411yescharacteristic
37C2×Q8×C127C2×C411yescharacteristic
38C2×D4×C127C2311yescharacteristic
39C22⋊C4×C127—22no—
40C22⋊C4×C127—22no—
41C42×C127C2×C411yescharacteristic
42C22×D127C22⋊C411yescharacteristic
43Dic254—41no—
44Dic254—42no—
45Dic254—41no—
46C22⋊D127—42no—
47C2×Dic127—22no—
48C2×Dic127—22no—
49C2×Dic127—21no—
50C2×Dic127—21no—
51C2×Dic127—42no—
52C2×Dic127C22⋊C412yes—
53C2×Dic127C22⋊C411yescharacteristic
54C4×D127—22no—
55C4×D127—42no—
56C23×C127C2×D412yes—
57Q8×C127—41no—
58D4×C127—41no—
59C2×C4×C127—21no—
60C2×C4×C127—21no—
61C2×C4×C127—42no—
62C2×C4×C127C22×C411yescharacteristic
63C22×C127—22no—
64C22×C127—42no—
65C22×C127C2×C22⋊C411yescharacteristic
66C2×D127—21no—
67C2×D127—41no—
68C4×C127—21no—
69C4×C127—41no—
70C4×C127—41no—
71C4×C127—82no—
72Dic127—22no—
73Dic127—41no—
74Dic127—41no—
75Dic127—82no—
76Dic127#9912yes—
77C2×C127—21no—
78C2×C127—42no—
79C2×C127C2×#9911yescharacteristic
80D127—41no—
81#2129—1271nomaximal, Sylow
82C127#212911yescharacteristic, complemented, Sylow
83#536—1271no—
84#410—1272no—
85#354—1271no—
86#354—1272no—
87#354D12711yescharacteristic, complemented
88C2×#41—1271no—
89C22×Q8—1271no—
90C4⋊Q8—1271no—
91C4⋊Q8—1271no—
92#124—1271no—
93#124C2×D12711yescharacteristic
94#123—2542no—
95C22⋊Q8—2542no—
96#117—1271no—
97#117—2542no—
98#99—1272no—
99#99—2542no—
100#99C2×D12712yes—
101#41—5082no—
102C2×Q8—1271no—
103C2×Q8—1271no—
104C2×Q8—5082no—
105C2×Q8—2541no—
106C2×Q8C4×D12711yescharacteristic
107C2×D4—1272no—
108C2×D4C22×D12711yescharacteristic
109C22×C4—1272no—
110C22×C4—1271no—
111C22×C4—2541no—
112C4⋊C4—5082no—
113C4⋊C4—2541no—
114C4⋊C4—2542no—
115C4⋊C4—2542no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—2541no—
119C22⋊C4—2542no—
120C22⋊C4—2542no—
121C42—1271no—
122C42—5082no—
123C42C4×D12711yescharacteristic
124C23—1271no—
125C23D4×D12712yes—
126Q8—41no—
127Q8—5081no—
128Q8—5081no—
129Q8—5082no—
130D4—41no—
131D4—5082no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—1271no—
136C2×C4—1272no—
137C2×C4—5082no—
138C2×C4—5082no—
139C2×C4—2542no—
140C2×C4—2542no—
141C2×C4—2542no—
142C2×C4—2541no—
143C2×C4—2541no—
144C2×C4C2×C4×D12711yescharacteristic
145C22—22no—
146C22—42no—
147C22—5081no—
148C22—2541no—
149C22C22⋊C4×D12711yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—1272no—
155C4—10162no—
156C4—5081no—
157C4—5081no—
158C4—2542no—
159C2—21no—
160C2—42no—
161C2—5081no—
162C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#354×C127C2
socleC2×C127C2×#99
prime radicalC2×#14523247C22
cosocleC2×D4×C127C23
hypercenter#354D127
hypocenterC127#2129
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #22549671C2semidirect
4. #22549670C2semidirect
5. #354×C127C2semidirect
7. C2×Q8×D127C4semidirect
8. #16861209C4semidirect
9. #16861207C4semidirect
13. #16861166C4semidirect
82. C127#2129semidirect
87. #354D127semidirect

Maximal subgroups

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

subgroupindex
#225504462
#225498482
#225496712
#225496702
#354×C1272
#2129127