Finite Group Databasebeta

← #30637361

Subgroups — #30637361

Order 16256.

200 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 73 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#22549933C211yesmaximal, characteristic, complemented
2#22549933C211yesmaximal, characteristic, complemented
3#22549618C212yesmaximal, complemented
4#22549616C212yesmaximal, complemented
5#529×C127C211yesmaximal, characteristic
6C4⋊(C127⋊C8)—21no—
7C4⋊(C127⋊C8)—21no—
8C22⋊(C127⋊C8)—21no—
9C22⋊(C127⋊C8)—21no—
10#16861120C2212yes—
11#16861120C2212yes—
12#126×C127—21no—
13#125×C127C2211yescharacteristic
14#124×C127C2211yescharacteristic
15#123×C127—21no—
16C22⋊Q8×C127C411yescharacteristic
17#121×C127C411yescharacteristic
18C4×Q8×C127C411yescharacteristic
19C4×D4×C127—21no—
20C4×D4×C127C411yescharacteristic
21#117×C127—21no—
22C2×C42×C127C2211yescharacteristic
23C2×C127⋊C8—22no—
24C2×C127⋊C8—22no—
25C2×Q8×C127C2×C411yescharacteristic
26C2×D4×C127—21no—
27C2×D4×C127C2×C411yescharacteristic
28C22×C4×C127—21no—
29C22×C4×C127C2×C411yescharacteristic
30C22×C4×C127D411yescharacteristic
31C22×C4×C127D411yescharacteristic
32C4⋊C4×C127—21no—
33C4⋊C4×C127—21no—
34C4⋊C4×C127—21no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127C2×C411yescharacteristic
37C4⋊C4×C127C2×C411yescharacteristic
38C22⋊C4×C127—21no—
39C22⋊C4×C127—21no—
40C22⋊C4×C127—21no—
41C22⋊C4×C127—21no—
42C22⋊C4×C127—21no—
43C42×C127—21no—
44C42×C127C2×C411yescharacteristic
45C42×C127D411yescharacteristic
46C42×C127D411yescharacteristic
47C42×C127C2311yescharacteristic
48C127⋊C8—42no—
49C127⋊C8—42no—
50C23×C127—21no—
51C23×C127C22⋊C411yescharacteristic
52Q8×C127—41no—
53D4×C127—41no—
54D4×C127—42no—
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—21no—
58C2×C4×C127—41no—
59C2×C4×C127—41no—
60C2×C4×C127—41no—
61C2×C4×C127—42no—
62C2×C4×C127C22⋊C411yescharacteristic
63C2×C4×C127C22⋊C411yescharacteristic
64C2×C4×C127C22⋊C411yescharacteristic
65C2×C4×C127C22×C411yescharacteristic
66C2×C4×C127C2×D411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C22×C127—21no—
69C22×C127—21no—
70C22×C127—21no—
71C22×C127—41no—
72C22×C127—41no—
73C22×C127—41no—
74C22×C127C2×C22⋊C411yescharacteristic
75C4×C127—21no—
76C4×C127—21no—
77C4×C127—21no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—41no—
82C4×C127—41no—
83C4×C127—41no—
84C2×C127—41no—
85C2×C127—81no—
86C2×C127#41311yescharacteristic
87C2×C127#42111yescharacteristic
88C2×C127#42111yescharacteristic
89#1535—1271nomaximal, Sylow
90C127#153511yescharacteristic, complemented, Sylow
91#529D12711yescharacteristic
92#432—1271no—
93#432—1271no—
94#330—1272no—
95#329—1272no—
96#126—21no—
97#125C2×D12711yescharacteristic
98#124C2×D12711yescharacteristic
99#123—21no—
100C22⋊Q8Dic12711yescharacteristic
101#121Dic12711yescharacteristic
102C4×Q8Dic12711yescharacteristic
103C4×D4—21no—
104C4×D4Dic12711yescharacteristic
105#117—21no—
106C2×C42C2×D12711yescharacteristic
107C4⋊C8—2541no—
108C4⋊C8—2541no—
109C22⋊C8—2541no—
110C22⋊C8—2541no—
111#97—1272no—
112#97—1272no—
113C2×Q8C2×Dic12711yescharacteristic
114C2×D4—21no—
115C2×D4C2×Dic12711yescharacteristic
116C22×C4—21no—
117C22×C4C2×Dic12711yescharacteristic
118C22×C4C22⋊D12711yescharacteristic
119C22×C4C22⋊D12711yescharacteristic
120C2×C8—2542no—
121C2×C8—2542no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4C2×Dic12711yescharacteristic
127C4⋊C4C2×Dic12711yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C42—21no—
134C42C2×Dic12711yescharacteristic
135C42C22⋊D12711yescharacteristic
136C42C22⋊D12711yescharacteristic
137C42C22×D12711yescharacteristic
138C23—21no—
139C23C22⋊Dic12711yescharacteristic
140Q8—41no—
141D4—41no—
142D4—42no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4C22⋊Dic12711yescharacteristic
151C2×C4C22⋊Dic12711yescharacteristic
152C2×C4C22⋊Dic12711yescharacteristic
153C2×C4C22×Dic12711yescharacteristic
154C2×C4C2×C22⋊D12711yescharacteristic
155C2×C4C2×C22⋊D12711yescharacteristic
156C8—5082no—
157C8—5082no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22C2×C22⋊Dic12711yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—41no—
174C2—41no—
175C2—81no—
176C2#2254986011yescharacteristic
177C2#2254989311yescharacteristic
178C2#2254989311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#529×C127C2
socleC22×C127C2×C22⋊C4
prime radical#121×C127C4
cosocleC42×C127C23
hypercenter#529D127
hypocenterC127#1535
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #22549933C2semidirect
2. #22549933C2semidirect
3. #22549618C2semidirect
4. #22549616C2semidirect
90. C127#1535semidirect

Maximal subgroups

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

subgroupindex
#225499332
#225496182
#225496162
#529×C1272
#1535127