Finite Group Databasebeta

← #30637337

Subgroups — #30637337

Order 16256.

216 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 77 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
2C4×#14523232C211yesmaximal, characteristic, complemented
3#22549618C212yesmaximal, complemented
4#22549616C212yesmaximal, complemented
5#529×C127C211yesmaximal, characteristic
6C2×#14523232—21no—
7C4⋊(C127⋊C8)—21no—
8C22⋊(C127⋊C8)—21no—
9#16861120C2212yes—
10#16861120C2212yes—
11C4×C127⋊C8—21no—
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
23#14523232—24no—
24C2×C127⋊C8—22no—
25C2×C127⋊C8—22no—
26C2×Q8×C127C2×C411yescharacteristic
27C2×D4×C127—21no—
28C2×D4×C127C2×C411yescharacteristic
29C22×C4×C127—21no—
30C22×C4×C127C2×C411yescharacteristic
31C22×C4×C127D411yescharacteristic
32C22×C4×C127D411yescharacteristic
33C4⋊C4×C127—21no—
34C4⋊C4×C127—21no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—21no—
37C4⋊C4×C127C2×C411yescharacteristic
38C4⋊C4×C127C2×C411yescharacteristic
39C22⋊C4×C127—21no—
40C22⋊C4×C127—21no—
41C22⋊C4×C127—21no—
42C22⋊C4×C127—21no—
43C22⋊C4×C127—21no—
44C42×C127—21no—
45C42×C127C2×C411yescharacteristic
46C42×C127D411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127C2311yescharacteristic
49C127⋊C8—24no—
50C127⋊C8—42no—
51C23×C127—21no—
52C23×C127C22⋊C411yescharacteristic
53Q8×C127—41no—
54D4×C127—41no—
55D4×C127—42no—
56C2×C4×C127—22no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127—42no—
63C2×C4×C127C22⋊C411yescharacteristic
64C2×C4×C127C22⋊C411yescharacteristic
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22×C411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C22×C127—21no—
70C22×C127—21no—
71C22×C127—21no—
72C22×C127—41no—
73C22×C127—41no—
74C22×C127—41no—
75C22×C127C2×C22⋊C411yescharacteristic
76C4×C127—21no—
77C4×C127—21no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—41no—
82C4×C127—41no—
83C4×C127—41no—
84C4×C127#10412yes—
85C2×C127—41no—
86C2×C127—81no—
87C2×C127#41311yescharacteristic
88C2×C127C2×#10411yescharacteristic
89C2×C127#42111yescharacteristic
90#1529—1271nomaximal, Sylow
91C127#152911yescharacteristic, complemented, Sylow
92#529D12711yescharacteristic
93#432—1271no—
94C4×M16—1271no—
95#330—1272no—
96#329—1272no—
97C2×M16—2541no—
98#126—21no—
99#125C2×D12711yescharacteristic
100#124C2×D12711yescharacteristic
101#123—21no—
102C22⋊Q8Dic12711yescharacteristic
103#121Dic12711yescharacteristic
104C4×Q8Dic12711yescharacteristic
105C4×D4—21no—
106C4×D4Dic12711yescharacteristic
107#117—21no—
108C2×C42C2×D12711yescharacteristic
109C4⋊C8—2541no—
110C22⋊C8—2541no—
111#97—1272no—
112#97—1272no—
113C4×C8—2541no—
114C2×Q8C2×Dic12711yescharacteristic
115C2×D4—21no—
116C2×D4C2×Dic12711yescharacteristic
117C22×C4—21no—
118C22×C4C2×Dic12711yescharacteristic
119C22×C4C22⋊D12711yescharacteristic
120C22×C4C22⋊D12711yescharacteristic
121M16—2544no—
122C2×C8—2542no—
123C2×C8—2542no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4C2×Dic12711yescharacteristic
129C4⋊C4C2×Dic12711yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C42—21no—
136C42C2×Dic12711yescharacteristic
137C42C22⋊D12711yescharacteristic
138C42C22⋊D12711yescharacteristic
139C42C22×D12711yescharacteristic
140C23—21no—
141C23C22⋊Dic12711yescharacteristic
142Q8—41no—
143D4—41no—
144D4—42no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4C22⋊Dic12711yescharacteristic
153C2×C4C22⋊Dic12711yescharacteristic
154C2×C4C22⋊Dic12711yescharacteristic
155C2×C4C22×Dic12711yescharacteristic
156C2×C4C2×C22⋊D12711yescharacteristic
157C2×C4C2×C22⋊D12711yescharacteristic
158C8—5082no—
159C8—2544no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22C2×C22⋊Dic12711yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—41no—
173C4—41no—
174C4—41no—
175C4#1686113612yes—
176C2—41no—
177C2—81no—
178C2#2254986011yescharacteristic
179C2C2×#1686113611yescharacteristic
180C2#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#1529
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. C4×#14523232C2semidirect
3. #22549618C2semidirect
4. #22549616C2semidirect
91. C127#1529semidirect

Maximal subgroups

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

subgroupindex
#225499332
C4×#145232322
#225496182
#225496162
#529×C1272
#1529127