Finite Group Databasebeta

← #30642008

Subgroups — #30642008

Order 16256.

840 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×D4×Dic127C213yesmaximal, complemented
2C2×(C22⋊D127)⋊C4C213yesmaximal, complemented
3#22549813C211yesmaximal, characteristic, complemented
4#22549812C213yesmaximal, complemented
5#22549784C213yesmaximal, complemented
6#22549721C211yesmaximal, characteristic, complemented
7#392×C127C211yesmaximal, characteristic, complemented
8C22×C22⋊D127C2213yes—
9C23×Dic127C2213yes—
10C22×C4×D127C2211yescharacteristic
11D4×Dic127—212no—
12(C22⋊D127)⋊C4—212no—
13C2×C4⋊Dic127C2213yes—
14C2×Dic127⋊C4C2213yes—
15C2×C22⋊Dic127C2216yes—
16C2×(C2×D127)⋊C4C2213yes—
17C2×C4×Dic127C2213yes—
18C127⋊#95C2213yes—
19C22×D4×C127C2213yes—
20C2×C22⋊C4×C127C2213yes—
21#95×C127C2211yescharacteristic
22C23×D127C2311yescharacteristic
23C2×C22⋊D127—26no—
24C2×C22⋊D127—26no—
25C2×C22⋊D127—26no—
26C22×Dic127—26no—
27C22×Dic127—212no—
28C22×Dic127C2313yes—
29C22×Dic127C2313yes—
30C22×Dic127C2311yescharacteristic
31C2×C4×D127—26no—
32C4⋊Dic127—26no—
33Dic127⋊C4—26no—
34C22⋊Dic127—212no—
35(C2×D127)⋊C4—26no—
36C4×Dic127D4112yescomplemented
37C24×C127C2313yes—
38C2×D4×C127—212no—
39C2×D4×C127—26no—
40C22×C4×C127C2311yescharacteristic
41C22×C4×C127C2313yes—
42C22⋊C4×C127—26no—
43C22×D127—23no—
44C22×D127—23no—
45C22×D127#4112yes—
46C22⋊D127—224no—
47C2×Dic127—23no—
48C2×Dic127—23no—
49C2×Dic127—23no—
50C2×Dic127—26no—
51C2×Dic127—412no—
52C2×Dic127C2×D416yes—
53C2×Dic127C2×D416yes—
54C4×D127—44no—
55C23×C127—23no—
56C23×C127—26no—
57C23×C127—26no—
58C23×C127—23no—
59C23×C127#4116yes—
60C23×C127C2411yescharacteristic
61D4×C127—224no—
62C2×C4×C127—26no—
63C2×C4×C127—23no—
64C2×C4×C127C2×D416yes—
65C22×C127—212no—
66C22×C127—26no—
67C22×C127—43no—
68C22×C127—46no—
69C22×C127—43no—
70C22×C127C22×D413yes—
71C22×C127C2×#4111yescharacteristic
72C22×C127C2×#4113yes—
73C2×D127—26no—
74C2×D127—43no—
75C2×D127—41no—
76C4×C127—26no—
77C4×C127—41no—
78Dic127—26no—
79Dic127—43no—
80Dic127#12714yes—
81C2×C127—46no—
82C2×C127#52513yes—
83C2×C127#52513yes—
84C2×C127C2×#12711yescharacteristic
85D127—42no—
86#2397—1271nomaximal, Sylow
87C127#239711yescharacteristic, complemented, Sylow
88C2×C4×D4—1273no—
89C2×C4×D4—1273no—
90#399—1273no—
91#399—1271no—
92#392—1273no—
93#392D12711yescharacteristic, complemented
94#376—1271no—
95C22×D4—1273no—
96C22×D4C2×D12713yes—
97C23×C4—1273no—
98C23×C4—1271no—
99C4×D4—25412no—
100C4×D4—25412no—
101C2×C4⋊C4—1273no—
102C2×C4⋊C4—1273no—
103C2×C22⋊C4—1276no—
104C2×C22⋊C4—1273no—
105C2×C22⋊C4C2×D12713yes—
106C2×C42—1273no—
107#95—1273no—
108#95C2×D12711yescharacteristic
109C24—1271no—
110C24C22×D12713yes—
111C2×D4—212no—
112C2×D4—26no—
113C2×D4—2546no—
114C2×D4—2546no—
115C2×D4—2546no—
116C22×C4—1271no—
117C22×C4—1273no—
118C22×C4—1273no—
119C22×C4—2546no—
120C22×C4—2546no—
121C22×C4—25412no—
122C22×C4C22×D12711yescharacteristic
123C22×C4C22×D12713yes—
124C4⋊C4—2546no—
125C4⋊C4—2546no—
126C22⋊C4—26no—
127C22⋊C4—25412no—
128C22⋊C4—2546no—
129C42—12712no—
130C23—23no—
131C23—26no—
132C23—23no—
133C23—26no—
134C23—1272no—
135C23—2543no—
136C23—2543no—
137C23#1452324716yes—
138C23C23×D12711yescharacteristic
139D4—224no—
140D4—25424no—
141C2×C4—26no—
142C2×C4—23no—
143C2×C4—1276no—
144C2×C4—1276no—
145C2×C4—5084no—
146C2×C4—50812no—
147C2×C4—2546no—
148C2×C4—2543no—
149C2×C4—2543no—
150C2×C4—2543no—
151C2×C4D4×D12716yes—
152C22—212no—
153C22—26no—
154C22—43no—
155C22—46no—
156C22—43no—
157C22—5083no—
158C22—5081no—
159C22—2546no—
160C22C2×D4×D12713yes—
161C22#41×D12711yescharacteristic
162C22C2×#1452324713yes—
163C4—26no—
164C4—41no—
165C4—1274no—
166C4—5083no—
167C4—2546no—
168C2—46no—
169C2—5082no—
170C2#2255038213yes—
171C2#2255038613yes—
172C2#127×D12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#392×C127C2
hypercenter#392D127
hypocenterC127#2397
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic127C2semidirect
2. C2×(C22⋊D127)⋊C4C2semidirect
3. #22549813C2semidirect
4. #22549812C2semidirect
5. #22549784C2semidirect
6. #22549721C2semidirect
7. #392×C127C2semidirect
36. C4×Dic127D4semidirect
87. C127#2397semidirect
93. #392D127semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1272
C2×(C22⋊D127)⋊C42
#225498132
#225498122
#225497842
#225497212
#392×C1272
#2397127