Finite Group Databasebeta

← #392×D127 #30641924

Subgroups — #392×D127 #30641924

Order 16256.

1812 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 223 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
1C22×D4×D127C213yesmaximal, complemented
2C2×C22⋊C4×D127C213yesmaximal, complemented
3#22549786C211yesmaximal, characteristic, complemented
4#22549785C213yesmaximal, complemented
5#22549784C213yesmaximal, complemented
6#95×D127C211yesmaximal, characteristic, complemented
7#392×C127C211yesmaximal, characteristic, complemented
8C24×D127C2213yes—
9C22×D508C2213yescomplemented
10C22×C22⋊D127C2216yescomplemented
11C2×D4×D127—224no—
12C2×D4×D127—212no—
13C22×C4×D127C2211yescharacteristic
14C22×C4×D127C2213yes—
15C2×C22⋊Dic127C2213yescomplemented
16C2×(C2×D127)⋊C4C2213yescomplemented
17C2×(C2×D127)⋊C4C2213yescomplemented
18C22⋊C4×D127—212no—
19C127⋊#95C2213yescomplemented
20C22×D4×C127C2213yescomplemented
21C2×C22⋊C4×C127C2213yescomplemented
22#95×C127C2211yescharacteristic, complemented
23C23×D127—212no—
24C23×D127—212no—
25C23×D127—26no—
26C23×D127—26no—
27C23×D127D4112yescomplemented
28C23×D127C2313yes—
29C23×D127C2311yescharacteristic
30C2×D508—212no—
31C2×D508—26no—
32C2×C22⋊D127—212no—
33C2×C22⋊D127—212no—
34C2×C22⋊D127—212no—
35D4×D127—296no—
36C22×Dic127C2311yescharacteristic
37C22×Dic127C2313yes—
38C2×C4×D127—212no—
39C2×C4×D127—26no—
40C2×C4×D127D4112yescomplemented
41C22⋊Dic127—26no—
42(C2×D127)⋊C4—26no—
43(C2×D127)⋊C4—26no—
44C24×C127C2313yes—
45C2×D4×C127—212no—
46C2×D4×C127—26no—
47C22×C4×C127C2311yescharacteristic
48C22×C4×C127C2313yes—
49C22⋊C4×C127—26no—
50C22×D127—23no—
51C22×D127—26no—
52C22×D127—26no—
53C22×D127—23no—
54C22×D127—248no—
55C22×D127—224no—
56C22×D127—412no—
57C22×D127—424no—
58C22×D127—412no—
59C22×D127C2×D416yes—
60C22×D127C2×D416yes—
61C22×D127C2×D416yescomplemented
62C22×D127#4112yes—
63D508—224no—
64C22⋊D127—248no—
65C2×Dic127—26no—
66C2×Dic127—23no—
67C2×Dic127C2×D416yescomplemented
68C4×D127—224no—
69C4×D127—44no—
70C23×C127—23no—
71C23×C127—26no—
72C23×C127—26no—
73C23×C127—23no—
74C23×C127C2×D416yescomplemented
75C23×C127C2411yescharacteristic
76D4×C127—224no—
77C2×C4×C127—26no—
78C2×C4×C127—23no—
79C2×C4×C127C2×D416yescomplemented
80C22×C127—212no—
81C22×C127—26no—
82C22×C127—43no—
83C22×C127—46no—
84C22×C127—43no—
85C22×C127C22×D413yes—
86C22×C127C22×D413yes—
87C22×C127C2×#4111yescharacteristic
88C2×D127—212no—
89C2×D127—26no—
90C2×D127—43no—
91C2×D127—46no—
92C2×D127—43no—
93C2×D127—448no—
94C2×D127#120112yes—
95C2×D127#121112yes—
96C2×D127#12714yes—
97C4×C127—26no—
98C4×C127—41no—
99Dic127—26no—
100Dic127—41no—
101C2×C127—46no—
102C2×C127C2×#12013yes—
103C2×C127C2×#12113yes—
104C2×C127C2×#12711yescharacteristic
105D127—46no—
106D127#39218yesdirect product factor
107C2×#392—1271nomaximal, Sylow
108C127C2×#39211yescharacteristic, complemented, Sylow
109C23×D4—1273no—
110C22×C22⋊C4—1273no—
111#392—1273no—
112#392—1273no—
113#392—1271no—
114#392D12711yescharacteristic, direct product factor
115C2×#95—1271no—
116C25—1273no—
117C22×D4—1276no—
118C22×D4—1273no—
119C22×D4—25412no—
120C22×D4—25424no—
121C22×D4C2×D12713yescomplemented
122C23×C4—1273no—
123C23×C4—1271no—
124C2×C22⋊C4—1273no—
125C2×C22⋊C4—1273no—
126C2×C22⋊C4—1273no—
127C2×C22⋊C4—25412no—
128C2×C22⋊C4C2×D12713yescomplemented
129#95—1273no—
130#95C2×D12711yescharacteristic, complemented
131C24—1271no—
132C24—12712no—
133C24—1273no—
134C24—2546no—
135C24—25412no—
136C24—25412no—
137C24—2546no—
138C24C22×D12713yes—
139C2×D4—212no—
140C2×D4—26no—
141C2×D4—25412no—
142C2×D4—25412no—
143C2×D4—25496no—
144C2×D4—25412no—
145C2×D4—25412no—
146C2×D4—2546no—
147C22×C4—1273no—
148C22×C4—12712no—
149C22×C4—1271no—
150C22×C4—2546no—
151C22×C4—25412no—
152C22×C4C22×D12711yescharacteristic
153C22×C4C22×D12713yes—
154C22⋊C4—26no—
155C22⋊C4—2546no—
156C22⋊C4—2546no—
157C22⋊C4—2546no—
158C23—23no—
159C23—26no—
160C23—26no—
161C23—23no—
162C23—1276no—
163C23—1276no—
164C23—1272no—
165C23—1276no—
166C23—50812no—
167C23—50824no—
168C23—50812no—
169C23—25448no—
170C23—25424no—
171C23—2543no—
172C23—2546no—
173C23—2546no—
174C23—2543no—
175C23D4×D12716yescomplemented
176C23C23×D12711yescharacteristic
177D4—224no—
178D4—25424no—
179D4—25448no—
180C2×C4—26no—
181C2×C4—23no—
182C2×C4—1276no—
183C2×C4—5084no—
184C2×C4—25424no—
185C2×C4—2543no—
186C2×C4—2546no—
187C2×C4D4×D12716yescomplemented
188C22—212no—
189C22—26no—
190C22—43no—
191C22—46no—
192C22—43no—
193C22—12712no—
194C22—12712no—
195C22—1274no—
196C22—50848no—
197C22—5083no—
198C22—5086no—
199C22—5083no—
200C22—25412no—
201C22—2546no—
202C22C2×D4×D12713yes—
203C22C2×D4×D12713yes—
204C22#41×D12711yescharacteristic
205C4—26no—
206C4—41no—
207C4—5081no—
208C4—2546no—
209C2—46no—
210C2—1278no—
211C2—5086no—
212C2#120×D12713yes—
213C2#121×D12713yes—
214C2#127×D12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D127
commutator, socle, cosocleC23×C127C24
Fitting#392×C127C2
hypercenter#392D127
hypocenterC127C2×#392
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×D4×D127C2semidirect
2. C2×C22⋊C4×D127C2semidirect
3. #22549786C2semidirect
4. #22549785C2semidirect
5. #22549784C2semidirect
6. #95×D127C2semidirect
7. #392×C127C2semidirect
9. C22×D508C22semidirect
10. C22×C22⋊D127C22semidirect
15. C2×C22⋊Dic127C22semidirect
16. C2×(C2×D127)⋊C4C22semidirect
17. C2×(C2×D127)⋊C4C22semidirect
19. C127⋊#95C22semidirect
20. C22×D4×C127C22semidirect
21. C2×C22⋊C4×C127C22semidirect
22. #95×C127C22semidirect
27. C23×D127D4semidirect
40. C2×C4×D127D4semidirect
61. C22×D127C2×D4semidirect
67. C2×Dic127C2×D4semidirect
74. C23×C127C2×D4semidirect
79. C2×C4×C127C2×D4semidirect
106. D127#392direct (N × H)
108. C127C2×#392semidirect
114. #392D127direct (N × H)
121. C22×D4C2×D127semidirect
128. C2×C22⋊C4C2×D127semidirect
130. #95C2×D127semidirect
175. C23D4×D127semidirect
187. C2×C4D4×D127semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1272
C2×C22⋊C4×D1272
#225497862
#225497852
#225497842
#95×D1272
#392×C1272
C2×#392127