Finite Group Databasebeta

← C127×#2328 #30635280

Subgroups — C127×#2328 #30635280

Order 16256.

740 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 296 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×#127×C127C211yesmaximal, characteristic
2C2×C4×D4×C127C212yesmaximal, complemented
3#390×C127C212yesmaximal, complemented
4#386×C127C218yesmaximal, complemented
5#382×C127C211yesmaximal, characteristic, complemented
6#376×C127C211yesmaximal, characteristic, complemented
7C22×D4×C127C2212yes—
8C22×D4×C127C2214yes—
9C23×C4×C127C2214yes—
10#127×C127C418yescomplemented
11C4×D4×C127—28no—
12C2×C4⋊C4×C127C2212yes—
13C2×C22⋊C4×C127C2214yes—
14C2×C22⋊C4×C127C2218yes—
15C2×C42×C127C2211yescharacteristic
16C2×C42×C127C2211yescharacteristic
17C2×C42×C127C2211yescharacteristic
18#95×C127C2214yes—
19#95×C127C2214yes—
20C24×C127C2314yes—
21C2×D4×C127—24no—
22C2×D4×C127—24no—
23C2×D4×C127—48no—
24C2×D4×C127C2×C418yescomplemented
25C2×D4×C127C2×C4116yescomplemented
26C22×C4×C127—28no—
27C22×C4×C127—48no—
28C22×C4×C127C2312yes—
29C22×C4×C127C2314yes—
30C22×C4×C127C2312yes—
31C22×C4×C127C2312yes—
32C22×C4×C127C2311yescharacteristic
33C4⋊C4×C127—24no—
34C22⋊C4×C127—28no—
35C22⋊C4×C127—48no—
36C42×C127—22no—
37C42×C127C2×C414yescomplemented
38C42×C127D414yescomplemented
39C23×C127—24no—
40C23×C127—24no—
41C23×C127—28no—
42C23×C127—28no—
43C23×C127C22×C418yes—
44C23×C127C2411yescharacteristic
45D4×C127—216no—
46D4×C127—416no—
47C2×C4×C127—24no—
48C2×C4×C127—24no—
49C2×C4×C127—24no—
50C2×C4×C127—22no—
51C2×C4×C127—24no—
52C2×C4×C127—28no—
53C2×C4×C127—416no—
54C2×C4×C127C22×C414yes—
55C2×C4×C127C22×C412yes—
56C2×C4×C127C2×D414yes—
57C2×C4×C127C2×D412yes—
58C2×C4×C127#4112yes—
59C22×C127—28no—
60C22×C127—216no—
61C22×C127—44no—
62C22×C127—44no—
63C22×C127—48no—
64C22×C127C23×C411yescharacteristic
65C22×C127C22×D411yescharacteristic
66C22×C127C2×#4111yescharacteristic
67C22×C127H(5,𝔽2)14yes—
68C4×C127—24no—
69C4×C127—24no—
70C4×C127—44no—
71C4×C127—42no—
72C4×C127C4×D414yes—
73C2×C127—48no—
74C2×C127C2×C4×D411yescharacteristic
75C2×C127#51812yes—
76C2×C127#53512yes—
77C2×C127#54012yes—
78#2328C12711yesmaximal, characteristic, direct product factor, Sylow
79C127#232811yescharacteristic, direct product factor, Sylow
80C2×#127C2×C12711yescharacteristic
81C2×C4×D4C2×C12712yescomplemented
82#390C2×C12712yescomplemented
83#386C2×C12718yescomplemented
84#382C2×C12711yescharacteristic, complemented
85#376C2×C12711yescharacteristic, complemented
86C22×D4C22×C12712yes—
87C22×D4C22×C12714yes—
88C23×C4C22×C12714yes—
89#127C4×C12718yescomplemented
90C4×D4—28no—
91C2×C4⋊C4C22×C12712yes—
92C2×C22⋊C4C22×C12714yes—
93C2×C22⋊C4C22×C12718yes—
94C2×C42C22×C12711yescharacteristic
95C2×C42C22×C12711yescharacteristic
96C2×C42C22×C12711yescharacteristic
97#95C22×C12714yes—
98#95C22×C12714yes—
99C24C23×C12714yes—
100C2×D4—24no—
101C2×D4—24no—
102C2×D4—48no—
103C2×D4C2×C4×C12718yescomplemented
104C2×D4C2×C4×C127116yescomplemented
105C22×C4—28no—
106C22×C4—48no—
107C22×C4C23×C12712yes—
108C22×C4C23×C12714yes—
109C22×C4C23×C12712yes—
110C22×C4C23×C12712yes—
111C22×C4C23×C12711yescharacteristic
112C4⋊C4—24no—
113C22⋊C4—28no—
114C22⋊C4—48no—
115C42—22no—
116C42C2×C4×C12714yescomplemented
117C42D4×C12714yescomplemented
118C23—24no—
119C23—24no—
120C23—28no—
121C23—28no—
122C23C22×C4×C12718yes—
123C23C24×C12711yescharacteristic
124D4—216no—
125D4—416no—
126C2×C4—24no—
127C2×C4—24no—
128C2×C4—24no—
129C2×C4—22no—
130C2×C4—24no—
131C2×C4—28no—
132C2×C4—416no—
133C2×C4C22×C4×C12714yes—
134C2×C4C22×C4×C12712yes—
135C2×C4C2×D4×C12714yes—
136C2×C4C2×D4×C12712yes—
137C2×C4#41×C12712yes—
138C22—28no—
139C22—216no—
140C22—44no—
141C22—44no—
142C22—48no—
143C22C23×C4×C12711yescharacteristic
144C22C22×D4×C12711yescharacteristic
145C22C2×#41×C12711yescharacteristic
146C22H(5,𝔽2)×C12714yes—
147C4—24no—
148C4—24no—
149C4—44no—
150C4—42no—
151C4C4×D4×C12714yes—
152C2—48no—
153C2C2×C4×D4×C12711yescharacteristic
154C2#518×C12712yes—
155C2#535×C12712yes—
156C2#540×C12712yes—
subgroupG/H
center, socleC23×C127C24
commutatorC22C23×C4×C127
Frattini, cosocleC23C24×C127
prime radicalC2×#127×C127C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×D4×C127C2semidirect
3. #390×C127C2semidirect
4. #386×C127C2semidirect
5. #382×C127C2semidirect
6. #376×C127C2semidirect
10. #127×C127C4semidirect
24. C2×D4×C127C2×C4semidirect
25. C2×D4×C127C2×C4semidirect
37. C42×C127C2×C4semidirect
38. C42×C127D4semidirect
78. #2328C127direct (N × H)
79. C127#2328direct (N × H)
81. C2×C4×D4C2×C127semidirect
82. #390C2×C127semidirect
83. #386C2×C127semidirect
84. #382C2×C127semidirect
85. #376C2×C127semidirect
89. #127C4×C127semidirect
103. C2×D4C2×C4×C127semidirect
104. C2×D4C2×C4×C127semidirect
116. C42C2×C4×C127semidirect
117. C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#127×C1272
C2×C4×D4×C1272
#390×C1272
#386×C1272
#382×C1272
#376×C1272
#2328127