Finite Group Databasebeta

← C2×#509×C127 #30636364

Subgroups — C2×#509×C127 #30636364

Order 16256.

400 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 184 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×#135×C127C211yesmaximal, characteristic, complemented
2C22×D8×C127C211yesmaximal, characteristic, complemented
3#509×C127C218yesmaximal, direct product factor
4C2×SD32×C127C212yesmaximal, complemented
5C2×D16×C127C212yesmaximal, complemented
6C2×M32×C127C211yesmaximal, characteristic, complemented
7C2×#41×C127—21no—
8C22×D4×C127—21no—
9#135×C127—22no—
10#135×C127C2214yescomplemented
11C2×Q16×C127C2211yescharacteristic, complemented
12C2×SD16×C127—21no—
13C2×D8×C127—21no—
14C2×D8×C127—22no—
15C2×D8×C127C2212yes—
16C2×D8×C127C2214yescomplemented
17C2×D8×C127C2211yescharacteristic, complemented
18C22×C8×C127C2211yescharacteristic
19SD32×C127C2218yescomplemented
20D16×C127C2218yescomplemented
21M32×C127C2214yescomplemented
22C2×C16×C127C2212yescomplemented
23C24×C127—41no—
24#41×C127—24no—
25#41×C127—42no—
26C2×Q8×C127—21no—
27C2×D4×C127—22no—
28C2×D4×C127—24no—
29C2×D4×C127—21no—
30C2×D4×C127—41no—
31C2×D4×C127—42no—
32C2×D4×C127—41no—
33C22×C4×C127—41no—
34C22×C4×C127D411yescharacteristic
35Q16×C127—21no—
36Q16×C127C2312yescomplemented
37SD16×C127—24no—
38D8×C127—22no—
39D8×C127—24no—
40D8×C127—21no—
41D8×C127C2314yes—
42D8×C127C2312yescomplemented
43C2×C8×C127D411yescharacteristic
44C2×C8×C127D412yes—
45C2×C8×C127C2311yescharacteristic
46C2×C8×C127C2312yes—
47C16×C127C2314yescomplemented
48C23×C127—44no—
49C23×C127—41no—
50C23×C127—42no—
51C23×C127—84no—
52C23×C127D811yescharacteristic
53Q8×C127—22no—
54Q8×C127—41no—
55D4×C127—24no—
56D4×C127—22no—
57D4×C127—44no—
58D4×C127—44no—
59D4×C127—42no—
60D4×C127—41no—
61C2×C4×C127—41no—
62C2×C4×C127—44no—
63C2×C4×C127D812yes—
64C2×C4×C127D811yescharacteristic
65C2×C4×C127C2×D412yes—
66C2×C4×C127C2×D411yescharacteristic
67C8×C127C2×D411yescharacteristic
68C8×C127C2×D412yes—
69C8×C127C2411yescharacteristic
70C22×C127—22no—
71C22×C127—42no—
72C22×C127—44no—
73C22×C127—88no—
74C22×C127—82no—
75C22×C127—84no—
76C22×C127C2×D812yes—
77C22×C127C2×D811yescharacteristic
78C4×C127—42no—
79C4×C127C2×D812yes—
80C4×C127C2×D811yescharacteristic
81C4×C127C22×D411yescharacteristic
82C2×C127—22no—
83C2×C127—82no—
84C2×C127—84no—
85C2×C127#50912yesdirect product factor
86C2×C127C22×D811yescharacteristic
87C2×#509C12711yesmaximal, characteristic, direct product factor, Sylow
88C127C2×#50911yescharacteristic, direct product factor, Sylow
89C2×#135C2×C12711yescharacteristic, complemented
90C22×D8C2×C12711yescharacteristic, complemented
91#509C2×C12718yesdirect product factor
92C2×SD32C2×C12712yescomplemented
93C2×D16C2×C12712yescomplemented
94C2×M32C2×C12711yescharacteristic, complemented
95C2×#41—21no—
96C22×D4—21no—
97#135—22no—
98#135C22×C12714yescomplemented
99C2×Q16C22×C12711yescharacteristic, complemented
100C2×SD16—21no—
101C2×D8—21no—
102C2×D8—22no—
103C2×D8C22×C12712yes—
104C2×D8C22×C12714yescomplemented
105C2×D8C22×C12711yescharacteristic, complemented
106C22×C8C22×C12711yescharacteristic
107SD32C22×C12718yescomplemented
108D16C22×C12718yescomplemented
109M32C22×C12714yescomplemented
110C2×C16C22×C12712yescomplemented
111C24—41no—
112#41—24no—
113#41—42no—
114C2×Q8—21no—
115C2×D4—22no—
116C2×D4—24no—
117C2×D4—21no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—42no—
121C22×C4—41no—
122C22×C4D4×C12711yescharacteristic
123Q16—21no—
124Q16C23×C12712yescomplemented
125SD16—24no—
126D8—22no—
127D8—24no—
128D8—21no—
129D8C23×C12714yes—
130D8C23×C12712yescomplemented
131C2×C8D4×C12711yescharacteristic
132C2×C8D4×C12712yes—
133C2×C8C23×C12711yescharacteristic
134C2×C8C23×C12712yes—
135C16C23×C12714yescomplemented
136C23—41no—
137C23—44no—
138C23—42no—
139C23—84no—
140C23D8×C12711yescharacteristic
141Q8—22no—
142Q8—41no—
143D4—24no—
144D4—22no—
145D4—44no—
146D4—44no—
147D4—42no—
148D4—41no—
149C2×C4—41no—
150C2×C4—44no—
151C2×C4D8×C12711yescharacteristic
152C2×C4D8×C12712yes—
153C2×C4C2×D4×C12711yescharacteristic
154C2×C4C2×D4×C12712yes—
155C8C2×D4×C12711yescharacteristic
156C8C2×D4×C12712yes—
157C8C24×C12711yescharacteristic
158C22—22no—
159C22—42no—
160C22—44no—
161C22—88no—
162C22—84no—
163C22—82no—
164C22C2×D8×C12712yes—
165C22C2×D8×C12711yescharacteristic
166C4—42no—
167C4C2×D8×C12712yes—
168C4C2×D8×C12711yescharacteristic
169C4C22×D4×C12711yescharacteristic
170C2—22no—
171C2—84no—
172C2—82no—
173C2#509×C12712yesdirect product factor
174C2C22×D8×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×D8
commutator, Frattini, cosocleC8C24×C127
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#135×C127C2semidirect
2. C22×D8×C127C2semidirect
3. #509×C127C2direct (N × H)
4. C2×SD32×C127C2semidirect
5. C2×D16×C127C2semidirect
6. C2×M32×C127C2semidirect
10. #135×C127C22semidirect
11. C2×Q16×C127C22semidirect
16. C2×D8×C127C22semidirect
17. C2×D8×C127C22semidirect
19. SD32×C127C22semidirect
20. D16×C127C22semidirect
21. M32×C127C22semidirect
22. C2×C16×C127C22semidirect
36. Q16×C127C23semidirect
42. D8×C127C23semidirect
47. C16×C127C23semidirect
85. C2×C127#509direct (N × H)
87. C2×#509C127direct (N × H)
88. C127C2×#509direct (N × H)
89. C2×#135C2×C127semidirect
90. C22×D8C2×C127semidirect
91. #509C2×C127direct (N × H)
92. C2×SD32C2×C127semidirect
93. C2×D16C2×C127semidirect
94. C2×M32C2×C127semidirect
98. #135C22×C127semidirect
99. C2×Q16C22×C127semidirect
104. C2×D8C22×C127semidirect
105. C2×D8C22×C127semidirect
107. SD32C22×C127semidirect
108. D16C22×C127semidirect
109. M32C22×C127semidirect
110. C2×C16C22×C127semidirect
124. Q16C23×C127semidirect
130. D8C23×C127semidirect
135. C16C23×C127semidirect
173. C2#509×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#135×C1272
C22×D8×C1272
#509×C1272
C2×SD32×C1272
C2×D16×C1272
C2×M32×C1272
C2×#509127