Finite Group Databasebeta

← C127×#2199 #30635151

Subgroups — C127×#2199 #30635151

Order 16256.

270 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 56 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
1H(3,𝔽4)×C127C211yesmaximal, characteristic, complemented
2B(4,𝔽2)×C127C212yesmaximal, complemented
3#455×C127C211yesmaximal, characteristic, complemented
4#354×C127C211yesmaximal, characteristic, complemented
5AΣL(1,𝔽16)×C127C212yesmaximal, complemented
6H(5,𝔽2)×C127C2211yescharacteristic
7#137×C127—21no—
8#124×C127—21no—
9#124×C127—22no—
10#124×C127—21no—
11#124×C127C2211yescharacteristic
12#120×C127—22no—
13#120×C127—22no—
14#120×C127C2212yes—
15#104×C127—21no—
16#100×C127C2211yescharacteristic
17#99×C127—22no—
18#99×C127C2212yes—
19C24×C127D412yescomplemented
20#41×C127—21no—
21#41×C127—41no—
22C2×Q8×C127—21no—
23C2×Q8×C127D411yescharacteristic, complemented
24C2×D4×C127—22no—
25C2×D4×C127—21no—
26C2×D4×C127—21no—
27C2×D4×C127—22no—
28C2×D4×C127—41no—
29C2×D4×C127—42no—
30C2×D4×C127C2311yescharacteristic
31Q16×C127—41no—
32SD16×C127—41no—
33M16×C127—21no—
34C22⋊C4×C127—22no—
35C22⋊C4×C127—22no—
36C22⋊C4×C127—22no—
37C22⋊C4×C127—22no—
38C22⋊C4×C127—22no—
39C22⋊C4×C127—42no—
40C22⋊C4×C127D412yescomplemented
41C42×C127—21no—
42C42×C127D411yescharacteristic, complemented
43C23×C127—22no—
44C23×C127—22no—
45C23×C127—42no—
46C23×C127—82no—
47C23×C127C2×D412yes—
48Q8×C127—21no—
49Q8×C127—41no—
50Q8×C127—81no—
51D4×C127—21no—
52D4×C127—41no—
53D4×C127—42no—
54D4×C127—41no—
55D4×C127—82no—
56D4×C127—81no—
57D4×C127—81no—
58D4×C127—84no—
59C2×C4×C127—22no—
60C2×C4×C127—21no—
61C2×C4×C127—22no—
62C2×C4×C127—21no—
63C2×C4×C127—41no—
64C2×C4×C127—42no—
65C2×C4×C127C2×D411yescharacteristic
66C8×C127—41no—
67C22×C127—22no—
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—42no—
71C22×C127—44no—
72C22×C127—41no—
73C22×C127—42no—
74C22×C127—82no—
75C22×C127—82no—
76C22×C127#12011yescharacteristic
77C4×C127—21no—
78C4×C127—42no—
79C4×C127—41no—
80C4×C127—41no—
81C4×C127—41no—
82C4×C127—82no—
83C2×C127—21no—
84C2×C127—42no—
85C2×C127—82no—
86C2×C127—81no—
87C2×C127B(4,𝔽2)11yescharacteristic
88#2199C12711yesmaximal, characteristic, direct product factor, Sylow
89C127#219911yescharacteristic, direct product factor, Sylow
90H(3,𝔽4)C2×C12711yescharacteristic, complemented
91B(4,𝔽2)C2×C12712yescomplemented
92#455C2×C12711yescharacteristic, complemented
93#354C2×C12711yescharacteristic, complemented
94AΣL(1,𝔽16)C2×C12712yescomplemented
95H(5,𝔽2)C22×C12711yescharacteristic
96#137—21no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C22×C12711yescharacteristic
101#120—22no—
102#120—22no—
103#120C22×C12712yes—
104#104—21no—
105#100C22×C12711yescharacteristic
106#99—22no—
107#99C22×C12712yes—
108C24D4×C12712yescomplemented
109#41—21no—
110#41—41no—
111C2×Q8—21no—
112C2×Q8D4×C12711yescharacteristic, complemented
113C2×D4—22no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—22no—
117C2×D4—41no—
118C2×D4—42no—
119C2×D4C23×C12711yescharacteristic
120Q16—41no—
121SD16—41no—
122M16—21no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—42no—
129C22⋊C4D4×C12712yescomplemented
130C42—21no—
131C42D4×C12711yescharacteristic, complemented
132C23—22no—
133C23—22no—
134C23—42no—
135C23—82no—
136C23C2×D4×C12712yes—
137Q8—21no—
138Q8—41no—
139Q8—81no—
140D4—21no—
141D4—41no—
142D4—42no—
143D4—41no—
144D4—82no—
145D4—81no—
146D4—81no—
147D4—84no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4C2×D4×C12711yescharacteristic
155C8—41no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—41no—
162C22—42no—
163C22—82no—
164C22—82no—
165C22#120×C12711yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—82no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—81no—
176C2B(4,𝔽2)×C12711yescharacteristic
subgroupG/H
center, socleC2×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C127
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽4)×C127C2semidirect
2. B(4,𝔽2)×C127C2semidirect
3. #455×C127C2semidirect
4. #354×C127C2semidirect
5. AΣL(1,𝔽16)×C127C2semidirect
19. C24×C127D4semidirect
23. C2×Q8×C127D4semidirect
40. C22⋊C4×C127D4semidirect
42. C42×C127D4semidirect
88. #2199C127direct (N × H)
89. C127#2199direct (N × H)
90. H(3,𝔽4)C2×C127semidirect
91. B(4,𝔽2)C2×C127semidirect
92. #455C2×C127semidirect
93. #354C2×C127semidirect
94. AΣL(1,𝔽16)C2×C127semidirect
108. C24D4×C127semidirect
112. C2×Q8D4×C127semidirect
129. C22⋊C4D4×C127semidirect
131. C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽4)×C1272
B(4,𝔽2)×C1272
#455×C1272
#354×C1272
AΣL(1,𝔽16)×C1272
#2199127