Finite Group Databasebeta

← C127×#1651 #30634603

Subgroups — C127×#1651 #30634603

Order 16256.

234 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 76 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
1D4×Q8×C127C211yesmaximal, characteristic
2#498×C127C211yesmaximal, characteristic, complemented
3#475×C127C211yesmaximal, characteristic, complemented
4#461×C127C211yesmaximal, characteristic
5#332×C127C211yesmaximal, characteristic, complemented
6Q8⋊C8×C127C211yesmaximal, characteristic, complemented
7D4⋊C8×C127C211yesmaximal, characteristic
8C22×Q8×C127—21no—
9C2×SD16×C127—21no—
10C4⋊Q8×C127—21no—
11C4⋊Q8×C127C2211yescharacteristic
12C4⋊Q8×C127C2211yescharacteristic
13C22⋊Q8×C127—21no—
14C22⋊Q8×C127—22no—
15C4×Q8×C127C2211yescharacteristic
16C4×D4×C127—21no—
17C4×D4×C127C2211yescharacteristic
18#106×C127—22no—
19#106×C127—21no—
20C4⋊C8×C127C2211yescharacteristic
21C4⋊C8×C127C2211yescharacteristic
22#103×C127—21no—
23#103×C127—21no—
24C4×C8×C127C2211yescharacteristic
25C2×Q8×C127—22no—
26C2×Q8×C127—21no—
27C2×Q8×C127—21no—
28C2×Q8×C127—42no—
29C2×Q8×C127—42no—
30C2×Q8×C127D411yescharacteristic
31C2×D4×C127D411yescharacteristic
32C22×C4×C127—22no—
33C22×C4×C127—21no—
34SD16×C127—42no—
35C2×C8×C127—21no—
36C2×C8×C127—21no—
37C2×C8×C127—21no—
38C4⋊C4×C127—22no—
39C4⋊C4×C127—21no—
40C4⋊C4×C127—21no—
41C4⋊C4×C127—21no—
42C4⋊C4×C127—21no—
43C4⋊C4×C127D411yescharacteristic
44C4⋊C4×C127D411yescharacteristic
45C4⋊C4×C127D412yes—
46C22⋊C4×C127—22no—
47C22⋊C4×C127—21no—
48C42×C127—21no—
49C42×C127C2311yescharacteristic
50C23×C127—21no—
51Q8×C127—21no—
52Q8×C127—44no—
53Q8×C127—41no—
54Q8×C127—41no—
55Q8×C127—41no—
56Q8×C127—41no—
57Q8×C127SD1612yescomplemented
58D4×C127—21no—
59D4×C127SD1612yes—
60C2×C4×C127—21no—
61C2×C4×C127—21no—
62C2×C4×C127—22no—
63C2×C4×C127—21no—
64C2×C4×C127—44no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—41no—
68C2×C4×C127C2×D411yescharacteristic
69C2×C4×C127C2×D411yescharacteristic
70C2×C4×C127C2×D411yescharacteristic
71C8×C127—22no—
72C8×C127—41no—
73C8×C127—41no—
74C22×C127—22no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127#12011yescharacteristic
78C4×C127—22no—
79C4×C127—21no—
80C4×C127—42no—
81C4×C127—41no—
82C4×C127—41no—
83C4×C127—81no—
84C4×C127C2×SD1611yescharacteristic
85C4×C127C2×SD1611yescharacteristic
86C4×C127AGL(1,ℤ/8)11yescharacteristic
87C4×C127#13711yescharacteristic
88C2×C127—42no—
89C2×C127#44811yescharacteristic
90C2×C127#45011yescharacteristic
91C2×C127#45611yescharacteristic
92#1651C12711yesmaximal, characteristic, direct product factor, Sylow
93C127#165111yescharacteristic, direct product factor, Sylow
94D4×Q8C2×C12711yescharacteristic
95#498C2×C12711yescharacteristic, complemented
96#475C2×C12711yescharacteristic, complemented
97#461C2×C12711yescharacteristic
98#332C2×C12711yescharacteristic, complemented
99Q8⋊C8C2×C12711yescharacteristic, complemented
100D4⋊C8C2×C12711yescharacteristic
101C22×Q8—21no—
102C2×SD16—21no—
103C4⋊Q8—21no—
104C4⋊Q8C22×C12711yescharacteristic
105C4⋊Q8C22×C12711yescharacteristic
106C22⋊Q8—21no—
107C22⋊Q8—22no—
108C4×Q8C22×C12711yescharacteristic
109C4×D4—21no—
110C4×D4C22×C12711yescharacteristic
111#106—21no—
112#106—22no—
113C4⋊C8C22×C12711yescharacteristic
114C4⋊C8C22×C12711yescharacteristic
115#103—21no—
116#103—21no—
117C4×C8C22×C12711yescharacteristic
118C2×Q8—22no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×Q8D4×C12711yescharacteristic
124C2×D4D4×C12711yescharacteristic
125C22×C4—22no—
126C22×C4—21no—
127SD16—42no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4D4×C12711yescharacteristic
137C4⋊C4D4×C12711yescharacteristic
138C4⋊C4D4×C12712yes—
139C22⋊C4—22no—
140C22⋊C4—21no—
141C42—21no—
142C42C23×C12711yescharacteristic
143C23—21no—
144Q8—21no—
145Q8—44no—
146Q8—41no—
147Q8—41no—
148Q8—41no—
149Q8—41no—
150Q8SD16×C12712yescomplemented
151D4—21no—
152D4SD16×C12712yes—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—44no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4C2×D4×C12711yescharacteristic
162C2×C4C2×D4×C12711yescharacteristic
163C2×C4C2×D4×C12711yescharacteristic
164C8—22no—
165C8—41no—
166C8—41no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22#120×C12711yescharacteristic
171C4—22no—
172C4—21no—
173C4—42no—
174C4—41no—
175C4—41no—
176C4—81no—
177C4C2×SD16×C12711yescharacteristic
178C4C2×SD16×C12711yescharacteristic
179C4AGL(1,ℤ/8)×C12711yescharacteristic
180C4#137×C12711yescharacteristic
181C2—42no—
182C2#448×C12711yescharacteristic
183C2#450×C12711yescharacteristic
184C2#456×C12711yescharacteristic
subgroupG/H
center, socleC22×C127#120
commutator, Frattini, cosocleC42C23×C127
prime radicalC2×D4×C127D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. #498×C127C2semidirect
3. #475×C127C2semidirect
5. #332×C127C2semidirect
6. Q8⋊C8×C127C2semidirect
57. Q8×C127SD16semidirect
92. #1651C127direct (N × H)
93. C127#1651direct (N × H)
95. #498C2×C127semidirect
96. #475C2×C127semidirect
98. #332C2×C127semidirect
99. Q8⋊C8C2×C127semidirect
150. Q8SD16×C127semidirect

Maximal subgroups

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

subgroupindex
D4×Q8×C1272
#498×C1272
#475×C1272
#461×C1272
#332×C1272
Q8⋊C8×C1272
D4⋊C8×C1272
#1651127