Finite Group Databasebeta

← C127×#7517 #44571190

Subgroups — C127×#7517 #44571190

Order 32512.

196 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 30 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
1C127×#1629C211yesmaximal, characteristic
2C127×#1410C211yesmaximal, characteristic, complemented
3C127×#1359C211yesmaximal, characteristic, complemented
4D42×C127C411yescharacteristic
5#492×C127C2211yescharacteristic
6#476×C127—21no—
7#364×C127—21no—
8#356×C127—21no—
9#331×C127C411yescharacteristic
10D4⋊C8×C127—21no—
11C22×D4×C127—41no—
12C2×SD16×C127—41no—
13C4⋊Q8×C127D411yescharacteristic
14#127×C127C2×C411yescharacteristic
15#121×C127—41no—
16#120×C127—41no—
17C4×D4×C127—21no—
18M32×C127—41no—
19#106×C127—41no—
20C4⋊C8×C127—21no—
21#102×C127—41no—
22#101×C127—41no—
23C4×C8×C127D411yescharacteristic
24C24×C127—41no—
25C2×Q8×C127—21no—
26C2×D4×C127—21no—
27C2×D4×C127—21no—
28C2×D4×C127—41no—
29C2×D4×C127—42no—
30C2×D4×C127—82no—
31C2×D4×C127—82no—
32C22×C4×C127—41no—
33SD16×C127—81no—
34SD16×C127—81no—
35M16×C127—81no—
36C2×C8×C127—21no—
37C2×C8×C127—41no—
38C4⋊C4×C127—21no—
39C4⋊C4×C127—41no—
40C22⋊C4×C127—41no—
41C22⋊C4×C127—41no—
42C42×C127C22⋊C411yescharacteristic
43C16×C127—81no—
44C23×C127—41no—
45C23×C127—41no—
46C23×C127—84no—
47C23×C127—161no—
48Q8×C127—81no—
49D4×C127—22no—
50D4×C127—41no—
51D4×C127—81no—
52D4×C127—81no—
53D4×C127—81no—
54D4×C127—84no—
55C2×C4×C127—21no—
56C2×C4×C127—41no—
57C2×C4×C127—41no—
58C2×C4×C127—81no—
59C2×C4×C127—81no—
60C2×C4×C127#9911yescharacteristic
61C8×C127—21no—
62C8×C127—21no—
63C8×C127—81no—
64C8×C127—161no—
65C22×C127—42no—
66C22×C127—81no—
67C22×C127—82no—
68C22×C127—81no—
69C22×C127—81no—
70C22×C127—162no—
71C22×C127—162no—
72C22×C127AΣL(1,𝔽16)11yescharacteristic
73C4×C127—21no—
74C4×C127—21no—
75C4×C127—21no—
76C4×C127—81no—
77C4×C127—81no—
78#7517C12711yesmaximal, characteristic, direct product factor, Sylow
79C2×C127—21no—
80C2×C127—82no—
81C2×C127—161no—
82C2×C127#140211yescharacteristic
83#1629C2×C12711yescharacteristic
84#1410C2×C12711yescharacteristic, complemented
85#1359C2×C12711yescharacteristic, complemented
86C127#751711yescharacteristic, direct product factor, Sylow
87D42C4×C12711yescharacteristic
88#492C22×C12711yescharacteristic
89#476—21no—
90#364—21no—
91#356—21no—
92#331C4×C12711yescharacteristic
93D4⋊C8—21no—
94C22×D4—41no—
95C2×SD16—41no—
96C4⋊Q8D4×C12711yescharacteristic
97#127C2×C4×C12711yescharacteristic
98#121—41no—
99#120—41no—
100C4×D4—21no—
101M32—41no—
102#106—41no—
103C4⋊C8—21no—
104#102—41no—
105#101—41no—
106C4×C8D4×C12711yescharacteristic
107C24—41no—
108C2×Q8—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—41no—
112C2×D4—42no—
113C2×D4—82no—
114C2×D4—82no—
115C22×C4—41no—
116SD16—81no—
117SD16—81no—
118M16—81no—
119C2×C8—21no—
120C2×C8—41no—
121C4⋊C4—21no—
122C4⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C42C22⋊C4×C12711yescharacteristic
126C16—81no—
127C23—41no—
128C23—41no—
129C23—84no—
130C23—161no—
131Q8—81no—
132D4—22no—
133D4—41no—
134D4—81no—
135D4—81no—
136D4—81no—
137D4—84no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4#99×C12711yescharacteristic
144C8—21no—
145C8—21no—
146C8—81no—
147C8—161no—
148C22—42no—
149C22—81no—
150C22—82no—
151C22—81no—
152C22—81no—
153C22—162no—
154C22—162no—
155C22AΣL(1,𝔽16)×C12711yescharacteristic
156C4—21no—
157C4—21no—
158C4—21no—
159C4—81no—
160C4—81no—
161C2—21no—
162C2—82no—
163C2—161no—
164C2C127×#140211yescharacteristic
subgroupG/H
center, socleC2×C127#1402
commutator#127C2×C4×C127
Frattini, cosocle#492C22×C127
prime radicalD42×C127C4
LCSG ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×C4×C127 ◁ C42×C127 ◁ #492×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1410C2semidirect
3. C127×#1359C2semidirect
78. #7517C127direct (N × H)
84. #1410C2×C127semidirect
85. #1359C2×C127semidirect
86. C127#7517direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#16292
C127×#14102
C127×#13592
#7517127