Finite Group Databasebeta

← C127×#7519 #44571192

Subgroups — C127×#7519 #44571192

Order 32512.

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

normal subgroups only

This group has 162 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×#1624C211yesmaximal, characteristic
2C127×#1409C211yesmaximal, characteristic, complemented
3C127×#1359C211yesmaximal, characteristic, complemented
4#555×C127C411yescharacteristic, complemented
5#492×C127C2211yescharacteristic
6#459×C127—21no—
7#364×C127—21no—
8#353×C127—21no—
9#333×C127C411yescharacteristic, complemented
10D4⋊C8×C127—21no—
11C2×SD16×C127—41no—
12C2×D8×C127—41no—
13C4⋊Q8×C127C2×C411yescharacteristic
14#127×C127D411yescharacteristic
15#124×C127—41no—
16#121×C127—41no—
17C4×D4×C127—21no—
18#117×C127—41no—
19M32×C127—41no—
20C4⋊C8×C127—21no—
21#102×C127—41no—
22#99×C127—41no—
23C4×C8×C127D411yescharacteristic
24C2×Q8×C127—21no—
25C2×D4×C127—21no—
26C2×D4×C127—21no—
27C2×D4×C127—41no—
28C2×D4×C127—41no—
29C22×C4×C127—41no—
30SD16×C127—81no—
31SD16×C127—81no—
32D8×C127—82no—
33C2×C8×C127—21no—
34C2×C8×C127—41no—
35C4⋊C4×C127—21no—
36C4⋊C4×C127—41no—
37C22⋊C4×C127—42no—
38C22⋊C4×C127—41no—
39C22⋊C4×C127—81no—
40C42×C127—41no—
41C42×C127C22⋊C411yescharacteristic
42C16×C127—81no—
43C23×C127—41no—
44C23×C127—41no—
45Q8×C127—81no—
46D4×C127—41no—
47D4×C127—41no—
48D4×C127—81no—
49D4×C127—81no—
50D4×C127—81no—
51D4×C127—161no—
52C2×C4×C127—21no—
53C2×C4×C127—41no—
54C2×C4×C127—41no—
55C2×C4×C127—81no—
56C2×C4×C127—81no—
57C2×C4×C127—161no—
58C2×C4×C127#9911yescharacteristic
59C8×C127—21no—
60C8×C127—21no—
61C8×C127—81no—
62C22×C127—81no—
63C22×C127—81no—
64C22×C127—81no—
65C22×C127—81no—
66C22×C127—161no—
67C22×C127AΣL(1,𝔽16)11yescharacteristic
68C4×C127—21no—
69C4×C127—21no—
70C4×C127—21no—
71C4×C127—81no—
72C4×C127—81no—
73C4×C127—321no—
74#7519C12711yesmaximal, characteristic, direct product factor, Sylow
75C2×C127—21no—
76C2×C127—161no—
77C2×C127—161no—
78C2×C127#140211yescharacteristic
79#1624C2×C12711yescharacteristic
80#1409C2×C12711yescharacteristic, complemented
81#1359C2×C12711yescharacteristic, complemented
82C127#751911yescharacteristic, direct product factor, Sylow
83#555C4×C12711yescharacteristic, complemented
84#492C22×C12711yescharacteristic
85#459—21no—
86#364—21no—
87#353—21no—
88#333C4×C12711yescharacteristic, complemented
89D4⋊C8—21no—
90C2×SD16—41no—
91C2×D8—41no—
92C4⋊Q8C2×C4×C12711yescharacteristic
93#127D4×C12711yescharacteristic
94#124—41no—
95#121—41no—
96C4×D4—21no—
97#117—41no—
98M32—41no—
99C4⋊C8—21no—
100#102—41no—
101#99—41no—
102C4×C8D4×C12711yescharacteristic
103C2×Q8—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—41no—
107C2×D4—41no—
108C22×C4—41no—
109SD16—81no—
110SD16—81no—
111D8—82no—
112C2×C8—21no—
113C2×C8—41no—
114C4⋊C4—21no—
115C4⋊C4—41no—
116C22⋊C4—42no—
117C22⋊C4—41no—
118C22⋊C4—81no—
119C42—41no—
120C42C22⋊C4×C12711yescharacteristic
121C16—81no—
122C23—41no—
123C23—41no—
124Q8—81no—
125D4—41no—
126D4—41no—
127D4—81no—
128D4—81no—
129D4—81no—
130D4—161no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—81no—
135C2×C4—81no—
136C2×C4—161no—
137C2×C4#99×C12711yescharacteristic
138C8—21no—
139C8—21no—
140C8—81no—
141C22—81no—
142C22—81no—
143C22—81no—
144C22—81no—
145C22—161no—
146C22AΣL(1,𝔽16)×C12711yescharacteristic
147C4—21no—
148C4—21no—
149C4—21no—
150C4—81no—
151C4—81no—
152C4—321no—
153C2—21no—
154C2—161no—
155C2—161no—
156C2C127×#140211yescharacteristic
subgroupG/H
center, socleC2×C127#1402
commutatorC4⋊Q8C2×C4×C127
Frattini, cosocle#492C22×C127
prime radicalC127×#1624C2
LCSG ▷ C4⋊Q8 ▷ 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×#1409C2semidirect
3. C127×#1359C2semidirect
4. #555×C127C4semidirect
9. #333×C127C4semidirect
74. #7519C127direct (N × H)
80. #1409C2×C127semidirect
81. #1359C2×C127semidirect
82. C127#7519direct (N × H)
83. #555C4×C127semidirect
88. #333C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#16242
C127×#14092
C127×#13592
#7519127