Finite Group Databasebeta

← C127×#7286 #44570959

Subgroups — C127×#7286 #44570959

Order 32512.

250 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 82 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×#1985C211yesmaximal, characteristic
2C127×#1461C212yesmaximal
3C2×C4×Q8×C127C2211yescharacteristic
4#431×C127—22no—
5C2×C4⋊C8×C127C411yescharacteristic
6C2×#97×C127C411yescharacteristic
7#336×C127—22no—
8#324×C127—22no—
9#324×C127C414yes—
10C22×Q8×C127Q811yescharacteristic
11C22×C8×C127—21no—
12C22×C8×C127—21no—
13C4×Q8×C127—41no—
14C4×Q8×C127—41no—
15C2×C4⋊C4×C127C2×C412yes—
16C2×C4⋊C4×C127D411yescharacteristic
17C2×C42×C127C2×C411yescharacteristic
18C2×C42×C127D412yes—
19C4⋊C8×C127—42no—
20C4⋊C8×C127—41no—
21C22⋊C8×C127—24no—
22#97×C127—41no—
23C4×C8×C127—42no—
24C2×Q8×C127—24no—
25C2×Q8×C127—41no—
26C22×C4×C127C4211yescharacteristic
27C22×C4×C127C22⋊C412yes—
28C22×C4×C127C22⋊C411yescharacteristic
29C22×C4×C127C4⋊C411yescharacteristic
30C22×C4×C127C4⋊C412yes—
31C2×C8×C127—22no—
32C2×C8×C127—41no—
33C2×C8×C127—41no—
34C2×C8×C127—44no—
35C2×C8×C127—42no—
36C2×C8×C127—41no—
37C4⋊C4×C127—42no—
38C4⋊C4×C127—41no—
39C42×C127—21no—
40C42×C127—22no—
41C42×C127—22no—
42C42×C127—21no—
43C23×C127#9511yescharacteristic
44Q8×C127—44no—
45C2×C4×C127—22no—
46C2×C4×C127—22no—
47C2×C4×C127—21no—
48C2×C4×C127—21no—
49C2×C4×C127—22no—
50C2×C4×C127—22no—
51C2×C4×C127—21no—
52C2×C4×C127—21no—
53C2×C4×C127—41no—
54C2×C4×C127—42no—
55C2×C4×C127#9911yescharacteristic
56C2×C4×C127#9911yescharacteristic
57C2×C4×C127#10814yes—
58C8×C127—42no—
59C8×C127—84no—
60C8×C127—81no—
61C22×C127—21no—
62C22×C127—21no—
63C22×C127#34112yes—
64C22×C127#34211yescharacteristic
65C4×C127—24no—
66C4×C127—21no—
67C4×C127—41no—
68C4×C127—42no—
69C4×C127—41no—
70C4×C127#35511yescharacteristic
71C4×C127#35611yescharacteristic
72#7286C12711yesmaximal, characteristic, direct product factor, Sylow
73C2×C127—22no—
74C2×C127#129811yescharacteristic
75C2×C127#139111yescharacteristic
76C2×C127#139411yescharacteristic
77#1985C2×C12711yescharacteristic
78#1461C2×C12712yes—
79C127#728611yescharacteristic, direct product factor, Sylow
80C2×C4×Q8C22×C12711yescharacteristic
81#431—22no—
82C2×C4⋊C8C4×C12711yescharacteristic
83C2×#97C4×C12711yescharacteristic
84#336—22no—
85#324—22no—
86#324C4×C12714yes—
87C22×Q8Q8×C12711yescharacteristic
88C22×C8—21no—
89C22×C8—21no—
90C4×Q8—41no—
91C4×Q8—41no—
92C2×C4⋊C4C2×C4×C12712yes—
93C2×C4⋊C4D4×C12711yescharacteristic
94C2×C42C2×C4×C12711yescharacteristic
95C2×C42D4×C12712yes—
96C4⋊C8—41no—
97C4⋊C8—42no—
98C22⋊C8—24no—
99#97—41no—
100C4×C8—42no—
101C2×Q8—24no—
102C2×Q8—41no—
103C22×C4C42×C12711yescharacteristic
104C22×C4C22⋊C4×C12711yescharacteristic
105C22×C4C22⋊C4×C12712yes—
106C22×C4C4⋊C4×C12711yescharacteristic
107C22×C4C4⋊C4×C12712yes—
108C2×C8—22no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—44no—
112C2×C8—42no—
113C2×C8—41no—
114C4⋊C4—42no—
115C4⋊C4—41no—
116C42—22no—
117C42—21no—
118C42—21no—
119C42—22no—
120C23#95×C12711yescharacteristic
121Q8—44no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4#99×C12711yescharacteristic
133C2×C4#99×C12711yescharacteristic
134C2×C4#108×C12714yes—
135C8—42no—
136C8—84no—
137C8—81no—
138C22—21no—
139C22—21no—
140C22#341×C12712yes—
141C22#342×C12711yescharacteristic
142C4—24no—
143C4—21no—
144C4—41no—
145C4—42no—
146C4—41no—
147C4#355×C12711yescharacteristic
148C4#356×C12711yescharacteristic
149C2—22no—
150C2C127×#129811yescharacteristic
151C2C127×#139111yescharacteristic
152C2C127×#139411yescharacteristic
subgroupG/H
centerC2×C4×C127#99
commutatorC22×C4C42×C127
Frattini, cosocleC2×C4×Q8C22×C127
socleC22×C127#342
prime radicalC23×C127#95
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×C4×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
72. #7286C127direct (N × H)
79. C127#7286direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#19852
C127×#14612
#7286127