Finite Group Databasebeta

← C127×#9778 #44573451

Subgroups — C127×#9778 #44573451

Order 32512.

390 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×#500×C127C211yesmaximal, characteristic
2C127×#1837C211yesmaximal, characteristic
3C127×#1549C212yesmaximal
4C127×#1548C212yesmaximal
5C127×#1452C211yesmaximal, characteristic
6C2×C4⋊Q8×C127C2212yes—
7#500×C127—22no—
8#500×C127C414yes—
9#431×C127C2211yescharacteristic
10#431×C127C2211yescharacteristic
11C2×#107×C127—22no—
12#423×C127C2212yes—
13C2×C4×C8×C127C2211yescharacteristic
14#341×C127—22no—
15#333×C127—24no—
16#332×C127—24no—
17#322×C127—22no—
18C22×Q8×C127—22no—
19C2×M16×C127—22no—
20C22×C8×C127D412yes—
21C4⋊Q8×C127—22no—
22C4⋊Q8×C127—22no—
23C4⋊Q8×C127—22no—
24C4⋊Q8×C127C2×C414yes—
25C2×C4⋊C4×C127—22no—
26C2×C4⋊C4×C127—22no—
27C2×C42×C127C2311yescharacteristic
28#107×C127—28no—
29C4⋊C8×C127—22no—
30C4⋊C8×C127—22no—
31C4⋊C8×C127D412yes—
32C4⋊C8×C127D412yes—
33C22⋊C8×C127—21no—
34C22⋊C8×C127—21no—
35C4×C8×C127—21no—
36C4×C8×C127—21no—
37C4×C8×C127C2×C412yes—
38C4×C8×C127D412yes—
39C2×Q8×C127—24no—
40C2×Q8×C127—42no—
41C2×Q8×C127—84no—
42C22×C4×C127—42no—
43C22×C4×C127C2×D411yescharacteristic
44C22×C4×C127#4111yescharacteristic
45C22×C4×C127#4111yescharacteristic
46M16×C127—44no—
47C2×C8×C127—24no—
48C2×C8×C127—22no—
49C2×C8×C127—22no—
50C2×C8×C127—42no—
51C2×C8×C127C22⋊C414yes—
52C4⋊C4×C127—24no—
53C4⋊C4×C127—24no—
54C4⋊C4×C127—42no—
55C4⋊C4×C127—42no—
56C42×C127C22×C411yescharacteristic
57C42×C127C2×D411yescharacteristic
58C42×C127C2×D411yescharacteristic
59C42×C127C2×D411yescharacteristic
60C23×C127#12411yescharacteristic
61Q8×C127—84no—
62Q8×C127—84no—
63C2×C4×C127—22no—
64C2×C4×C127—22no—
65C2×C4×C127—44no—
66C2×C4×C127—84no—
67C2×C4×C127C2×C22⋊C411yescharacteristic
68C2×C4×C127C4×D411yescharacteristic
69C2×C4×C127C4×D411yescharacteristic
70C2×C4×C127#12111yescharacteristic
71C2×C4×C127#12111yescharacteristic
72C2×C4×C127#12711yescharacteristic
73C2×C4×C127#13714yes—
74C8×C127—24no—
75C8×C127—42no—
76C8×C127—42no—
77C8×C127—82no—
78C22×C127—21no—
79C22×C127—21no—
80C22×C127#39011yescharacteristic
81C22×C127#49112yes—
82C4×C127—21no—
83C4×C127—21no—
84C4×C127—84no—
85C4×C127#41912yes—
86C4×C127#47212yes—
87C4×C127#47312yes—
88C4×C127#49712yes—
89#9778C12711yesmaximal, characteristic, direct product factor, Sylow
90C2×C127—22no—
91C2×C127#196011yescharacteristic
92C2×C127#197811yescharacteristic
93C2×C127#197911yescharacteristic
94C2×#500C2×C12711yescharacteristic
95#1837C2×C12711yescharacteristic
96#1549C2×C12712yes—
97#1548C2×C12712yes—
98#1452C2×C12711yescharacteristic
99C127#977811yescharacteristic, direct product factor, Sylow
100C2×C4⋊Q8C22×C12712yes—
101#500—22no—
102#500C4×C12714yes—
103#431C22×C12711yescharacteristic
104#431C22×C12711yescharacteristic
105C2×#107—22no—
106#423C22×C12712yes—
107C2×C4×C8C22×C12711yescharacteristic
108#341—22no—
109#333—24no—
110#332—24no—
111#322—22no—
112C22×Q8—22no—
113C2×M16—22no—
114C22×C8D4×C12712yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8C2×C4×C12714yes—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C42C23×C12711yescharacteristic
122#107—28no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8D4×C12712yes—
126C4⋊C8D4×C12712yes—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C4×C8—21no—
130C4×C8—21no—
131C4×C8C2×C4×C12712yes—
132C4×C8D4×C12712yes—
133C2×Q8—24no—
134C2×Q8—42no—
135C2×Q8—84no—
136C22×C4—42no—
137C22×C4C2×D4×C12711yescharacteristic
138C22×C4#41×C12711yescharacteristic
139C22×C4#41×C12711yescharacteristic
140M16—44no—
141C2×C8—24no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—42no—
145C2×C8C22⋊C4×C12714yes—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C42C22×C4×C12711yescharacteristic
151C42C2×D4×C12711yescharacteristic
152C42C2×D4×C12711yescharacteristic
153C42C2×D4×C12711yescharacteristic
154C23#124×C12711yescharacteristic
155Q8—84no—
156Q8—84no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—44no—
160C2×C4—84no—
161C2×C4C2×C22⋊C4×C12711yescharacteristic
162C2×C4C4×D4×C12711yescharacteristic
163C2×C4C4×D4×C12711yescharacteristic
164C2×C4#121×C12711yescharacteristic
165C2×C4#121×C12711yescharacteristic
166C2×C4#127×C12711yescharacteristic
167C2×C4#137×C12714yes—
168C8—24no—
169C8—42no—
170C8—42no—
171C8—82no—
172C22—21no—
173C22—21no—
174C22#390×C12711yescharacteristic
175C22#491×C12712yes—
176C4—21no—
177C4—21no—
178C4—84no—
179C4#419×C12712yes—
180C4#472×C12712yes—
181C4#473×C12712yes—
182C4#497×C12712yes—
183C2—22no—
184C2C127×#196011yescharacteristic
185C2C127×#197811yescharacteristic
186C2C127×#197911yescharacteristic
subgroupG/H
center, socleC22×C127#390
commutatorC42C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC23×C127#124
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
89. #9778C127direct (N × H)
99. C127#9778direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#500×C1272
C127×#18372
C127×#15492
C127×#15482
C127×#14522
#9778127