Finite Group Databasebeta

← C127×#11468 #44575141

Subgroups — C127×#11468 #44575141

Order 32512.

398 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2929C211yesmaximal, characteristic
2C2×C22⋊C16×C127C212yesmaximal, complemented
3C127×#1323C212yesmaximal, complemented
4C127×#1315C212yesmaximal, complemented
5C2×#131×C127C411yescharacteristic
6C23×C8×C127C2211yescharacteristic
7#525×C127C411yescharacteristic
8C22×C16×C127—22no—
9#435×C127—22no—
10C8×D4×C127—22no—
11#424×C127C2211yescharacteristic
12#408×C127C2211yescharacteristic
13#407×C127C411yescharacteristic
14#405×C127C411yescharacteristic
15C22⋊C16×C127—22no—
16C22⋊C16×C127C2214yes—
17C2×#41×C127C2×C411yescharacteristic
18C23×C4×C127C2×C411yescharacteristic
19#131×C127—42no—
20C2×M16×C127—21no—
21C2×M16×C127C2×C411yescharacteristic
22C2×M16×C127C2×C411yescharacteristic
23C22×C8×C127—21no—
24C22×C8×C127—42no—
25C22×C8×C127C2×C411yescharacteristic
26C22×C8×C127D412yes—
27C22×C8×C127D412yes—
28C22×C8×C127C2311yescharacteristic
29#123×C127—21no—
30C22⋊Q8×C127C812yes—
31#121×C127C812yes—
32#120×C127—21no—
33C4×D4×C127—22no—
34#117×C127C2×C411yescharacteristic
35C2×C16×C127—24no—
36C2×C16×C127—44no—
37C4⋊C8×C127—22no—
38C22⋊C8×C127—22no—
39C22⋊C8×C127—22no—
40#97×C127—21no—
41C4×C8×C127—21no—
42C24×C127M1611yescharacteristic
43#41×C127—41no—
44#41×C127—41no—
45C2×Q8×C127C2×C811yescharacteristic
46C2×D4×C127—22no—
47C2×D4×C127—21no—
48C2×D4×C127C2×C811yescharacteristic
49C22×C4×C127—21no—
50C22×C4×C127—41no—
51C22×C4×C127—41no—
52C22×C4×C127C22⋊C412yes—
53C22×C4×C127C2×C812yes—
54C22×C4×C127M1611yescharacteristic
55C22×C4×C127C22×C411yescharacteristic
56M16×C127—42no—
57M16×C127—41no—
58M16×C127—41no—
59C2×C8×C127—24no—
60C2×C8×C127—22no—
61C2×C8×C127—21no—
62C2×C8×C127—21no—
63C2×C8×C127—42no—
64C2×C8×C127—41no—
65C2×C8×C127—42no—
66C2×C8×C127C22⋊C411yescharacteristic
67C2×C8×C127C22⋊C411yescharacteristic
68C2×C8×C127C2×D412yes—
69C4⋊C4×C127—22no—
70C4⋊C4×C127C2×C812yes—
71C22⋊C4×C127—22no—
72C22⋊C4×C127—21no—
73C22⋊C4×C127—22no—
74C42×C127—21no—
75C16×C127—44no—
76C23×C127—24no—
77C23×C127—21no—
78C23×C127—41no—
79C23×C127C22⋊C812yes—
80C23×C127C2×M1611yescharacteristic
81Q8×C127—41no—
82D4×C127—41no—
83D4×C127—42no—
84D4×C127—44no—
85C2×C4×C127—21no—
86C2×C4×C127—22no—
87C2×C4×C127—21no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—21no—
91C2×C4×C127—42no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127C22⋊C812yes—
96C2×C4×C127C2×C22⋊C411yescharacteristic
97C2×C4×C127C22×C811yescharacteristic
98C8×C127—22no—
99C8×C127—21no—
100C8×C127—41no—
101C8×C127—41no—
102C8×C127#9912yes—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—22no—
106C22×C127—21no—
107C22×C127—44no—
108C22×C127—42no—
109C22×C127—41no—
110C22×C127#35014yes—
111C22×C127C2×C22⋊C811yescharacteristic
112C4×C127—22no—
113C4×C127—21no—
114C4×C127—42no—
115C4×C127—41no—
116C4×C127C2×#9911yescharacteristic
117C4×C127#41311yescharacteristic
118#11468C12711yesmaximal, characteristic, direct product factor, Sylow
119C2×C127—24no—
120C2×C127—41no—
121C2×C127—81no—
122C2×C127#145911yescharacteristic
123C2×C127C2×#35012yes—
124#2929C2×C12711yescharacteristic
125C2×C22⋊C16C2×C12712yescomplemented
126#1323C2×C12712yescomplemented
127#1315C2×C12712yescomplemented
128C127#1146811yescharacteristic, direct product factor, Sylow
129C2×#131C4×C12711yescharacteristic
130C23×C8C22×C12711yescharacteristic
131#525C4×C12711yescharacteristic
132C22×C16—22no—
133#435—22no—
134C8×D4—22no—
135#424C22×C12711yescharacteristic
136#408C22×C12711yescharacteristic
137#407C4×C12711yescharacteristic
138#405C4×C12711yescharacteristic
139C22⋊C16—22no—
140C22⋊C16C22×C12714yes—
141C2×#41C2×C4×C12711yescharacteristic
142C23×C4C2×C4×C12711yescharacteristic
143#131—42no—
144C2×M16—21no—
145C2×M16C2×C4×C12711yescharacteristic
146C2×M16C2×C4×C12711yescharacteristic
147C22×C8—21no—
148C22×C8—42no—
149C22×C8C2×C4×C12711yescharacteristic
150C22×C8D4×C12712yes—
151C22×C8D4×C12712yes—
152C22×C8C23×C12711yescharacteristic
153#123—21no—
154C22⋊Q8C8×C12712yes—
155#121C8×C12712yes—
156#120—21no—
157C4×D4—22no—
158#117C2×C4×C12711yescharacteristic
159C2×C16—24no—
160C2×C16—44no—
161C4⋊C8—22no—
162C22⋊C8—22no—
163C22⋊C8—22no—
164#97—21no—
165C4×C8—21no—
166C24M16×C12711yescharacteristic
167#41—41no—
168#41—41no—
169C2×Q8C2×C8×C12711yescharacteristic
170C2×D4—22no—
171C2×D4—21no—
172C2×D4C2×C8×C12711yescharacteristic
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4C22⋊C4×C12712yes—
177C22×C4C2×C8×C12712yes—
178C22×C4M16×C12711yescharacteristic
179C22×C4C22×C4×C12711yescharacteristic
180M16—42no—
181M16—41no—
182M16—41no—
183C2×C8—24no—
184C2×C8—22no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8C22⋊C4×C12711yescharacteristic
191C2×C8C22⋊C4×C12711yescharacteristic
192C2×C8C2×D4×C12712yes—
193C4⋊C4—22no—
194C4⋊C4C2×C8×C12712yes—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C22⋊C4—22no—
198C42—21no—
199C16—44no—
200C23—24no—
201C23—21no—
202C23—41no—
203C23C22⋊C8×C12712yes—
204C23C2×M16×C12711yescharacteristic
205Q8—41no—
206D4—41no—
207D4—42no—
208D4—44no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4C22⋊C8×C12712yes—
220C2×C4C2×C22⋊C4×C12711yescharacteristic
221C2×C4C22×C8×C12711yescharacteristic
222C8—22no—
223C8—21no—
224C8—41no—
225C8—41no—
226C8#99×C12712yes—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—21no—
231C22—44no—
232C22—42no—
233C22—41no—
234C22#350×C12714yes—
235C22C2×C22⋊C8×C12711yescharacteristic
236C4—22no—
237C4—21no—
238C4—42no—
239C4—41no—
240C4C2×#99×C12711yescharacteristic
241C4#413×C12711yescharacteristic
242C2—24no—
243C2—41no—
244C2—81no—
245C2C127×#145911yescharacteristic
246C2C2×#350×C12712yes—
subgroupG/H
centerC2×C8×C127C22⋊C4
commutatorC2×C4C22×C8×C127
Frattini, cosocleC22×C8C23×C127
socleC22×C127C2×C22⋊C8
prime radical#525×C127C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C16×C127C2semidirect
3. C127×#1323C2semidirect
4. C127×#1315C2semidirect
118. #11468C127direct (N × H)
125. C2×C22⋊C16C2×C127semidirect
126. #1323C2×C127semidirect
127. #1315C2×C127semidirect
128. C127#11468direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#29292
C2×C22⋊C16×C1272
C127×#13232
C127×#13152
#11468127