Finite Group Databasebeta

← C127×#13438 #44577111

Subgroups — C127×#13438 #44577111

Order 32512.

406 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3202C211yesmaximal, characteristic
2C127×#2123C212yesmaximal
3C127×#1909C212yesmaximal
4C127×#1858C211yesmaximal, characteristic
5C127×#1391C211yesmaximal, characteristic
6#568×C127C2211yescharacteristic
7#536×C127C2212yes—
8#485×C127—22no—
9#470×C127—21no—
10#467×C127—21no—
11#451×C127—22no—
12#430×C127—21no—
13#419×C127—22no—
14#413×C127C2211yescharacteristic
15#410×C127C2212yes—
16#408×C127C2211yescharacteristic
17#352×C127—24no—
18#341×C127—21no—
19H(3,ℤ/4)×C127—22no—
20C2×#41×C127C2311yescharacteristic
21C22×Q8×C127D412yes—
22C2×Q16×C127—21no—
23C2×Q16×C127—21no—
24#131×C127—42no—
25C2×M16×C127—22no—
26C2×M16×C127—21no—
27C2×M16×C127—21no—
28C2×M16×C127D411yescharacteristic
29C22×C8×C127D411yescharacteristic
30C4⋊Q8×C127—22no—
31#124×C127—22no—
32#123×C127—22no—
33#123×C127—22no—
34C22⋊Q8×C127—22no—
35C22⋊Q8×C127—22no—
36#117×C127—22no—
37#117×C127D412yes—
38#108×C127—42no—
39#107×C127—21no—
40#106×C127—21no—
41#103×C127—22no—
42#103×C127—22no—
43#103×C127—44no—
44#101×C127—21no—
45#100×C127D412yes—
46#99×C127—22no—
47#99×C127—22no—
48C22⋊C8×C127—22no—
49#41×C127—41no—
50#41×C127—41no—
51C2×Q8×C127—22no—
52C2×Q8×C127—22no—
53C2×Q8×C127—42no—
54C2×Q8×C127—42no—
55C2×Q8×C127#4111yescharacteristic
56C2×D4×C127C2×D411yescharacteristic
57C2×D4×C127#4112yes—
58C22×C4×C127—22no—
59C22×C4×C127C2×D411yescharacteristic
60C22×C4×C127C2×D412yes—
61Q16×C127—81no—
62Q16×C127—81no—
63M16×C127—22no—
64M16×C127—22no—
65M16×C127—41no—
66M16×C127—42no—
67M16×C127—41no—
68M16×C127—41no—
69C2×C8×C127—24no—
70C2×C8×C127—21no—
71C2×C8×C127—21no—
72C2×C8×C127—41no—
73C2×C8×C127—41no—
74C2×C8×C127—41no—
75C4⋊C4×C127—22no—
76C4⋊C4×C127—42no—
77C4⋊C4×C127—42no—
78C4⋊C4×C127—42no—
79C22⋊C4×C127—22no—
80C22⋊C4×C127—42no—
81C22⋊C4×C127—42no—
82C22⋊C4×C127—42no—
83C42×C127—22no—
84C42×C127—42no—
85C23×C127#12112yes—
86C23×C127#12411yescharacteristic
87Q8×C127—41no—
88Q8×C127—44no—
89Q8×C127—42no—
90Q8×C127—42no—
91D4×C127—41no—
92D4×C127—42no—
93C2×C4×C127—22no—
94C2×C4×C127—21no—
95C2×C4×C127—22no—
96C2×C4×C127—21no—
97C2×C4×C127—42no—
98C2×C4×C127—42no—
99C2×C4×C127—82no—
100C2×C4×C127—82no—
101C2×C4×C127#12011yescharacteristic
102C2×C4×C127#12111yescharacteristic
103C2×C4×C127#12312yes—
104C8×C127—42no—
105C8×C127—42no—
106C8×C127—41no—
107C8×C127—41no—
108C22×C127—22no—
109C22×C127—21no—
110C22×C127—42no—
111C22×C127—41no—
112C22×C127#39411yescharacteristic
113C4×C127—21no—
114C4×C127—22no—
115C4×C127—82no—
116C4×C127—82no—
117C4×C127—162no—
118C4×C127B(4,𝔽2)11yescharacteristic
119C4×C127#45811yescharacteristic
120#13438C12711yesmaximal, characteristic, direct product factor, Sylow
121C2×C127—21no—
122C2×C127—42no—
123C2×C127—41no—
124C2×C127#204411yescharacteristic
125#3202C2×C12711yescharacteristic
126#2123C2×C12712yes—
127#1909C2×C12712yes—
128#1858C2×C12711yescharacteristic
129#1391C2×C12711yescharacteristic
130C127#1343811yescharacteristic, direct product factor, Sylow
131#568C22×C12711yescharacteristic
132#536C22×C12712yes—
133#485—22no—
134#470—21no—
135#467—21no—
136#451—22no—
137#430—21no—
138#419—22no—
139#413C22×C12711yescharacteristic
140#410C22×C12712yes—
141#408C22×C12711yescharacteristic
142#352—24no—
143#341—21no—
144H(3,ℤ/4)—22no—
145C2×#41C23×C12711yescharacteristic
146C22×Q8D4×C12712yes—
147C2×Q16—21no—
148C2×Q16—21no—
149#131—42no—
150C2×M16—22no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16D4×C12711yescharacteristic
154C22×C8D4×C12711yescharacteristic
155C4⋊Q8—22no—
156#124—22no—
157#123—22no—
158#123—22no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161#117—22no—
162#117D4×C12712yes—
163#108—42no—
164#107—21no—
165#106—21no—
166#103—22no—
167#103—22no—
168#103—44no—
169#101—21no—
170#100D4×C12712yes—
171#99—22no—
172#99—22no—
173C22⋊C8—22no—
174#41—41no—
175#41—41no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8#41×C12711yescharacteristic
181C2×D4C2×D4×C12711yescharacteristic
182C2×D4#41×C12712yes—
183C22×C4—22no—
184C22×C4C2×D4×C12711yescharacteristic
185C22×C4C2×D4×C12712yes—
186Q16—81no—
187Q16—81no—
188M16—22no—
189M16—22no—
190M16—41no—
191M16—42no—
192M16—41no—
193M16—41no—
194C2×C8—24no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C22⋊C4—22no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—22no—
209C42—42no—
210C23#121×C12712yes—
211C23#124×C12711yescharacteristic
212Q8—41no—
213Q8—44no—
214Q8—42no—
215Q8—42no—
216D4—41no—
217D4—42no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4#120×C12711yescharacteristic
227C2×C4#121×C12711yescharacteristic
228C2×C4#123×C12712yes—
229C8—42no—
230C8—42no—
231C8—41no—
232C8—41no—
233C22—22no—
234C22—21no—
235C22—42no—
236C22—41no—
237C22#394×C12711yescharacteristic
238C4—21no—
239C4—22no—
240C4—82no—
241C4—82no—
242C4—162no—
243C4B(4,𝔽2)×C12711yescharacteristic
244C4#458×C12711yescharacteristic
245C2—21no—
246C2—42no—
247C2—41no—
248C2C127×#204411yescharacteristic
subgroupG/H
center, socleC2×C127#2044
commutator, Frattini, cosocleC2×#41C23×C127
prime radicalC2×#41×C127C23
LCSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
120. #13438C127direct (N × H)
130. C127#13438direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32022
C127×#21232
C127×#19092
C127×#18582
C127×#13912
#13438127