Finite Group Databasebeta

← C127×#13436 #44577109

Subgroups — C127×#13436 #44577109

Order 32512.

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

normal subgroups only

This group has 438 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×#3199C211yesmaximal, characteristic
2C127×#2122C212yesmaximal, complemented
3C127×#1910C212yesmaximal, complemented
4C127×#1858C211yesmaximal, characteristic, complemented
5C127×#1391C211yesmaximal, characteristic, complemented
6#568×C127C2211yescharacteristic
7#535×C127C2212yes—
8#482×C127—22no—
9#469×C127—21no—
10#466×C127—21no—
11#447×C127—22no—
12#430×C127—21no—
13#418×C127—22no—
14#413×C127C2211yescharacteristic
15#410×C127C2212yes—
16#408×C127C2211yescharacteristic
17AΣL(1,𝔽16)×C127—24no—
18#341×C127—21no—
19H(3,ℤ/4)×C127—22no—
20C2×#41×C127C2311yescharacteristic
21C22×D4×C127D412yes—
22C2×D8×C127—21no—
23C2×D8×C127—21no—
24#131×C127—42no—
25C2×M16×C127—22no—
26C2×M16×C127—21no—
27C2×M16×C127—21no—
28C2×M16×C127D411yescharacteristic
29C22×C8×C127D411yescharacteristic
30#127×C127—22no—
31#124×C127—22no—
32#121×C127—22no—
33#121×C127—22no—
34#120×C127—22no—
35#120×C127—22no—
36#117×C127—22no—
37#117×C127D412yes—
38#108×C127—42no—
39#107×C127—21no—
40#106×C127—21no—
41#102×C127—22no—
42#102×C127—22no—
43#102×C127—44no—
44#101×C127—21no—
45#100×C127D412yes—
46#99×C127—22no—
47#99×C127—22no—
48C22⋊C8×C127—22no—
49C24×C127—22no—
50#41×C127—41no—
51#41×C127—41no—
52C2×Q8×C127#4111yescharacteristic
53C2×D4×C127—22no—
54C2×D4×C127—22no—
55C2×D4×C127—42no—
56C2×D4×C127—42no—
57C2×D4×C127—42no—
58C2×D4×C127—42no—
59C2×D4×C127C2×D411yescharacteristic
60C2×D4×C127#4112yes—
61C22×C4×C127C2×D411yescharacteristic
62C22×C4×C127C2×D412yes—
63D8×C127—81no—
64D8×C127—81no—
65M16×C127—22no—
66M16×C127—22no—
67M16×C127—41no—
68M16×C127—42no—
69M16×C127—41no—
70M16×C127—41no—
71C2×C8×C127—24no—
72C2×C8×C127—21no—
73C2×C8×C127—21no—
74C2×C8×C127—41no—
75C2×C8×C127—41no—
76C2×C8×C127—41no—
77C4⋊C4×C127—22no—
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—42no—
86C23×C127—82no—
87C23×C127—162no—
88C23×C127#12112yes—
89C23×C127#12411yescharacteristic
90Q8×C127—41no—
91D4×C127—41no—
92D4×C127—42no—
93D4×C127—44no—
94D4×C127—42no—
95D4×C127—42no—
96D4×C127—162no—
97D4×C127—162no—
98C2×C4×C127—22no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
101C2×C4×C127—21no—
102C2×C4×C127—42no—
103C2×C4×C127—82no—
104C2×C4×C127#12011yescharacteristic
105C2×C4×C127#12111yescharacteristic
106C2×C4×C127#12312yes—
107C8×C127—42no—
108C8×C127—42no—
109C8×C127—41no—
110C8×C127—41no—
111C22×C127—22no—
112C22×C127—21no—
113C22×C127—41no—
114C22×C127—42no—
115C22×C127—82no—
116C22×C127—82no—
117C22×C127—82no—
118C22×C127—164no—
119C22×C127#39411yescharacteristic
120C4×C127—21no—
121C4×C127—22no—
122C4×C127—82no—
123C4×C127—162no—
124C4×C127B(4,𝔽2)11yescharacteristic
125C4×C127#45811yescharacteristic
126#13436C12711yesmaximal, characteristic, direct product factor, Sylow
127C2×C127—21no—
128C2×C127—42no—
129C2×C127—41no—
130C2×C127—162no—
131C2×C127#204411yescharacteristic
132#3199C2×C12711yescharacteristic
133#2122C2×C12712yescomplemented
134#1910C2×C12712yescomplemented
135#1858C2×C12711yescharacteristic, complemented
136#1391C2×C12711yescharacteristic, complemented
137C127#1343611yescharacteristic, direct product factor, Sylow
138#568C22×C12711yescharacteristic
139#535C22×C12712yes—
140#482—22no—
141#469—21no—
142#466—21no—
143#447—22no—
144#430—21no—
145#418—22no—
146#413C22×C12711yescharacteristic
147#410C22×C12712yes—
148#408C22×C12711yescharacteristic
149AΣL(1,𝔽16)—24no—
150#341—21no—
151H(3,ℤ/4)—22no—
152C2×#41C23×C12711yescharacteristic
153C22×D4D4×C12712yes—
154C2×D8—21no—
155C2×D8—21no—
156#131—42no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16D4×C12711yescharacteristic
161C22×C8D4×C12711yescharacteristic
162#127—22no—
163#124—22no—
164#121—22no—
165#121—22no—
166#120—22no—
167#120—22no—
168#117—22no—
169#117D4×C12712yes—
170#108—42no—
171#107—21no—
172#106—21no—
173#102—22no—
174#102—22no—
175#102—44no—
176#101—21no—
177#100D4×C12712yes—
178#99—22no—
179#99—22no—
180C22⋊C8—22no—
181C24—22no—
182#41—41no—
183#41—41no—
184C2×Q8#41×C12711yescharacteristic
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4C2×D4×C12711yescharacteristic
192C2×D4#41×C12712yes—
193C22×C4C2×D4×C12711yescharacteristic
194C22×C4C2×D4×C12712yes—
195D8—81no—
196D8—81no—
197M16—22no—
198M16—22no—
199M16—41no—
200M16—42no—
201M16—41no—
202M16—41no—
203C2×C8—24no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C4⋊C4—22no—
210C4⋊C4—42no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C42—22no—
216C42—42no—
217C23—42no—
218C23—82no—
219C23—162no—
220C23#121×C12712yes—
221C23#124×C12711yescharacteristic
222Q8—41no—
223D4—41no—
224D4—42no—
225D4—44no—
226D4—42no—
227D4—42no—
228D4—162no—
229D4—162no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4#120×C12711yescharacteristic
237C2×C4#121×C12711yescharacteristic
238C2×C4#123×C12712yes—
239C8—42no—
240C8—42no—
241C8—41no—
242C8—41no—
243C22—22no—
244C22—21no—
245C22—41no—
246C22—42no—
247C22—82no—
248C22—82no—
249C22—82no—
250C22—164no—
251C22#394×C12711yescharacteristic
252C4—21no—
253C4—22no—
254C4—82no—
255C4—162no—
256C4B(4,𝔽2)×C12711yescharacteristic
257C4#458×C12711yescharacteristic
258C2—21no—
259C2—42no—
260C2—41no—
261C2—162no—
262C2C127×#204411yescharacteristic
subgroupG/H
center, socleC2×C127#2044
commutator, Frattini, cosocleC2×#41C23×C127
prime radicalC127×#3199C2
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
2. C127×#2122C2semidirect
3. C127×#1910C2semidirect
4. C127×#1858C2semidirect
5. C127×#1391C2semidirect
126. #13436C127direct (N × H)
133. #2122C2×C127semidirect
134. #1910C2×C127semidirect
135. #1858C2×C127semidirect
136. #1391C2×C127semidirect
137. C127#13436direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31992
C127×#21222
C127×#19102
C127×#18582
C127×#13912
#13436127