Finite Group Databasebeta

← C127×#13500 #44577173

Subgroups — C127×#13500 #44577173

Order 32512.

276 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2158C211yesmaximal, characteristic, complemented
2C2×#350×C127C211yesmaximal, characteristic
3C127×#1814C211yesmaximal, characteristic
4C127×#1812C211yesmaximal, characteristic
5C127×#1389C211yesmaximal, characteristic, complemented
6C127×#1384C211yesmaximal, characteristic, complemented
7C127×#1382C211yesmaximal, characteristic, complemented
8C2×#131×C127C2211yescharacteristic
9C2×M32×C127C2211yescharacteristic
10C22×C16×C127C2211yescharacteristic
11C2×#108×C127C2211yescharacteristic
12#427×C127C2211yescharacteristic
13#427×C127C2211yescharacteristic
14#427×C127C2211yescharacteristic
15#421×C127—21no—
16#410×C127—21no—
17#367×C127—21no—
18#366×C127—21no—
19#350×C127—21no—
20#350×C127—21no—
21#344×C127—21no—
22H(3,ℤ/4)×C127—21no—
23C2×#41×C127D411yescharacteristic
24#131×C127—21no—
25#131×C127—21no—
26#131×C127—21no—
27#131×C127—21no—
28C2×M16×C127—21no—
29C2×M16×C127—21no—
30C2×M16×C127D411yescharacteristic
31C22×C8×C127—21no—
32C22×C8×C127C2311yescharacteristic
33#117×C127—21no—
34#117×C127—21no—
35#117×C127—21no—
36M32×C127—21no—
37M32×C127—21no—
38C2×C16×C127—21no—
39C2×C16×C127—21no—
40C2×C16×C127—21no—
41C2×C16×C127D411yescharacteristic
42C2×C16×C127D411yescharacteristic
43#108×C127—21no—
44#108×C127—21no—
45#107×C127—21no—
46#107×C127—21no—
47#107×C127—21no—
48#106×C127—21no—
49#106×C127—21no—
50#106×C127—21no—
51#104×C127—42no—
52#99×C127—42no—
53#41×C127—21no—
54#41×C127—21no—
55#41×C127—21no—
56#41×C127—21no—
57C2×Q8×C127Q1611yescharacteristic
58C2×D4×C127—21no—
59C2×D4×C127Q1611yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127C2×D411yescharacteristic
62M16×C127—21no—
63M16×C127—21no—
64M16×C127—41no—
65M16×C127—41no—
66M16×C127—41no—
67M16×C127—41no—
68C2×C8×C127—21no—
69C2×C8×C127—41no—
70C2×C8×C127—41no—
71C2×C8×C127—41no—
72C2×C8×C127C2×D411yescharacteristic
73C2×C8×C127#4111yescharacteristic
74C2×C8×C127#4111yescharacteristic
75C2×C8×C127#4111yescharacteristic
76C2×C8×C127#4111yescharacteristic
77C2×C8×C127#4111yescharacteristic
78C4⋊C4×C127—41no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127—41no—
81C22⋊C4×C127—41no—
82C22⋊C4×C127—41no—
83C22⋊C4×C127—41no—
84C42×C127—41no—
85C42×C127—41no—
86C42×C127—41no—
87C16×C127—21no—
88C16×C127—21no—
89C16×C127—41no—
90C23×C127—21no—
91C23×C127AGL(1,ℤ/8)11yescharacteristic
92Q8×C127—21no—
93Q8×C127—21no—
94D4×C127—21no—
95D4×C127—21no—
96D4×C127—41no—
97D4×C127—41no—
98C2×C4×C127—21no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—81no—
102C2×C4×C127—81no—
103C2×C4×C127—81no—
104C2×C4×C127#12111yescharacteristic
105C2×C4×C127#12311yescharacteristic
106C2×C4×C127#12311yescharacteristic
107C2×C4×C127C2×Q1611yescharacteristic
108C2×C4×C127#13511yescharacteristic
109C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
110C8×C127—41no—
111C8×C127—81no—
112C8×C127#12311yescharacteristic
113C8×C127#12411yescharacteristic
114C8×C127#12611yescharacteristic
115C8×C127#12611yescharacteristic
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—81no—
119C22×C127#46911yescharacteristic
120C22×C127#48211yescharacteristic
121C22×C127#48411yescharacteristic
122C4×C127—41no—
123C4×C127—161no—
124C4×C127—161no—
125C4×C127—161no—
126C4×C127#39711yescharacteristic
127C4×C127#46711yescharacteristic
128C4×C127#48211yescharacteristic
129C4×C127#48411yescharacteristic
130#13500C12711yesmaximal, characteristic, direct product factor, Sylow
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—21no—
134C2×C127—81no—
135C2×C127#207611yescharacteristic
136#2158C2×C12711yescharacteristic, complemented
137C2×#350C2×C12711yescharacteristic
138#1814C2×C12711yescharacteristic
139#1812C2×C12711yescharacteristic
140#1389C2×C12711yescharacteristic, complemented
141#1384C2×C12711yescharacteristic, complemented
142#1382C2×C12711yescharacteristic, complemented
143C127#1350011yescharacteristic, direct product factor, Sylow
144C2×#131C22×C12711yescharacteristic
145C2×M32C22×C12711yescharacteristic
146C22×C16C22×C12711yescharacteristic
147C2×#108C22×C12711yescharacteristic
148#427C22×C12711yescharacteristic
149#427C22×C12711yescharacteristic
150#427C22×C12711yescharacteristic
151#421—21no—
152#410—21no—
153#367—21no—
154#366—21no—
155#350—21no—
156#350—21no—
157#344—21no—
158H(3,ℤ/4)—21no—
159C2×#41D4×C12711yescharacteristic
160#131—21no—
161#131—21no—
162#131—21no—
163#131—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16D4×C12711yescharacteristic
167C22×C8—21no—
168C22×C8C23×C12711yescharacteristic
169#117—21no—
170#117—21no—
171#117—21no—
172M32—21no—
173M32—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16—21no—
177C2×C16D4×C12711yescharacteristic
178C2×C16D4×C12711yescharacteristic
179#108—21no—
180#108—21no—
181#107—21no—
182#107—21no—
183#107—21no—
184#106—21no—
185#106—21no—
186#106—21no—
187#104—42no—
188#99—42no—
189#41—21no—
190#41—21no—
191#41—21no—
192#41—21no—
193C2×Q8Q16×C12711yescharacteristic
194C2×D4—21no—
195C2×D4Q16×C12711yescharacteristic
196C22×C4—21no—
197C22×C4C2×D4×C12711yescharacteristic
198M16—21no—
199M16—21no—
200M16—41no—
201M16—41no—
202M16—41no—
203M16—41no—
204C2×C8—21no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8C2×D4×C12711yescharacteristic
209C2×C8#41×C12711yescharacteristic
210C2×C8#41×C12711yescharacteristic
211C2×C8#41×C12711yescharacteristic
212C2×C8#41×C12711yescharacteristic
213C2×C8#41×C12711yescharacteristic
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C42—41no—
221C42—41no—
222C42—41no—
223C16—21no—
224C16—21no—
225C16—41no—
226C23—21no—
227C23AGL(1,ℤ/8)×C12711yescharacteristic
228Q8—21no—
229Q8—21no—
230D4—21no—
231D4—21no—
232D4—41no—
233D4—41no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4#121×C12711yescharacteristic
241C2×C4#123×C12711yescharacteristic
242C2×C4#123×C12711yescharacteristic
243C2×C4C2×Q16×C12711yescharacteristic
244C2×C4#135×C12711yescharacteristic
245C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
246C8—41no—
247C8—81no—
248C8#123×C12711yescharacteristic
249C8#124×C12711yescharacteristic
250C8#126×C12711yescharacteristic
251C8#126×C12711yescharacteristic
252C22—41no—
253C22—41no—
254C22—81no—
255C22#469×C12711yescharacteristic
256C22#482×C12711yescharacteristic
257C22#484×C12711yescharacteristic
258C4—41no—
259C4—161no—
260C4—161no—
261C4—161no—
262C4#397×C12711yescharacteristic
263C4#467×C12711yescharacteristic
264C4#482×C12711yescharacteristic
265C4#484×C12711yescharacteristic
266C2—21no—
267C2—21no—
268C2—21no—
269C2—81no—
270C2C127×#207611yescharacteristic
subgroupG/H
centerC4×C127#482
commutator, Frattini, cosocleC22×C8C23×C127
socleC2×C127#2076
prime radicalC2×#41×C127D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C2×#131×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2158C2semidirect
5. C127×#1389C2semidirect
6. C127×#1384C2semidirect
7. C127×#1382C2semidirect
130. #13500C127direct (N × H)
136. #2158C2×C127semidirect
140. #1389C2×C127semidirect
141. #1384C2×C127semidirect
142. #1382C2×C127semidirect
143. C127#13500direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#21582
C2×#350×C1272
C127×#18142
C127×#18122
C127×#13892
C127×#13842
C127×#13822
#13500127