Finite Group Databasebeta

← C127×#13675 #44577348

Subgroups — C127×#13675 #44577348

Order 32512.

408 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3292C211yesmaximal, characteristic
2C127×#2198C212yesmaximal, complemented
3C127×#1620C211yesmaximal, characteristic, complemented
4C127×#1408C211yesmaximal, characteristic, complemented
5C127×#1402C212yesmaximal, complemented
6#576×C127—22no—
7#559×C127C2211yescharacteristic, complemented
8#493×C127C2211yescharacteristic
9#472×C127—22no—
10#460×C127—21no—
11#458×C127—22no—
12#453×C127—21no—
13#453×C127C2212yes—
14#443×C127—21no—
15#353×C127C2212yescomplemented
16#352×C127—22no—
17#331×C127C2211yescharacteristic, complemented
18Q8⋊C8×C127—21no—
19H(5,𝔽2)×C127—21no—
20H(5,𝔽2)×C127D412yes—
21AGL(1,ℤ/8)×C127—22no—
22AGL(1,ℤ/8)×C127—22no—
23AGL(1,ℤ/8)×C127—42no—
24AGL(1,ℤ/8)×C127—42no—
25#135×C127—21no—
26#135×C127—42no—
27C2×SD16×C127—41no—
28C2×D8×C127—22no—
29C2×D8×C127—42no—
30#131×C127—22no—
31#127×C127D411yescharacteristic
32#127×C127C2311yescharacteristic
33#124×C127—41no—
34#123×C127—22no—
35#123×C127—42no—
36C4×Q8×C127—21no—
37C4×D4×C127—41no—
38#108×C127—21no—
39C4⋊C8×C127—21no—
40#104×C127—22no—
41#102×C127—41no—
42#100×C127—21no—
43#100×C127D412yes—
44#99×C127—22no—
45#99×C127—42no—
46C4×C8×C127D411yescharacteristic
47#41×C127—22no—
48#41×C127—42no—
49#41×C127—42no—
50C2×Q8×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127—42no—
55C2×D4×C127—42no—
56C2×D4×C127—42no—
57C2×D4×C127—81no—
58C2×D4×C127C2×D412yes—
59C22×C4×C127—22no—
60Q16×C127—21no—
61SD16×C127—22no—
62SD16×C127—42no—
63SD16×C127—42no—
64SD16×C127—82no—
65D8×C127—21no—
66D8×C127—42no—
67D8×C127—42no—
68D8×C127—42no—
69D8×C127—42no—
70M16×C127—22no—
71M16×C127—42no—
72C2×C8×C127—21no—
73C2×C8×C127—42no—
74C2×C8×C127—41no—
75C4⋊C4×C127—21no—
76C4⋊C4×C127—41no—
77C4⋊C4×C127—82no—
78C22⋊C4×C127—22no—
79C22⋊C4×C127—42no—
80C22⋊C4×C127—42no—
81C22⋊C4×C127—82no—
82C42×C127—41no—
83C42×C127C2×D411yescharacteristic
84C23×C127—22no—
85C23×C127—42no—
86C23×C127—41no—
87C23×C127—81no—
88Q8×C127—22no—
89Q8×C127—41no—
90Q8×C127—41no—
91D4×C127—22no—
92D4×C127—42no—
93D4×C127—42no—
94D4×C127—42no—
95D4×C127—81no—
96D4×C127—81no—
97D4×C127—81no—
98D4×C127—81no—
99D4×C127—82no—
100D4×C127—82no—
101C2×C4×C127—21no—
102C2×C4×C127—42no—
103C2×C4×C127—42no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—81no—
107C2×C4×C127—82no—
108C2×C4×C127—82no—
109C2×C4×C127#12011yescharacteristic
110C8×C127—21no—
111C8×C127—21no—
112C8×C127—42no—
113C8×C127—81no—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127—82no—
117C22×C127—81no—
118C22×C127—161no—
119C22×C127—162no—
120C22×C127—321no—
121C22×C127B(4,𝔽2)11yescharacteristic
122C4×C127—21no—
123C4×C127—21no—
124C4×C127—21no—
125C4×C127—42no—
126C4×C127—81no—
127C4×C127—81no—
128C4×C127—162no—
129#13675C12711yesmaximal, characteristic, direct product factor, Sylow
130C2×C127—21no—
131C2×C127—82no—
132C2×C127—82no—
133C2×C127—161no—
134C2×C127#219611yescharacteristic
135#3292C2×C12711yescharacteristic
136#2198C2×C12712yescomplemented
137#1620C2×C12711yescharacteristic, complemented
138#1408C2×C12711yescharacteristic, complemented
139#1402C2×C12712yescomplemented
140C127#1367511yescharacteristic, direct product factor, Sylow
141#576—22no—
142#559C22×C12711yescharacteristic, complemented
143#493C22×C12711yescharacteristic
144#472—22no—
145#460—21no—
146#458—22no—
147#453—21no—
148#453C22×C12712yes—
149#443—21no—
150#353C22×C12712yescomplemented
151#352—22no—
152#331C22×C12711yescharacteristic, complemented
153Q8⋊C8—21no—
154H(5,𝔽2)—21no—
155H(5,𝔽2)D4×C12712yes—
156AGL(1,ℤ/8)—22no—
157AGL(1,ℤ/8)—22no—
158AGL(1,ℤ/8)—42no—
159AGL(1,ℤ/8)—42no—
160#135—21no—
161#135—42no—
162C2×SD16—41no—
163C2×D8—22no—
164C2×D8—42no—
165#131—22no—
166#127D4×C12711yescharacteristic
167#127C23×C12711yescharacteristic
168#124—41no—
169#123—22no—
170#123—42no—
171C4×Q8—21no—
172C4×D4—41no—
173#108—21no—
174C4⋊C8—21no—
175#104—22no—
176#102—41no—
177#100—21no—
178#100D4×C12712yes—
179#99—22no—
180#99—42no—
181C4×C8D4×C12711yescharacteristic
182#41—22no—
183#41—42no—
184#41—42no—
185C2×Q8—21no—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—81no—
193C2×D4C2×D4×C12712yes—
194C22×C4—22no—
195Q16—21no—
196SD16—22no—
197SD16—42no—
198SD16—42no—
199SD16—82no—
200D8—21no—
201D8—42no—
202D8—42no—
203D8—42no—
204D8—42no—
205M16—22no—
206M16—42no—
207C2×C8—21no—
208C2×C8—42no—
209C2×C8—41no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C22⋊C4—22no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C42—41no—
218C42C2×D4×C12711yescharacteristic
219C23—22no—
220C23—42no—
221C23—41no—
222C23—81no—
223Q8—22no—
224Q8—41no—
225Q8—41no—
226D4—22no—
227D4—42no—
228D4—42no—
229D4—42no—
230D4—81no—
231D4—81no—
232D4—81no—
233D4—81no—
234D4—82no—
235D4—82no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4#120×C12711yescharacteristic
245C8—21no—
246C8—21no—
247C8—42no—
248C8—81no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22—81no—
253C22—161no—
254C22—162no—
255C22—321no—
256C22B(4,𝔽2)×C12711yescharacteristic
257C4—21no—
258C4—21no—
259C4—21no—
260C4—42no—
261C4—81no—
262C4—81no—
263C4—162no—
264C2—21no—
265C2—82no—
266C2—82no—
267C2—161no—
268C2C127×#219611yescharacteristic
subgroupG/H
center, socleC2×C127#2196
commutator, Frattini, cosocle#127C23×C127
prime radicalC127×#3292C2
LCSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×C4×C127 ◁ #127×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2198C2semidirect
3. C127×#1620C2semidirect
4. C127×#1408C2semidirect
5. C127×#1402C2semidirect
7. #559×C127C22semidirect
15. #353×C127C22semidirect
17. #331×C127C22semidirect
129. #13675C127direct (N × H)
136. #2198C2×C127semidirect
137. #1620C2×C127semidirect
138. #1408C2×C127semidirect
139. #1402C2×C127semidirect
140. C127#13675direct (N × H)
142. #559C22×C127semidirect
150. #353C22×C127semidirect
152. #331C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32922
C127×#21982
C127×#16202
C127×#14082
C127×#14022
#13675127