Finite Group Databasebeta

← C127×#12314 #44575987

Subgroups — C127×#12314 #44575987

Order 32512.

384 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3291C211yesmaximal, characteristic, complemented
2C127×#2213C212yesmaximal, complemented
3C127×#1721C211yesmaximal, characteristic, complemented
4C127×#1359C211yesmaximal, characteristic, complemented
5C127×#1336C212yesmaximal, complemented
6#577×C127—22no—
7#509×C127—22no—
8#493×C127—21no—
9#493×C127C2211yescharacteristic, complemented
10#492×C127C2211yescharacteristic
11#486×C127—21no—
12#471×C127—22no—
13#456×C127—21no—
14#453×C127—21no—
15#443×C127C2212yes—
16#364×C127C2212yescomplemented
17#361×C127—22no—
18#322×C127C2211yescharacteristic, complemented
192−1+4×C127—21no—
20H(5,𝔽2)×C127—21no—
21#137×C127—22no—
22#137×C127—42no—
23AGL(1,ℤ/8)×C127—22no—
24AGL(1,ℤ/8)×C127—42no—
25#135×C127—42no—
26#135×C127D412yes—
27C2×SD16×C127—22no—
28C2×SD16×C127—42no—
29C2×D8×C127—22no—
30C2×D8×C127—42no—
31#131×C127—22no—
32C4⋊Q8×C127D411yescharacteristic
33#127×C127—21no—
34#127×C127D411yescharacteristic
35SD32×C127—42no—
36D16×C127—42no—
37M32×C127—22no—
38#108×C127D412yes—
39#104×C127—22no—
40#102×C127—42no—
41#101×C127—21no—
42#100×C127—21no—
43C4×C8×C127—21no—
44C4×C8×C127C2311yescharacteristic
45#41×C127—22no—
46#41×C127—42no—
47#41×C127—42no—
48#41×C127—81no—
49C2×Q8×C127—21no—
50C2×Q8×C127—42no—
51C2×D4×C127—21no—
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127—41no—
55C2×D4×C127—42no—
56C2×D4×C127—41no—
57C2×D4×C127—81no—
58Q16×C127—42no—
59Q16×C127D812yescomplemented
60SD16×C127—22no—
61SD16×C127—42no—
62SD16×C127—42no—
63SD16×C127—42no—
64SD16×C127—42no—
65D8×C127—42no—
66D8×C127—42no—
67D8×C127—42no—
68D8×C127—84no—
69D8×C127D812yescomplemented
70M16×C127—22no—
71M16×C127—42no—
72C2×C8×C127—22no—
73C2×C8×C127—42no—
74C2×C8×C127C2×D412yes—
75C4⋊C4×C127—41no—
76C42×C127C2×D411yescharacteristic
77C16×C127—42no—
78C23×C127—41no—
79C23×C127—81no—
80C23×C127—81no—
81Q8×C127—22no—
82Q8×C127—42no—
83Q8×C127—81no—
84D4×C127—22no—
85D4×C127—42no—
86D4×C127—42no—
87D4×C127—82no—
88D4×C127—82no—
89D4×C127—82no—
90D4×C127—82no—
91D4×C127—81no—
92D4×C127—82no—
93C2×C4×C127—21no—
94C2×C4×C127—42no—
95C2×C4×C127—41no—
96C2×C4×C127—81no—
97C2×C4×C127—82no—
98C2×C4×C127#12011yescharacteristic
99C8×C127—42no—
100C8×C127—42no—
101C8×C127C2×D812yes—
102C8×C127AGL(1,ℤ/8)12yes—
103C22×C127—42no—
104C22×C127—81no—
105C22×C127—161no—
106C22×C127—161no—
107C22×C127—161no—
108C22×C127—161no—
109C22×C127—321no—
110C22×C127#45311yescharacteristic
111C4×C127—22no—
112C4×C127—42no—
113C4×C127—81no—
114C4×C127#44712yes—
115#12314C12711yesmaximal, characteristic, direct product factor, Sylow
116C2×C127—21no—
117C2×C127—82no—
118C2×C127—161no—
119C2×C127—321no—
120C2×C127#161911yescharacteristic
121#3291C2×C12711yescharacteristic, complemented
122#2213C2×C12712yescomplemented
123#1721C2×C12711yescharacteristic, complemented
124#1359C2×C12711yescharacteristic, complemented
125#1336C2×C12712yescomplemented
126C127#1231411yescharacteristic, direct product factor, Sylow
127#577—22no—
128#509—22no—
129#493—21no—
130#493C22×C12711yescharacteristic, complemented
131#492C22×C12711yescharacteristic
132#486—21no—
133#471—22no—
134#456—21no—
135#453—21no—
136#443C22×C12712yes—
137#364C22×C12712yescomplemented
138#361—22no—
139#322C22×C12711yescharacteristic, complemented
1402−1+4—21no—
141H(5,𝔽2)—21no—
142#137—22no—
143#137—42no—
144AGL(1,ℤ/8)—22no—
145AGL(1,ℤ/8)—42no—
146#135—42no—
147#135D4×C12712yes—
148C2×SD16—22no—
149C2×SD16—42no—
150C2×D8—22no—
151C2×D8—42no—
152#131—22no—
153C4⋊Q8D4×C12711yescharacteristic
154#127—21no—
155#127D4×C12711yescharacteristic
156SD32—42no—
157D16—42no—
158M32—22no—
159#108D4×C12712yes—
160#104—22no—
161#102—42no—
162#101—21no—
163#100—21no—
164C4×C8—21no—
165C4×C8C23×C12711yescharacteristic
166#41—22no—
167#41—42no—
168#41—42no—
169#41—81no—
170C2×Q8—21no—
171C2×Q8—42no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—81no—
179Q16—42no—
180Q16D8×C12712yescomplemented
181SD16—22no—
182SD16—42no—
183SD16—42no—
184SD16—42no—
185SD16—42no—
186D8—42no—
187D8—42no—
188D8—42no—
189D8—84no—
190D8D8×C12712yescomplemented
191M16—22no—
192M16—42no—
193C2×C8—22no—
194C2×C8—42no—
195C2×C8C2×D4×C12712yes—
196C4⋊C4—41no—
197C42C2×D4×C12711yescharacteristic
198C16—42no—
199C23—41no—
200C23—81no—
201C23—81no—
202Q8—22no—
203Q8—42no—
204Q8—81no—
205D4—22no—
206D4—42no—
207D4—42no—
208D4—82no—
209D4—82no—
210D4—82no—
211D4—82no—
212D4—81no—
213D4—82no—
214C2×C4—21no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4#120×C12711yescharacteristic
220C8—42no—
221C8—42no—
222C8C2×D8×C12712yes—
223C8AGL(1,ℤ/8)×C12712yes—
224C22—42no—
225C22—81no—
226C22—161no—
227C22—161no—
228C22—161no—
229C22—161no—
230C22—321no—
231C22#453×C12711yescharacteristic
232C4—22no—
233C4—42no—
234C4—81no—
235C4#447×C12712yes—
236C2—21no—
237C2—82no—
238C2—161no—
239C2—321no—
240C2C127×#161911yescharacteristic
subgroupG/H
center, socleC2×C127#1619
commutator, Frattini, cosocleC4×C8C23×C127
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3291C2semidirect
2. C127×#2213C2semidirect
3. C127×#1721C2semidirect
4. C127×#1359C2semidirect
5. C127×#1336C2semidirect
9. #493×C127C22semidirect
16. #364×C127C22semidirect
18. #322×C127C22semidirect
59. Q16×C127D8semidirect
69. D8×C127D8semidirect
115. #12314C127direct (N × H)
121. #3291C2×C127semidirect
122. #2213C2×C127semidirect
123. #1721C2×C127semidirect
124. #1359C2×C127semidirect
125. #1336C2×C127semidirect
126. C127#12314direct (N × H)
130. #493C22×C127semidirect
137. #364C22×C127semidirect
139. #322C22×C127semidirect
180. Q16D8×C127semidirect
190. D8D8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32912
C127×#22132
C127×#17212
C127×#13592
C127×#13362
#12314127