Finite Group Databasebeta

← C127×#13669 #44577342

Subgroups — C127×#13669 #44577342

Order 32512.

288 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3415C211yesmaximal, characteristic, complemented
2C127×#2263C211yesmaximal, characteristic, complemented
3C2×D32×C127C211yesmaximal, characteristic, complemented
4C127×#2221C211yesmaximal, characteristic, complemented
5C127×#1419C211yesmaximal, characteristic, complemented
6C127×#1417C211yesmaximal, characteristic, complemented
7C127×#1401C211yesmaximal, characteristic, complemented
8#576×C127—21no—
9#509×C127—22no—
10#509×C127—21no—
11#508×C127—21no—
12#508×C127C2211yescharacteristic
13C2×D16×C127—21no—
14C2×D16×C127C2211yescharacteristic
15C2×D16×C127C2211yescharacteristic, complemented
16#504×C127C2211yescharacteristic
17#472×C127—21no—
18SD64×C127—21no—
19D32×C127—22no—
20D32×C127—21no—
21M64×C127C2211yescharacteristic, complemented
22C2×C32×C127C2211yescharacteristic, complemented
23#368×C127C2211yescharacteristic, complemented
24#361×C127—21no—
25H(5,𝔽2)×C127—41no—
26AGL(1,ℤ/8)×C127—42no—
27AGL(1,ℤ/8)×C127—41no—
28AGL(1,ℤ/8)×C127—41no—
29#135×C127—22no—
30#135×C127—21no—
31C2×D8×C127—22no—
32C2×D8×C127—21no—
33C2×D8×C127—21no—
34#131×C127D411yescharacteristic
35Q32×C127D411yescharacteristic
36SD32×C127—22no—
37SD32×C127—21no—
38D16×C127—22no—
39D16×C127—21no—
40D16×C127—21no—
41D16×C127—21no—
42D16×C127D411yescharacteristic
43D16×C127D412yes—
44M32×C127—21no—
45M32×C127D411yescharacteristic
46C2×C16×C127—21no—
47C2×C16×C127C2311yescharacteristic
48#108×C127—21no—
49#100×C127—41no—
50C32×C127—21no—
51C32×C127—21no—
52#41×C127—42no—
53#41×C127—41no—
54#41×C127—81no—
55#41×C127D811yescharacteristic
56C2×D4×C127—42no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C2×D4×C127—82no—
60C2×D4×C127—81no—
61Q16×C127—21no—
62SD16×C127—42no—
63SD16×C127—41no—
64SD16×C127—81no—
65D8×C127—21no—
66D8×C127—22no—
67D8×C127—42no—
68D8×C127—41no—
69D8×C127—41no—
70D8×C127—41no—
71D8×C127—81no—
72M16×C127—21no—
73M16×C127—41no—
74M16×C127D811yescharacteristic
75C2×C8×C127—21no—
76C2×C8×C127C2×D411yescharacteristic
77C16×C127—21no—
78C16×C127C2×D411yescharacteristic
79C16×C127C2×D411yescharacteristic
80C23×C127—82no—
81C23×C127—81no—
82C23×C127—81no—
83Q8×C127—41no—
84Q8×C127D1611yescharacteristic
85D4×C127—21no—
86D4×C127—41no—
87D4×C127—42no—
88D4×C127—82no—
89D4×C127—82no—
90D4×C127—81no—
91D4×C127—81no—
92D4×C127—81no—
93D4×C127—81no—
94D4×C127D1611yescharacteristic
95C2×C4×C127—21no—
96C2×C4×C127—82no—
97C2×C4×C127—81no—
98C2×C4×C127C2×D811yescharacteristic
99C8×C127—21no—
100C8×C127—81no—
101C8×C127#12011yescharacteristic
102C8×C127AGL(1,ℤ/8)11yescharacteristic
103C22×C127—21no—
104C22×C127—82no—
105C22×C127—81no—
106C22×C127—161no—
107C22×C127—321no—
108C22×C127—321no—
109C22×C127—322no—
110C22×C127#50911yescharacteristic
111C4×C127—21no—
112C4×C127—81no—
113C4×C127#44711yescharacteristic
114C4×C127C2×D1611yescharacteristic
115#13669C12711yesmaximal, characteristic, direct product factor, Sylow
116C2×C127—21no—
117C2×C127—41no—
118C2×C127—162no—
119C2×C127—161no—
120C2×C127—321no—
121C2×C127#218411yescharacteristic
122#3415C2×C12711yescharacteristic, complemented
123#2263C2×C12711yescharacteristic, complemented
124C2×D32C2×C12711yescharacteristic, complemented
125#2221C2×C12711yescharacteristic, complemented
126#1419C2×C12711yescharacteristic, complemented
127#1417C2×C12711yescharacteristic, complemented
128#1401C2×C12711yescharacteristic, complemented
129C127#1366911yescharacteristic, direct product factor, Sylow
130#576—21no—
131#509—22no—
132#509—21no—
133#508—21no—
134#508C22×C12711yescharacteristic
135C2×D16—21no—
136C2×D16C22×C12711yescharacteristic
137C2×D16C22×C12711yescharacteristic, complemented
138#504C22×C12711yescharacteristic
139#472—21no—
140SD64—21no—
141D32—22no—
142D32—21no—
143M64C22×C12711yescharacteristic, complemented
144C2×C32C22×C12711yescharacteristic, complemented
145#368C22×C12711yescharacteristic, complemented
146#361—21no—
147H(5,𝔽2)—41no—
148AGL(1,ℤ/8)—42no—
149AGL(1,ℤ/8)—41no—
150AGL(1,ℤ/8)—41no—
151#135—22no—
152#135—21no—
153C2×D8—22no—
154C2×D8—21no—
155C2×D8—21no—
156#131D4×C12711yescharacteristic
157Q32D4×C12711yescharacteristic
158SD32—22no—
159SD32—21no—
160D16—22no—
161D16—21no—
162D16—21no—
163D16—21no—
164D16D4×C12711yescharacteristic
165D16D4×C12712yes—
166M32—21no—
167M32D4×C12711yescharacteristic
168C2×C16—21no—
169C2×C16C23×C12711yescharacteristic
170#108—21no—
171#100—41no—
172C32—21no—
173C32—21no—
174#41—42no—
175#41—41no—
176#41—81no—
177#41D8×C12711yescharacteristic
178C2×D4—42no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—82no—
182C2×D4—81no—
183Q16—21no—
184SD16—42no—
185SD16—41no—
186SD16—81no—
187D8—21no—
188D8—22no—
189D8—42no—
190D8—41no—
191D8—41no—
192D8—41no—
193D8—81no—
194M16—21no—
195M16—41no—
196M16D8×C12711yescharacteristic
197C2×C8—21no—
198C2×C8C2×D4×C12711yescharacteristic
199C16—21no—
200C16C2×D4×C12711yescharacteristic
201C16C2×D4×C12711yescharacteristic
202C23—82no—
203C23—81no—
204C23—81no—
205Q8—41no—
206Q8D16×C12711yescharacteristic
207D4—21no—
208D4—41no—
209D4—42no—
210D4—82no—
211D4—82no—
212D4—81no—
213D4—81no—
214D4—81no—
215D4—81no—
216D4D16×C12711yescharacteristic
217C2×C4—21no—
218C2×C4—82no—
219C2×C4—81no—
220C2×C4C2×D8×C12711yescharacteristic
221C8—21no—
222C8—81no—
223C8#120×C12711yescharacteristic
224C8AGL(1,ℤ/8)×C12711yescharacteristic
225C22—21no—
226C22—82no—
227C22—81no—
228C22—161no—
229C22—321no—
230C22—321no—
231C22—322no—
232C22#509×C12711yescharacteristic
233C4—21no—
234C4—81no—
235C4#447×C12711yescharacteristic
236C4C2×D16×C12711yescharacteristic
237C2—21no—
238C2—41no—
239C2—162no—
240C2—161no—
241C2—321no—
242C2C127×#218411yescharacteristic
subgroupG/H
center, socleC2×C127#2184
commutator, Frattini, cosocleC2×C16C23×C127
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ #131×C127 ◁ #504×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3415C2semidirect
2. C127×#2263C2semidirect
3. C2×D32×C127C2semidirect
4. C127×#2221C2semidirect
5. C127×#1419C2semidirect
6. C127×#1417C2semidirect
7. C127×#1401C2semidirect
15. C2×D16×C127C22semidirect
21. M64×C127C22semidirect
22. C2×C32×C127C22semidirect
23. #368×C127C22semidirect
115. #13669C127direct (N × H)
122. #3415C2×C127semidirect
123. #2263C2×C127semidirect
124. C2×D32C2×C127semidirect
125. #2221C2×C127semidirect
126. #1419C2×C127semidirect
127. #1417C2×C127semidirect
128. #1401C2×C127semidirect
129. C127#13669direct (N × H)
137. C2×D16C22×C127semidirect
143. M64C22×C127semidirect
144. C2×C32C22×C127semidirect
145. #368C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34152
C127×#22632
C2×D32×C1272
C127×#22212
C127×#14192
C127×#14172
C127×#14012
#13669127