Finite Group Databasebeta

← C127×#12309 #44575982

Subgroups — C127×#12309 #44575982

Order 32512.

280 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3256C211yesmaximal, characteristic
2C127×#2239C211yesmaximal, characteristic, complemented
3C127×#2238C211yesmaximal, characteristic
4C127×#2102C211yesmaximal, characteristic, complemented
5C127×#1340C211yesmaximal, characteristic, complemented
6C127×#1339C211yesmaximal, characteristic
7C127×#1327C211yesmaximal, characteristic, complemented
8#529×C127—21no—
9#479×C127—21no—
10#478×C127—21no—
11#470×C127C2211yescharacteristic
12#469×C127C2211yescharacteristic
13#468×C127—21no—
14#464×C127—21no—
15#463×C127—21no—
16#430×C127C2211yescharacteristic
17#423×C127C2211yescharacteristic
18#365×C127—21no—
19#365×C127—21no—
20#362×C127—21no—
21#361×C127—21no—
22#349×C127C2211yescharacteristic
23#349×C127C2211yescharacteristic
24#344×C127—21no—
25#339×C127—21no—
26H(3,ℤ/4)×C127C2211yescharacteristic
27C2×Q16×C127D411yescharacteristic
28C2×SD16×C127—21no—
29C2×D8×C127D411yescharacteristic
30C2×M16×C127—21no—
31C2×M16×C127D411yescharacteristic
32C2×M16×C127C2311yescharacteristic
33#126×C127—41no—
34#125×C127—21no—
35#124×C127—21no—
36#123×C127—41no—
37C22⋊Q8×C127—21no—
38#121×C127—21no—
39C4×Q8×C127—21no—
40C4×D4×C127—21no—
41C4×D4×C127—41no—
42#117×C127—21no—
43#117×C127—41no—
44C2×C42×C127D411yescharacteristic
45M32×C127—21no—
46M32×C127—21no—
47#108×C127—21no—
48#108×C127—21no—
49#107×C127—21no—
50#106×C127D411yescharacteristic
51#106×C127D411yescharacteristic
52C4⋊C8×C127—21no—
53C4⋊C8×C127—21no—
54#103×C127—21no—
55#103×C127—21no—
56#102×C127—21no—
57#102×C127—21no—
58C2×Q8×C127—21no—
59C2×D4×C127—21no—
60C2×D4×C127—41no—
61C22×C4×C127—21no—
62C22×C4×C127—41no—
63C22×C4×C127C2×D411yescharacteristic
64Q16×C127—41no—
65SD16×C127—81no—
66D8×C127—41no—
67M16×C127—41no—
68M16×C127—41no—
69M16×C127—41no—
70M16×C127—41no—
71C2×C8×C127—21no—
72C2×C8×C127—41no—
73C2×C8×C127C2×D411yescharacteristic
74C2×C8×C127C2×D411yescharacteristic
75C4⋊C4×C127—21no—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—41no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127—41no—
81C4⋊C4×C127—41no—
82C22⋊C4×C127—41no—
83C22⋊C4×C127—41no—
84C22⋊C4×C127—41no—
85C22⋊C4×C127—41no—
86C22⋊C4×C127—41no—
87C22⋊C4×C127—41no—
88C42×C127—21no—
89C42×C127—41no—
90C42×C127—41no—
91C42×C127SD1611yescharacteristic
92C42×C127SD1611yescharacteristic
93C16×C127—41no—
94C16×C127—41no—
95C23×C127—41no—
96C23×C127AGL(1,ℤ/8)11yescharacteristic
97Q8×C127—41no—
98Q8×C127—41no—
99D4×C127—41no—
100D4×C127—41no—
101D4×C127—82no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—81no—
110C2×C4×C127—82no—
111C2×C4×C127—81no—
112C2×C4×C127#12011yescharacteristic
113C2×C4×C127C2×SD1611yescharacteristic
114C8×C127—21no—
115C8×C127—21no—
116C8×C127—41no—
117C8×C127—81no—
118C22×C127—21no—
119C22×C127—41no—
120C22×C127—81no—
121C22×C127—81no—
122C22×C127—81no—
123C22×C127#45011yescharacteristic
124C4×C127—21no—
125C4×C127—41no—
126C4×C127—41no—
127C4×C127—81no—
128C4×C127—81no—
129C4×C127—81no—
130C4×C127—161no—
131C4×C127#45411yescharacteristic
132C4×C127#45811yescharacteristic
133#12309C12711yesmaximal, characteristic, direct product factor, Sylow
134C2×C127—21no—
135C2×C127—41no—
136C2×C127—161no—
137C2×C127#161611yescharacteristic
138#3256C2×C12711yescharacteristic
139#2239C2×C12711yescharacteristic, complemented
140#2238C2×C12711yescharacteristic
141#2102C2×C12711yescharacteristic, complemented
142#1340C2×C12711yescharacteristic, complemented
143#1339C2×C12711yescharacteristic
144#1327C2×C12711yescharacteristic, complemented
145C127#1230911yescharacteristic, direct product factor, Sylow
146#529—21no—
147#479—21no—
148#478—21no—
149#470C22×C12711yescharacteristic
150#469C22×C12711yescharacteristic
151#468—21no—
152#464—21no—
153#463—21no—
154#430C22×C12711yescharacteristic
155#423C22×C12711yescharacteristic
156#365—21no—
157#365—21no—
158#362—21no—
159#361—21no—
160#349C22×C12711yescharacteristic
161#349C22×C12711yescharacteristic
162#344—21no—
163#339—21no—
164H(3,ℤ/4)C22×C12711yescharacteristic
165C2×Q16D4×C12711yescharacteristic
166C2×SD16—21no—
167C2×D8D4×C12711yescharacteristic
168C2×M16—21no—
169C2×M16D4×C12711yescharacteristic
170C2×M16C23×C12711yescharacteristic
171#126—41no—
172#125—21no—
173#124—21no—
174#123—41no—
175C22⋊Q8—21no—
176#121—21no—
177C4×Q8—21no—
178C4×D4—21no—
179C4×D4—41no—
180#117—21no—
181#117—41no—
182C2×C42D4×C12711yescharacteristic
183M32—21no—
184M32—21no—
185#108—21no—
186#108—21no—
187#107—21no—
188#106D4×C12711yescharacteristic
189#106D4×C12711yescharacteristic
190C4⋊C8—21no—
191C4⋊C8—21no—
192#103—21no—
193#103—21no—
194#102—21no—
195#102—21no—
196C2×Q8—21no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—21no—
200C22×C4—41no—
201C22×C4C2×D4×C12711yescharacteristic
202Q16—41no—
203SD16—81no—
204D8—41no—
205M16—41no—
206M16—41no—
207M16—41no—
208M16—41no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8C2×D4×C12711yescharacteristic
212C2×C8C2×D4×C12711yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C42—21no—
227C42—41no—
228C42—41no—
229C42SD16×C12711yescharacteristic
230C42SD16×C12711yescharacteristic
231C16—41no—
232C16—41no—
233C23—41no—
234C23AGL(1,ℤ/8)×C12711yescharacteristic
235Q8—41no—
236Q8—41no—
237D4—41no—
238D4—41no—
239D4—82no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4#120×C12711yescharacteristic
251C2×C4C2×SD16×C12711yescharacteristic
252C8—21no—
253C8—21no—
254C8—41no—
255C8—81no—
256C22—21no—
257C22—41no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22#450×C12711yescharacteristic
262C4—21no—
263C4—41no—
264C4—41no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—161no—
269C4#454×C12711yescharacteristic
270C4#458×C12711yescharacteristic
271C2—21no—
272C2—41no—
273C2—161no—
274C2C127×#161611yescharacteristic
subgroupG/H
center, socleC2×C127#1616
commutator, Frattini, cosocleC2×M16C23×C127
prime radical#469×C127C22
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2239C2semidirect
4. C127×#2102C2semidirect
5. C127×#1340C2semidirect
7. C127×#1327C2semidirect
133. #12309C127direct (N × H)
139. #2239C2×C127semidirect
141. #2102C2×C127semidirect
142. #1340C2×C127semidirect
144. #1327C2×C127semidirect
145. C127#12309direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32562
C127×#22392
C127×#22382
C127×#21022
C127×#13402
C127×#13392
C127×#13272
#12309127