Finite Group Databasebeta

← C127×#12306 #44575979

Subgroups — C127×#12306 #44575979

Order 32512.

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

normal subgroups only

This group has 274 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×#3263C211yesmaximal, characteristic
2C127×#2239C211yesmaximal, characteristic, complemented
3C127×#2238C211yesmaximal, characteristic
4C127×#2103C211yesmaximal, characteristic, complemented
5C127×#1342C211yesmaximal, characteristic, complemented
6C127×#1341C211yesmaximal, characteristic
7C127×#1328C211yesmaximal, characteristic, complemented
8#542×C127—21no—
9#476×C127—21no—
10#475×C127—21no—
11#470×C127C2211yescharacteristic
12#469×C127C2211yescharacteristic
13#465×C127—21no—
14#464×C127—21no—
15#463×C127—21no—
16#430×C127C2211yescharacteristic
17#424×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#341×C127—21no—
26#338×C127C2211yescharacteristic
27C2×Q16×C127D411yescharacteristic
28C2×SD16×C127—21no—
29C2×D8×C127D411yescharacteristic
30C2×M16×C127—21no—
31C2×M16×C127—21no—
32C2×M16×C127C2311yescharacteristic
33C22×C8×C127D411yescharacteristic
34C4⋊Q8×C127—21no—
35#126×C127—41no—
36#124×C127—21no—
37#124×C127—41no—
38#123×C127—41no—
39C22⋊Q8×C127—21no—
40C22⋊Q8×C127—41no—
41#121×C127—21no—
42C4×Q8×C127—21no—
43C4×D4×C127—21no—
44#117×C127D411yescharacteristic
45M32×C127—21no—
46M32×C127—21no—
47#108×C127—21no—
48#108×C127—21no—
49#106×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×Q8×C127—41no—
60C2×D4×C127—21no—
61C2×D4×C127—41no—
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—
71M16×C127—41no—
72C2×C8×C127—21no—
73C2×C8×C127—41no—
74C2×C8×C127—41no—
75C2×C8×C127—41no—
76C2×C8×C127C2×D411yescharacteristic
77C2×C8×C127C2×D411yescharacteristic
78C4⋊C4×C127—21no—
79C4⋊C4×C127—21no—
80C4⋊C4×C127—21no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C22⋊C4×C127—41no—
85C22⋊C4×C127—41no—
86C22⋊C4×C127—41no—
87C22⋊C4×C127—41no—
88C22⋊C4×C127—41no—
89C22⋊C4×C127SD1611yescharacteristic
90C22⋊C4×C127SD1611yescharacteristic
91C42×C127—21no—
92C42×C127—41no—
93C16×C127—41no—
94C16×C127—41no—
95C23×C127—41no—
96C23×C127C2×SD1611yescharacteristic
97Q8×C127—41no—
98Q8×C127—41no—
99Q8×C127—161no—
100D4×C127—41no—
101D4×C127—41no—
102D4×C127—161no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—161no—
111C2×C4×C127#12011yescharacteristic
112C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
113C8×C127—21no—
114C8×C127—21no—
115C8×C127—41no—
116C8×C127—81no—
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#45411yescharacteristic
131C4×C127#45811yescharacteristic
132#12306C12711yesmaximal, characteristic, direct product factor, Sylow
133C2×C127—21no—
134C2×C127—41no—
135C2×C127—161no—
136C2×C127#161611yescharacteristic
137#3263C2×C12711yescharacteristic
138#2239C2×C12711yescharacteristic, complemented
139#2238C2×C12711yescharacteristic
140#2103C2×C12711yescharacteristic, complemented
141#1342C2×C12711yescharacteristic, complemented
142#1341C2×C12711yescharacteristic
143#1328C2×C12711yescharacteristic, complemented
144C127#1230611yescharacteristic, direct product factor, Sylow
145#542—21no—
146#476—21no—
147#475—21no—
148#470C22×C12711yescharacteristic
149#469C22×C12711yescharacteristic
150#465—21no—
151#464—21no—
152#463—21no—
153#430C22×C12711yescharacteristic
154#424C22×C12711yescharacteristic
155#365—21no—
156#365—21no—
157#362—21no—
158#361—21no—
159#349C22×C12711yescharacteristic
160#349C22×C12711yescharacteristic
161#344—21no—
162#341—21no—
163#338C22×C12711yescharacteristic
164C2×Q16D4×C12711yescharacteristic
165C2×SD16—21no—
166C2×D8D4×C12711yescharacteristic
167C2×M16—21no—
168C2×M16—21no—
169C2×M16C23×C12711yescharacteristic
170C22×C8D4×C12711yescharacteristic
171C4⋊Q8—21no—
172#126—41no—
173#124—21no—
174#124—41no—
175#123—41no—
176C22⋊Q8—21no—
177C22⋊Q8—41no—
178#121—21no—
179C4×Q8—21no—
180C4×D4—21no—
181#117D4×C12711yescharacteristic
182M32—21no—
183M32—21no—
184#108—21no—
185#108—21no—
186#106—21no—
187#106D4×C12711yescharacteristic
188#106D4×C12711yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191#103—21no—
192#103—21no—
193#102—21no—
194#102—21no—
195C2×Q8—21no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—41no—
200C22×C4C2×D4×C12711yescharacteristic
201Q16—41no—
202SD16—81no—
203D8—41no—
204M16—41no—
205M16—41no—
206M16—41no—
207M16—41no—
208M16—41no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8C2×D4×C12711yescharacteristic
214C2×C8C2×D4×C12711yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4SD16×C12711yescharacteristic
227C22⋊C4SD16×C12711yescharacteristic
228C42—21no—
229C42—41no—
230C16—41no—
231C16—41no—
232C23—41no—
233C23C2×SD16×C12711yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—161no—
237D4—41no—
238D4—41no—
239D4—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—161no—
248C2×C4#120×C12711yescharacteristic
249C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
250C8—21no—
251C8—21no—
252C8—41no—
253C8—81no—
254C8—81no—
255C22—21no—
256C22—41no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22#450×C12711yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4#454×C12711yescharacteristic
268C4#458×C12711yescharacteristic
269C2—21no—
270C2—41no—
271C2—161no—
272C2C127×#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×#2103C2semidirect
5. C127×#1342C2semidirect
7. C127×#1328C2semidirect
132. #12306C127direct (N × H)
138. #2239C2×C127semidirect
140. #2103C2×C127semidirect
141. #1342C2×C127semidirect
143. #1328C2×C127semidirect
144. C127#12306direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32632
C127×#22392
C127×#22382
C127×#21032
C127×#13422
C127×#13412
C127×#13282
#12306127