Finite Group Databasebeta

← C127×#12111 #44575784

Subgroups — C127×#12111 #44575784

Order 32512.

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

normal subgroups only

This group has 422 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×#3199C211yesmaximal, characteristic
2C127×#2185C212yesmaximal, complemented
3C127×#2020C211yesmaximal, characteristic, complemented
4C127×#1349C212yesmaximal, complemented
5C127×#1315C211yesmaximal, characteristic, complemented
6C2×#137×C127—21no—
7C22×Q16×C127C2211yescharacteristic, complemented
8#535×C127—21no—
9C2×SD32×C127—22no—
10#482×C127—22no—
11#469×C127—21no—
12#466×C127C2212yes—
13#447×C127—22no—
14C2×#101×C127—21no—
15#408×C127C2211yescharacteristic
16#358×C127—22no—
17C22⋊C16×C127C2212yescomplemented
18#341×C127C2211yescharacteristic, complemented
19C2×#41×C127D411yescharacteristic
20C22×Q8×C127—21no—
21C22×D4×C127—21no—
22#137×C127—42no—
23#137×C127—42no—
24C2×Q16×C127—22no—
25C2×Q16×C127—42no—
26C2×Q16×C127—41no—
27C2×SD16×C127—41no—
28C2×D8×C127—21no—
29C2×D8×C127D412yes—
30C2×M16×C127—21no—
31C2×M16×C127D411yescharacteristic
32C22×C8×C127C2311yescharacteristic
33#127×C127—41no—
34#124×C127—41no—
35#121×C127—22no—
36#121×C127—22no—
37#120×C127—41no—
38#120×C127—41no—
39#117×C127—21no—
40SD32×C127—44no—
41C2×C16×C127—22no—
42#107×C127D412yes—
43#106×C127—21no—
44#102×C127—22no—
45#102×C127—22no—
46#101×C127—42no—
47C22⋊C8×C127—22no—
48C24×C127—41no—
49#41×C127—41no—
50#41×C127—41no—
51C2×Q8×C127—22no—
52C2×Q8×C127—41no—
53C2×Q8×C127—41no—
54C2×Q8×C127—41no—
55C2×Q8×C127—41no—
56C2×Q8×C127—41no—
57C2×Q8×C127D811yescharacteristic
58C2×D4×C127—21no—
59C2×D4×C127—22no—
60C2×D4×C127—41no—
61C2×D4×C127—42no—
62C2×D4×C127—42no—
63C2×D4×C127—81no—
64C2×D4×C127—81no—
65C2×D4×C127D811yescharacteristic
66C22×C4×C127—21no—
67C22×C4×C127—41no—
68C22×C4×C127C2×D411yescharacteristic
69Q16×C127—24no—
70Q16×C127—42no—
71Q16×C127—44no—
72SD16×C127—44no—
73D8×C127—42no—
74D8×C127—81no—
75M16×C127—22no—
76M16×C127—41no—
77M16×C127—41no—
78C2×C8×C127—21no—
79C2×C8×C127—41no—
80C2×C8×C127—41no—
81C2×C8×C127C2×D412yes—
82C4⋊C4×C127—22no—
83C22⋊C4×C127—42no—
84C22⋊C4×C127—41no—
85C22⋊C4×C127—42no—
86C42×C127—41no—
87C16×C127—42no—
88C23×C127—21no—
89C23×C127—42no—
90C23×C127—82no—
91C23×C127—162no—
92C23×C127AGL(1,ℤ/8)11yescharacteristic
93Q8×C127—41no—
94Q8×C127—42no—
95Q8×C127—42no—
96Q8×C127—81no—
97Q8×C127—81no—
98D4×C127—41no—
99D4×C127—42no—
100D4×C127—42no—
101D4×C127—81no—
102D4×C127—82no—
103D4×C127—162no—
104D4×C127—162no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—42no—
111C2×C4×C127—42no—
112C2×C4×C127—81no—
113C2×C4×C127—81no—
114C2×C4×C127#12011yescharacteristic
115C2×C4×C127C2×D811yescharacteristic
116C8×C127—22no—
117C8×C127—42no—
118C8×C127—41no—
119C22×C127—21no—
120C22×C127—22no—
121C22×C127—41no—
122C22×C127—42no—
123C22×C127—82no—
124C22×C127—82no—
125C22×C127—82no—
126C22×C127—164no—
127C22×C127#44711yescharacteristic
128C4×C127—21no—
129C4×C127—41no—
130C4×C127—82no—
131C4×C127—82no—
132C4×C127#45311yescharacteristic
133C4×C127B(4,𝔽2)11yescharacteristic
134#12111C12711yesmaximal, characteristic, direct product factor, Sylow
135C2×C127—41no—
136C2×C127—81no—
137C2×C127—162no—
138C2×C127#159511yescharacteristic
139C2×C127#219412yes—
140#3199C2×C12711yescharacteristic
141#2185C2×C12712yescomplemented
142#2020C2×C12711yescharacteristic, complemented
143#1349C2×C12712yescomplemented
144#1315C2×C12711yescharacteristic, complemented
145C127#1211111yescharacteristic, direct product factor, Sylow
146C2×#137—21no—
147C22×Q16C22×C12711yescharacteristic, complemented
148#535—21no—
149C2×SD32—22no—
150#482—22no—
151#469—21no—
152#466C22×C12712yes—
153#447—22no—
154C2×#101—21no—
155#408C22×C12711yescharacteristic
156#358—22no—
157C22⋊C16C22×C12712yescomplemented
158#341C22×C12711yescharacteristic, complemented
159C2×#41D4×C12711yescharacteristic
160C22×Q8—21no—
161C22×D4—21no—
162#137—42no—
163#137—42no—
164C2×Q16—22no—
165C2×Q16—42no—
166C2×Q16—41no—
167C2×SD16—41no—
168C2×D8—21no—
169C2×D8D4×C12712yes—
170C2×M16—21no—
171C2×M16D4×C12711yescharacteristic
172C22×C8C23×C12711yescharacteristic
173#127—41no—
174#124—41no—
175#121—22no—
176#121—22no—
177#120—41no—
178#120—41no—
179#117—21no—
180SD32—44no—
181C2×C16—22no—
182#107D4×C12712yes—
183#106—21no—
184#102—22no—
185#102—22no—
186#101—42no—
187C22⋊C8—22no—
188C24—41no—
189#41—41no—
190#41—41no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8D8×C12711yescharacteristic
198C2×D4—21no—
199C2×D4—22no—
200C2×D4—41no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—81no—
204C2×D4—81no—
205C2×D4D8×C12711yescharacteristic
206C22×C4—21no—
207C22×C4—41no—
208C22×C4C2×D4×C12711yescharacteristic
209Q16—24no—
210Q16—42no—
211Q16—44no—
212SD16—44no—
213D8—42no—
214D8—81no—
215M16—22no—
216M16—41no—
217M16—41no—
218C2×C8—21no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8C2×D4×C12712yes—
222C4⋊C4—22no—
223C22⋊C4—42no—
224C22⋊C4—41no—
225C22⋊C4—42no—
226C42—41no—
227C16—42no—
228C23—21no—
229C23—42no—
230C23—82no—
231C23—162no—
232C23AGL(1,ℤ/8)×C12711yescharacteristic
233Q8—41no—
234Q8—42no—
235Q8—42no—
236Q8—81no—
237Q8—81no—
238D4—41no—
239D4—42no—
240D4—42no—
241D4—81no—
242D4—82no—
243D4—162no—
244D4—162no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4#120×C12711yescharacteristic
255C2×C4C2×D8×C12711yescharacteristic
256C8—22no—
257C8—42no—
258C8—41no—
259C22—21no—
260C22—22no—
261C22—41no—
262C22—42no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22—164no—
267C22#447×C12711yescharacteristic
268C4—21no—
269C4—41no—
270C4—82no—
271C4—82no—
272C4#453×C12711yescharacteristic
273C4B(4,𝔽2)×C12711yescharacteristic
274C2—41no—
275C2—81no—
276C2—162no—
277C2C127×#159511yescharacteristic
278C2C127×#219412yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#3199C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #408×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2185C2semidirect
3. C127×#2020C2semidirect
4. C127×#1349C2semidirect
5. C127×#1315C2semidirect
7. C22×Q16×C127C22semidirect
17. C22⋊C16×C127C22semidirect
18. #341×C127C22semidirect
134. #12111C127direct (N × H)
141. #2185C2×C127semidirect
142. #2020C2×C127semidirect
143. #1349C2×C127semidirect
144. #1315C2×C127semidirect
145. C127#12111direct (N × H)
147. C22×Q16C22×C127semidirect
157. C22⋊C16C22×C127semidirect
158. #341C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31992
C127×#21852
C127×#20202
C127×#13492
C127×#13152
#12111127