Finite Group Databasebeta

← C127×#12332 #44576005

Subgroups — C127×#12332 #44576005

Order 32512.

362 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3391C211yesmaximal, characteristic
2C127×#2248C211yesmaximal, characteristic, complemented
3C127×#2208C211yesmaximal, characteristic, complemented
4C127×#1670C211yesmaximal, characteristic, complemented
5C127×#1361C211yesmaximal, characteristic, complemented
6C127×Q8⋊C16C211yesmaximal, characteristic, complemented
7C127×#1331C211yesmaximal, characteristic, complemented
8#550×C127—21no—
9C2×SD32×C127—21no—
10C2×SD32×C127—22no—
11#494×C127C2211yescharacteristic
12#493×C127—21no—
13#493×C127C2211yescharacteristic
14#475×C127—21no—
15#459×C127—22no—
16#459×C127—21no—
17C8×Q8×C127C2211yescharacteristic
18C4×D8×C127—21no—
19C4×D8×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic
21#358×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#332×C127—21no—
25C2×#41×C127—41no—
26C2×Q16×C127—21no—
27C2×Q16×C127—21no—
28C2×D8×C127—22no—
29C2×D8×C127—21no—
30C2×D8×C127D411yescharacteristic
31C2×D8×C127D412yes—
32C4⋊Q8×C127—21no—
33#127×C127—22no—
34#127×C127—21no—
35#121×C127—42no—
36#121×C127—41no—
37C4×Q8×C127D411yescharacteristic
38C4×D4×C127—22no—
39C4×D4×C127—21no—
40SD32×C127—44no—
41SD32×C127—42no—
42C2×C16×C127—21no—
43C2×C16×C127—21no—
44#107×C127D411yescharacteristic
45#106×C127—41no—
46C4⋊C8×C127—21no—
47C4⋊C8×C127—21no—
48C4⋊C8×C127D411yescharacteristic
49#103×C127—41no—
50#102×C127—22no—
51#102×C127—21no—
52C4×C8×C127—21no—
53C4×C8×C127C2311yescharacteristic
54#41×C127—82no—
55#41×C127—82no—
56C2×Q8×C127—41no—
57C2×Q8×C127D811yescharacteristic
58C2×D4×C127—21no—
59C2×D4×C127—22no—
60C2×D4×C127—42no—
61C2×D4×C127—42no—
62C2×D4×C127—41no—
63C2×D4×C127—41no—
64C22×C4×C127—42no—
65C22×C4×C127—41no—
66Q16×C127—42no—
67Q16×C127—41no—
68Q16×C127—41no—
69D8×C127—21no—
70D8×C127—21no—
71D8×C127—42no—
72D8×C127—41no—
73D8×C127—41no—
74D8×C127—82no—
75C2×C8×C127—21no—
76C2×C8×C127—41no—
77C2×C8×C127C2×D411yescharacteristic
78C2×C8×C127C2×D411yescharacteristic
79C4⋊C4×C127—21no—
80C4⋊C4×C127—21no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127D811yescharacteristic
84C22⋊C4×C127—42no—
85C22⋊C4×C127—41no—
86C42×C127—21no—
87C42×C127C2×D411yescharacteristic
88C16×C127—22no—
89C16×C127—41no—
90C23×C127—42no—
91C23×C127—41no—
92Q8×C127—21no—
93Q8×C127—81no—
94Q8×C127—81no—
95Q8×C127SD3212yescomplemented
96D4×C127—41no—
97D4×C127—42no—
98D4×C127—42no—
99D4×C127—41no—
100D4×C127—84no—
101D4×C127—81no—
102D4×C127—81no—
103D4×C127—161no—
104D4×C127—162no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—41no—
108C2×C4×C127—84no—
109C2×C4×C127—81no—
110C2×C4×C127—81no—
111C2×C4×C127—81no—
112C2×C4×C127#12011yescharacteristic
113C2×C4×C127C2×D811yescharacteristic
114C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
115C8×C127—21no—
116C8×C127—41no—
117C8×C127—81no—
118C8×C127#13511yescharacteristic
119C8×C127AGL(1,ℤ/8)11yescharacteristic
120C22×C127—81no—
121C22×C127—82no—
122C22×C127—82no—
123C22×C127—81no—
124C22×C127—82no—
125C22×C127—81no—
126C22×C127#44711yescharacteristic
127C4×C127—22no—
128C4×C127—21no—
129C4×C127—41no—
130C4×C127—81no—
131C4×C127—161no—
132C4×C127#44911yescharacteristic
133C4×C127#45311yescharacteristic
134C4×C127C2×SD3211yescharacteristic
135C4×C127#50911yescharacteristic
136#12332C12711yesmaximal, characteristic, direct product factor, Sylow
137C2×C127—161no—
138C2×C127—162no—
139C2×C127#162111yescharacteristic
140C2×C127#218711yescharacteristic
141C2×C127#219411yescharacteristic
142#3391C2×C12711yescharacteristic
143#2248C2×C12711yescharacteristic, complemented
144#2208C2×C12711yescharacteristic, complemented
145#1670C2×C12711yescharacteristic, complemented
146#1361C2×C12711yescharacteristic, complemented
147Q8⋊C16C2×C12711yescharacteristic, complemented
148#1331C2×C12711yescharacteristic, complemented
149C127#1233211yescharacteristic, direct product factor, Sylow
150#550—21no—
151C2×SD32—21no—
152C2×SD32—22no—
153#494C22×C12711yescharacteristic
154#493—21no—
155#493C22×C12711yescharacteristic
156#475—21no—
157#459—22no—
158#459—21no—
159C8×Q8C22×C12711yescharacteristic
160C4×D8—21no—
161C4×D8C22×C12711yescharacteristic
162C4⋊C16C22×C12711yescharacteristic
163#358—21no—
164C4×C16C22×C12711yescharacteristic
165#335C22×C12711yescharacteristic
166#332—21no—
167C2×#41—41no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×D8—22no—
171C2×D8—21no—
172C2×D8D4×C12711yescharacteristic
173C2×D8D4×C12712yes—
174C4⋊Q8—21no—
175#127—22no—
176#127—21no—
177#121—42no—
178#121—41no—
179C4×Q8D4×C12711yescharacteristic
180C4×D4—22no—
181C4×D4—21no—
182SD32—44no—
183SD32—42no—
184C2×C16—21no—
185C2×C16—21no—
186#107D4×C12711yescharacteristic
187#106—41no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8D4×C12711yescharacteristic
191#103—41no—
192#102—22no—
193#102—21no—
194C4×C8—21no—
195C4×C8C23×C12711yescharacteristic
196#41—82no—
197#41—82no—
198C2×Q8—41no—
199C2×Q8D8×C12711yescharacteristic
200C2×D4—21no—
201C2×D4—22no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—41no—
206C22×C4—42no—
207C22×C4—41no—
208Q16—42no—
209Q16—41no—
210Q16—41no—
211D8—21no—
212D8—21no—
213D8—42no—
214D8—41no—
215D8—41no—
216D8—82no—
217C2×C8—21no—
218C2×C8—41no—
219C2×C8C2×D4×C12711yescharacteristic
220C2×C8C2×D4×C12711yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4D8×C12711yescharacteristic
226C22⋊C4—42no—
227C22⋊C4—41no—
228C42—21no—
229C42C2×D4×C12711yescharacteristic
230C16—22no—
231C16—41no—
232C23—42no—
233C23—41no—
234Q8—21no—
235Q8—81no—
236Q8—81no—
237Q8SD32×C12712yescomplemented
238D4—41no—
239D4—42no—
240D4—42no—
241D4—41no—
242D4—84no—
243D4—81no—
244D4—81no—
245D4—161no—
246D4—162no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—84no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4#120×C12711yescharacteristic
255C2×C4C2×D8×C12711yescharacteristic
256C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
257C8—21no—
258C8—41no—
259C8—81no—
260C8#135×C12711yescharacteristic
261C8AGL(1,ℤ/8)×C12711yescharacteristic
262C22—81no—
263C22—82no—
264C22—82no—
265C22—81no—
266C22—82no—
267C22—81no—
268C22#447×C12711yescharacteristic
269C4—22no—
270C4—21no—
271C4—41no—
272C4—81no—
273C4—161no—
274C4#449×C12711yescharacteristic
275C4#453×C12711yescharacteristic
276C4C2×SD32×C12711yescharacteristic
277C4#509×C12711yescharacteristic
278C2—161no—
279C2—162no—
280C2C127×#162111yescharacteristic
281C2C127×#218711yescharacteristic
282C2C127×#219411yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3391C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2248C2semidirect
3. C127×#2208C2semidirect
4. C127×#1670C2semidirect
5. C127×#1361C2semidirect
6. C127×Q8⋊C16C2semidirect
7. C127×#1331C2semidirect
95. Q8×C127SD32semidirect
136. #12332C127direct (N × H)
143. #2248C2×C127semidirect
144. #2208C2×C127semidirect
145. #1670C2×C127semidirect
146. #1361C2×C127semidirect
147. Q8⋊C16C2×C127semidirect
148. #1331C2×C127semidirect
149. C127#12332direct (N × H)
237. Q8SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#33912
C127×#22482
C127×#22082
C127×#16702
C127×#13612
C127×Q8⋊C162
C127×#13312
#12332127