Finite Group Databasebeta

← C127×#12117 #44575790

Subgroups — C127×#12117 #44575790

Order 32512.

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

normal subgroups only

This group has 424 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×#3199C211yesmaximal, characteristic, complemented
2C127×#2190C212yesmaximal, complemented
3C127×#2017C211yesmaximal, characteristic, complemented
4C127×#1341C212yesmaximal, complemented
5C127×#1321C211yesmaximal, characteristic, complemented
6#575×C127—21no—
7#575×C127C2211yescharacteristic, complemented
8#535×C127—21no—
9#509×C127—22no—
10#482×C127—22no—
11#469×C127C2212yes—
12#466×C127—21no—
13#447×C127—22no—
14#408×C127—21no—
15#408×C127C2211yescharacteristic
16#360×C127—22no—
17#349×C127C2212yescomplemented
18#338×C127C2211yescharacteristic, complemented
19C2×#41×C127—21no—
20C2×#41×C127D411yescharacteristic
21C22×D4×C127—21no—
22#137×C127—41no—
23#137×C127—41no—
24#137×C127—41no—
25AGL(1,ℤ/8)×C127—41no—
26AGL(1,ℤ/8)×C127—41no—
27AGL(1,ℤ/8)×C127—41no—
28#135×C127—22no—
29#135×C127—42no—
30C2×D8×C127—21no—
31C2×D8×C127D412yes—
32C2×M16×C127—21no—
33C2×M16×C127C2311yescharacteristic
34C22×C8×C127D411yescharacteristic
35#127×C127—41no—
36#124×C127—41no—
37#121×C127—22no—
38#121×C127—22no—
39#120×C127—41no—
40#120×C127—41no—
41#117×C127—21no—
42SD32×C127—42no—
43D16×C127—42no—
44M32×C127—22no—
45#107×C127—21no—
46#106×C127D412yes—
47#102×C127—22no—
48#102×C127—22no—
49C22⋊C8×C127—22no—
50C22⋊C8×C127—42no—
51C24×C127—41no—
52#41×C127—41no—
53#41×C127—41no—
54#41×C127—41no—
55#41×C127—41no—
56#41×C127—41no—
57#41×C127—41no—
58C2×Q8×C127—21no—
59C2×Q8×C127D811yescharacteristic
60C2×D4×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—22no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127—42no—
66C2×D4×C127—42no—
67C2×D4×C127—81no—
68C2×D4×C127—81no—
69C2×D4×C127D811yescharacteristic
70C22×C4×C127—21no—
71C22×C4×C127—41no—
72C22×C4×C127C2×D411yescharacteristic
73Q16×C127—22no—
74Q16×C127—42no—
75SD16×C127—42no—
76SD16×C127—42no—
77SD16×C127—42no—
78D8×C127—22no—
79D8×C127—42no—
80D8×C127—42no—
81D8×C127—81no—
82M16×C127—41no—
83M16×C127—41no—
84M16×C127—41no—
85C2×C8×C127—22no—
86C2×C8×C127—21no—
87C2×C8×C127—41no—
88C2×C8×C127C2×D412yes—
89C4⋊C4×C127—22no—
90C22⋊C4×C127—42no—
91C22⋊C4×C127—41no—
92C22⋊C4×C127—42no—
93C42×C127—41no—
94C16×C127—42no—
95C23×C127—21no—
96C23×C127—41no—
97C23×C127—42no—
98C23×C127—82no—
99C23×C127—162no—
100C23×C127AGL(1,ℤ/8)11yescharacteristic
101Q8×C127—41no—
102Q8×C127—41no—
103Q8×C127—41no—
104D4×C127—41no—
105D4×C127—41no—
106D4×C127—42no—
107D4×C127—41no—
108D4×C127—42no—
109D4×C127—81no—
110D4×C127—82no—
111D4×C127—81no—
112D4×C127—81no—
113D4×C127—162no—
114D4×C127—162no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—42no—
122C2×C4×C127—81no—
123C2×C4×C127—81no—
124C2×C4×C127#12011yescharacteristic
125C2×C4×C127C2×D811yescharacteristic
126C8×C127—22no—
127C8×C127—42no—
128C8×C127—41no—
129C22×C127—21no—
130C22×C127—41no—
131C22×C127—42no—
132C22×C127—41no—
133C22×C127—82no—
134C22×C127—82no—
135C22×C127—82no—
136C22×C127—81no—
137C22×C127—82no—
138C22×C127—164no—
139C22×C127#44711yescharacteristic
140C4×C127—21no—
141C4×C127—41no—
142C4×C127—82no—
143C4×C127—81no—
144C4×C127#45311yescharacteristic
145C4×C127B(4,𝔽2)11yescharacteristic
146#12117C12711yesmaximal, characteristic, direct product factor, Sylow
147C2×C127—21no—
148C2×C127—41no—
149C2×C127—81no—
150C2×C127—161no—
151C2×C127—162no—
152C2×C127#159511yescharacteristic
153#3199C2×C12711yescharacteristic, complemented
154#2190C2×C12712yescomplemented
155#2017C2×C12711yescharacteristic, complemented
156#1341C2×C12712yescomplemented
157#1321C2×C12711yescharacteristic, complemented
158C127#1211711yescharacteristic, direct product factor, Sylow
159#575—21no—
160#575C22×C12711yescharacteristic, complemented
161#535—21no—
162#509—22no—
163#482—22no—
164#469C22×C12712yes—
165#466—21no—
166#447—22no—
167#408—21no—
168#408C22×C12711yescharacteristic
169#360—22no—
170#349C22×C12712yescomplemented
171#338C22×C12711yescharacteristic, complemented
172C2×#41—21no—
173C2×#41D4×C12711yescharacteristic
174C22×D4—21no—
175#137—41no—
176#137—41no—
177#137—41no—
178AGL(1,ℤ/8)—41no—
179AGL(1,ℤ/8)—41no—
180AGL(1,ℤ/8)—41no—
181#135—22no—
182#135—42no—
183C2×D8—21no—
184C2×D8D4×C12712yes—
185C2×M16—21no—
186C2×M16C23×C12711yescharacteristic
187C22×C8D4×C12711yescharacteristic
188#127—41no—
189#124—41no—
190#121—22no—
191#121—22no—
192#120—41no—
193#120—41no—
194#117—21no—
195SD32—42no—
196D16—42no—
197M32—22no—
198#107—21no—
199#106D4×C12712yes—
200#102—22no—
201#102—22no—
202C22⋊C8—22no—
203C22⋊C8—42no—
204C24—41no—
205#41—41no—
206#41—41no—
207#41—41no—
208#41—41no—
209#41—41no—
210#41—41no—
211C2×Q8—21no—
212C2×Q8D8×C12711yescharacteristic
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—81no—
221C2×D4—81no—
222C2×D4D8×C12711yescharacteristic
223C22×C4—21no—
224C22×C4—41no—
225C22×C4C2×D4×C12711yescharacteristic
226Q16—22no—
227Q16—42no—
228SD16—42no—
229SD16—42no—
230SD16—42no—
231D8—22no—
232D8—42no—
233D8—42no—
234D8—81no—
235M16—41no—
236M16—41no—
237M16—41no—
238C2×C8—22no—
239C2×C8—21no—
240C2×C8—41no—
241C2×C8C2×D4×C12712yes—
242C4⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C42—41no—
247C16—42no—
248C23—21no—
249C23—41no—
250C23—42no—
251C23—82no—
252C23—162no—
253C23AGL(1,ℤ/8)×C12711yescharacteristic
254Q8—41no—
255Q8—41no—
256Q8—41no—
257D4—41no—
258D4—41no—
259D4—42no—
260D4—41no—
261D4—42no—
262D4—81no—
263D4—82no—
264D4—81no—
265D4—81no—
266D4—162no—
267D4—162no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4#120×C12711yescharacteristic
278C2×C4C2×D8×C12711yescharacteristic
279C8—22no—
280C8—42no—
281C8—41no—
282C22—21no—
283C22—41no—
284C22—42no—
285C22—41no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—81no—
290C22—82no—
291C22—164no—
292C22#447×C12711yescharacteristic
293C4—21no—
294C4—41no—
295C4—82no—
296C4—81no—
297C4#453×C12711yescharacteristic
298C4B(4,𝔽2)×C12711yescharacteristic
299C2—21no—
300C2—41no—
301C2—81no—
302C2—162no—
303C2—161no—
304C2C127×#159511yescharacteristic
subgroupG/H
center, socleC2×C127#1595
commutator, Frattini, cosocleC2×M16C23×C127
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
1. C127×#3199C2semidirect
2. C127×#2190C2semidirect
3. C127×#2017C2semidirect
4. C127×#1341C2semidirect
5. C127×#1321C2semidirect
7. #575×C127C22semidirect
17. #349×C127C22semidirect
18. #338×C127C22semidirect
146. #12117C127direct (N × H)
153. #3199C2×C127semidirect
154. #2190C2×C127semidirect
155. #2017C2×C127semidirect
156. #1341C2×C127semidirect
157. #1321C2×C127semidirect
158. C127#12117direct (N × H)
160. #575C22×C127semidirect
170. #349C22×C127semidirect
171. #338C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31992
C127×#21902
C127×#20172
C127×#13412
C127×#13212
#12117127