Finite Group Databasebeta

← C127×#12156 #44575829

Subgroups — C127×#12156 #44575829

Order 32512.

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

normal subgroups only

This group has 484 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×#2819C211yesmaximal, characteristic, complemented
2C127×#2040C211yesmaximal, characteristic, complemented
3C127×#1598C212yesmaximal, complemented
4C127×#1345C212yesmaximal, complemented
5C127×#1324C211yesmaximal, characteristic, complemented
6#537×C127C2211yescharacteristic
7C2×#124×C127—21no—
8#447×C127—22no—
9C2×#104×C127—21no—
10#417×C127—21no—
11#394×C127—21no—
12#394×C127C2212yes—
13#393×C127—22no—
14#392×C127—22no—
15#382×C127C2211yescharacteristic
16#343×C127C2211yescharacteristic
17#327×C127—22no—
18#324×C127C2212yes—
19C2×#41×C127—21no—
20C22×Q8×C127—21no—
21C22×D4×C127—21no—
22C22×D4×C127—21no—
23C22×D4×C127D412yes—
24C2×D8×C127—42no—
25C2×M16×C127—21no—
26#124×C127—42no—
27#124×C127—42no—
28C22⋊Q8×C127—41no—
29#121×C127—22no—
30#121×C127—41no—
31C2×C4⋊C4×C127D411yescharacteristic
32C2×C4⋊C4×C127C2311yescharacteristic
33C2×C22⋊C4×C127—22no—
34C2×C22⋊C4×C127—22no—
35C2×C22⋊C4×C127—22no—
36C2×C22⋊C4×C127—22no—
37C2×C42×C127D411yescharacteristic
38#104×C127—42no—
39#103×C127—41no—
40#102×C127—42no—
41#102×C127—41no—
42C22⋊C8×C127—22no—
43#95×C127—21no—
44#95×C127—22no—
45#95×C127D412yes—
46C24×C127—22no—
47#41×C127—41no—
48#41×C127—41no—
49#41×C127—41no—
50#41×C127—41no—
51C2×Q8×C127—21no—
52C2×Q8×C127—42no—
53C2×Q8×C127—81no—
54C2×Q8×C127—81no—
55C2×D4×C127—21no—
56C2×D4×C127—22no—
57C2×D4×C127—42no—
58C2×D4×C127—42no—
59C2×D4×C127—41no—
60C2×D4×C127—82no—
61C2×D4×C127—81no—
62C2×D4×C127—81no—
63C2×D4×C127—82no—
64C2×D4×C127—81no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—22no—
68C22×C4×C127—41no—
69C22×C4×C127C2×D411yescharacteristic
70C22×C4×C127C2×D412yes—
71D8×C127—84no—
72M16×C127—41no—
73M16×C127—41no—
74C2×C8×C127—42no—
75C2×C8×C127—41no—
76C4⋊C4×C127—22no—
77C4⋊C4×C127—41no—
78C4⋊C4×C127—41no—
79C22⋊C4×C127—44no—
80C22⋊C4×C127—42no—
81C22⋊C4×C127—42no—
82C22⋊C4×C127—42no—
83C22⋊C4×C127—82no—
84C22⋊C4×C127—82no—
85C42×C127—21no—
86C42×C127D812yescomplemented
87C23×C127—42no—
88C23×C127—42no—
89C23×C127—42no—
90C23×C127—42no—
91C23×C127—42no—
92C23×C127—42no—
93C23×C127—41no—
94C23×C127—42no—
95C23×C127#12011yescharacteristic
96Q8×C127—41no—
97Q8×C127—41no—
98Q8×C127—84no—
99D4×C127—41no—
100D4×C127—41no—
101D4×C127—44no—
102D4×C127—82no—
103D4×C127—82no—
104D4×C127—84no—
105D4×C127—84no—
106D4×C127—81no—
107D4×C127—81no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—81no—
122C2×C4×C127—81no—
123C2×C4×C127C2×D811yescharacteristic
124C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
125C8×C127—81no—
126C8×C127—82no—
127C22×C127—21no—
128C22×C127—21no—
129C22×C127—82no—
130C22×C127—82no—
131C22×C127—82no—
132C22×C127—82no—
133C22×C127—81no—
134C22×C127—82no—
135C22×C127—82no—
136C22×C127—81no—
137C22×C127—82no—
138C22×C127—81no—
139C22×C127#44711yescharacteristic
140C22×C127#45411yescharacteristic
141C22×C127B(4,𝔽2)11yescharacteristic
142C4×C127—21no—
143C4×C127—21no—
144C4×C127—42no—
145C4×C127—42no—
146C4×C127—82no—
147C4×C127—81no—
148C4×C127—81no—
149#12156C12711yesmaximal, characteristic, direct product factor, Sylow
150C2×C127—21no—
151C2×C127—21no—
152C2×C127—161no—
153C2×C127—162no—
154C2×C127#159811yescharacteristic
155C2×C127#219911yescharacteristic
156C2×C127#220211yescharacteristic
157#2819C2×C12711yescharacteristic, complemented
158#2040C2×C12711yescharacteristic, complemented
159#1598C2×C12712yescomplemented
160#1345C2×C12712yescomplemented
161#1324C2×C12711yescharacteristic, complemented
162C127#1215611yescharacteristic, direct product factor, Sylow
163#537C22×C12711yescharacteristic
164C2×#124—21no—
165#447—22no—
166C2×#104—21no—
167#417—21no—
168#394—21no—
169#394C22×C12712yes—
170#393—22no—
171#392—22no—
172#382C22×C12711yescharacteristic
173#343C22×C12711yescharacteristic
174#327—22no—
175#324C22×C12712yes—
176C2×#41—21no—
177C22×Q8—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4D4×C12712yes—
181C2×D8—42no—
182C2×M16—21no—
183#124—42no—
184#124—42no—
185C22⋊Q8—41no—
186#121—22no—
187#121—41no—
188C2×C4⋊C4D4×C12711yescharacteristic
189C2×C4⋊C4C23×C12711yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C42D4×C12711yescharacteristic
195#104—42no—
196#103—41no—
197#102—42no—
198#102—41no—
199C22⋊C8—22no—
200#95—21no—
201#95—22no—
202#95D4×C12712yes—
203C24—22no—
204#41—41no—
205#41—41no—
206#41—41no—
207#41—41no—
208C2×Q8—21no—
209C2×Q8—42no—
210C2×Q8—81no—
211C2×Q8—81no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—82no—
218C2×D4—81no—
219C2×D4—81no—
220C2×D4—82no—
221C2×D4—81no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4C2×D4×C12711yescharacteristic
227C22×C4C2×D4×C12712yes—
228D8—84no—
229M16—41no—
230M16—41no—
231C2×C8—42no—
232C2×C8—41no—
233C4⋊C4—22no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—44no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—21no—
243C42D8×C12712yescomplemented
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—41no—
251C23—42no—
252C23#120×C12711yescharacteristic
253Q8—41no—
254Q8—41no—
255Q8—84no—
256D4—41no—
257D4—41no—
258D4—44no—
259D4—82no—
260D4—82no—
261D4—84no—
262D4—84no—
263D4—81no—
264D4—81no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4C2×D8×C12711yescharacteristic
281C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
282C8—81no—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—81no—
291C22—82no—
292C22—82no—
293C22—81no—
294C22—82no—
295C22—81no—
296C22#447×C12711yescharacteristic
297C22#454×C12711yescharacteristic
298C22B(4,𝔽2)×C12711yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—42no—
303C4—82no—
304C4—81no—
305C4—81no—
306C2—21no—
307C2—21no—
308C2—162no—
309C2—161no—
310C2C127×#159811yescharacteristic
311C2C127×#219911yescharacteristic
312C2C127×#220211yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2819C2semidirect
2. C127×#2040C2semidirect
3. C127×#1598C2semidirect
4. C127×#1345C2semidirect
5. C127×#1324C2semidirect
86. C42×C127D8semidirect
149. #12156C127direct (N × H)
157. #2819C2×C127semidirect
158. #2040C2×C127semidirect
159. #1598C2×C127semidirect
160. #1345C2×C127semidirect
161. #1324C2×C127semidirect
162. C127#12156direct (N × H)
243. C42D8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#28192
C127×#20402
C127×#15982
C127×#13452
C127×#13242
#12156127