Finite Group Databasebeta

← C127×#12208 #44575881

Subgroups — C127×#12208 #44575881

Order 32512.

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

normal subgroups only

This group has 664 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×#2679C211yesmaximal, characteristic
2C127×#2027C211yesmaximal, characteristic, complemented
3C127×#1599C212yesmaximal, complemented
4C127×#1345C212yesmaximal, complemented
5C127×#1318C211yesmaximal, characteristic, complemented
6#544×C127C2211yescharacteristic, complemented
7C2×#120×C127—22no—
8C2×#120×C127—21no—
9#450×C127—22no—
10#418×C127—21no—
11#394×C127—22no—
12#394×C127C2212yes—
13#390×C127—21no—
14#390×C127C2211yescharacteristic
15#343×C127C2211yescharacteristic, complemented
16#339×C127—21no—
17#328×C127—22no—
18#324×C127C2212yescomplemented
19C25×C127—21no—
20C22×D4×C127—21no—
21C22×D4×C127—22no—
22C22×D4×C127D411yescharacteristic
23C22×D4×C127D412yes—
24C2×SD16×C127—42no—
25C2×M16×C127—21no—
26C4⋊Q8×C127—41no—
27#125×C127—41no—
28C22⋊Q8×C127—22no—
29#120×C127—48no—
30#120×C127—42no—
31#120×C127—42no—
32#117×C127—21no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127C2311yescharacteristic
35C2×C22⋊C4×C127—22no—
36C2×C22⋊C4×C127—21no—
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127—22no—
40C2×C42×C127D411yescharacteristic
41#102×C127—41no—
42#102×C127—41no—
43#102×C127—42no—
44C22⋊C8×C127—22no—
45#95×C127D412yes—
46C24×C127—22no—
47C24×C127—21no—
48C24×C127—44no—
49C24×C127—42no—
50C24×C127—82no—
51C24×C127—81no—
52C24×C127—81no—
53C2×Q8×C127—22no—
54C2×D4×C127—24no—
55C2×D4×C127—21no—
56C2×D4×C127—41no—
57C2×D4×C127—42no—
58C2×D4×C127—41no—
59C2×D4×C127—44no—
60C2×D4×C127—44no—
61C2×D4×C127—44no—
62C2×D4×C127—81no—
63C2×D4×C127—81no—
64C2×D4×C127—82no—
65C2×D4×C127SD1612yescomplemented
66C22×C4×C127—21no—
67C22×C4×C127—22no—
68C22×C4×C127C2×D411yescharacteristic
69C22×C4×C127C2×D412yes—
70SD16×C127—84no—
71M16×C127—41no—
72M16×C127—41no—
73C2×C8×C127—42no—
74C2×C8×C127—41no—
75C4⋊C4×C127—21no—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—22no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—42no—
80C4⋊C4×C127—81no—
81C22⋊C4×C127—44no—
82C22⋊C4×C127—44no—
83C22⋊C4×C127—44no—
84C22⋊C4×C127—42no—
85C22⋊C4×C127—41no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C42×C127—41no—
89C42×C127—41no—
90C23×C127—24no—
91C23×C127—22no—
92C23×C127—42no—
93C23×C127—42no—
94C23×C127—42no—
95C23×C127—42no—
96C23×C127—42no—
97C23×C127—42no—
98C23×C127—41no—
99C23×C127—42no—
100C23×C127—41no—
101C23×C127—42no—
102C23×C127—88no—
103C23×C127—84no—
104C23×C127—88no—
105C23×C127—84no—
106C23×C127—84no—
107C23×C127#12011yescharacteristic
108Q8×C127—82no—
109D4×C127—42no—
110D4×C127—44no—
111D4×C127—81no—
112D4×C127—84no—
113D4×C127—82no—
114D4×C127—88no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127C2×SD1611yescharacteristic
128C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
129C8×C127—81no—
130C8×C127—82no—
131C22×C127—21no—
132C22×C127—21no—
133C22×C127—44no—
134C22×C127—44no—
135C22×C127—44no—
136C22×C127—42no—
137C22×C127—44no—
138C22×C127—82no—
139C22×C127—82no—
140C22×C127—82no—
141C22×C127—816no—
142C22×C127—82no—
143C22×C127—81no—
144C22×C127—81no—
145C22×C127—82no—
146C22×C127#45011yescharacteristic
147C22×C127#45511yescharacteristic
148C22×C127B(4,𝔽2)11yescharacteristic
149C4×C127—21no—
150C4×C127—21no—
151C4×C127—42no—
152C4×C127—42no—
153C4×C127—82no—
154C4×C127—81no—
155C4×C127—81no—
156#12208C12711yesmaximal, characteristic, direct product factor, Sylow
157C2×C127—21no—
158C2×C127—21no—
159C2×C127—82no—
160C2×C127—84no—
161C2×C127#160111yescharacteristic
162C2×C127#219611yescharacteristic
163C2×C127#220211yescharacteristic
164#2679C2×C12711yescharacteristic
165#2027C2×C12711yescharacteristic, complemented
166#1599C2×C12712yescomplemented
167#1345C2×C12712yescomplemented
168#1318C2×C12711yescharacteristic, complemented
169C127#1220811yescharacteristic, direct product factor, Sylow
170#544C22×C12711yescharacteristic, complemented
171C2×#120—22no—
172C2×#120—21no—
173#450—22no—
174#418—21no—
175#394—22no—
176#394C22×C12712yes—
177#390—21no—
178#390C22×C12711yescharacteristic
179#343C22×C12711yescharacteristic, complemented
180#339—21no—
181#328—22no—
182#324C22×C12712yescomplemented
183C25—21no—
184C22×D4—21no—
185C22×D4—22no—
186C22×D4D4×C12711yescharacteristic
187C22×D4D4×C12712yes—
188C2×SD16—42no—
189C2×M16—21no—
190C4⋊Q8—41no—
191#125—41no—
192C22⋊Q8—22no—
193#120—48no—
194#120—42no—
195#120—42no—
196#117—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C23×C12711yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C42D4×C12711yescharacteristic
205#102—41no—
206#102—41no—
207#102—42no—
208C22⋊C8—22no—
209#95D4×C12712yes—
210C24—22no—
211C24—21no—
212C24—44no—
213C24—42no—
214C24—82no—
215C24—81no—
216C24—81no—
217C2×Q8—22no—
218C2×D4—24no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—42no—
222C2×D4—41no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—81no—
227C2×D4—81no—
228C2×D4—82no—
229C2×D4SD16×C12712yescomplemented
230C22×C4—21no—
231C22×C4—22no—
232C22×C4C2×D4×C12711yescharacteristic
233C22×C4C2×D4×C12712yes—
234SD16—84no—
235M16—41no—
236M16—41no—
237C2×C8—42no—
238C2×C8—41no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—81no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—42no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C42—41no—
253C42—41no—
254C23—24no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—41no—
263C23—42no—
264C23—41no—
265C23—42no—
266C23—88no—
267C23—84no—
268C23—88no—
269C23—84no—
270C23—84no—
271C23#120×C12711yescharacteristic
272Q8—82no—
273D4—42no—
274D4—44no—
275D4—81no—
276D4—84no—
277D4—82no—
278D4—88no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4C2×SD16×C12711yescharacteristic
292C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
293C8—81no—
294C8—82no—
295C22—21no—
296C22—21no—
297C22—44no—
298C22—44no—
299C22—44no—
300C22—42no—
301C22—44no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—816no—
306C22—82no—
307C22—81no—
308C22—81no—
309C22—82no—
310C22#450×C12711yescharacteristic
311C22#455×C12711yescharacteristic
312C22B(4,𝔽2)×C12711yescharacteristic
313C4—21no—
314C4—21no—
315C4—42no—
316C4—42no—
317C4—82no—
318C4—81no—
319C4—81no—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—84no—
324C2C127×#160111yescharacteristic
325C2C127×#219611yescharacteristic
326C2C127×#220211yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC127×#2679C2
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
2. C127×#2027C2semidirect
3. C127×#1599C2semidirect
4. C127×#1345C2semidirect
5. C127×#1318C2semidirect
6. #544×C127C22semidirect
15. #343×C127C22semidirect
18. #324×C127C22semidirect
65. C2×D4×C127SD16semidirect
156. #12208C127direct (N × H)
165. #2027C2×C127semidirect
166. #1599C2×C127semidirect
167. #1345C2×C127semidirect
168. #1318C2×C127semidirect
169. C127#12208direct (N × H)
170. #544C22×C127semidirect
179. #343C22×C127semidirect
182. #324C22×C127semidirect
229. C2×D4SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#26792
C127×#20272
C127×#15992
C127×#13452
C127×#13182
#12208127