Finite Group Databasebeta

← C127×#13331 #44577004

Subgroups — C127×#13331 #44577004

Order 32512.

650 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C2×#509×C127C212yesmaximal, complemented
2C2×#493×C127C211yesmaximal, characteristic, complemented
3C2×#361×C127C212yesmaximal, complemented
4C127×#1974C211yesmaximal, characteristic, complemented
5C127×#1376C211yesmaximal, characteristic, complemented
6C2×#135×C127C2211yescharacteristic
7C22×D8×C127—21no—
8C22×D8×C127C2212yes—
9C2×#127×C127—21no—
10#509×C127—24no—
11#509×C127—24no—
12C2×SD32×C127—22no—
13C2×D16×C127—22no—
14C2×M32×C127C2212yes—
15#493×C127—24no—
16C2×#108×C127C2211yescharacteristic
17C2×#104×C127—21no—
18#408×C127—21no—
19C2×C4×C8×C127C2211yescharacteristic
20#361×C127—24no—
21C2×#41×C127—21no—
22C22×D4×C127—22no—
23C22×D4×C127—42no—
24#135×C127—22no—
25#135×C127—22no—
26C2×Q16×C127D411yescharacteristic, complemented
27C2×SD16×C127—21no—
28C2×D8×C127—24no—
29C2×D8×C127—22no—
30C2×D8×C127—22no—
31C2×D8×C127—22no—
32C2×D8×C127—22no—
33C2×D8×C127—22no—
34C2×D8×C127—42no—
35C2×D8×C127—42no—
36C2×D8×C127D414yes—
37C2×D8×C127D411yescharacteristic, complemented
38C2×M16×C127—21no—
39C22×C8×C127D411yescharacteristic
40C22×C8×C127C2311yescharacteristic
41#127×C127—24no—
42#127×C127—42no—
43C2×C42×C127D411yescharacteristic
44SD32×C127—28no—
45D16×C127—28no—
46M32×C127—22no—
47M32×C127D414yescomplemented
48C2×C16×C127—22no—
49#108×C127—22no—
50#104×C127—42no—
51C22⋊C8×C127—42no—
52C4×C8×C127—22no—
53C24×C127—42no—
54#41×C127—41no—
55#41×C127—41no—
56#41×C127—41no—
57#41×C127—41no—
58C2×Q8×C127—21no—
59C2×D4×C127—24no—
60C2×D4×C127—21no—
61C2×D4×C127—48no—
62C2×D4×C127—41no—
63C2×D4×C127—42no—
64C2×D4×C127—42no—
65C2×D4×C127—42no—
66C2×D4×C127—42no—
67C2×D4×C127—42no—
68C2×D4×C127—88no—
69C22×C4×C127—21no—
70C22×C4×C127—41no—
71C22×C4×C127C2×D411yescharacteristic
72Q16×C127—22no—
73SD16×C127—42no—
74D8×C127—22no—
75D8×C127—24no—
76D8×C127—24no—
77D8×C127—48no—
78D8×C127—42no—
79D8×C127—42no—
80M16×C127—41no—
81M16×C127—41no—
82C2×C8×C127—21no—
83C2×C8×C127—21no—
84C2×C8×C127—41no—
85C2×C8×C127D812yes—
86C2×C8×C127C2×D411yescharacteristic
87C2×C8×C127C2×D412yes—
88C2×C8×C127C2×D412yes—
89C2×C8×C127#4111yescharacteristic
90C42×C127—21no—
91C42×C127D812yes—
92C16×C127—24no—
93C23×C127—44no—
94C23×C127—41no—
95C23×C127—88no—
96C23×C127—82no—
97C23×C127—82no—
98C23×C127AGL(1,ℤ/8)11yescharacteristic
99Q8×C127—41no—
100Q8×C127—41no—
101D4×C127—44no—
102D4×C127—41no—
103D4×C127—44no—
104D4×C127—41no—
105D4×C127—816no—
106D4×C127—81no—
107D4×C127—81no—
108D4×C127—82no—
109D4×C127—82no—
110C2×C4×C127—24no—
111C2×C4×C127—22no—
112C2×C4×C127—41no—
113C2×C4×C127—81no—
114C2×C4×C127—81no—
115C2×C4×C127#12011yescharacteristic
116C2×C4×C127#12111yescharacteristic
117C2×C4×C127#12711yescharacteristic
118C2×C4×C127C2×D811yescharacteristic
119C2×C4×C127C2×D811yescharacteristic
120C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
121C8×C127—22no—
122C8×C127—81no—
123C8×C127#12012yes—
124C8×C127#12112yes—
125C22×C127—22no—
126C22×C127—88no—
127C22×C127—82no—
128C22×C127—81no—
129C22×C127—84no—
130C22×C127—164no—
131C22×C127—164no—
132C22×C127#44711yescharacteristic
133C22×C127#45911yescharacteristic
134C22×C127#49611yescharacteristic
135C4×C127—24no—
136C4×C127—81no—
137C4×C127#39211yescharacteristic
138C4×C127#44711yescharacteristic
139C4×C127#45911yescharacteristic
140C4×C127#49311yescharacteristic
141#13331C12711yesmaximal, characteristic, direct product factor, Sylow
142C2×C127—21no—
143C2×C127—21no—
144C2×C127—161no—
145C2×C127—164no—
146C2×C127#201111yescharacteristic
147C2×C127#221312yes—
148C2×#509C2×C12712yescomplemented
149C2×#493C2×C12711yescharacteristic, complemented
150C2×#361C2×C12712yescomplemented
151#1974C2×C12711yescharacteristic, complemented
152#1376C2×C12711yescharacteristic, complemented
153C127#1333111yescharacteristic, direct product factor, Sylow
154C2×#135C22×C12711yescharacteristic
155C22×D8—21no—
156C22×D8C22×C12712yes—
157C2×#127—21no—
158#509—24no—
159#509—24no—
160C2×SD32—22no—
161C2×D16—22no—
162C2×M32C22×C12712yes—
163#493—24no—
164C2×#108C22×C12711yescharacteristic
165C2×#104—21no—
166#408—21no—
167C2×C4×C8C22×C12711yescharacteristic
168#361—24no—
169C2×#41—21no—
170C22×D4—22no—
171C22×D4—42no—
172#135—22no—
173#135—22no—
174C2×Q16D4×C12711yescharacteristic, complemented
175C2×SD16—21no—
176C2×D8—24no—
177C2×D8—22no—
178C2×D8—22no—
179C2×D8—22no—
180C2×D8—22no—
181C2×D8—22no—
182C2×D8—42no—
183C2×D8—42no—
184C2×D8D4×C12714yes—
185C2×D8D4×C12711yescharacteristic, complemented
186C2×M16—21no—
187C22×C8D4×C12711yescharacteristic
188C22×C8C23×C12711yescharacteristic
189#127—24no—
190#127—42no—
191C2×C42D4×C12711yescharacteristic
192SD32—28no—
193D16—28no—
194M32—22no—
195M32D4×C12714yescomplemented
196C2×C16—22no—
197#108—22no—
198#104—42no—
199C22⋊C8—42no—
200C4×C8—22no—
201C24—42no—
202#41—41no—
203#41—41no—
204#41—41no—
205#41—41no—
206C2×Q8—21no—
207C2×D4—24no—
208C2×D4—21no—
209C2×D4—48no—
210C2×D4—41no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—88no—
217C22×C4—21no—
218C22×C4—41no—
219C22×C4C2×D4×C12711yescharacteristic
220Q16—22no—
221SD16—42no—
222D8—22no—
223D8—24no—
224D8—24no—
225D8—48no—
226D8—42no—
227D8—42no—
228M16—41no—
229M16—41no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—41no—
233C2×C8D8×C12712yes—
234C2×C8C2×D4×C12711yescharacteristic
235C2×C8C2×D4×C12712yes—
236C2×C8C2×D4×C12712yes—
237C2×C8#41×C12711yescharacteristic
238C42—21no—
239C42D8×C12712yes—
240C16—24no—
241C23—44no—
242C23—41no—
243C23—88no—
244C23—82no—
245C23—82no—
246C23AGL(1,ℤ/8)×C12711yescharacteristic
247Q8—41no—
248Q8—41no—
249D4—44no—
250D4—41no—
251D4—44no—
252D4—41no—
253D4—816no—
254D4—81no—
255D4—81no—
256D4—82no—
257D4—82no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4#120×C12711yescharacteristic
264C2×C4#121×C12711yescharacteristic
265C2×C4#127×C12711yescharacteristic
266C2×C4C2×D8×C12711yescharacteristic
267C2×C4C2×D8×C12711yescharacteristic
268C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
269C8—22no—
270C8—81no—
271C8#120×C12712yes—
272C8#121×C12712yes—
273C22—22no—
274C22—88no—
275C22—82no—
276C22—81no—
277C22—84no—
278C22—164no—
279C22—164no—
280C22#447×C12711yescharacteristic
281C22#459×C12711yescharacteristic
282C22#496×C12711yescharacteristic
283C4—24no—
284C4—81no—
285C4#392×C12711yescharacteristic
286C4#447×C12711yescharacteristic
287C4#459×C12711yescharacteristic
288C4#493×C12711yescharacteristic
289C2—21no—
290C2—21no—
291C2—164no—
292C2—161no—
293C2C127×#201111yescharacteristic
294C2C127×#221312yes—
subgroupG/H
center, socleC22×C127#459
commutator, Frattini, cosocleC22×C8C23×C127
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#509×C127C2semidirect
2. C2×#493×C127C2semidirect
3. C2×#361×C127C2semidirect
4. C127×#1974C2semidirect
5. C127×#1376C2semidirect
26. C2×Q16×C127D4semidirect
37. C2×D8×C127D4semidirect
47. M32×C127D4semidirect
141. #13331C127direct (N × H)
148. C2×#509C2×C127semidirect
149. C2×#493C2×C127semidirect
150. C2×#361C2×C127semidirect
151. #1974C2×C127semidirect
152. #1376C2×C127semidirect
153. C127#13331direct (N × H)
174. C2×Q16D4×C127semidirect
185. C2×D8D4×C127semidirect
195. M32D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#509×C1272
C2×#493×C1272
C2×#361×C1272
C127×#19742
C127×#13762
#13331127