Finite Group Databasebeta

← C127×#12303 #44575976

Subgroups — C127×#12303 #44575976

Order 32512.

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

normal subgroups only

This group has 322 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×#3204C211yesmaximal, characteristic
2C127×#2239C211yesmaximal, characteristic, complemented
3C127×#2238C211yesmaximal, characteristic, complemented
4C127×#2098C211yesmaximal, characteristic, complemented
5C127×#1340C211yesmaximal, characteristic, complemented
6C127×#1339C211yesmaximal, characteristic, complemented
7C127×#1321C211yesmaximal, characteristic, complemented
8#537×C127—21no—
9#483×C127—21no—
10#481×C127—21no—
11#470×C127C2211yescharacteristic
12#469×C127C2211yescharacteristic
13#465×C127—21no—
14#452×C127—21no—
15#449×C127—21no—
16#430×C127C2211yescharacteristic
17#424×C127—21no—
18#421×C127—21no—
19#408×C127C2211yescharacteristic
20#365×C127—21no—
21#365×C127—21no—
22#362×C127—21no—
23#361×C127—21no—
24#349×C127C2211yescharacteristic
25#349×C127C2211yescharacteristic
26H(3,ℤ/4)×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C2×#41×C127D411yescharacteristic
29C2×Q16×C127D411yescharacteristic
30C2×SD16×C127—21no—
31C2×D8×C127D411yescharacteristic
32C2×M16×C127—21no—
33C2×M16×C127C2311yescharacteristic
34C22×C8×C127D411yescharacteristic
35C22⋊Q8×C127—21no—
36C22⋊Q8×C127—21no—
37C22⋊Q8×C127—41no—
38#121×C127—21no—
39#121×C127—21no—
40#121×C127—41no—
41#121×C127—41no—
42#121×C127—41no—
43#117×C127—21no—
44C2×C4⋊C4×C127—21no—
45M32×C127—21no—
46M32×C127—21no—
47#108×C127—21no—
48#108×C127—21no—
49#106×C127—21no—
50#106×C127D411yescharacteristic
51#106×C127D411yescharacteristic
52C4⋊C8×C127—42no—
53#104×C127—42no—
54#103×C127—21no—
55#103×C127—21no—
56#102×C127—21no—
57#102×C127—21no—
58C22⋊C8×C127—21no—
59C22⋊C8×C127—21no—
60#41×C127—41no—
61#41×C127—41no—
62#41×C127—81no—
63#41×C127—81no—
64C2×Q8×C127—21no—
65C2×Q8×C127SD1611yescharacteristic
66C2×D4×C127—21no—
67C2×D4×C127—21no—
68C2×D4×C127—41no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C2×D4×C127SD1611yescharacteristic
72C22×C4×C127—21no—
73C22×C4×C127—41no—
74C22×C4×C127—41no—
75C22×C4×C127C2×D411yescharacteristic
76Q16×C127—41no—
77SD16×C127—81no—
78D8×C127—41no—
79M16×C127—41no—
80M16×C127—41no—
81M16×C127—41no—
82C2×C8×C127—21no—
83C2×C8×C127—41no—
84C2×C8×C127—41no—
85C2×C8×C127C2×D411yescharacteristic
86C2×C8×C127C2×D411yescharacteristic
87C4⋊C4×C127—21no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—41no—
92C4⋊C4×C127—41no—
93C22⋊C4×C127—41no—
94C22⋊C4×C127—41no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C42×C127—22no—
99C16×C127—41no—
100C16×C127—41no—
101C23×C127—21no—
102C23×C127—41no—
103C23×C127AGL(1,ℤ/8)11yescharacteristic
104Q8×C127—41no—
105Q8×C127—41no—
106Q8×C127—41no—
107D4×C127—41no—
108D4×C127—41no—
109D4×C127—41no—
110D4×C127—81no—
111D4×C127—82no—
112D4×C127—161no—
113D4×C127—161no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—42no—
120C2×C4×C127—41no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—82no—
124C2×C4×C127—161no—
125C2×C4×C127#12011yescharacteristic
126C2×C4×C127C2×SD1611yescharacteristic
127C8×C127—21no—
128C8×C127—21no—
129C8×C127—41no—
130C8×C127—81no—
131C22×C127—21no—
132C22×C127—41no—
133C22×C127—41no—
134C22×C127—41no—
135C22×C127—41no—
136C22×C127—81no—
137C22×C127—81no—
138C22×C127—81no—
139C22×C127#45011yescharacteristic
140C4×C127—21no—
141C4×C127—41no—
142C4×C127—81no—
143C4×C127—81no—
144C4×C127—82no—
145C4×C127—81no—
146C4×C127#45311yescharacteristic
147C4×C127#45811yescharacteristic
148#12303C12711yesmaximal, characteristic, direct product factor, Sylow
149C2×C127—21no—
150C2×C127—41no—
151C2×C127—81no—
152C2×C127—161no—
153C2×C127#161311yescharacteristic
154#3204C2×C12711yescharacteristic
155#2239C2×C12711yescharacteristic, complemented
156#2238C2×C12711yescharacteristic, complemented
157#2098C2×C12711yescharacteristic, complemented
158#1340C2×C12711yescharacteristic, complemented
159#1339C2×C12711yescharacteristic, complemented
160#1321C2×C12711yescharacteristic, complemented
161C127#1230311yescharacteristic, direct product factor, Sylow
162#537—21no—
163#483—21no—
164#481—21no—
165#470C22×C12711yescharacteristic
166#469C22×C12711yescharacteristic
167#465—21no—
168#452—21no—
169#449—21no—
170#430C22×C12711yescharacteristic
171#424—21no—
172#421—21no—
173#408C22×C12711yescharacteristic
174#365—21no—
175#365—21no—
176#362—21no—
177#361—21no—
178#349C22×C12711yescharacteristic
179#349C22×C12711yescharacteristic
180H(3,ℤ/4)C22×C12711yescharacteristic
181C2×#41—21no—
182C2×#41D4×C12711yescharacteristic
183C2×Q16D4×C12711yescharacteristic
184C2×SD16—21no—
185C2×D8D4×C12711yescharacteristic
186C2×M16—21no—
187C2×M16C23×C12711yescharacteristic
188C22×C8D4×C12711yescharacteristic
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—41no—
192#121—21no—
193#121—21no—
194#121—41no—
195#121—41no—
196#121—41no—
197#117—21no—
198C2×C4⋊C4—21no—
199M32—21no—
200M32—21no—
201#108—21no—
202#108—21no—
203#106—21no—
204#106D4×C12711yescharacteristic
205#106D4×C12711yescharacteristic
206C4⋊C8—42no—
207#104—42no—
208#103—21no—
209#103—21no—
210#102—21no—
211#102—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214#41—41no—
215#41—41no—
216#41—81no—
217#41—81no—
218C2×Q8—21no—
219C2×Q8SD16×C12711yescharacteristic
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4SD16×C12711yescharacteristic
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4C2×D4×C12711yescharacteristic
230Q16—41no—
231SD16—81no—
232D8—41no—
233M16—41no—
234M16—41no—
235M16—41no—
236C2×C8—21no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8C2×D4×C12711yescharacteristic
240C2×C8C2×D4×C12711yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C42—22no—
253C16—41no—
254C16—41no—
255C23—21no—
256C23—41no—
257C23AGL(1,ℤ/8)×C12711yescharacteristic
258Q8—41no—
259Q8—41no—
260Q8—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—81no—
265D4—82no—
266D4—161no—
267D4—161no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—161no—
279C2×C4#120×C12711yescharacteristic
280C2×C4C2×SD16×C12711yescharacteristic
281C8—21no—
282C8—21no—
283C8—41no—
284C8—81no—
285C22—21no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22#450×C12711yescharacteristic
294C4—21no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—82no—
299C4—81no—
300C4#453×C12711yescharacteristic
301C4#458×C12711yescharacteristic
302C2—21no—
303C2—41no—
304C2—81no—
305C2—161no—
306C2C127×#161311yescharacteristic
subgroupG/H
center, socleC2×C127#1613
commutator, Frattini, cosocleC2×M16C23×C127
prime radicalC127×#3204C2
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
2. C127×#2239C2semidirect
3. C127×#2238C2semidirect
4. C127×#2098C2semidirect
5. C127×#1340C2semidirect
6. C127×#1339C2semidirect
7. C127×#1321C2semidirect
148. #12303C127direct (N × H)
155. #2239C2×C127semidirect
156. #2238C2×C127semidirect
157. #2098C2×C127semidirect
158. #1340C2×C127semidirect
159. #1339C2×C127semidirect
160. #1321C2×C127semidirect
161. C127#12303direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32042
C127×#22392
C127×#22382
C127×#20982
C127×#13402
C127×#13392
C127×#13212
#12303127