Finite Group Databasebeta

← C127×#13341 #44577014

Subgroups — C127×#13341 #44577014

Order 32512.

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

normal subgroups only

This group has 594 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
2C127×#3150C211yesmaximal, characteristic, complemented
3C2×#360×C127C212yesmaximal, complemented
4C127×#1972C211yesmaximal, characteristic, complemented
5C127×#1377C211yesmaximal, characteristic, complemented
6C2×#135×C127—21no—
7C2×#135×C127C2212yes—
8C22×D8×C127C2211yescharacteristic
9#532×C127—21no—
10#509×C127—24no—
11#509×C127—24no—
12C2×SD32×C127—22no—
13C2×D16×C127—22no—
14C2×M32×C127C2212yes—
15#497×C127—22no—
16#496×C127—22no—
17C2×#106×C127C2211yescharacteristic
18C2×#102×C127—21no—
19C2×#97×C127C2211yescharacteristic
20#390×C127—21no—
21#360×C127—24no—
22C2×#41×C127—22no—
23C22×D4×C127—21no—
24#135×C127—22no—
25#135×C127—22no—
26#135×C127—22no—
27#135×C127—22no—
28#135×C127—42no—
29#135×C127—42no—
30C2×Q16×C127—21no—
31C2×Q16×C127D412yes—
32C2×SD16×C127—22no—
33C2×SD16×C127—22no—
34C2×D8×C127—21no—
35C2×D8×C127—22no—
36C2×D8×C127—22no—
37C2×D8×C127—21no—
38C2×D8×C127D412yes—
39C2×D8×C127D412yescomplemented
40C22×C8×C127D411yescharacteristic
41C22×C8×C127C2311yescharacteristic
42C4⋊Q8×C127—21no—
43#127×C127—21no—
44#124×C127—22no—
45#121×C127—42no—
46C4×D4×C127—42no—
47C2×C4⋊C4×C127—21no—
48C2×C22⋊C4×C127—42no—
49C2×C42×C127D411yescharacteristic
50SD32×C127—28no—
51D16×C127—28no—
52M32×C127—22no—
53M32×C127D414yescomplemented
54C2×C16×C127—22no—
55#106×C127—22no—
56#102×C127—42no—
57#97×C127—22no—
58C24×C127—41no—
59#41×C127—42no—
60#41×C127—42no—
61#41×C127—42no—
62#41×C127—42no—
63C2×Q8×C127—22no—
64C2×D4×C127—22no—
65C2×D4×C127—22no—
66C2×D4×C127—42no—
67C2×D4×C127—42no—
68C2×D4×C127—41no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C22×C4×C127—42no—
74C22×C4×C127—41no—
75C22×C4×C127D812yes—
76C22×C4×C127C2×D411yescharacteristic
77Q16×C127—22no—
78Q16×C127—22no—
79Q16×C127—42no—
80SD16×C127—44no—
81SD16×C127—42no—
82SD16×C127—42no—
83D8×C127—22no—
84D8×C127—24no—
85D8×C127—22no—
86D8×C127—42no—
87D8×C127—42no—
88C2×C8×C127—21no—
89C2×C8×C127—21no—
90C2×C8×C127D812yes—
91C2×C8×C127C2×D411yescharacteristic
92C2×C8×C127C2×D412yes—
93C2×C8×C127C2×D412yes—
94C2×C8×C127#4111yescharacteristic
95C4⋊C4×C127—42no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—41no—
98C22⋊C4×C127—44no—
99C22⋊C4×C127—84no—
100C42×C127—21no—
101C42×C127—21no—
102C16×C127—24no—
103C23×C127—42no—
104C23×C127—42no—
105C23×C127—82no—
106C23×C127—82no—
107C23×C127—81no—
108C23×C127—81no—
109C23×C127C2×D811yescharacteristic
110Q8×C127—42no—
111Q8×C127—42no—
112D4×C127—42no—
113D4×C127—42no—
114D4×C127—42no—
115D4×C127—42no—
116D4×C127—84no—
117D4×C127—82no—
118D4×C127—82no—
119D4×C127—81no—
120D4×C127—81no—
121C2×C4×C127—22no—
122C2×C4×C127—42no—
123C2×C4×C127—42no—
124C2×C4×C127—82no—
125C2×C4×C127—82no—
126C2×C4×C127—82no—
127C2×C4×C127—81no—
128C2×C4×C127#12011yescharacteristic
129C2×C4×C127#12111yescharacteristic
130C2×C4×C127#12711yescharacteristic
131C2×C4×C127C2×D811yescharacteristic
132C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
134C8×C127—22no—
135C8×C127#12012yes—
136C8×C127#12112yes—
137C22×C127—22no—
138C22×C127—84no—
139C22×C127—84no—
140C22×C127—82no—
141C22×C127—82no—
142C22×C127—161no—
143C22×C127—161no—
144C22×C127—162no—
145C22×C127#44711yescharacteristic
146C22×C127#45911yescharacteristic
147C22×C127#49311yescharacteristic
148C4×C127—42no—
149C4×C127—82no—
150C4×C127—161no—
151C4×C127#39211yescharacteristic
152C4×C127#44711yescharacteristic
153C4×C127#45911yescharacteristic
154C4×C127#49611yescharacteristic
155#13341C12711yesmaximal, characteristic, direct product factor, Sylow
156C2×C127—21no—
157C2×C127—21no—
158C2×C127—162no—
159C2×C127—162no—
160C2×C127#201111yescharacteristic
161C2×C127#219012yes—
162C2×#509C2×C12712yescomplemented
163#3150C2×C12711yescharacteristic, complemented
164C2×#360C2×C12712yescomplemented
165#1972C2×C12711yescharacteristic, complemented
166#1377C2×C12711yescharacteristic, complemented
167C127#1334111yescharacteristic, direct product factor, Sylow
168C2×#135—21no—
169C2×#135C22×C12712yes—
170C22×D8C22×C12711yescharacteristic
171#532—21no—
172#509—24no—
173#509—24no—
174C2×SD32—22no—
175C2×D16—22no—
176C2×M32C22×C12712yes—
177#497—22no—
178#496—22no—
179C2×#106C22×C12711yescharacteristic
180C2×#102—21no—
181C2×#97C22×C12711yescharacteristic
182#390—21no—
183#360—24no—
184C2×#41—22no—
185C22×D4—21no—
186#135—22no—
187#135—22no—
188#135—22no—
189#135—22no—
190#135—42no—
191#135—42no—
192C2×Q16—21no—
193C2×Q16D4×C12712yes—
194C2×SD16—22no—
195C2×SD16—22no—
196C2×D8—21no—
197C2×D8—22no—
198C2×D8—22no—
199C2×D8—21no—
200C2×D8D4×C12712yes—
201C2×D8D4×C12712yescomplemented
202C22×C8D4×C12711yescharacteristic
203C22×C8C23×C12711yescharacteristic
204C4⋊Q8—21no—
205#127—21no—
206#124—22no—
207#121—42no—
208C4×D4—42no—
209C2×C4⋊C4—21no—
210C2×C22⋊C4—42no—
211C2×C42D4×C12711yescharacteristic
212SD32—28no—
213D16—28no—
214M32—22no—
215M32D4×C12714yescomplemented
216C2×C16—22no—
217#106—22no—
218#102—42no—
219#97—22no—
220C24—41no—
221#41—42no—
222#41—42no—
223#41—42no—
224#41—42no—
225C2×Q8—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4D8×C12712yes—
238C22×C4C2×D4×C12711yescharacteristic
239Q16—22no—
240Q16—22no—
241Q16—42no—
242SD16—44no—
243SD16—42no—
244SD16—42no—
245D8—22no—
246D8—24no—
247D8—22no—
248D8—42no—
249D8—42no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8D8×C12712yes—
253C2×C8C2×D4×C12711yescharacteristic
254C2×C8C2×D4×C12712yes—
255C2×C8C2×D4×C12712yes—
256C2×C8#41×C12711yescharacteristic
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—44no—
261C22⋊C4—84no—
262C42—21no—
263C42—21no—
264C16—24no—
265C23—42no—
266C23—42no—
267C23—82no—
268C23—82no—
269C23—81no—
270C23—81no—
271C23C2×D8×C12711yescharacteristic
272Q8—42no—
273Q8—42no—
274D4—42no—
275D4—42no—
276D4—42no—
277D4—42no—
278D4—84no—
279D4—82no—
280D4—82no—
281D4—81no—
282D4—81no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4#120×C12711yescharacteristic
291C2×C4#121×C12711yescharacteristic
292C2×C4#127×C12711yescharacteristic
293C2×C4C2×D8×C12711yescharacteristic
294C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
295C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
296C8—22no—
297C8#120×C12712yes—
298C8#121×C12712yes—
299C22—22no—
300C22—84no—
301C22—84no—
302C22—82no—
303C22—82no—
304C22—161no—
305C22—161no—
306C22—162no—
307C22#447×C12711yescharacteristic
308C22#459×C12711yescharacteristic
309C22#493×C12711yescharacteristic
310C4—42no—
311C4—82no—
312C4—161no—
313C4#392×C12711yescharacteristic
314C4#447×C12711yescharacteristic
315C4#459×C12711yescharacteristic
316C4#496×C12711yescharacteristic
317C2—21no—
318C2—21no—
319C2—162no—
320C2—162no—
321C2C127×#201111yescharacteristic
322C2C127×#219012yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×#97×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#509×C127C2semidirect
2. C127×#3150C2semidirect
3. C2×#360×C127C2semidirect
4. C127×#1972C2semidirect
5. C127×#1377C2semidirect
39. C2×D8×C127D4semidirect
53. M32×C127D4semidirect
155. #13341C127direct (N × H)
162. C2×#509C2×C127semidirect
163. #3150C2×C127semidirect
164. C2×#360C2×C127semidirect
165. #1972C2×C127semidirect
166. #1377C2×C127semidirect
167. C127#13341direct (N × H)
201. C2×D8D4×C127semidirect
215. 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
C127×#31502
C2×#360×C1272
C127×#19722
C127×#13772
#13341127