Finite Group Databasebeta

← C127×#13340 #44577013

Subgroups — C127×#13340 #44577013

Order 32512.

552 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 118 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×#3414C212yesmaximal, complemented
2C127×#3150C211yesmaximal, characteristic, complemented
3C127×#2143C212yesmaximal, complemented
4C127×#1975C211yesmaximal, characteristic, complemented
5C127×#1375C211yesmaximal, characteristic, complemented
6C2×#135×C127—21no—
7C2×#135×C127C2212yes—
8C2×#135×C127C2211yescharacteristic
9#532×C127—21no—
10#510×C127—22no—
11#510×C127—22no—
12#509×C127—22no—
13#509×C127—22no—
14#508×C127—24no—
15C2×M32×C127C2212yes—
16#497×C127—22no—
17#496×C127—22no—
18C2×#108×C127C2211yescharacteristic
19C2×#104×C127—21no—
20#408×C127—21no—
21C2×#97×C127C2211yescharacteristic
22#359×C127—24no—
23C2×#41×C127—22no—
24C2×#41×C127—21no—
25#135×C127—22no—
26#135×C127—22no—
27#135×C127—22no—
28#135×C127—22no—
29#135×C127—22no—
30#135×C127—22no—
31#135×C127—42no—
32#135×C127—42no—
33C2×Q16×C127—21no—
34C2×Q16×C127D412yes—
35C2×Q16×C127D411yescharacteristic, complemented
36C2×SD16×C127—22no—
37C2×SD16×C127—22no—
38C2×SD16×C127—21no—
39C2×D8×C127—21no—
40C2×D8×C127D412yes—
41C2×D8×C127D411yescharacteristic, complemented
42C2×M16×C127—21no—
43C22×C8×C127D411yescharacteristic
44C22×C8×C127C2311yescharacteristic
45C4⋊Q8×C127—21no—
46#127×C127—21no—
47#124×C127—22no—
48#121×C127—42no—
49C4×D4×C127—42no—
50C2×C42×C127D411yescharacteristic
51Q32×C127—24no—
52SD32×C127—24no—
53SD32×C127—24no—
54D16×C127—24no—
55M32×C127—22no—
56M32×C127—22no—
57C2×C16×C127D414yescomplemented
58#108×C127—22no—
59#104×C127—42no—
60C22⋊C8×C127—42no—
61#97×C127—22no—
62#41×C127—41no—
63#41×C127—42no—
64#41×C127—41no—
65#41×C127—42no—
66#41×C127—41no—
67#41×C127—42no—
68#41×C127—42no—
69#41×C127—41no—
70C2×Q8×C127—22no—
71C2×Q8×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127—21no—
74C2×D4×C127—42no—
75C2×D4×C127—42no—
76C2×D4×C127—41no—
77C22×C4×C127—21no—
78C22×C4×C127—41no—
79C22×C4×C127—42no—
80C22×C4×C127C2×D411yescharacteristic
81Q16×C127—22no—
82Q16×C127—22no—
83Q16×C127—22no—
84Q16×C127—42no—
85SD16×C127—44no—
86SD16×C127—42no—
87SD16×C127—42no—
88SD16×C127—42no—
89D8×C127—22no—
90D8×C127—22no—
91D8×C127—22no—
92D8×C127—42no—
93M16×C127—41no—
94M16×C127—41no—
95C2×C8×C127—21no—
96C2×C8×C127—21no—
97C2×C8×C127—41no—
98C2×C8×C127D812yes—
99C2×C8×C127C2×D411yescharacteristic
100C2×C8×C127C2×D412yes—
101C2×C8×C127C2×D412yes—
102C2×C8×C127#4111yescharacteristic
103C4⋊C4×C127—42no—
104C22⋊C4×C127—44no—
105C42×C127—21no—
106C42×C127D812yes—
107C16×C127—24no—
108C23×C127—42no—
109C23×C127—41no—
110C23×C127AGL(1,ℤ/8)11yescharacteristic
111Q8×C127—42no—
112Q8×C127—41no—
113Q8×C127—42no—
114Q8×C127—41no—
115D4×C127—42no—
116D4×C127—41no—
117D4×C127—42no—
118D4×C127—41no—
119D4×C127—84no—
120D4×C127—81no—
121D4×C127—82no—
122D4×C127—82no—
123D4×C127—81no—
124C2×C4×C127—22no—
125C2×C4×C127—42no—
126C2×C4×C127—41no—
127C2×C4×C127—81no—
128C2×C4×C127—81no—
129C2×C4×C127—82no—
130C2×C4×C127—81no—
131C2×C4×C127—82no—
132C2×C4×C127#12011yescharacteristic
133C2×C4×C127#12111yescharacteristic
134C2×C4×C127#12711yescharacteristic
135C2×C4×C127C2×D811yescharacteristic
136C2×C4×C127C2×D811yescharacteristic
137C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
138C8×C127—22no—
139C8×C127—81no—
140C8×C127#12012yes—
141C8×C127#12112yes—
142C22×C127—41no—
143C22×C127—84no—
144C22×C127—82no—
145C22×C127—81no—
146C22×C127—82no—
147C22×C127#44711yescharacteristic
148C22×C127#45911yescharacteristic
149C22×C127#49611yescharacteristic
150C4×C127—42no—
151C4×C127—82no—
152C4×C127—81no—
153C4×C127#39211yescharacteristic
154C4×C127#44711yescharacteristic
155C4×C127#45911yescharacteristic
156C4×C127#49311yescharacteristic
157#13340C12711yesmaximal, characteristic, direct product factor, Sylow
158C2×C127—21no—
159C2×C127—21no—
160C2×C127—21no—
161C2×C127—162no—
162C2×C127—161no—
163C2×C127#201111yescharacteristic
164#3414C2×C12712yescomplemented
165#3150C2×C12711yescharacteristic, complemented
166#2143C2×C12712yescomplemented
167#1975C2×C12711yescharacteristic, complemented
168#1375C2×C12711yescharacteristic, complemented
169C127#1334011yescharacteristic, direct product factor, Sylow
170C2×#135—21no—
171C2×#135C22×C12712yes—
172C2×#135C22×C12711yescharacteristic
173#532—21no—
174#510—22no—
175#510—22no—
176#509—22no—
177#509—22no—
178#508—24no—
179C2×M32C22×C12712yes—
180#497—22no—
181#496—22no—
182C2×#108C22×C12711yescharacteristic
183C2×#104—21no—
184#408—21no—
185C2×#97C22×C12711yescharacteristic
186#359—24no—
187C2×#41—22no—
188C2×#41—21no—
189#135—22no—
190#135—22no—
191#135—22no—
192#135—22no—
193#135—22no—
194#135—22no—
195#135—42no—
196#135—42no—
197C2×Q16—21no—
198C2×Q16D4×C12712yes—
199C2×Q16D4×C12711yescharacteristic, complemented
200C2×SD16—22no—
201C2×SD16—22no—
202C2×SD16—21no—
203C2×D8—21no—
204C2×D8D4×C12712yes—
205C2×D8D4×C12711yescharacteristic, complemented
206C2×M16—21no—
207C22×C8D4×C12711yescharacteristic
208C22×C8C23×C12711yescharacteristic
209C4⋊Q8—21no—
210#127—21no—
211#124—22no—
212#121—42no—
213C4×D4—42no—
214C2×C42D4×C12711yescharacteristic
215Q32—24no—
216SD32—24no—
217SD32—24no—
218D16—24no—
219M32—22no—
220M32—22no—
221C2×C16D4×C12714yescomplemented
222#108—22no—
223#104—42no—
224C22⋊C8—42no—
225#97—22no—
226#41—41no—
227#41—42no—
228#41—42no—
229#41—41no—
230#41—41no—
231#41—42no—
232#41—42no—
233#41—41no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×D4—22no—
237C2×D4—21no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4C2×D4×C12711yescharacteristic
245Q16—22no—
246Q16—22no—
247Q16—22no—
248Q16—42no—
249SD16—44no—
250SD16—42no—
251SD16—42no—
252SD16—42no—
253D8—22no—
254D8—22no—
255D8—22no—
256D8—42no—
257M16—41no—
258M16—41no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—41no—
262C2×C8D8×C12712yes—
263C2×C8C2×D4×C12711yescharacteristic
264C2×C8C2×D4×C12712yes—
265C2×C8C2×D4×C12712yes—
266C2×C8#41×C12711yescharacteristic
267C4⋊C4—42no—
268C22⋊C4—44no—
269C42—21no—
270C42D8×C12712yes—
271C16—24no—
272C23—42no—
273C23—41no—
274C23AGL(1,ℤ/8)×C12711yescharacteristic
275Q8—42no—
276Q8—41no—
277Q8—42no—
278Q8—41no—
279D4—42no—
280D4—41no—
281D4—42no—
282D4—41no—
283D4—84no—
284D4—82no—
285D4—81no—
286D4—81no—
287D4—82no—
288C2×C4—22no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4#120×C12711yescharacteristic
297C2×C4#121×C12711yescharacteristic
298C2×C4#127×C12711yescharacteristic
299C2×C4C2×D8×C12711yescharacteristic
300C2×C4C2×D8×C12711yescharacteristic
301C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
302C8—22no—
303C8—81no—
304C8#120×C12712yes—
305C8#121×C12712yes—
306C22—41no—
307C22—84no—
308C22—82no—
309C22—81no—
310C22—82no—
311C22#447×C12711yescharacteristic
312C22#459×C12711yescharacteristic
313C22#496×C12711yescharacteristic
314C4—42no—
315C4—82no—
316C4—81no—
317C4#392×C12711yescharacteristic
318C4#447×C12711yescharacteristic
319C4#459×C12711yescharacteristic
320C4#493×C12711yescharacteristic
321C2—21no—
322C2—21no—
323C2—21no—
324C2—162no—
325C2—161no—
326C2C127×#201111yescharacteristic
subgroupG/H
centerC4×C127#493
commutator, Frattini, cosocleC22×C8C23×C127
socleC2×C127#2011
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C22×C4×C127 ◁ C2×#97×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3414C2semidirect
2. C127×#3150C2semidirect
3. C127×#2143C2semidirect
4. C127×#1975C2semidirect
5. C127×#1375C2semidirect
35. C2×Q16×C127D4semidirect
41. C2×D8×C127D4semidirect
57. C2×C16×C127D4semidirect
157. #13340C127direct (N × H)
164. #3414C2×C127semidirect
165. #3150C2×C127semidirect
166. #2143C2×C127semidirect
167. #1975C2×C127semidirect
168. #1375C2×C127semidirect
169. C127#13340direct (N × H)
199. C2×Q16D4×C127semidirect
205. C2×D8D4×C127semidirect
221. C2×C16D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34142
C127×#31502
C127×#21432
C127×#19752
C127×#13752
#13340127