Finite Group Databasebeta

← C127×#13317 #44576990

Subgroups — C127×#13317 #44576990

Order 32512.

552 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#510×C127C211yesmaximal, characteristic, complemented
2C127×#3054C211yesmaximal, characteristic
3C2×#470×C127C211yesmaximal, characteristic, complemented
4C2×#360×C127C211yesmaximal, characteristic, complemented
5C127×#2141C211yesmaximal, characteristic, complemented
6C2×#349×C127C211yesmaximal, characteristic, complemented
7C127×#1386C211yesmaximal, characteristic, complemented
8C2×#135×C127C2211yescharacteristic
9C22×Q16×C127C2211yescharacteristic
10C22×M16×C127C2211yescharacteristic
11#525×C127—21no—
12C2×C22⋊Q8×C127—21no—
13#510×C127—22no—
14#510×C127—22no—
15C2×Q32×C127—21no—
16C2×SD32×C127—21no—
17C2×M32×C127C2211yescharacteristic
18C2×M32×C127C2211yescharacteristic
19#470×C127—21no—
20#470×C127—22no—
21#470×C127—22no—
22#469×C127—21no—
23#468×C127—22no—
24#427×C127C2211yescharacteristic
25C2×#106×C127C2211yescharacteristic
26#416×C127—21no—
27C2×#103×C127—21no—
28#360×C127—22no—
29#358×C127—22no—
30#349×C127—22no—
31C2×#41×C127—21no—
32C22×Q8×C127—21no—
33C23×C4×C127D411yescharacteristic
34#135×C127—21no—
35#135×C127—21no—
36#135×C127—21no—
37#135×C127—21no—
38C2×Q16×C127—21no—
39C2×Q16×C127—21no—
40C2×Q16×C127—21no—
41C2×Q16×C127—21no—
42C2×Q16×C127—21no—
43C2×Q16×C127D411yescharacteristic
44C2×Q16×C127D412yescomplemented
45C2×SD16×C127—21no—
46C2×D8×C127D411yescharacteristic
47C2×M16×C127—21no—
48C2×M16×C127—21no—
49C2×M16×C127—21no—
50C2×M16×C127—21no—
51C2×M16×C127—21no—
52C2×M16×C127—21no—
53C22×C8×C127D411yescharacteristic
54C22×C8×C127C2311yescharacteristic
55#123×C127—41no—
56C22⋊Q8×C127—24no—
57C22⋊Q8×C127—22no—
58C22⋊Q8×C127—42no—
59#121×C127—22no—
60#120×C127—41no—
61C4×D4×C127—42no—
62#117×C127—21no—
63C2×C4⋊C4×C127—21no—
64C2×C4⋊C4×C127—41no—
65C2×C22⋊C4×C127—41no—
66Q32×C127—24no—
67SD32×C127—24no—
68M32×C127—21no—
69M32×C127—21no—
70M32×C127—21no—
71M32×C127D412yescomplemented
72C2×C16×C127—21no—
73C2×C16×C127—21no—
74#107×C127D412yes—
75#106×C127—21no—
76#106×C127—21no—
77#106×C127D412yescomplemented
78#103×C127—24no—
79#103×C127—22no—
80#102×C127—22no—
81C24×C127D811yescharacteristic
82#41×C127—41no—
83#41×C127—41no—
84#41×C127—41no—
85#41×C127—41no—
86C2×Q8×C127—21no—
87C2×Q8×C127—22no—
88C2×Q8×C127—41no—
89C2×Q8×C127—41no—
90C2×Q8×C127—41no—
91C2×Q8×C127—41no—
92C2×Q8×C127—41no—
93C2×D4×C127—21no—
94C2×D4×C127—42no—
95C2×D4×C127—41no—
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127—21no—
99C22×C4×C127—21no—
100C22×C4×C127—21no—
101C22×C4×C127—21no—
102C22×C4×C127—41no—
103C22×C4×C127—41no—
104C22×C4×C127—41no—
105C22×C4×C127D811yescharacteristic
106C22×C4×C127C2×D411yescharacteristic
107Q16×C127—22no—
108Q16×C127—21no—
109Q16×C127—22no—
110Q16×C127—21no—
111Q16×C127—41no—
112Q16×C127—41no—
113SD16×C127—41no—
114SD16×C127—41no—
115D8×C127—21no—
116D8×C127—21no—
117M16×C127—41no—
118M16×C127—41no—
119M16×C127—41no—
120M16×C127—41no—
121C2×C8×C127—21no—
122C2×C8×C127—21no—
123C2×C8×C127—21no—
124C2×C8×C127C2×D411yescharacteristic
125C2×C8×C127C2×D411yescharacteristic
126C2×C8×C127C2×D411yescharacteristic
127C2×C8×C127C2×D411yescharacteristic
128C2×C8×C127C2×D411yescharacteristic
129C2×C8×C127#4111yescharacteristic
130C4⋊C4×C127—22no—
131C4⋊C4×C127—22no—
132C4⋊C4×C127—44no—
133C4⋊C4×C127—42no—
134C4⋊C4×C127—41no—
135C22⋊C4×C127—42no—
136C22⋊C4×C127—44no—
137C22⋊C4×C127—42no—
138C22⋊C4×C127—41no—
139C42×C127—41no—
140C16×C127—22no—
141C16×C127—41no—
142C23×C127—21no—
143C23×C127—21no—
144C23×C127—21no—
145C23×C127—42no—
146C23×C127—41no—
147C23×C127C2×D811yescharacteristic
148Q8×C127—41no—
149Q8×C127—42no—
150Q8×C127—42no—
151Q8×C127—41no—
152Q8×C127—81no—
153Q8×C127—81no—
154D4×C127—41no—
155D4×C127—41no—
156D4×C127—84no—
157D4×C127—81no—
158D4×C127—81no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—41no—
163C2×C4×C127—41no—
164C2×C4×C127—41no—
165C2×C4×C127—41no—
166C2×C4×C127—42no—
167C2×C4×C127—42no—
168C2×C4×C127—42no—
169C2×C4×C127—41no—
170C2×C4×C127—82no—
171C2×C4×C127—81no—
172C2×C4×C127—81no—
173C2×C4×C127—81no—
174C2×C4×C127—81no—
175C2×C4×C127#12011yescharacteristic
176C2×C4×C127#12011yescharacteristic
177C2×C4×C127#12111yescharacteristic
178C2×C4×C127#13511yescharacteristic
179C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
181C8×C127—41no—
182C8×C127#12011yescharacteristic
183C8×C127#12111yescharacteristic
184C8×C127#12111yescharacteristic
185C8×C127#12711yescharacteristic
186C22×C127—22no—
187C22×C127—42no—
188C22×C127—42no—
189C22×C127—41no—
190C22×C127—81no—
191C22×C127—82no—
192C22×C127—81no—
193C22×C127#44711yescharacteristic
194C22×C127#44711yescharacteristic
195C22×C127#46611yescharacteristic
196C4×C127—41no—
197C4×C127—82no—
198C4×C127—82no—
199C4×C127—81no—
200C4×C127—82no—
201C4×C127#39211yescharacteristic
202C4×C127#44911yescharacteristic
203C4×C127#44911yescharacteristic
204C4×C127#46911yescharacteristic
205#13317C12711yesmaximal, characteristic, direct product factor, Sylow
206C2×C127—21no—
207C2×C127—21no—
208C2×C127—81no—
209C2×C127—161no—
210C2×C127#199911yescharacteristic
211C2×C127#219112yes—
212C2×#510C2×C12711yescharacteristic, complemented
213#3054C2×C12711yescharacteristic
214C2×#470C2×C12711yescharacteristic, complemented
215C2×#360C2×C12711yescharacteristic, complemented
216#2141C2×C12711yescharacteristic, complemented
217C2×#349C2×C12711yescharacteristic, complemented
218#1386C2×C12711yescharacteristic, complemented
219C127#1331711yescharacteristic, direct product factor, Sylow
220C2×#135C22×C12711yescharacteristic
221C22×Q16C22×C12711yescharacteristic
222C22×M16C22×C12711yescharacteristic
223#525—21no—
224C2×C22⋊Q8—21no—
225#510—22no—
226#510—22no—
227C2×Q32—21no—
228C2×SD32—21no—
229C2×M32C22×C12711yescharacteristic
230C2×M32C22×C12711yescharacteristic
231#470—21no—
232#470—22no—
233#470—22no—
234#469—21no—
235#468—22no—
236#427C22×C12711yescharacteristic
237C2×#106C22×C12711yescharacteristic
238#416—21no—
239C2×#103—21no—
240#360—22no—
241#358—22no—
242#349—22no—
243C2×#41—21no—
244C22×Q8—21no—
245C23×C4D4×C12711yescharacteristic
246#135—21no—
247#135—21no—
248#135—21no—
249#135—21no—
250C2×Q16—21no—
251C2×Q16—21no—
252C2×Q16—21no—
253C2×Q16—21no—
254C2×Q16—21no—
255C2×Q16D4×C12711yescharacteristic
256C2×Q16D4×C12712yescomplemented
257C2×SD16—21no—
258C2×D8D4×C12711yescharacteristic
259C2×M16—21no—
260C2×M16—21no—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—21no—
264C2×M16—21no—
265C22×C8D4×C12711yescharacteristic
266C22×C8C23×C12711yescharacteristic
267#123—41no—
268C22⋊Q8—24no—
269C22⋊Q8—22no—
270C22⋊Q8—42no—
271#121—22no—
272#120—41no—
273C4×D4—42no—
274#117—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—41no—
277C2×C22⋊C4—41no—
278Q32—24no—
279SD32—24no—
280M32—21no—
281M32—21no—
282M32—21no—
283M32D4×C12712yescomplemented
284C2×C16—21no—
285C2×C16—21no—
286#107D4×C12712yes—
287#106—21no—
288#106—21no—
289#106D4×C12712yescomplemented
290#103—24no—
291#103—22no—
292#102—22no—
293C24D8×C12711yescharacteristic
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—21no—
306C2×D4—42no—
307C2×D4—41no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4D8×C12711yescharacteristic
318C22×C4C2×D4×C12711yescharacteristic
319Q16—22no—
320Q16—21no—
321Q16—22no—
322Q16—21no—
323Q16—41no—
324Q16—41no—
325SD16—41no—
326SD16—41no—
327D8—21no—
328D8—21no—
329M16—41no—
330M16—41no—
331M16—41no—
332M16—41no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8C2×D4×C12711yescharacteristic
337C2×C8C2×D4×C12711yescharacteristic
338C2×C8C2×D4×C12711yescharacteristic
339C2×C8C2×D4×C12711yescharacteristic
340C2×C8C2×D4×C12711yescharacteristic
341C2×C8#41×C12711yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—41no—
351C42—41no—
352C16—22no—
353C16—41no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—42no—
358C23—41no—
359C23C2×D8×C12711yescharacteristic
360Q8—41no—
361Q8—42no—
362Q8—42no—
363Q8—41no—
364Q8—81no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—84no—
369D4—81no—
370D4—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4#120×C12711yescharacteristic
388C2×C4#120×C12711yescharacteristic
389C2×C4#121×C12711yescharacteristic
390C2×C4#135×C12711yescharacteristic
391C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
392C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
393C8—41no—
394C8#120×C12711yescharacteristic
395C8#121×C12711yescharacteristic
396C8#121×C12711yescharacteristic
397C8#127×C12711yescharacteristic
398C22—22no—
399C22—42no—
400C22—42no—
401C22—41no—
402C22—81no—
403C22—82no—
404C22—81no—
405C22#447×C12711yescharacteristic
406C22#447×C12711yescharacteristic
407C22#466×C12711yescharacteristic
408C4—41no—
409C4—82no—
410C4—82no—
411C4—81no—
412C4—82no—
413C4#392×C12711yescharacteristic
414C4#449×C12711yescharacteristic
415C4#449×C12711yescharacteristic
416C4#469×C12711yescharacteristic
417C2—21no—
418C2—21no—
419C2—81no—
420C2—161no—
421C2C127×#199911yescharacteristic
422C2C127×#219112yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#3054C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#510×C127C2semidirect
3. C2×#470×C127C2semidirect
4. C2×#360×C127C2semidirect
5. C127×#2141C2semidirect
6. C2×#349×C127C2semidirect
7. C127×#1386C2semidirect
44. C2×Q16×C127D4semidirect
71. M32×C127D4semidirect
77. #106×C127D4semidirect
205. #13317C127direct (N × H)
212. C2×#510C2×C127semidirect
214. C2×#470C2×C127semidirect
215. C2×#360C2×C127semidirect
216. #2141C2×C127semidirect
217. C2×#349C2×C127semidirect
218. #1386C2×C127semidirect
219. C127#13317direct (N × H)
256. C2×Q16D4×C127semidirect
283. M32D4×C127semidirect
289. #106D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#510×C1272
C127×#30542
C2×#470×C1272
C2×#360×C1272
C127×#21412
C2×#349×C1272
C127×#13862
#13317127