Finite Group Databasebeta

← C127×#10558 #44574231

Subgroups — C127×#10558 #44574231

Order 32512.

474 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3187C211yesmaximal, characteristic
2C127×#1901C211yesmaximal, characteristic, complemented
3C127×#1900C211yesmaximal, characteristic
4C127×#1851C211yesmaximal, characteristic, complemented
5C127×#1508C211yesmaximal, characteristic, complemented
6C127×#1507C211yesmaximal, characteristic
7C127×#1293C211yesmaximal, characteristic, complemented
8C22×M16×C127C2211yescharacteristic
9C2×C22⋊Q8×C127C2211yescharacteristic
10C2×#121×C127C2211yescharacteristic
11#483×C127—21no—
12#481×C127—21no—
13#465×C127—21no—
14#465×C127—21no—
15#450×C127—21no—
16#448×C127—21no—
17C2×C4⋊C8×C127—21no—
18C2×#101×C127—21no—
19C2×#100×C127—21no—
20#407×C127C2211yescharacteristic
21#407×C127C2211yescharacteristic
22C2×C22⋊C8×C127C2211yescharacteristic
23C4×C22⋊C4×C127C2211yescharacteristic
24#343×C127—21no—
25#343×C127—21no—
26#336×C127—21no—
27#328×C127—21no—
28#328×C127—21no—
29#327×C127—21no—
30#327×C127—21no—
31C22×Q8×C127D411yescharacteristic
32C22×D4×C127—21no—
33C22×D4×C127D411yescharacteristic
34C23×C4×C127C2311yescharacteristic
35C2×SD16×C127—21no—
36C2×SD16×C127—21no—
37C2×M16×C127—21no—
38C2×M16×C127—21no—
39C2×M16×C127—21no—
40C2×M16×C127—21no—
41C2×M16×C127—42no—
42C22×C8×C127—21no—
43C22×C8×C127D411yescharacteristic
44C22×C8×C127D411yescharacteristic
45C22⋊Q8×C127—21no—
46C22⋊Q8×C127—21no—
47C22⋊Q8×C127—21no—
48C22⋊Q8×C127—21no—
49#121×C127—21no—
50#121×C127—21no—
51#121×C127—21no—
52#121×C127—21no—
53C2×C4⋊C4×C127—21no—
54C2×C4⋊C4×C127D411yescharacteristic
55C2×C4⋊C4×C127D411yescharacteristic
56C2×C22⋊C4×C127—21no—
57C2×C22⋊C4×C127—21no—
58C2×C22⋊C4×C127—21no—
59C2×C42×C127—21no—
60#106×C127—21no—
61#106×C127—21no—
62C4⋊C8×C127—42no—
63#103×C127—21no—
64#103×C127—21no—
65#102×C127—21no—
66#102×C127—21no—
67#101×C127—42no—
68#100×C127—42no—
69C22⋊C8×C127—21no—
70C22⋊C8×C127—21no—
71C22⋊C8×C127—21no—
72C22⋊C8×C127—21no—
73C22⋊C8×C127—21no—
74C22⋊C8×C127—21no—
75#95×C127D411yescharacteristic
76#95×C127D411yescharacteristic
77C24×C127—21no—
78C24×C127#4111yescharacteristic
79C2×Q8×C127—21no—
80C2×Q8×C127—21no—
81C2×Q8×C127—41no—
82C2×Q8×C127—41no—
83C2×D4×C127—21no—
84C2×D4×C127—21no—
85C2×D4×C127—41no—
86C2×D4×C127—41no—
87C2×D4×C127—41no—
88C2×D4×C127—41no—
89C2×D4×C127—81no—
90C2×D4×C127—81no—
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127—21no—
94C22×C4×C127—21no—
95C22×C4×C127—21no—
96C22×C4×C127—41no—
97C22×C4×C127—41no—
98C22×C4×C127C2×D411yescharacteristic
99C22×C4×C127C2×D411yescharacteristic
100C22×C4×C127C2×D411yescharacteristic
101C22×C4×C127C2×D411yescharacteristic
102C22×C4×C127#4111yescharacteristic
103C22×C4×C127#4111yescharacteristic
104SD16×C127—81no—
105SD16×C127—81no—
106M16×C127—22no—
107M16×C127—22no—
108M16×C127—41no—
109M16×C127—41no—
110M16×C127—41no—
111M16×C127—41no—
112M16×C127—41no—
113M16×C127—41no—
114C2×C8×C127—22no—
115C2×C8×C127—21no—
116C2×C8×C127—21no—
117C2×C8×C127—41no—
118C2×C8×C127—41no—
119C2×C8×C127—41no—
120C2×C8×C127—41no—
121C2×C8×C127—41no—
122C2×C8×C127—41no—
123C4⋊C4×C127—21no—
124C4⋊C4×C127—21no—
125C4⋊C4×C127—21no—
126C4⋊C4×C127—21no—
127C4⋊C4×C127—41no—
128C4⋊C4×C127—41no—
129C22⋊C4×C127—42no—
130C22⋊C4×C127—41no—
131C22⋊C4×C127—41no—
132C22⋊C4×C127—41no—
133C22⋊C4×C127—41no—
134C42×C127—41no—
135C42×C127—41no—
136C23×C127—21no—
137C23×C127—21no—
138C23×C127—41no—
139C23×C127—41no—
140C23×C127—42no—
141C23×C127—41no—
142C23×C127—81no—
143C23×C127—81no—
144C23×C127#12111yescharacteristic
145C23×C127#12311yescharacteristic
146C23×C127#12311yescharacteristic
147Q8×C127—41no—
148Q8×C127—41no—
149Q8×C127—81no—
150D4×C127—41no—
151D4×C127—41no—
152D4×C127—81no—
153D4×C127—82no—
154D4×C127—81no—
155D4×C127—81no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—21no—
162C2×C4×C127—22no—
163C2×C4×C127—41no—
164C2×C4×C127—41no—
165C2×C4×C127—41no—
166C2×C4×C127—41no—
167C2×C4×C127—41no—
168C2×C4×C127—41no—
169C2×C4×C127—41no—
170C2×C4×C127—42no—
171C2×C4×C127—41no—
172C2×C4×C127—41no—
173C2×C4×C127—41no—
174C2×C4×C127—41no—
175C2×C4×C127—81no—
176C2×C4×C127—81no—
177C2×C4×C127#12011yescharacteristic
178C2×C4×C127#12111yescharacteristic
179C2×C4×C127#12111yescharacteristic
180C2×C4×C127#12411yescharacteristic
181C8×C127—42no—
182C8×C127—41no—
183C8×C127—41no—
184C8×C127—81no—
185C8×C127—81no—
186C22×C127—21no—
187C22×C127—21no—
188C22×C127—21no—
189C22×C127—21no—
190C22×C127—21no—
191C22×C127—21no—
192C22×C127—21no—
193C22×C127—21no—
194C22×C127—81no—
195C22×C127—81no—
196C22×C127—81no—
197C22×C127—81no—
198C22×C127—81no—
199C22×C127—162no—
200C22×C127#39411yescharacteristic
201C22×C127#47111yescharacteristic
202C22×C127#47111yescharacteristic
203C4×C127—21no—
204C4×C127—21no—
205C4×C127—21no—
206C4×C127—42no—
207C4×C127—81no—
208C4×C127—81no—
209C4×C127—81no—
210C4×C127—81no—
211C4×C127B(4,𝔽2)11yescharacteristic
212C4×C127#45811yescharacteristic
213#10558C12711yesmaximal, characteristic, direct product factor, Sylow
214C2×C127—21no—
215C2×C127—21no—
216C2×C127—41no—
217C2×C127—41no—
218C2×C127—161no—
219C2×C127#204411yescharacteristic
220C2×C127#204611yescharacteristic
221C2×C127#205011yescharacteristic
222#3187C2×C12711yescharacteristic
223#1901C2×C12711yescharacteristic, complemented
224#1900C2×C12711yescharacteristic
225#1851C2×C12711yescharacteristic, complemented
226#1508C2×C12711yescharacteristic, complemented
227#1507C2×C12711yescharacteristic
228#1293C2×C12711yescharacteristic, complemented
229C127#1055811yescharacteristic, direct product factor, Sylow
230C22×M16C22×C12711yescharacteristic
231C2×C22⋊Q8C22×C12711yescharacteristic
232C2×#121C22×C12711yescharacteristic
233#483—21no—
234#481—21no—
235#465—21no—
236#465—21no—
237#450—21no—
238#448—21no—
239C2×C4⋊C8—21no—
240C2×#101—21no—
241C2×#100—21no—
242#407C22×C12711yescharacteristic
243#407C22×C12711yescharacteristic
244C2×C22⋊C8C22×C12711yescharacteristic
245C4×C22⋊C4C22×C12711yescharacteristic
246#343—21no—
247#343—21no—
248#336—21no—
249#328—21no—
250#328—21no—
251#327—21no—
252#327—21no—
253C22×Q8D4×C12711yescharacteristic
254C22×D4—21no—
255C22×D4D4×C12711yescharacteristic
256C23×C4C23×C12711yescharacteristic
257C2×SD16—21no—
258C2×SD16—21no—
259C2×M16—21no—
260C2×M16—21no—
261C2×M16—21no—
262C2×M16—21no—
263C2×M16—42no—
264C22×C8—21no—
265C22×C8D4×C12711yescharacteristic
266C22×C8D4×C12711yescharacteristic
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271#121—21no—
272#121—21no—
273#121—21no—
274#121—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4D4×C12711yescharacteristic
277C2×C4⋊C4D4×C12711yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C42—21no—
282#106—21no—
283#106—21no—
284C4⋊C8—42no—
285#103—21no—
286#103—21no—
287#102—21no—
288#102—21no—
289#101—42no—
290#100—42no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297#95D4×C12711yescharacteristic
298#95D4×C12711yescharacteristic
299C24—21no—
300C24#41×C12711yescharacteristic
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—81no—
312C2×D4—81no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4C2×D4×C12711yescharacteristic
321C22×C4C2×D4×C12711yescharacteristic
322C22×C4C2×D4×C12711yescharacteristic
323C22×C4C2×D4×C12711yescharacteristic
324C22×C4#41×C12711yescharacteristic
325C22×C4#41×C12711yescharacteristic
326SD16—81no—
327SD16—81no—
328M16—22no—
329M16—22no—
330M16—41no—
331M16—41no—
332M16—41no—
333M16—41no—
334M16—41no—
335M16—41no—
336C2×C8—22no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C42—41no—
357C42—41no—
358C23—21no—
359C23—21no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23—41no—
364C23—81no—
365C23—81no—
366C23#121×C12711yescharacteristic
367C23#123×C12711yescharacteristic
368C23#123×C12711yescharacteristic
369Q8—41no—
370Q8—41no—
371Q8—81no—
372D4—41no—
373D4—41no—
374D4—81no—
375D4—82no—
376D4—81no—
377D4—81no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4#120×C12711yescharacteristic
400C2×C4#121×C12711yescharacteristic
401C2×C4#121×C12711yescharacteristic
402C2×C4#124×C12711yescharacteristic
403C8—42no—
404C8—41no—
405C8—41no—
406C8—81no—
407C8—81no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—162no—
422C22#394×C12711yescharacteristic
423C22#471×C12711yescharacteristic
424C22#471×C12711yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—42no—
429C4—81no—
430C4—81no—
431C4—81no—
432C4—81no—
433C4B(4,𝔽2)×C12711yescharacteristic
434C4#458×C12711yescharacteristic
435C2—21no—
436C2—21no—
437C2—41no—
438C2—41no—
439C2—161no—
440C2C127×#204411yescharacteristic
441C2C127×#204611yescharacteristic
442C2C127×#205011yescharacteristic
subgroupG/H
center, socleC22×C127#394
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC2×#121×C127C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1901C2semidirect
4. C127×#1851C2semidirect
5. C127×#1508C2semidirect
7. C127×#1293C2semidirect
213. #10558C127direct (N × H)
223. #1901C2×C127semidirect
225. #1851C2×C127semidirect
226. #1508C2×C127semidirect
228. #1293C2×C127semidirect
229. C127#10558direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31872
C127×#19012
C127×#19002
C127×#18512
C127×#15082
C127×#15072
C127×#12932
#10558127