Finite Group Databasebeta

← C127×#12129 #44575802

Subgroups — C127×#12129 #44575802

Order 32512.

456 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2784C211yesmaximal, characteristic, complemented
2C127×#2006C211yesmaximal, characteristic
3C127×#1616C211yesmaximal, characteristic, complemented
4C127×#1604C211yesmaximal, characteristic, complemented
5C127×#1354C211yesmaximal, characteristic, complemented
6C127×#1353C211yesmaximal, characteristic, complemented
7C127×#1318C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127—21no—
9#537×C127C2211yescharacteristic
10#481×C127—21no—
11#448×C127—21no—
12C2×#104×C127—21no—
13#400×C127C2211yescharacteristic
14#396×C127—21no—
15#395×C127C2211yescharacteristic
16#393×C127—21no—
17#390×C127C2211yescharacteristic
18#387×C127—21no—
19#386×C127—21no—
20#385×C127—21no—
21#380×C127—21no—
22#343×C127C2211yescharacteristic
23#327×C127—21no—
24#327×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C22×Q8×C127D411yescharacteristic
29C22×D4×C127D411yescharacteristic
30C23×C4×C127—21no—
31AGL(1,ℤ/8)×C127—42no—
32AGL(1,ℤ/8)×C127—42no—
33C2×SD16×C127—41no—
34C2×SD16×C127—41no—
35C2×D8×C127—41no—
36C2×M16×C127—21no—
37C22⋊Q8×C127—41no—
38#121×C127—21no—
39#121×C127—21no—
40#121×C127—41no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127D411yescharacteristic
45C2×C4⋊C4×C127C2311yescharacteristic
46C2×C22⋊C4×C127—21no—
47C2×C22⋊C4×C127—21no—
48C2×C22⋊C4×C127—21no—
49C2×C22⋊C4×C127—21no—
50C2×C22⋊C4×C127—21no—
51C2×C42×C127D411yescharacteristic
52#106×C127—41no—
53#104×C127—42no—
54#103×C127—41no—
55#102×C127—41no—
56C22⋊C8×C127—21no—
57C22⋊C8×C127—21no—
58#95×C127—21no—
59#95×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127—21no—
63#95×C127D411yescharacteristic
64#95×C127D411yescharacteristic
65C24×C127—21no—
66#41×C127—41no—
67#41×C127—41no—
68#41×C127—41no—
69#41×C127—41no—
70C2×Q8×C127—21no—
71C2×Q8×C127—41no—
72C2×Q8×C127—81no—
73C2×D4×C127—21no—
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127—21no—
77C2×D4×C127—41no—
78C2×D4×C127—41no—
79C2×D4×C127—41no—
80C2×D4×C127SD1612yescomplemented
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—21no—
85C22×C4×C127—21no—
86C22×C4×C127—41no—
87C22×C4×C127—42no—
88C22×C4×C127—81no—
89C22×C4×C127—81no—
90C22×C4×C127C2×D411yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92C22×C4×C127C2×D411yescharacteristic
93SD16×C127—44no—
94SD16×C127—82no—
95D8×C127—44no—
96M16×C127—22no—
97M16×C127—41no—
98C2×C8×C127—41no—
99C2×C8×C127—41no—
100C2×C8×C127—41no—
101C4⋊C4×C127—21no—
102C4⋊C4×C127—21no—
103C4⋊C4×C127—41no—
104C4⋊C4×C127—41no—
105C4⋊C4×C127—42no—
106C4⋊C4×C127—81no—
107C4⋊C4×C127—81no—
108C22⋊C4×C127—42no—
109C22⋊C4×C127—41no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—81no—
112C22⋊C4×C127—81no—
113C22⋊C4×C127—81no—
114C22⋊C4×C127—81no—
115C42×C127—41no—
116C23×C127—22no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127#12011yescharacteristic
125Q8×C127—41no—
126Q8×C127—41no—
127Q8×C127—42no—
128Q8×C127—81no—
129D4×C127—42no—
130D4×C127—41no—
131D4×C127—41no—
132D4×C127—81no—
133D4×C127—81no—
134D4×C127—81no—
135D4×C127—81no—
136D4×C127—81no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
139C2×C4×C127—22no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—41no—
145C2×C4×C127—41no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—41no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—41no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—41no—
156C2×C4×C127—41no—
157C2×C4×C127—41no—
158C2×C4×C127—41no—
159C2×C4×C127—81no—
160C2×C4×C127—81no—
161C2×C4×C127—84no—
162C2×C4×C127C2×SD1611yescharacteristic
163C2×C4×C127#13711yescharacteristic
164C8×C127—42no—
165C8×C127—81no—
166C8×C127—81no—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—42no—
170C22×C127—42no—
171C22×C127—42no—
172C22×C127—81no—
173C22×C127—81no—
174C22×C127—81no—
175C22×C127—81no—
176C22×C127—81no—
177C22×C127—81no—
178C22×C127—81no—
179C22×C127#44811yescharacteristic
180C22×C127#45311yescharacteristic
181C22×C127B(4,𝔽2)11yescharacteristic
182C4×C127—21no—
183C4×C127—21no—
184C4×C127—42no—
185C4×C127—41no—
186C4×C127—41no—
187C4×C127—81no—
188C4×C127—81no—
189C4×C127—81no—
190C4×C127—81no—
191C4×C127—81no—
192#12129C12711yesmaximal, characteristic, direct product factor, Sylow
193C2×C127—21no—
194C2×C127—21no—
195C2×C127—82no—
196C2×C127—161no—
197C2×C127#159611yescharacteristic
198C2×C127#220011yescharacteristic
199C2×C127#220411yescharacteristic
200#2784C2×C12711yescharacteristic, complemented
201#2006C2×C12711yescharacteristic
202#1616C2×C12711yescharacteristic, complemented
203#1604C2×C12711yescharacteristic, complemented
204#1354C2×C12711yescharacteristic, complemented
205#1353C2×C12711yescharacteristic, complemented
206#1318C2×C12711yescharacteristic, complemented
207C127#1212911yescharacteristic, direct product factor, Sylow
208C2×AGL(1,ℤ/8)—21no—
209#537C22×C12711yescharacteristic
210#481—21no—
211#448—21no—
212C2×#104—21no—
213#400C22×C12711yescharacteristic
214#396—21no—
215#395C22×C12711yescharacteristic
216#393—21no—
217#390C22×C12711yescharacteristic
218#387—21no—
219#386—21no—
220#385—21no—
221#380—21no—
222#343C22×C12711yescharacteristic
223#327—21no—
224#327—21no—
225#324C22×C12711yescharacteristic
226#324C22×C12711yescharacteristic
227C2×#41—21no—
228C22×Q8D4×C12711yescharacteristic
229C22×D4D4×C12711yescharacteristic
230C23×C4—21no—
231AGL(1,ℤ/8)—42no—
232AGL(1,ℤ/8)—42no—
233C2×SD16—41no—
234C2×SD16—41no—
235C2×D8—41no—
236C2×M16—21no—
237C22⋊Q8—41no—
238#121—21no—
239#121—21no—
240#121—41no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4D4×C12711yescharacteristic
245C2×C4⋊C4C23×C12711yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42D4×C12711yescharacteristic
252#106—41no—
253#104—42no—
254#103—41no—
255#102—41no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95D4×C12711yescharacteristic
264#95D4×C12711yescharacteristic
265C24—21no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270C2×Q8—21no—
271C2×Q8—41no—
272C2×Q8—81no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4SD16×C12712yescomplemented
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4C2×D4×C12711yescharacteristic
291C22×C4C2×D4×C12711yescharacteristic
292C22×C4C2×D4×C12711yescharacteristic
293SD16—44no—
294SD16—82no—
295D8—44no—
296M16—22no—
297M16—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C22⋊C4—42no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C42—41no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#120×C12711yescharacteristic
325Q8—41no—
326Q8—41no—
327Q8—42no—
328Q8—81no—
329D4—42no—
330D4—41no—
331D4—41no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—84no—
362C2×C4C2×SD16×C12711yescharacteristic
363C2×C4#137×C12711yescharacteristic
364C8—42no—
365C8—81no—
366C8—81no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22#448×C12711yescharacteristic
380C22#453×C12711yescharacteristic
381C22B(4,𝔽2)×C12711yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C2—21no—
393C2—21no—
394C2—82no—
395C2—161no—
396C2C127×#159611yescharacteristic
397C2C127×#220011yescharacteristic
398C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC127×#2006C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2784C2semidirect
3. C127×#1616C2semidirect
4. C127×#1604C2semidirect
5. C127×#1354C2semidirect
6. C127×#1353C2semidirect
7. C127×#1318C2semidirect
80. C2×D4×C127SD16semidirect
192. #12129C127direct (N × H)
200. #2784C2×C127semidirect
202. #1616C2×C127semidirect
203. #1604C2×C127semidirect
204. #1354C2×C127semidirect
205. #1353C2×C127semidirect
206. #1318C2×C127semidirect
207. C127#12129direct (N × H)
280. C2×D4SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#27842
C127×#20062
C127×#16162
C127×#16042
C127×#13542
C127×#13532
C127×#13182
#12129127