Finite Group Databasebeta

← C127×#12234 #44575907

Subgroups — C127×#12234 #44575907

Order 32512.

448 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2680C211yesmaximal, characteristic
2C127×#2060C211yesmaximal, characteristic, complemented
3C127×#1605C211yesmaximal, characteristic, complemented
4C127×#1599C211yesmaximal, characteristic
5C127×#1354C211yesmaximal, characteristic, complemented
6C127×#1345C211yesmaximal, characteristic
7C127×#1325C211yesmaximal, characteristic, complemented
8#544×C127C2211yescharacteristic
9C2×#123×C127—21no—
10C2×C22⋊Q8×C127—21no—
11#451×C127—21no—
12#450×C127—21no—
13#428×C127—21no—
14#398×C127—21no—
15#397×C127—21no—
16#395×C127C2211yescharacteristic
17#394×C127C2211yescharacteristic
18#388×C127—21no—
19#385×C127—21no—
20#384×C127C2211yescharacteristic
21#343×C127C2211yescharacteristic
22#339×C127—21no—
23#328×C127—21no—
24#328×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C22×Q8×C127D411yescharacteristic
28C22×D4×C127D411yescharacteristic
29C23×C4×C127—21no—
30C2×Q16×C127—41no—
31C2×SD16×C127—41no—
32C2×M16×C127—21no—
33C4⋊Q8×C127—41no—
34#125×C127—41no—
35#123×C127—44no—
36C22⋊Q8×C127—21no—
37C22⋊Q8×C127—21no—
38C22⋊Q8×C127—44no—
39#117×C127—21no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127—21no—
45C2×C4⋊C4×C127D411yescharacteristic
46C2×C4⋊C4×C127C2311yescharacteristic
47C2×C22⋊C4×C127—21no—
48C2×C22⋊C4×C127—21no—
49C2×C22⋊C4×C127—21no—
50C2×C22⋊C4×C127—21no—
51C2×C22⋊C4×C127—21no—
52C2×C42×C127D411yescharacteristic
53#107×C127—41no—
54#106×C127—41no—
55#103×C127—41no—
56#102×C127—41no—
57C22⋊C8×C127—21no—
58C22⋊C8×C127—21no—
59#95×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127D411yescharacteristic
63#95×C127D411yescharacteristic
64C24×C127—21no—
65C2×Q8×C127—21no—
66C2×Q8×C127—21no—
67C2×Q8×C127—42no—
68C2×Q8×C127—41no—
69C2×D4×C127—41no—
70C2×D4×C127—42no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—42no—
77C22×C4×C127—42no—
78C22×C4×C127—81no—
79C22×C4×C127C2×D411yescharacteristic
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82Q16×C127—82no—
83SD16×C127—82no—
84M16×C127—22no—
85M16×C127—41no—
86C2×C8×C127—41no—
87C2×C8×C127—41no—
88C2×C8×C127—41no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—21no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—42no—
96C4⋊C4×C127—42no—
97C4⋊C4×C127—42no—
98C4⋊C4×C127—42no—
99C4⋊C4×C127—42no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C22⋊C4×C127—24no—
103C22⋊C4×C127—42no—
104C22⋊C4×C127—42no—
105C22⋊C4×C127—42no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C42×C127—21no—
110C42×C127—21no—
111C42×C127—41no—
112C23×C127—22no—
113C23×C127—41no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127#12011yescharacteristic
120Q8×C127—42no—
121Q8×C127—81no—
122Q8×C127—81no—
123Q8×C127—81no—
124D4×C127—42no—
125D4×C127—81no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—22no—
129C2×C4×C127—22no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
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—84no—
150C2×C4×C127#13511yescharacteristic
151C2×C4×C127#13711yescharacteristic
152C8×C127—42no—
153C8×C127—81no—
154C8×C127—81no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—42no—
158C22×C127—42no—
159C22×C127—42no—
160C22×C127—81no—
161C22×C127—81no—
162C22×C127—81no—
163C22×C127—81no—
164C22×C127#45211yescharacteristic
165C22×C127#45611yescharacteristic
166C22×C127B(4,𝔽2)11yescharacteristic
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—42no—
170C4×C127—41no—
171C4×C127—41no—
172C4×C127—42no—
173C4×C127—81no—
174C4×C127—81no—
175C4×C127—81no—
176C4×C127—81no—
177C4×C127—81no—
178#12234C12711yesmaximal, characteristic, direct product factor, Sylow
179C2×C127—21no—
180C2×C127—21no—
181C2×C127—82no—
182C2×C127#160311yescharacteristic
183C2×C127#219611yescharacteristic
184C2×C127#220311yescharacteristic
185#2680C2×C12711yescharacteristic
186#2060C2×C12711yescharacteristic, complemented
187#1605C2×C12711yescharacteristic, complemented
188#1599C2×C12711yescharacteristic
189#1354C2×C12711yescharacteristic, complemented
190#1345C2×C12711yescharacteristic
191#1325C2×C12711yescharacteristic, complemented
192C127#1223411yescharacteristic, direct product factor, Sylow
193#544C22×C12711yescharacteristic
194C2×#123—21no—
195C2×C22⋊Q8—21no—
196#451—21no—
197#450—21no—
198#428—21no—
199#398—21no—
200#397—21no—
201#395C22×C12711yescharacteristic
202#394C22×C12711yescharacteristic
203#388—21no—
204#385—21no—
205#384C22×C12711yescharacteristic
206#343C22×C12711yescharacteristic
207#339—21no—
208#328—21no—
209#328—21no—
210#324C22×C12711yescharacteristic
211#324C22×C12711yescharacteristic
212C22×Q8D4×C12711yescharacteristic
213C22×D4D4×C12711yescharacteristic
214C23×C4—21no—
215C2×Q16—41no—
216C2×SD16—41no—
217C2×M16—21no—
218C4⋊Q8—41no—
219#125—41no—
220#123—44no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—44no—
224#117—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4D4×C12711yescharacteristic
231C2×C4⋊C4C23×C12711yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42D4×C12711yescharacteristic
238#107—41no—
239#106—41no—
240#103—41no—
241#102—41no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95D4×C12711yescharacteristic
248#95D4×C12711yescharacteristic
249C24—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×D4—41no—
255C2×D4—42no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4C2×D4×C12711yescharacteristic
265C22×C4C2×D4×C12711yescharacteristic
266C22×C4C2×D4×C12711yescharacteristic
267Q16—82no—
268SD16—82no—
269M16—22no—
270M16—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C22⋊C4—24no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42—21no—
295C42—21no—
296C42—41no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#120×C12711yescharacteristic
305Q8—42no—
306Q8—81no—
307Q8—81no—
308Q8—81no—
309D4—42no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—84no—
335C2×C4#135×C12711yescharacteristic
336C2×C4#137×C12711yescharacteristic
337C8—42no—
338C8—81no—
339C8—81no—
340C22—21no—
341C22—21no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#452×C12711yescharacteristic
350C22#456×C12711yescharacteristic
351C22B(4,𝔽2)×C12711yescharacteristic
352C4—21no—
353C4—21no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—82no—
366C2C127×#160311yescharacteristic
367C2C127×#219611yescharacteristic
368C2C127×#220311yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC22×D4×C127D4
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
2. C127×#2060C2semidirect
3. C127×#1605C2semidirect
5. C127×#1354C2semidirect
7. C127×#1325C2semidirect
178. #12234C127direct (N × H)
186. #2060C2×C127semidirect
187. #1605C2×C127semidirect
189. #1354C2×C127semidirect
191. #1325C2×C127semidirect
192. C127#12234direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#26802
C127×#20602
C127×#16052
C127×#15992
C127×#13542
C127×#13452
C127×#13252
#12234127