Finite Group Databasebeta

← C127×#12226 #44575899

Subgroups — C127×#12226 #44575899

Order 32512.

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

normal subgroups only

This group has 510 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×#2455C211yesmaximal, characteristic
2C127×#2062C211yesmaximal, characteristic, complemented
3C127×#1614C211yesmaximal, characteristic, complemented
4C127×#1596C211yesmaximal, characteristic
5C127×#1352C211yesmaximal, characteristic, complemented
6C127×#1343C211yesmaximal, characteristic
7C127×#1326C211yesmaximal, characteristic, complemented
8#543×C127C2211yescharacteristic
9C2×#123×C127—21no—
10C2×C4×D4×C127—21no—
11#484×C127—21no—
12#450×C127—21no—
13#399×C127C2211yescharacteristic
14#394×C127—21no—
15#392×C127C2211yescharacteristic
16#389×C127—21no—
17#388×C127—21no—
18#385×C127—21no—
19#385×C127—21no—
20#385×C127C2211yescharacteristic
21#342×C127—21no—
22#342×C127C2211yescharacteristic
23#328×C127—21no—
24#328×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C22×D4×C127—21no—
28C22×D4×C127D411yescharacteristic
29C23×C4×C127—21no—
30C23×C4×C127D411yescharacteristic
31C2×SD16×C127—41no—
32#123×C127—44no—
33C22⋊Q8×C127—21no—
34C22⋊Q8×C127—21no—
35C22⋊Q8×C127—41no—
36C22⋊Q8×C127—41no—
37C4×D4×C127—44no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127—41no—
43C2×C4⋊C4×C127D411yescharacteristic
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
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×C22⋊C4×C127D411yescharacteristic
52C2×C22⋊C4×C127C2311yescharacteristic
53C2×C42×C127—21no—
54#107×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—41no—
62#95×C127D411yescharacteristic
63#95×C127D411yescharacteristic
64C24×C127—21no—
65C24×C127C2×D411yescharacteristic
66C2×Q8×C127—21no—
67C2×Q8×C127—21no—
68C2×D4×C127—42no—
69C2×D4×C127—41no—
70C2×D4×C127—42no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—42no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127—42no—
82C22×C4×C127—42no—
83C22×C4×C127—41no—
84C22×C4×C127—41no—
85C22×C4×C127—42no—
86C22×C4×C127—81no—
87C22×C4×C127C2×D411yescharacteristic
88C22×C4×C127C2×D411yescharacteristic
89SD16×C127—82no—
90C2×C8×C127—41no—
91C2×C8×C127—41no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—42no—
96C4⋊C4×C127—42no—
97C4⋊C4×C127—42no—
98C4⋊C4×C127—42no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—82no—
102C22⋊C4×C127—24no—
103C22⋊C4×C127—22no—
104C22⋊C4×C127—21no—
105C22⋊C4×C127—21no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—42no—
108C22⋊C4×C127—42no—
109C22⋊C4×C127—42no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—42no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—41no—
114C42×C127—42no—
115C23×C127—21no—
116C23×C127—21no—
117C23×C127—22no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—41no—
126C23×C127#12011yescharacteristic
127C23×C127#13511yescharacteristic
128C23×C127#13711yescharacteristic
129Q8×C127—81no—
130Q8×C127—81no—
131D4×C127—42no—
132D4×C127—44no—
133D4×C127—44no—
134D4×C127—81no—
135C2×C4×C127—22no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
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—42no—
149C2×C4×C127—42no—
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—81no—
159C2×C4×C127—84no—
160C2×C4×C127—81no—
161C2×C4×C127—81no—
162C2×C4×C127—81no—
163C2×C4×C127—81no—
164C8×C127—81no—
165C8×C127—81no—
166C22×C127—21no—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—42no—
173C22×C127—42no—
174C22×C127—41no—
175C22×C127—41no—
176C22×C127—42no—
177C22×C127—81no—
178C22×C127—81no—
179C22×C127—81no—
180C22×C127—81no—
181C22×C127—81no—
182C22×C127#45211yescharacteristic
183C22×C127#45611yescharacteristic
184C22×C127B(4,𝔽2)11yescharacteristic
185C4×C127—42no—
186C4×C127—42no—
187C4×C127—41no—
188C4×C127—42no—
189C4×C127—41no—
190C4×C127—81no—
191C4×C127—82no—
192C4×C127—81no—
193C4×C127—81no—
194C4×C127—81no—
195#12226C12711yesmaximal, characteristic, direct product factor, Sylow
196C2×C127—21no—
197C2×C127—21no—
198C2×C127—41no—
199C2×C127—41no—
200C2×C127—82no—
201C2×C127#160311yescharacteristic
202C2×C127#219611yescharacteristic
203C2×C127#219811yescharacteristic
204#2455C2×C12711yescharacteristic
205#2062C2×C12711yescharacteristic, complemented
206#1614C2×C12711yescharacteristic, complemented
207#1596C2×C12711yescharacteristic
208#1352C2×C12711yescharacteristic, complemented
209#1343C2×C12711yescharacteristic
210#1326C2×C12711yescharacteristic, complemented
211C127#1222611yescharacteristic, direct product factor, Sylow
212#543C22×C12711yescharacteristic
213C2×#123—21no—
214C2×C4×D4—21no—
215#484—21no—
216#450—21no—
217#399C22×C12711yescharacteristic
218#394—21no—
219#392C22×C12711yescharacteristic
220#389—21no—
221#388—21no—
222#385—21no—
223#385—21no—
224#385C22×C12711yescharacteristic
225#342—21no—
226#342C22×C12711yescharacteristic
227#328—21no—
228#328—21no—
229#323C22×C12711yescharacteristic
230#323C22×C12711yescharacteristic
231C22×D4—21no—
232C22×D4D4×C12711yescharacteristic
233C23×C4—21no—
234C23×C4D4×C12711yescharacteristic
235C2×SD16—41no—
236#123—44no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C4×D4—44no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4D4×C12711yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4D4×C12711yescharacteristic
256C2×C22⋊C4C23×C12711yescharacteristic
257C2×C42—21no—
258#107—41no—
259#103—41no—
260#102—41no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95D4×C12711yescharacteristic
267#95D4×C12711yescharacteristic
268C24—21no—
269C24C2×D4×C12711yescharacteristic
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4C2×D4×C12711yescharacteristic
292C22×C4C2×D4×C12711yescharacteristic
293SD16—82no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—42no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#120×C12711yescharacteristic
331C23#135×C12711yescharacteristic
332C23#137×C12711yescharacteristic
333Q8—81no—
334Q8—81no—
335D4—42no—
336D4—44no—
337D4—44no—
338D4—81no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
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—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—84no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C8—81no—
369C8—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#452×C12711yescharacteristic
387C22#456×C12711yescharacteristic
388C22B(4,𝔽2)×C12711yescharacteristic
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—41no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—82no—
404C2C127×#160311yescharacteristic
405C2C127×#219611yescharacteristic
406C2C127×#219811yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radical#392×C127C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2062C2semidirect
3. C127×#1614C2semidirect
5. C127×#1352C2semidirect
7. C127×#1326C2semidirect
195. #12226C127direct (N × H)
205. #2062C2×C127semidirect
206. #1614C2×C127semidirect
208. #1352C2×C127semidirect
210. #1326C2×C127semidirect
211. C127#12226direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#24552
C127×#20622
C127×#16142
C127×#15962
C127×#13522
C127×#13432
C127×#13262
#12226127