Finite Group Databasebeta

← C127×#12160 #44575833

Subgroups — C127×#12160 #44575833

Order 32512.

396 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3250C211yesmaximal, characteristic, complemented
2C127×#2187C211yesmaximal, characteristic, complemented
3C127×#2184C211yesmaximal, characteristic
4C127×#2054C211yesmaximal, characteristic, complemented
5C127×#1348C211yesmaximal, characteristic, complemented
6C127×#1347C211yesmaximal, characteristic, complemented
7C127×#1322C211yesmaximal, characteristic, complemented
8C22×D8×C127C2211yescharacteristic
9#529×C127—21no—
10C2×SD32×C127—21no—
11C2×D16×C127—21no—
12#479×C127—21no—
13#478×C127—21no—
14#467×C127C2211yescharacteristic
15#466×C127C2211yescharacteristic
16#465×C127—21no—
17#464×C127—21no—
18#463×C127—21no—
19C2×C4⋊C8×C127C2211yescharacteristic
20C2×#102×C127—21no—
21#390×C127—21no—
22#357×C127—21no—
23#357×C127—21no—
24C22⋊C16×C127C2211yescharacteristic
25C22⋊C16×C127C2211yescharacteristic
26#340×C127C2211yescharacteristic
27C22×D4×C127—21no—
28C2×Q16×C127D411yescharacteristic
29C2×SD16×C127—21no—
30C2×D8×C127—21no—
31C2×D8×C127—21no—
32C2×D8×C127—42no—
33C2×D8×C127D411yescharacteristic
34C22×C8×C127D411yescharacteristic
35C22×C8×C127C2311yescharacteristic
36#126×C127—41no—
37#125×C127—21no—
38#124×C127—21no—
39#123×C127—41no—
40C22⋊Q8×C127—21no—
41#121×C127—21no—
42C4×Q8×C127—21no—
43C4×D4×C127—21no—
44C4×D4×C127—41no—
45#117×C127—41no—
46C2×C4⋊C4×C127—21no—
47C2×C22⋊C4×C127—41no—
48C2×C22⋊C4×C127—41no—
49C2×C42×C127D411yescharacteristic
50SD32×C127—42no—
51D16×C127—42no—
52C2×C16×C127—21no—
53C2×C16×C127—21no—
54#107×C127D411yescharacteristic
55#107×C127D411yescharacteristic
56#106×C127—21no—
57C4⋊C8×C127—21no—
58C4⋊C8×C127—21no—
59#103×C127—21no—
60#103×C127—21no—
61#102×C127—21no—
62#102×C127—21no—
63#102×C127—42no—
64C24×C127—41no—
65C2×Q8×C127—21no—
66C2×D4×C127—22no—
67C2×D4×C127—21no—
68C2×D4×C127—41no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—41no—
74C22×C4×C127—41no—
75C22×C4×C127—41no—
76C22×C4×C127D811yescharacteristic
77C22×C4×C127D811yescharacteristic
78C22×C4×C127C2×D411yescharacteristic
79Q16×C127—41no—
80SD16×C127—81no—
81D8×C127—22no—
82D8×C127—22no—
83D8×C127—41no—
84D8×C127—41no—
85D8×C127—41no—
86C2×C8×C127—21no—
87C2×C8×C127—41no—
88C2×C8×C127—41no—
89C2×C8×C127C2×D411yescharacteristic
90C2×C8×C127C2×D411yescharacteristic
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—41no—
98C4⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—41no—
104C22⋊C4×C127—82no—
105C22⋊C4×C127—82no—
106C42×C127—21no—
107C42×C127—21no—
108C42×C127—41no—
109C16×C127—41no—
110C16×C127—41no—
111C23×C127—42no—
112C23×C127—41no—
113C23×C127—81no—
114C23×C127—81no—
115C23×C127—81no—
116C23×C127—81no—
117C23×C127—81no—
118C23×C127—81no—
119C23×C127C2×D811yescharacteristic
120Q8×C127—41no—
121Q8×C127—41no—
122D4×C127—42no—
123D4×C127—41no—
124D4×C127—42no—
125D4×C127—41no—
126D4×C127—81no—
127D4×C127—81no—
128D4×C127—81no—
129D4×C127—81no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—82no—
141C2×C4×C127—81no—
142C2×C4×C127—81no—
143C2×C4×C127—81no—
144C2×C4×C127#12011yescharacteristic
145C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
146C8×C127—21no—
147C8×C127—21no—
148C8×C127—41no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—82no—
152C22×C127—82no—
153C22×C127—82no—
154C22×C127—81no—
155C22×C127—81no—
156C22×C127—81no—
157C22×C127—161no—
158C22×C127—161no—
159C22×C127—161no—
160C22×C127—161no—
161C22×C127#44711yescharacteristic
162C22×C127#50811yescharacteristic
163C22×C127#50911yescharacteristic
164C4×C127—21no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127—81no—
168C4×C127—81no—
169C4×C127—81no—
170C4×C127—161no—
171C4×C127#45411yescharacteristic
172C4×C127B(4,𝔽2)11yescharacteristic
173#12160C12711yesmaximal, characteristic, direct product factor, Sylow
174C2×C127—21no—
175C2×C127—21no—
176C2×C127—161no—
177C2×C127—162no—
178C2×C127#159811yescharacteristic
179C2×C127#218611yescharacteristic
180C2×C127#219011yescharacteristic
181#3250C2×C12711yescharacteristic, complemented
182#2187C2×C12711yescharacteristic, complemented
183#2184C2×C12711yescharacteristic
184#2054C2×C12711yescharacteristic, complemented
185#1348C2×C12711yescharacteristic, complemented
186#1347C2×C12711yescharacteristic, complemented
187#1322C2×C12711yescharacteristic, complemented
188C127#1216011yescharacteristic, direct product factor, Sylow
189C22×D8C22×C12711yescharacteristic
190#529—21no—
191C2×SD32—21no—
192C2×D16—21no—
193#479—21no—
194#478—21no—
195#467C22×C12711yescharacteristic
196#466C22×C12711yescharacteristic
197#465—21no—
198#464—21no—
199#463—21no—
200C2×C4⋊C8C22×C12711yescharacteristic
201C2×#102—21no—
202#390—21no—
203#357—21no—
204#357—21no—
205C22⋊C16C22×C12711yescharacteristic
206C22⋊C16C22×C12711yescharacteristic
207#340C22×C12711yescharacteristic
208C22×D4—21no—
209C2×Q16D4×C12711yescharacteristic
210C2×SD16—21no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—42no—
214C2×D8D4×C12711yescharacteristic
215C22×C8D4×C12711yescharacteristic
216C22×C8C23×C12711yescharacteristic
217#126—41no—
218#125—21no—
219#124—21no—
220#123—41no—
221C22⋊Q8—21no—
222#121—21no—
223C4×Q8—21no—
224C4×D4—21no—
225C4×D4—41no—
226#117—41no—
227C2×C4⋊C4—21no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—41no—
230C2×C42D4×C12711yescharacteristic
231SD32—42no—
232D16—42no—
233C2×C16—21no—
234C2×C16—21no—
235#107D4×C12711yescharacteristic
236#107D4×C12711yescharacteristic
237#106—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240#103—21no—
241#103—21no—
242#102—21no—
243#102—21no—
244#102—42no—
245C24—41no—
246C2×Q8—21no—
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4D8×C12711yescharacteristic
258C22×C4D8×C12711yescharacteristic
259C22×C4C2×D4×C12711yescharacteristic
260Q16—41no—
261SD16—81no—
262D8—22no—
263D8—22no—
264D8—41no—
265D8—41no—
266D8—41no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8C2×D4×C12711yescharacteristic
271C2×C8C2×D4×C12711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—82no—
286C22⋊C4—82no—
287C42—21no—
288C42—21no—
289C42—41no—
290C16—41no—
291C16—41no—
292C23—42no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23C2×D8×C12711yescharacteristic
301Q8—41no—
302Q8—41no—
303D4—42no—
304D4—41no—
305D4—42no—
306D4—41no—
307D4—81no—
308D4—81no—
309D4—81no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4#120×C12711yescharacteristic
326C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
327C8—21no—
328C8—21no—
329C8—41no—
330C22—21no—
331C22—21no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—161no—
339C22—161no—
340C22—161no—
341C22—161no—
342C22#447×C12711yescharacteristic
343C22#508×C12711yescharacteristic
344C22#509×C12711yescharacteristic
345C4—21no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—161no—
352C4#454×C12711yescharacteristic
353C4B(4,𝔽2)×C12711yescharacteristic
354C2—21no—
355C2—21no—
356C2—162no—
357C2—161no—
358C2C127×#159811yescharacteristic
359C2C127×#218611yescharacteristic
360C2C127×#219011yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#2184C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3250C2semidirect
2. C127×#2187C2semidirect
4. C127×#2054C2semidirect
5. C127×#1348C2semidirect
6. C127×#1347C2semidirect
7. C127×#1322C2semidirect
173. #12160C127direct (N × H)
181. #3250C2×C127semidirect
182. #2187C2×C127semidirect
184. #2054C2×C127semidirect
185. #1348C2×C127semidirect
186. #1347C2×C127semidirect
187. #1322C2×C127semidirect
188. C127#12160direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32502
C127×#21872
C127×#21842
C127×#20542
C127×#13482
C127×#13472
C127×#13222
#12160127