Finite Group Databasebeta

← C127×#12166 #44575839

Subgroups — C127×#12166 #44575839

Order 32512.

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

normal subgroups only

This group has 362 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×#3262C211yesmaximal, characteristic, complemented
2C127×#2187C211yesmaximal, characteristic, complemented
3C127×#2184C211yesmaximal, characteristic
4C127×#2045C211yesmaximal, characteristic, complemented
5C127×#1350C211yesmaximal, characteristic, complemented
6C127×#1349C211yesmaximal, characteristic, complemented
7C127×#1323C211yesmaximal, characteristic, complemented
8C22×D8×C127C2211yescharacteristic
9#542×C127—21no—
10C2×SD32×C127—21no—
11C2×D16×C127—21no—
12#477×C127—21no—
13#474×C127—21no—
14#468×C127—21no—
15#467×C127C2211yescharacteristic
16#466×C127C2211yescharacteristic
17#464×C127—21no—
18#463×C127—21no—
19#424×C127C2211yescharacteristic
20#418×C127—21no—
21C2×#100×C127—21no—
22#357×C127—21no—
23#357×C127—21no—
24C22⋊C16×C127C2211yescharacteristic
25C22⋊C16×C127C2211yescharacteristic
26#341×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
34C2×M16×C127—21no—
35C2×M16×C127D411yescharacteristic
36C22×C8×C127C2311yescharacteristic
37C4⋊Q8×C127—21no—
38#126×C127—41no—
39#124×C127—21no—
40#124×C127—41no—
41#123×C127—41no—
42C22⋊Q8×C127—21no—
43C22⋊Q8×C127—41no—
44#121×C127—21no—
45C4×Q8×C127—21no—
46C4×D4×C127—21no—
47#117×C127D411yescharacteristic
48SD32×C127—42no—
49D16×C127—42no—
50C2×C16×C127—21no—
51C2×C16×C127—21no—
52#107×C127—21no—
53#107×C127D411yescharacteristic
54#107×C127D411yescharacteristic
55C4⋊C8×C127—21no—
56C4⋊C8×C127—21no—
57#103×C127—21no—
58#103×C127—21no—
59#102×C127—21no—
60#102×C127—21no—
61#102×C127—41no—
62#102×C127—41no—
63#100×C127—42no—
64C24×C127—41no—
65C2×Q8×C127—21no—
66C2×Q8×C127—41no—
67C2×D4×C127—22no—
68C2×D4×C127—21no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C22×C4×C127—41no—
76C22×C4×C127C2×D411yescharacteristic
77Q16×C127—41no—
78SD16×C127—81no—
79D8×C127—22no—
80D8×C127—22no—
81D8×C127—41no—
82D8×C127—41no—
83D8×C127—41no—
84M16×C127—41no—
85M16×C127—41no—
86M16×C127—41no—
87C2×C8×C127—21no—
88C2×C8×C127—41no—
89C2×C8×C127—41no—
90C2×C8×C127C2×D411yescharacteristic
91C2×C8×C127C2×D411yescharacteristic
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127D811yescharacteristic
98C4⋊C4×C127D811yescharacteristic
99C22⋊C4×C127—21no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—41no—
104C22⋊C4×C127—41no—
105C42×C127—21no—
106C42×C127—41no—
107C16×C127—41no—
108C16×C127—41no—
109C23×C127—42no—
110C23×C127—41no—
111C23×C127—81no—
112C23×C127—81no—
113C23×C127—161no—
114C23×C127—161no—
115C23×C127AGL(1,ℤ/8)11yescharacteristic
116Q8×C127—41no—
117Q8×C127—41no—
118Q8×C127—161no—
119D4×C127—42no—
120D4×C127—41no—
121D4×C127—42no—
122D4×C127—41no—
123D4×C127—81no—
124D4×C127—81no—
125D4×C127—161no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—41no—
133C2×C4×C127—161no—
134C2×C4×C127#12011yescharacteristic
135C2×C4×C127C2×D811yescharacteristic
136C8×C127—21no—
137C8×C127—21no—
138C8×C127—41no—
139C8×C127—81no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—82no—
144C22×C127—81no—
145C22×C127—81no—
146C22×C127—81no—
147C22×C127—162no—
148C22×C127—162no—
149C22×C127—161no—
150C22×C127—161no—
151C22×C127#44711yescharacteristic
152C4×C127—21no—
153C4×C127—41no—
154C4×C127—41no—
155C4×C127—81no—
156C4×C127—81no—
157C4×C127—81no—
158C4×C127#45411yescharacteristic
159C4×C127B(4,𝔽2)11yescharacteristic
160#12166C12711yesmaximal, characteristic, direct product factor, Sylow
161C2×C127—41no—
162C2×C127—161no—
163C2×C127—162no—
164C2×C127#159811yescharacteristic
165C2×C127#219211yescharacteristic
166C2×C127#219311yescharacteristic
167#3262C2×C12711yescharacteristic, complemented
168#2187C2×C12711yescharacteristic, complemented
169#2184C2×C12711yescharacteristic
170#2045C2×C12711yescharacteristic, complemented
171#1350C2×C12711yescharacteristic, complemented
172#1349C2×C12711yescharacteristic, complemented
173#1323C2×C12711yescharacteristic, complemented
174C127#1216611yescharacteristic, direct product factor, Sylow
175C22×D8C22×C12711yescharacteristic
176#542—21no—
177C2×SD32—21no—
178C2×D16—21no—
179#477—21no—
180#474—21no—
181#468—21no—
182#467C22×C12711yescharacteristic
183#466C22×C12711yescharacteristic
184#464—21no—
185#463—21no—
186#424C22×C12711yescharacteristic
187#418—21no—
188C2×#100—21no—
189#357—21no—
190#357—21no—
191C22⋊C16C22×C12711yescharacteristic
192C22⋊C16C22×C12711yescharacteristic
193#341C22×C12711yescharacteristic
194C22×D4—21no—
195C2×Q16D4×C12711yescharacteristic
196C2×SD16—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—42no—
200C2×D8D4×C12711yescharacteristic
201C2×M16—21no—
202C2×M16D4×C12711yescharacteristic
203C22×C8C23×C12711yescharacteristic
204C4⋊Q8—21no—
205#126—41no—
206#124—21no—
207#124—41no—
208#123—41no—
209C22⋊Q8—21no—
210C22⋊Q8—41no—
211#121—21no—
212C4×Q8—21no—
213C4×D4—21no—
214#117D4×C12711yescharacteristic
215SD32—42no—
216D16—42no—
217C2×C16—21no—
218C2×C16—21no—
219#107—21no—
220#107D4×C12711yescharacteristic
221#107D4×C12711yescharacteristic
222C4⋊C8—21no—
223C4⋊C8—21no—
224#103—21no—
225#103—21no—
226#102—21no—
227#102—21no—
228#102—41no—
229#102—41no—
230#100—42no—
231C24—41no—
232C2×Q8—21no—
233C2×Q8—41no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C22×C4—41no—
243C22×C4C2×D4×C12711yescharacteristic
244Q16—41no—
245SD16—81no—
246D8—22no—
247D8—22no—
248D8—41no—
249D8—41no—
250D8—41no—
251M16—41no—
252M16—41no—
253M16—41no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8C2×D4×C12711yescharacteristic
258C2×C8C2×D4×C12711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4D8×C12711yescharacteristic
265C4⋊C4D8×C12711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C42—21no—
273C42—41no—
274C16—41no—
275C16—41no—
276C23—42no—
277C23—41no—
278C23—81no—
279C23—81no—
280C23—161no—
281C23—161no—
282C23AGL(1,ℤ/8)×C12711yescharacteristic
283Q8—41no—
284Q8—41no—
285Q8—161no—
286D4—42no—
287D4—41no—
288D4—42no—
289D4—41no—
290D4—81no—
291D4—81no—
292D4—161no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—161no—
301C2×C4#120×C12711yescharacteristic
302C2×C4C2×D8×C12711yescharacteristic
303C8—21no—
304C8—21no—
305C8—41no—
306C8—81no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—82no—
311C22—81no—
312C22—81no—
313C22—81no—
314C22—162no—
315C22—162no—
316C22—161no—
317C22—161no—
318C22#447×C12711yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4#454×C12711yescharacteristic
326C4B(4,𝔽2)×C12711yescharacteristic
327C2—41no—
328C2—161no—
329C2—162no—
330C2C127×#159811yescharacteristic
331C2C127×#219211yescharacteristic
332C2C127×#219311yescharacteristic
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 ◁ #424×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3262C2semidirect
2. C127×#2187C2semidirect
4. C127×#2045C2semidirect
5. C127×#1350C2semidirect
6. C127×#1349C2semidirect
7. C127×#1323C2semidirect
160. #12166C127direct (N × H)
167. #3262C2×C127semidirect
168. #2187C2×C127semidirect
170. #2045C2×C127semidirect
171. #1350C2×C127semidirect
172. #1349C2×C127semidirect
173. #1323C2×C127semidirect
174. C127#12166direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32622
C127×#21872
C127×#21842
C127×#20452
C127×#13502
C127×#13492
C127×#13232
#12166127