Finite Group Databasebeta

← C127×#12128 #44575801

Subgroups — C127×#12128 #44575801

Order 32512.

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

normal subgroups only

This group has 408 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×#2799C211yesmaximal, characteristic, complemented
2C127×#2007C211yesmaximal, characteristic
3C127×#1616C211yesmaximal, characteristic
4C127×#1604C211yesmaximal, characteristic
5C127×#1354C211yesmaximal, characteristic, complemented
6C127×#1353C211yesmaximal, characteristic, complemented
7C127×#1319C211yesmaximal, characteristic, complemented
8C2×#137×C127—21no—
9#537×C127C2211yescharacteristic
10#481×C127—21no—
11#448×C127—21no—
12C2×#104×C127—21no—
13#400×C127—21no—
14#400×C127C2211yescharacteristic
15#398×C127—21no—
16#395×C127C2211yescharacteristic
17#391×C127—21no—
18#391×C127—21no—
19#391×C127C2211yescharacteristic
20#387×C127—21no—
21#387×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×Q8×C127D411yescharacteristic
30#137×C127—42no—
31#137×C127—42no—
32C2×Q16×C127—41no—
33C2×SD16×C127—41no—
34C2×SD16×C127—41no—
35C2×M16×C127—21no—
36C22⋊Q8×C127—41no—
37#121×C127—21no—
38#121×C127—21no—
39#121×C127—41no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127D411yescharacteristic
45C2×C4⋊C4×C127C2311yescharacteristic
46C2×C42×C127—21no—
47C2×C42×C127D411yescharacteristic
48#106×C127—41no—
49#104×C127—42no—
50#103×C127—41no—
51#102×C127—41no—
52C22⋊C8×C127—21no—
53C22⋊C8×C127—21no—
54#95×C127—21no—
55#95×C127—21no—
56#95×C127—21no—
57#95×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
65#41×C127—41no—
66#41×C127—41no—
67#41×C127—41no—
68#41×C127—41no—
69C2×Q8×C127—21no—
70C2×Q8×C127—21no—
71C2×Q8×C127—42no—
72C2×Q8×C127—41no—
73C2×Q8×C127—41no—
74C2×Q8×C127—41no—
75C2×Q8×C127SD1612yescomplemented
76C2×D4×C127—21no—
77C2×D4×C127—21no—
78C2×D4×C127—21no—
79C2×D4×C127—41no—
80C22×C4×C127—21no—
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×C127C2×D411yescharacteristic
88C22×C4×C127C2×D411yescharacteristic
89C22×C4×C127C2×D411yescharacteristic
90Q16×C127—44no—
91SD16×C127—44no—
92SD16×C127—82no—
93M16×C127—22no—
94M16×C127—41no—
95C2×C8×C127—41no—
96C2×C8×C127—41no—
97C2×C8×C127—41no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127—21no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—42no—
103C4⋊C4×C127—42no—
104C4⋊C4×C127—81no—
105C4⋊C4×C127—81no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C42×C127—41no—
109C42×C127—81no—
110C23×C127—41no—
111C23×C127#12011yescharacteristic
112Q8×C127—42no—
113Q8×C127—41no—
114Q8×C127—41no—
115Q8×C127—42no—
116Q8×C127—81no—
117Q8×C127—81no—
118D4×C127—41no—
119D4×C127—41no—
120D4×C127—81no—
121D4×C127—81no—
122D4×C127—81no—
123D4×C127—81no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—22no—
127C2×C4×C127—22no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
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—42no—
143C2×C4×C127—41no—
144C2×C4×C127—41no—
145C2×C4×C127—41no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—81no—
149C2×C4×C127—81no—
150C2×C4×C127C2×SD1611yescharacteristic
151C2×C4×C127#13711yescharacteristic
152C8×C127—42no—
153C8×C127—81no—
154C8×C127—81no—
155C22×C127—21no—
156C22×C127—21no—
157C22×C127—81no—
158C22×C127—81no—
159C22×C127—81no—
160C22×C127#44811yescharacteristic
161C22×C127#45311yescharacteristic
162C22×C127B(4,𝔽2)11yescharacteristic
163C4×C127—21no—
164C4×C127—21no—
165C4×C127—42no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127—42no—
169C4×C127—81no—
170C4×C127—81no—
171C4×C127—81no—
172C4×C127—81no—
173C4×C127—81no—
174#12128C12711yesmaximal, characteristic, direct product factor, Sylow
175C2×C127—21no—
176C2×C127—21no—
177C2×C127—161no—
178C2×C127#159611yescharacteristic
179C2×C127#219711yescharacteristic
180C2×C127#220311yescharacteristic
181#2799C2×C12711yescharacteristic, complemented
182#2007C2×C12711yescharacteristic
183#1616C2×C12711yescharacteristic
184#1604C2×C12711yescharacteristic
185#1354C2×C12711yescharacteristic, complemented
186#1353C2×C12711yescharacteristic, complemented
187#1319C2×C12711yescharacteristic, complemented
188C127#1212811yescharacteristic, direct product factor, Sylow
189C2×#137—21no—
190#537C22×C12711yescharacteristic
191#481—21no—
192#448—21no—
193C2×#104—21no—
194#400—21no—
195#400C22×C12711yescharacteristic
196#398—21no—
197#395C22×C12711yescharacteristic
198#391—21no—
199#391—21no—
200#391C22×C12711yescharacteristic
201#387—21no—
202#387—21no—
203#343C22×C12711yescharacteristic
204#327—21no—
205#327—21no—
206#324C22×C12711yescharacteristic
207#324C22×C12711yescharacteristic
208C2×#41—21no—
209C22×Q8D4×C12711yescharacteristic
210C22×Q8D4×C12711yescharacteristic
211#137—42no—
212#137—42no—
213C2×Q16—41no—
214C2×SD16—41no—
215C2×SD16—41no—
216C2×M16—21no—
217C22⋊Q8—41no—
218#121—21no—
219#121—21no—
220#121—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4D4×C12711yescharacteristic
226C2×C4⋊C4C23×C12711yescharacteristic
227C2×C42—21no—
228C2×C42D4×C12711yescharacteristic
229#106—41no—
230#104—42no—
231#103—41no—
232#102—41no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95D4×C12711yescharacteristic
245#95D4×C12711yescharacteristic
246#41—41no—
247#41—41no—
248#41—41no—
249#41—41no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8SD16×C12712yescomplemented
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4C2×D4×C12711yescharacteristic
269C22×C4C2×D4×C12711yescharacteristic
270C22×C4C2×D4×C12711yescharacteristic
271Q16—44no—
272SD16—44no—
273SD16—82no—
274M16—22no—
275M16—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C42—41no—
290C42—81no—
291C23—41no—
292C23#120×C12711yescharacteristic
293Q8—42no—
294Q8—41no—
295Q8—41no—
296Q8—42no—
297Q8—81no—
298Q8—81no—
299D4—41no—
300D4—41no—
301D4—81no—
302D4—81no—
303D4—81no—
304D4—81no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4C2×SD16×C12711yescharacteristic
332C2×C4#137×C12711yescharacteristic
333C8—42no—
334C8—81no—
335C8—81no—
336C22—21no—
337C22—21no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#448×C12711yescharacteristic
342C22#453×C12711yescharacteristic
343C22B(4,𝔽2)×C12711yescharacteristic
344C4—21no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—42no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—81no—
355C2—21no—
356C2—21no—
357C2—161no—
358C2C127×#159611yescharacteristic
359C2C127×#219711yescharacteristic
360C2C127×#220311yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radical#537×C127C22
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×#2799C2semidirect
5. C127×#1354C2semidirect
6. C127×#1353C2semidirect
7. C127×#1319C2semidirect
75. C2×Q8×C127SD16semidirect
174. #12128C127direct (N × H)
181. #2799C2×C127semidirect
185. #1354C2×C127semidirect
186. #1353C2×C127semidirect
187. #1319C2×C127semidirect
188. C127#12128direct (N × H)
256. C2×Q8SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#27992
C127×#20072
C127×#16162
C127×#16042
C127×#13542
C127×#13532
C127×#13192
#12128127