Finite Group Databasebeta

← C127×#13573 #44577246

Subgroups — C127×#13573 #44577246

Order 32512.

470 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#456×C127C211yesmaximal, characteristic
2C2×#356×C127C211yesmaximal, characteristic, complemented
3C127×#1558C211yesmaximal, characteristic, complemented
4C127×#1405C214yesmaximal, complemented
5C2×2−1+4×C127C411yescharacteristic
6C2×#137×C127—21no—
7C2×C4⋊Q8×C127C2211yescharacteristic
8#456×C127—21no—
9#456×C127—21no—
10#456×C127C412yes—
11#456×C127C2212yes—
12#432×C127—21no—
13C2×#104×C127—21no—
14C2×#101×C127—21no—
15C2×#101×C127C411yescharacteristic
16#356×C127—21no—
17#356×C127C2212yescomplemented
18#333×C127—21no—
19#333×C127C2212yescomplemented
202−1+4×C127—21no—
212−1+4×C127—41no—
222−1+4×C127—42no—
232−1+4×C127C2×C412yes—
24C2×#41×C127—21no—
25C2×#41×C127—41no—
26C2×#41×C127—41no—
27C22×Q8×C127—41no—
28C22×Q8×C127C2×C411yescharacteristic
29C22×Q8×C127D411yescharacteristic
30#137×C127—22no—
31#137×C127—22no—
32#137×C127—42no—
33C2×Q16×C127—41no—
34C2×SD16×C127—41no—
35C2×M16×C127—21no—
36C2×M16×C127—41no—
37C4⋊Q8×C127—41no—
38C4⋊Q8×C127C2×C411yescharacteristic
39C4⋊Q8×C127D411yescharacteristic
40C4⋊Q8×C127D411yescharacteristic
41C4⋊Q8×C127C2311yescharacteristic
42C2×C4⋊C4×C127—41no—
43C2×C42×C127D411yescharacteristic
44C4⋊C8×C127—22no—
45#104×C127—22no—
46#104×C127—22no—
47#101×C127—21no—
48#101×C127—22no—
49#101×C127—41no—
50#101×C127C2×C412yes—
51C22⋊C8×C127—41no—
52#97×C127—41no—
53#41×C127—22no—
54#41×C127—22no—
55#41×C127—41no—
56#41×C127—42no—
57#41×C127—42no—
58#41×C127—42no—
59#41×C127—42no—
60#41×C127—44no—
61#41×C127—42no—
62#41×C127—42no—
63#41×C127—41no—
64C2×Q8×C127—21no—
65C2×Q8×C127—21no—
66C2×Q8×C127—21no—
67C2×Q8×C127—41no—
68C2×Q8×C127—41no—
69C2×Q8×C127—41no—
70C2×Q8×C127—41no—
71C2×Q8×C127—42no—
72C2×Q8×C127—42no—
73C2×Q8×C127—44no—
74C2×Q8×C127—41no—
75C2×Q8×C127C22⋊C411yescharacteristic
76C2×Q8×C127C22⋊C411yescharacteristic
77C2×Q8×C127C22×C411yescharacteristic
78C2×Q8×C127C2×D411yescharacteristic
79C2×D4×C127—21no—
80C2×D4×C127—41no—
81C2×D4×C127—41no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—21no—
85C22×C4×C127—41no—
86C22×C4×C127—42no—
87C22×C4×C127C22⋊C411yescharacteristic
88Q16×C127—42no—
89Q16×C127—42no—
90SD16×C127—42no—
91SD16×C127—42no—
92M16×C127—22no—
93M16×C127—41no—
94M16×C127—42no—
95M16×C127—81no—
96C2×C8×C127—41no—
97C2×C8×C127—42no—
98C2×C8×C127—81no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C42×C127—21no—
104C42×C127C22⋊C411yescharacteristic
105C42×C127C2×D411yescharacteristic
106C23×C127—41no—
107C23×C127#9911yescharacteristic
108Q8×C127—22no—
109Q8×C127—41no—
110Q8×C127—41no—
111Q8×C127—41no—
112Q8×C127—41no—
113Q8×C127—42no—
114Q8×C127—41no—
115Q8×C127—42no—
116Q8×C127—42no—
117Q8×C127—81no—
118D4×C127—22no—
119D4×C127—41no—
120D4×C127—42no—
121D4×C127—42no—
122D4×C127—42no—
123D4×C127—42no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—42no—
133C2×C4×C127—42no—
134C2×C4×C127—42no—
135C2×C4×C127—44no—
136C2×C4×C127—44no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127#9911yescharacteristic
143C2×C4×C127C2×C22⋊C411yescharacteristic
144C8×C127—42no—
145C8×C127—82no—
146C8×C127—82no—
147C22×C127—41no—
148C22×C127—42no—
149C22×C127—82no—
150C22×C127AΣL(1,𝔽16)11yescharacteristic
151C22×C127AΣL(1,𝔽16)11yescharacteristic
152C22×C127C2×#9911yescharacteristic
153C4×C127—22no—
154C4×C127—21no—
155C4×C127—21no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—41no—
161C4×C127—42no—
162#13573C12711yesmaximal, characteristic, direct product factor, Sylow
163C2×C127—21no—
164C2×C127—21no—
165C2×C127—21no—
166C2×C127—82no—
167C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
168C2×#456C2×C12711yescharacteristic
169C2×#356C2×C12711yescharacteristic, complemented
170#1558C2×C12711yescharacteristic, complemented
171#1405C2×C12714yescomplemented
172C127#1357311yescharacteristic, direct product factor, Sylow
173C2×2−1+4C4×C12711yescharacteristic
174C2×#137—21no—
175C2×C4⋊Q8C22×C12711yescharacteristic
176#456—21no—
177#456—21no—
178#456C4×C12712yes—
179#456C22×C12712yes—
180#432—21no—
181C2×#104—21no—
182C2×#101—21no—
183C2×#101C4×C12711yescharacteristic
184#356—21no—
185#356C22×C12712yescomplemented
186#333—21no—
187#333C22×C12712yescomplemented
1882−1+4—21no—
1892−1+4—41no—
1902−1+4—42no—
1912−1+4C2×C4×C12712yes—
192C2×#41—21no—
193C2×#41—41no—
194C2×#41—41no—
195C22×Q8—41no—
196C22×Q8C2×C4×C12711yescharacteristic
197C22×Q8D4×C12711yescharacteristic
198#137—22no—
199#137—22no—
200#137—42no—
201C2×Q16—41no—
202C2×SD16—41no—
203C2×M16—21no—
204C2×M16—41no—
205C4⋊Q8—41no—
206C4⋊Q8C2×C4×C12711yescharacteristic
207C4⋊Q8D4×C12711yescharacteristic
208C4⋊Q8D4×C12711yescharacteristic
209C4⋊Q8C23×C12711yescharacteristic
210C2×C4⋊C4—41no—
211C2×C42D4×C12711yescharacteristic
212C4⋊C8—22no—
213#104—22no—
214#104—22no—
215#101—21no—
216#101—22no—
217#101—41no—
218#101C2×C4×C12712yes—
219C22⋊C8—41no—
220#97—41no—
221#41—22no—
222#41—22no—
223#41—41no—
224#41—42no—
225#41—42no—
226#41—42no—
227#41—42no—
228#41—44no—
229#41—42no—
230#41—42no—
231#41—41no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8—41no—
243C2×Q8C22⋊C4×C12711yescharacteristic
244C2×Q8C22⋊C4×C12711yescharacteristic
245C2×Q8C22×C4×C12711yescharacteristic
246C2×Q8C2×D4×C12711yescharacteristic
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4C22⋊C4×C12711yescharacteristic
256Q16—42no—
257Q16—42no—
258SD16—42no—
259SD16—42no—
260M16—22no—
261M16—41no—
262M16—42no—
263M16—81no—
264C2×C8—41no—
265C2×C8—42no—
266C2×C8—81no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C42—21no—
272C42C22⋊C4×C12711yescharacteristic
273C42C2×D4×C12711yescharacteristic
274C23—41no—
275C23#99×C12711yescharacteristic
276Q8—22no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281Q8—42no—
282Q8—41no—
283Q8—42no—
284Q8—42no—
285Q8—81no—
286D4—22no—
287D4—41no—
288D4—42no—
289D4—42no—
290D4—42no—
291D4—42no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4#99×C12711yescharacteristic
311C2×C4C2×C22⋊C4×C12711yescharacteristic
312C8—42no—
313C8—82no—
314C8—82no—
315C22—41no—
316C22—42no—
317C22—82no—
318C22AΣL(1,𝔽16)×C12711yescharacteristic
319C22AΣL(1,𝔽16)×C12711yescharacteristic
320C22C2×#99×C12711yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—42no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—82no—
334C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
subgroupG/H
center, socleC2×C127C2×AΣL(1,𝔽16)
commutatorC2×Q8C22×C4×C127
Frattini, cosocleC4⋊Q8C23×C127
prime radicalC2×2−1+4×C127C4
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C23×C127 ◁ C22×C4×C127 ◁ C2×C4⋊Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#356×C127C2semidirect
3. C127×#1558C2semidirect
4. C127×#1405C2semidirect
17. #356×C127C22semidirect
19. #333×C127C22semidirect
162. #13573C127direct (N × H)
169. C2×#356C2×C127semidirect
170. #1558C2×C127semidirect
171. #1405C2×C127semidirect
172. C127#13573direct (N × H)
185. #356C22×C127semidirect
187. #333C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#456×C1272
C2×#356×C1272
C127×#15582
C127×#14052
#13573127