Finite Group Databasebeta

← C127×#19074 #44582747

Subgroups — C127×#19074 #44582747

Order 32512.

1072 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 230 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×#3549C211yesmaximal, characteristic
2C127×#3152C211yesmaximal, characteristic, complemented
3C127×#3141C212yesmaximal, complemented
4C127×#1659C218yesmaximal, complemented
5C127×#1525C212yesmaximal, complemented
6C127×#1525C211yesmaximal, characteristic, complemented
7C2×#137×C127—22no—
8H(3,𝔽4)×C127C2214yes—
9#555×C127—22no—
10#555×C127—21no—
11D42×C127—22no—
12D42×C127—21no—
13#535×C127—22no—
14#535×C127—24no—
15#532×C127C2212yes—
16#532×C127C2211yescharacteristic
17C2×C4⋊Q8×C127C2211yescharacteristic, complemented
18#497×C127C2214yescomplemented
19#494×C127—21no—
20#492×C127—21no—
21#489×C127C2218yescomplemented
22#487×C127—22no—
23#486×C127—22no—
24#417×C127—24no—
25C4×M16×C127C2212yescomplemented
26C4×M16×C127C2211yescharacteristic, complemented
27#329×C127C2218yescomplemented
28#329×C127C2214yescomplemented
29C2×#41×C127D412yes—
30C2×#41×C127D414yes—
31C22×Q8×C127—21no—
32C22×D4×C127—22no—
33C22×D4×C127—24no—
34#137×C127—48no—
35C2×Q16×C127—24no—
36C2×SD16×C127—24no—
37C2×M16×C127—21no—
38C2×M16×C127—22no—
39C4⋊Q8×C127—22no—
40C4⋊Q8×C127—21no—
41C4⋊Q8×C127D412yes—
42C4⋊Q8×C127D411yescharacteristic
43C4⋊Q8×C127C2312yes—
44#127×C127—24no—
45#127×C127D412yes—
46#127×C127D411yescharacteristic
47#124×C127—28no—
48#124×C127—24no—
49#124×C127C2312yes—
50#124×C127C2314yes—
51#121×C127—24no—
52#121×C127—24no—
53#121×C127—24no—
54#121×C127—22no—
55#121×C127—22no—
56#121×C127—22no—
57#120×C127—24no—
58#120×C127—28no—
59#120×C127—24no—
60C4×D4×C127—22no—
61C4×D4×C127—22no—
62C4×D4×C127—22no—
63#117×C127—22no—
64#117×C127—24no—
65C2×C4⋊C4×C127—22no—
66C2×C42×C127C2311yescharacteristic
67#103×C127—28no—
68#102×C127—28no—
69#97×C127C2312yes—
70#97×C127C2314yes—
71C4×C8×C127—22no—
72C4×C8×C127—21no—
73C24×C127—24no—
74#41×C127—44no—
75#41×C127—48no—
76C2×Q8×C127—22no—
77C2×Q8×C127—44no—
78C2×Q8×C127—41no—
79C2×Q8×C127C2×D414yes—
80C2×Q8×C127C2×D412yes—
81C2×D4×C127—24no—
82C2×D4×C127—28no—
83C2×D4×C127—22no—
84C2×D4×C127—24no—
85C2×D4×C127—22no—
86C2×D4×C127—22no—
87C2×D4×C127—22no—
88C2×D4×C127—84no—
89C2×D4×C127—88no—
90C2×D4×C127C2×D414yes—
91C2×D4×C127C2×D412yes—
92C22×C4×C127—22no—
93C22×C4×C127—24no—
94C22×C4×C127—41no—
95C22×C4×C127C2×D412yes—
96C22×C4×C127C2×D411yescharacteristic
97Q16×C127—48no—
98SD16×C127—48no—
99M16×C127—24no—
100M16×C127—44no—
101C2×C8×C127—22no—
102C2×C8×C127—24no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127—22no—
105C4⋊C4×C127—22no—
106C4⋊C4×C127—24no—
107C4⋊C4×C127—42no—
108C22⋊C4×C127—28no—
109C22⋊C4×C127—24no—
110C22⋊C4×C127—24no—
111C22⋊C4×C127—24no—
112C22⋊C4×C127—24no—
113C42×C127—24no—
114C42×C127C2×D412yes—
115C42×C127C2×D411yescharacteristic
116C42×C127C2411yescharacteristic
117C23×C127—22no—
118C23×C127—24no—
119C23×C127—84no—
120C23×C127—88no—
121C23×C127#12011yescharacteristic
122Q8×C127—42no—
123Q8×C127—44no—
124Q8×C127—44no—
125Q8×C127—82no—
126D4×C127—42no—
127D4×C127—44no—
128D4×C127—44no—
129D4×C127—44no—
130D4×C127—44no—
131D4×C127—82no—
132D4×C127—84no—
133D4×C127—84no—
134D4×C127—88no—
135C2×C4×C127—22no—
136C2×C4×C127—24no—
137C2×C4×C127—22no—
138C2×C4×C127—24no—
139C2×C4×C127—44no—
140C2×C4×C127—48no—
141C2×C4×C127—42no—
142C2×C4×C127—82no—
143C2×C4×C127#12011yescharacteristic
144C2×C4×C127#12012yes—
145C2×C4×C127C22×D412yes—
146C2×C4×C127C22×D411yescharacteristic
147C8×C127—24no—
148C8×C127—44no—
149C22×C127—21no—
150C22×C127—22no—
151C22×C127—42no—
152C22×C127—44no—
153C22×C127—44no—
154C22×C127—44no—
155C22×C127—44no—
156C22×C127—168no—
157C22×C127C2×#12011yescharacteristic
158C4×C127—21no—
159C4×C127—22no—
160C4×C127—42no—
161C4×C127—44no—
162C4×C127—82no—
163C4×C127#45312yes—
164C4×C127#45514yes—
165#19074C12711yesmaximal, characteristic, direct product factor, Sylow
166C2×C127—41no—
167C2×C127—82no—
168C2×C127—84no—
169C2×C127C2×#45311yescharacteristic
170C2×C127C2×#45512yes—
171#3549C2×C12711yescharacteristic
172#3152C2×C12711yescharacteristic, complemented
173#3141C2×C12712yescomplemented
174#1659C2×C12718yescomplemented
175#1525C2×C12712yescomplemented
176#1525C2×C12711yescharacteristic, complemented
177C127#1907411yescharacteristic, direct product factor, Sylow
178C2×#137—22no—
179H(3,𝔽4)C22×C12714yes—
180#555—22no—
181#555—21no—
182D42—22no—
183D42—21no—
184#535—22no—
185#535—24no—
186#532C22×C12712yes—
187#532C22×C12711yescharacteristic
188C2×C4⋊Q8C22×C12711yescharacteristic, complemented
189#497C22×C12714yescomplemented
190#494—21no—
191#492—21no—
192#489C22×C12718yescomplemented
193#487—22no—
194#486—22no—
195#417—24no—
196C4×M16C22×C12712yescomplemented
197C4×M16C22×C12711yescharacteristic, complemented
198#329C22×C12714yescomplemented
199#329C22×C12718yescomplemented
200C2×#41D4×C12712yes—
201C2×#41D4×C12714yes—
202C22×Q8—21no—
203C22×D4—22no—
204C22×D4—24no—
205#137—48no—
206C2×Q16—24no—
207C2×SD16—24no—
208C2×M16—21no—
209C2×M16—22no—
210C4⋊Q8—22no—
211C4⋊Q8—21no—
212C4⋊Q8D4×C12712yes—
213C4⋊Q8D4×C12711yescharacteristic
214C4⋊Q8C23×C12712yes—
215#127—24no—
216#127D4×C12712yes—
217#127D4×C12711yescharacteristic
218#124—28no—
219#124—24no—
220#124C23×C12714yes—
221#124C23×C12712yes—
222#121—24no—
223#121—24no—
224#121—24no—
225#121—22no—
226#121—22no—
227#121—22no—
228#120—24no—
229#120—28no—
230#120—24no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234#117—22no—
235#117—24no—
236C2×C4⋊C4—22no—
237C2×C42C23×C12711yescharacteristic
238#103—28no—
239#102—28no—
240#97C23×C12714yes—
241#97C23×C12712yes—
242C4×C8—22no—
243C4×C8—21no—
244C24—24no—
245#41—44no—
246#41—48no—
247C2×Q8—22no—
248C2×Q8—44no—
249C2×Q8—41no—
250C2×Q8C2×D4×C12714yes—
251C2×Q8C2×D4×C12712yes—
252C2×D4—24no—
253C2×D4—28no—
254C2×D4—22no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—84no—
260C2×D4—88no—
261C2×D4C2×D4×C12714yes—
262C2×D4C2×D4×C12712yes—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—41no—
266C22×C4C2×D4×C12712yes—
267C22×C4C2×D4×C12711yescharacteristic
268Q16—48no—
269SD16—48no—
270M16—24no—
271M16—44no—
272C2×C8—22no—
273C2×C8—24no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—42no—
279C22⋊C4—28no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C42—24no—
285C42C2×D4×C12712yes—
286C42C2×D4×C12711yescharacteristic
287C42C24×C12711yescharacteristic
288C23—22no—
289C23—24no—
290C23—84no—
291C23—88no—
292C23#120×C12711yescharacteristic
293Q8—42no—
294Q8—44no—
295Q8—44no—
296Q8—82no—
297D4—42no—
298D4—44no—
299D4—44no—
300D4—44no—
301D4—44no—
302D4—82no—
303D4—84no—
304D4—84no—
305D4—88no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—44no—
311C2×C4—48no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4#120×C12711yescharacteristic
315C2×C4#120×C12712yes—
316C2×C4C22×D4×C12712yes—
317C2×C4C22×D4×C12711yescharacteristic
318C8—24no—
319C8—44no—
320C22—21no—
321C22—22no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—168no—
328C22C2×#120×C12711yescharacteristic
329C4—21no—
330C4—22no—
331C4—42no—
332C4—44no—
333C4—82no—
334C4#453×C12712yes—
335C4#455×C12714yes—
336C2—41no—
337C2—82no—
338C2—84no—
339C2C2×#453×C12711yescharacteristic
340C2C2×#455×C12712yes—
subgroupG/H
center, socleC22×C127C2×#120
commutator, Frattini, cosocleC42C24×C127
prime radicalC127×#3549C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#3152C2semidirect
3. C127×#3141C2semidirect
4. C127×#1659C2semidirect
5. C127×#1525C2semidirect
6. C127×#1525C2semidirect
17. C2×C4⋊Q8×C127C22semidirect
18. #497×C127C22semidirect
21. #489×C127C22semidirect
25. C4×M16×C127C22semidirect
26. C4×M16×C127C22semidirect
27. #329×C127C22semidirect
28. #329×C127C22semidirect
165. #19074C127direct (N × H)
172. #3152C2×C127semidirect
173. #3141C2×C127semidirect
174. #1659C2×C127semidirect
175. #1525C2×C127semidirect
176. #1525C2×C127semidirect
177. C127#19074direct (N × H)
188. C2×C4⋊Q8C22×C127semidirect
189. #497C22×C127semidirect
192. #489C22×C127semidirect
196. C4×M16C22×C127semidirect
197. C4×M16C22×C127semidirect
198. #329C22×C127semidirect
199. #329C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35492
C127×#31522
C127×#31412
C127×#16592
C127×#15252
#19074127