Finite Group Databasebeta

← C127×#11900 #44575573

Subgroups — C127×#11900 #44575573

Order 32512.

452 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2563C211yesmaximal, characteristic
2C127×#1799C211yesmaximal, characteristic
3C127×#1459C211yesmaximal, characteristic
4C127×#1351C214yesmaximal, complemented
5C2×C4⋊Q8×C127C411yescharacteristic, complemented
6C2×C4×D4×C127C2211yescharacteristic
7C2×#99×C127—21no—
8C2×C22⋊C8×C127—21no—
9#393×C127C412yescomplemented
10#393×C127C2212yes—
11#390×C127—21no—
12#382×C127—21no—
13#380×C127—21no—
14#380×C127—21no—
15#380×C127—21no—
16C4×C4⋊C4×C127C411yescharacteristic, complemented
17#342×C127C2212yes—
18#324×C127—21no—
19#323×C127C2212yes—
20C22×Q8×C127C2×C412yes—
21C22×D4×C127D411yescharacteristic
22C23×C4×C127D411yescharacteristic
23C23×C4×C127D411yescharacteristic
24C22×C8×C127—41no—
25C4⋊Q8×C127—41no—
26C4⋊Q8×C127—41no—
27C4×D4×C127—41no—
28C4×D4×C127—41no—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127—41no—
33C2×C4⋊C4×C127D411yescharacteristic
34C2×C22⋊C4×C127—22no—
35C2×C22⋊C4×C127—22no—
36C2×C22⋊C4×C127—22no—
37C2×C22⋊C4×C127C2×C411yescharacteristic
38C2×C22⋊C4×C127C2311yescharacteristic
39C2×C42×C127—21no—
40C2×C42×C127C2×C411yescharacteristic
41#99×C127—42no—
42C22⋊C8×C127—22no—
43C22⋊C8×C127—41no—
44#95×C127—22no—
45#95×C127—22no—
46#95×C127—41no—
47#95×C127C2×C412yes—
48C24×C127C22⋊C411yescharacteristic
49C24×C127C2×D411yescharacteristic
50C2×Q8×C127—42no—
51C2×Q8×C127—82no—
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127SD1612yes—
55C2×D4×C127Q1612yes—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127—22no—
61C22×C4×C127—22no—
62C22×C4×C127—42no—
63C22×C4×C127—42no—
64C22×C4×C127—42no—
65C22×C4×C127C22⋊C411yescharacteristic
66C22×C4×C127C22⋊C411yescharacteristic
67C22×C4×C127C22⋊C411yescharacteristic
68C22×C4×C127C22×C411yescharacteristic
69C22×C4×C127C2×D411yescharacteristic
70C2×C8×C127—42no—
71C2×C8×C127—82no—
72C4⋊C4×C127—41no—
73C4⋊C4×C127—42no—
74C4⋊C4×C127—42no—
75C4⋊C4×C127—41no—
76C4⋊C4×C127—41no—
77C4⋊C4×C127—82no—
78C22⋊C4×C127—41no—
79C22⋊C4×C127—41no—
80C22⋊C4×C127—42no—
81C22⋊C4×C127—42no—
82C22⋊C4×C127—82no—
83C22⋊C4×C127—82no—
84C42×C127—21no—
85C42×C127—21no—
86C42×C127—42no—
87C23×C127—21no—
88C23×C127—21no—
89C23×C127—21no—
90C23×C127—21no—
91C23×C127—41no—
92C23×C127—41no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127#10312yes—
96C23×C127C2×C22⋊C411yescharacteristic
97C23×C127C2×SD1611yescharacteristic
98C23×C127C2×Q1611yescharacteristic
99Q8×C127—84no—
100D4×C127—42no—
101D4×C127—42no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—42no—
109C2×C4×C127—42no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127—42no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—81no—
120C2×C4×C127—81no—
121C2×C4×C127—82no—
122C2×C4×C127—82no—
123C2×C4×C127—82no—
124C2×C4×C127#9911yescharacteristic
125C2×C4×C127#9911yescharacteristic
126C2×C4×C127#10312yes—
127C8×C127—82no—
128C22×C127—21no—
129C22×C127—21no—
130C22×C127—21no—
131C22×C127—22no—
132C22×C127—21no—
133C22×C127—42no—
134C22×C127—41no—
135C22×C127—41no—
136C22×C127—41no—
137C22×C127—41no—
138C22×C127—41no—
139C22×C127—41no—
140C22×C127—81no—
141C22×C127—81no—
142C22×C127C2×#9911yescharacteristic
143C22×C127C2×#10311yescharacteristic
144C22×C127#42111yescharacteristic
145C4×C127—21no—
146C4×C127—22no—
147C4×C127—41no—
148C4×C127—41no—
149C4×C127—41no—
150C4×C127—82no—
151C4×C127—82no—
152C4×C127—162no—
153C4×C127#35412yes—
154#11900C12711yesmaximal, characteristic, direct product factor, Sylow
155C2×C127—21no—
156C2×C127—21no—
157C2×C127—41no—
158C2×C127—42no—
159C2×C127—41no—
160C2×C127#152011yescharacteristic
161C2×C127#212211yescharacteristic
162C2×C127C2×#35411yescharacteristic
163#2563C2×C12711yescharacteristic
164#1799C2×C12711yescharacteristic
165#1459C2×C12711yescharacteristic
166#1351C2×C12714yescomplemented
167C127#1190011yescharacteristic, direct product factor, Sylow
168C2×C4⋊Q8C4×C12711yescharacteristic, complemented
169C2×C4×D4C22×C12711yescharacteristic
170C2×#99—21no—
171C2×C22⋊C8—21no—
172#393C4×C12712yescomplemented
173#393C22×C12712yes—
174#390—21no—
175#382—21no—
176#380—21no—
177#380—21no—
178#380—21no—
179C4×C4⋊C4C4×C12711yescharacteristic, complemented
180#342C22×C12712yes—
181#324—21no—
182#323C22×C12712yes—
183C22×Q8C2×C4×C12712yes—
184C22×D4D4×C12711yescharacteristic
185C23×C4D4×C12711yescharacteristic
186C23×C4D4×C12711yescharacteristic
187C22×C8—41no—
188C4⋊Q8—41no—
189C4⋊Q8—41no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4D4×C12711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C2×C4×C12711yescharacteristic
201C2×C22⋊C4C23×C12711yescharacteristic
202C2×C42—21no—
203C2×C42C2×C4×C12711yescharacteristic
204#99—42no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207#95—22no—
208#95—22no—
209#95—41no—
210#95C2×C4×C12712yes—
211C24C22⋊C4×C12711yescharacteristic
212C24C2×D4×C12711yescharacteristic
213C2×Q8—42no—
214C2×Q8—82no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4SD16×C12712yes—
218C2×D4Q16×C12712yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4C22⋊C4×C12711yescharacteristic
229C22×C4C22⋊C4×C12711yescharacteristic
230C22×C4C22⋊C4×C12711yescharacteristic
231C22×C4C22×C4×C12711yescharacteristic
232C22×C4C2×D4×C12711yescharacteristic
233C2×C8—42no—
234C2×C8—82no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—82no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C42—21no—
248C42—21no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23#103×C12712yes—
259C23C2×C22⋊C4×C12711yescharacteristic
260C23C2×SD16×C12711yescharacteristic
261C23C2×Q16×C12711yescharacteristic
262Q8—84no—
263D4—42no—
264D4—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4#99×C12711yescharacteristic
288C2×C4#99×C12711yescharacteristic
289C2×C4#103×C12712yes—
290C8—82no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—21no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22—81no—
305C22C2×#99×C12711yescharacteristic
306C22C2×#103×C12711yescharacteristic
307C22#421×C12711yescharacteristic
308C4—21no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C4—82no—
315C4—162no—
316C4#354×C12712yes—
317C2—21no—
318C2—21no—
319C2—41no—
320C2—42no—
321C2—41no—
322C2C127×#152011yescharacteristic
323C2C127×#212211yescharacteristic
324C2C2×#354×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC22×D4×C127D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1351C2semidirect
5. C2×C4⋊Q8×C127C4semidirect
9. #393×C127C4semidirect
16. C4×C4⋊C4×C127C4semidirect
154. #11900C127direct (N × H)
166. #1351C2×C127semidirect
167. C127#11900direct (N × H)
168. C2×C4⋊Q8C4×C127semidirect
172. #393C4×C127semidirect
179. C4×C4⋊C4C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#25632
C127×#17992
C127×#14592
C127×#13512
#11900127