Finite Group Databasebeta

← C127×#11878 #44575551

Subgroups — C127×#11878 #44575551

Order 32512.

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

normal subgroups only

This group has 472 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×#2547C211yesmaximal, characteristic
2C127×#1799C211yesmaximal, characteristic
3C127×#1459C211yesmaximal, characteristic
4C127×#1344C214yesmaximal, complemented
5C2×C4⋊Q8×C127C411yescharacteristic, complemented
6C2×#124×C127—21no—
7C2×C4×D4×C127C2211yescharacteristic
8C2×#99×C127—21no—
9C2×C22⋊C8×C127—21no—
10#397×C127C412yescomplemented
11#397×C127C2212yes—
12#391×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×C127—21no—
21C22×D4×C127D411yescharacteristic
22C23×C4×C127D411yescharacteristic
23C23×C4×C127D411yescharacteristic
24C22×C8×C127—41no—
25C4⋊Q8×C127—41no—
26C4⋊Q8×C127—41no—
27#124×C127—42no—
28#124×C127—42no—
29C4×D4×C127—41no—
30C4×D4×C127—41no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—41no—
34C2×C4⋊C4×C127C2×C412yes—
35C2×C4⋊C4×C127D411yescharacteristic
36C2×C22⋊C4×C127—22no—
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127C2×C411yescharacteristic
40C2×C22⋊C4×C127C2311yescharacteristic
41C2×C42×C127—21no—
42C2×C42×C127C2×C411yescharacteristic
43#99×C127—42no—
44C22⋊C8×C127—22no—
45C22⋊C8×C127—41no—
46#95×C127—22no—
47#95×C127—22no—
48#95×C127—41no—
49#95×C127C2×C412yes—
50C24×C127C22⋊C411yescharacteristic
51C24×C127C2×D411yescharacteristic
52C2×Q8×C127—42no—
53C2×Q8×C127—42no—
54C2×Q8×C127—42no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127D812yes—
58C2×D4×C127SD1612yes—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—42no—
66C22×C4×C127—42no—
67C22×C4×C127—42no—
68C22×C4×C127C22⋊C411yescharacteristic
69C22×C4×C127C22⋊C411yescharacteristic
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22×C411yescharacteristic
72C22×C4×C127C2×D411yescharacteristic
73C2×C8×C127—42no—
74C2×C8×C127—82no—
75C4⋊C4×C127—41no—
76C4⋊C4×C127—42no—
77C4⋊C4×C127—42no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127—82no—
81C22⋊C4×C127—41no—
82C22⋊C4×C127—41no—
83C22⋊C4×C127—44no—
84C22⋊C4×C127—42no—
85C22⋊C4×C127—42no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C42×C127—21no—
89C42×C127—21no—
90C42×C127—42no—
91C23×C127—21no—
92C23×C127—21no—
93C23×C127—21no—
94C23×C127—21no—
95C23×C127—41no—
96C23×C127—41no—
97C23×C127—41no—
98C23×C127—41no—
99C23×C127#10212yes—
100C23×C127C2×C22⋊C411yescharacteristic
101C23×C127C2×D811yescharacteristic
102C23×C127C2×SD1611yescharacteristic
103Q8×C127—84no—
104D4×C127—42no—
105D4×C127—42no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—22no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127—42no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—81no—
124C2×C4×C127—81no—
125C2×C4×C127—82no—
126C2×C4×C127—82no—
127C2×C4×C127—82no—
128C2×C4×C127#9911yescharacteristic
129C2×C4×C127#9911yescharacteristic
130C2×C4×C127#10212yes—
131C8×C127—82no—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—21no—
135C22×C127—22no—
136C22×C127—21no—
137C22×C127—42no—
138C22×C127—41no—
139C22×C127—41no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127—41no—
144C22×C127—81no—
145C22×C127—81no—
146C22×C127C2×#9911yescharacteristic
147C22×C127C2×#10211yescharacteristic
148C22×C127#42111yescharacteristic
149C4×C127—21no—
150C4×C127—22no—
151C4×C127—41no—
152C4×C127—41no—
153C4×C127—41no—
154C4×C127—82no—
155C4×C127—82no—
156C4×C127—162no—
157C4×C127#35412yes—
158#11878C12711yesmaximal, characteristic, direct product factor, Sylow
159C2×C127—21no—
160C2×C127—21no—
161C2×C127—41no—
162C2×C127—42no—
163C2×C127—41no—
164C2×C127#151911yescharacteristic
165C2×C127#212311yescharacteristic
166C2×C127C2×#35411yescharacteristic
167#2547C2×C12711yescharacteristic
168#1799C2×C12711yescharacteristic
169#1459C2×C12711yescharacteristic
170#1344C2×C12714yescomplemented
171C127#1187811yescharacteristic, direct product factor, Sylow
172C2×C4⋊Q8C4×C12711yescharacteristic, complemented
173C2×#124—21no—
174C2×C4×D4C22×C12711yescharacteristic
175C2×#99—21no—
176C2×C22⋊C8—21no—
177#397C4×C12712yescomplemented
178#397C22×C12712yes—
179#391—21no—
180#380—21no—
181#380—21no—
182#380—21no—
183C4×C4⋊C4C4×C12711yescharacteristic, complemented
184#342C22×C12712yes—
185#324—21no—
186#323C22×C12712yes—
187C22×Q8—21no—
188C22×D4D4×C12711yescharacteristic
189C23×C4D4×C12711yescharacteristic
190C23×C4D4×C12711yescharacteristic
191C22×C8—41no—
192C4⋊Q8—41no—
193C4⋊Q8—41no—
194#124—42no—
195#124—42no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C2×C4×C12712yes—
202C2×C4⋊C4D4×C12711yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C2×C4×C12711yescharacteristic
207C2×C22⋊C4C23×C12711yescharacteristic
208C2×C42—21no—
209C2×C42C2×C4×C12711yescharacteristic
210#99—42no—
211C22⋊C8—22no—
212C22⋊C8—41no—
213#95—22no—
214#95—22no—
215#95—41no—
216#95C2×C4×C12712yes—
217C24C22⋊C4×C12711yescharacteristic
218C24C2×D4×C12711yescharacteristic
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4D8×C12712yes—
225C2×D4SD16×C12712yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4C22⋊C4×C12711yescharacteristic
236C22×C4C22⋊C4×C12711yescharacteristic
237C22×C4C22⋊C4×C12711yescharacteristic
238C22×C4C22×C4×C12711yescharacteristic
239C22×C4C2×D4×C12711yescharacteristic
240C2×C8—42no—
241C2×C8—82no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—82no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—21no—
256C42—21no—
257C42—42no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#102×C12712yes—
267C23C2×C22⋊C4×C12711yescharacteristic
268C23C2×D8×C12711yescharacteristic
269C23C2×SD16×C12711yescharacteristic
270Q8—84no—
271D4—42no—
272D4—42no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#99×C12711yescharacteristic
296C2×C4#99×C12711yescharacteristic
297C2×C4#102×C12712yes—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22C2×#99×C12711yescharacteristic
314C22C2×#102×C12711yescharacteristic
315C22#421×C12711yescharacteristic
316C4—21no—
317C4—22no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—82no—
322C4—82no—
323C4—162no—
324C4#354×C12712yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—42no—
329C2—41no—
330C2C127×#151911yescharacteristic
331C2C127×#212311yescharacteristic
332C2C2×#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×#1344C2semidirect
5. C2×C4⋊Q8×C127C4semidirect
10. #397×C127C4semidirect
16. C4×C4⋊C4×C127C4semidirect
158. #11878C127direct (N × H)
170. #1344C2×C127semidirect
171. C127#11878direct (N × H)
172. C2×C4⋊Q8C4×C127semidirect
177. #397C4×C127semidirect
183. 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×#25472
C127×#17992
C127×#14592
C127×#13442
#11878127