Finite Group Databasebeta

← C127×#11822 #44575495

Subgroups — C127×#11822 #44575495

Order 32512.

434 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2395C211yesmaximal, characteristic
2C127×#1799C211yesmaximal, characteristic
3C127×#1466C211yesmaximal, characteristic
4C127×#1352C212yesmaximal, complemented
5C127×#1344C212yesmaximal, complemented
6C2×C4×D4×C127C2211yescharacteristic
7C2×#117×C127C411yescharacteristic, complemented
8#432×C127—21no—
9C2×#99×C127—21no—
10#400×C127C411yescharacteristic, complemented
11#399×C127C2211yescharacteristic
12#398×C127C411yescharacteristic, complemented
13#397×C127—21no—
14#397×C127C2211yescharacteristic
15#394×C127—21no—
16#380×C127—21no—
17C4×C4⋊C4×C127—21no—
18C4×C22⋊C4×C127—21no—
19C4×C22⋊C4×C127C411yescharacteristic, complemented
20#342×C127C2212yes—
21#323×C127—21no—
22#323×C127C2212yes—
23C22×D4×C127D411yescharacteristic
24C23×C4×C127C2×C411yescharacteristic
25C23×C4×C127D411yescharacteristic
26C4×D4×C127—41no—
27C4×D4×C127—41no—
28#117×C127—41no—
29#117×C127—41no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127—41no—
33C2×C4⋊C4×C127C2×C411yescharacteristic
34C2×C4⋊C4×C127C2×C411yescharacteristic
35C2×C4⋊C4×C127C2×C411yescharacteristic
36C2×C22⋊C4×C127—21no—
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127D411yescharacteristic
42C2×C22⋊C4×C127C2311yescharacteristic
43C2×C42×C127—21no—
44C2×C42×C127—21no—
45C2×C42×C127D411yescharacteristic
46C4⋊C8×C127—41no—
47#99×C127—42no—
48C22⋊C8×C127—22no—
49#97×C127—41no—
50#95×C127—21no—
51#95×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—41no—
55#95×C127C2×C411yescharacteristic
56#95×C127C2×C411yescharacteristic
57C24×C127C22⋊C411yescharacteristic
58C24×C127C2×D411yescharacteristic
59C2×D4×C127—21no—
60C2×D4×C127—21no—
61C2×D4×C127—41no—
62C2×D4×C127—41no—
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—42no—
72C22×C4×C127—42no—
73C22×C4×C127—42no—
74C22×C4×C127C22⋊C411yescharacteristic
75C22×C4×C127C22⋊C411yescharacteristic
76C22×C4×C127C22⋊C411yescharacteristic
77C22×C4×C127C22×C411yescharacteristic
78C22×C4×C127C2×D411yescharacteristic
79C2×C8×C127—42no—
80C4⋊C4×C127—41no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—42no—
84C4⋊C4×C127—42no—
85C4⋊C4×C127—82no—
86C22⋊C4×C127—41no—
87C22⋊C4×C127—41no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—42no—
90C22⋊C4×C127—42no—
91C22⋊C4×C127—41no—
92C22⋊C4×C127—41no—
93C22⋊C4×C127—42no—
94C22⋊C4×C127—42no—
95C42×C127—21no—
96C42×C127—21no—
97C42×C127—42no—
98C42×C127—41no—
99C42×C127—41no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127—21no—
103C23×C127—21no—
104C23×C127—21no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127—41no—
109C23×C127C2×C22⋊C411yescharacteristic
110C23×C127AGL(1,ℤ/8)11yescharacteristic
111C23×C127#13711yescharacteristic
112D4×C127—42no—
113D4×C127—42no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—42no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
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—81no—
140C2×C4×C127—82no—
141C2×C4×C127—82no—
142C2×C4×C127—82no—
143C2×C4×C127#9911yescharacteristic
144C2×C4×C127#9911yescharacteristic
145C8×C127—82no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—81no—
158C22×C127—81no—
159C22×C127—81no—
160C22×C127C2×#9911yescharacteristic
161C22×C127#41711yescharacteristic
162C22×C127#42111yescharacteristic
163C4×C127—21no—
164C4×C127—42no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127—41no—
168C4×C127—41no—
169C4×C127—81no—
170C4×C127—81no—
171C4×C127—81no—
172C4×C127—162no—
173C4×C127#35212yes—
174#11822C12711yesmaximal, characteristic, direct product factor, Sylow
175C2×C127—21no—
176C2×C127—21no—
177C2×C127—41no—
178C2×C127—41no—
179C2×C127—81no—
180C2×C127#151311yescharacteristic
181C2×C127C2×#35211yescharacteristic
182C2×C127#212611yescharacteristic
183#2395C2×C12711yescharacteristic
184#1799C2×C12711yescharacteristic
185#1466C2×C12711yescharacteristic
186#1352C2×C12712yescomplemented
187#1344C2×C12712yescomplemented
188C127#1182211yescharacteristic, direct product factor, Sylow
189C2×C4×D4C22×C12711yescharacteristic
190C2×#117C4×C12711yescharacteristic, complemented
191#432—21no—
192C2×#99—21no—
193#400C4×C12711yescharacteristic, complemented
194#399C22×C12711yescharacteristic
195#398C4×C12711yescharacteristic, complemented
196#397—21no—
197#397C22×C12711yescharacteristic
198#394—21no—
199#380—21no—
200C4×C4⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4C4×C12711yescharacteristic, complemented
203#342C22×C12712yes—
204#323—21no—
205#323C22×C12712yes—
206C22×D4D4×C12711yescharacteristic
207C23×C4C2×C4×C12711yescharacteristic
208C23×C4D4×C12711yescharacteristic
209C4×D4—41no—
210C4×D4—41no—
211#117—41no—
212#117—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4C2×C4×C12711yescharacteristic
217C2×C4⋊C4C2×C4×C12711yescharacteristic
218C2×C4⋊C4C2×C4×C12711yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4D4×C12711yescharacteristic
225C2×C22⋊C4C23×C12711yescharacteristic
226C2×C42—21no—
227C2×C42—21no—
228C2×C42D4×C12711yescharacteristic
229C4⋊C8—41no—
230#99—42no—
231C22⋊C8—22no—
232#97—41no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—41no—
238#95C2×C4×C12711yescharacteristic
239#95C2×C4×C12711yescharacteristic
240C24C22⋊C4×C12711yescharacteristic
241C24C2×D4×C12711yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C22⋊C4×C12711yescharacteristic
258C22×C4C22⋊C4×C12711yescharacteristic
259C22×C4C22⋊C4×C12711yescharacteristic
260C22×C4C22×C4×C12711yescharacteristic
261C22×C4C2×D4×C12711yescharacteristic
262C2×C8—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—82no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—41no—
282C42—41no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C2×C22⋊C4×C12711yescharacteristic
293C23AGL(1,ℤ/8)×C12711yescharacteristic
294C23#137×C12711yescharacteristic
295D4—42no—
296D4—42no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
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—81no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4#99×C12711yescharacteristic
327C2×C4#99×C12711yescharacteristic
328C8—82no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C2×#99×C12711yescharacteristic
344C22#417×C12711yescharacteristic
345C22#421×C12711yescharacteristic
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—162no—
356C4#352×C12712yes—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—41no—
361C2—81no—
362C2C127×#151311yescharacteristic
363C2C2×#352×C12711yescharacteristic
364C2C127×#212611yescharacteristic
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×#1352C2semidirect
5. C127×#1344C2semidirect
7. C2×#117×C127C4semidirect
10. #400×C127C4semidirect
12. #398×C127C4semidirect
19. C4×C22⋊C4×C127C4semidirect
174. #11822C127direct (N × H)
186. #1352C2×C127semidirect
187. #1344C2×C127semidirect
188. C127#11822direct (N × H)
190. C2×#117C4×C127semidirect
193. #400C4×C127semidirect
195. #398C4×C127semidirect
202. C4×C22⋊C4C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#23952
C127×#17992
C127×#14662
C127×#13522
C127×#13442
#11822127