Finite Group Databasebeta

← C127×#11694 #44575367

Subgroups — C127×#11694 #44575367

Order 32512.

468 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2621C211yesmaximal, characteristic
2C127×#1740C211yesmaximal, characteristic
3C127×#1458C211yesmaximal, characteristic
4C127×#1344C214yesmaximal
5C2×C4⋊Q8×C127C411yescharacteristic, complemented
6C2×C22⋊Q8×C127—21no—
7C2×#117×C127—21no—
8C2×#117×C127C2211yescharacteristic
9#431×C127—21no—
10#398×C127—21no—
11#398×C127C412yescomplemented
12#397×C127—21no—
13#397×C127C2212yes—
14#391×C127—21no—
15#382×C127C411yescharacteristic, complemented
16#342×C127—21no—
17#342×C127C2212yes—
18#324×C127—21no—
19#323×C127C2212yes—
20C22×Q8×C127—21no—
21C23×C4×C127D411yescharacteristic
22C4⋊Q8×C127—41no—
23C4⋊Q8×C127—41no—
24C22⋊Q8×C127—42no—
25C22⋊Q8×C127—42no—
26#117×C127—24no—
27#117×C127—41no—
28#117×C127—41no—
29#117×C127—42no—
30C2×C4⋊C4×C127—22no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—41no—
34C2×C4⋊C4×C127C2×C412yes—
35C2×C4⋊C4×C127C2×C411yescharacteristic
36C2×C4⋊C4×C127D411yescharacteristic
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127D411yescharacteristic
40C2×C22⋊C4×C127C2311yescharacteristic
41C2×C42×C127—41no—
42C2×C42×C127C2×C411yescharacteristic
43C2×C42×C127D411yescharacteristic
44C4⋊C8×C127—41no—
45C22⋊C8×C127—22no—
46C4×C8×C127—41no—
47#95×C127—21no—
48#95×C127—22no—
49#95×C127—22no—
50#95×C127C2×C412yes—
51C24×C127C2×D411yescharacteristic
52C2×Q8×C127—41no—
53C2×Q8×C127—41no—
54C2×Q8×C127—42no—
55C2×Q8×C127—42no—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127—22no—
59C22×C4×C127—22no—
60C22×C4×C127—22no—
61C22×C4×C127—42no—
62C22×C4×C127C22⋊C411yescharacteristic
63C22×C4×C127C22⋊C411yescharacteristic
64C22×C4×C127C22⋊C411yescharacteristic
65C22×C4×C127C22⋊C411yescharacteristic
66C22×C4×C127D812yes—
67C22×C4×C127SD1612yes—
68C22×C4×C127C22×C411yescharacteristic
69C22×C4×C127C2×D411yescharacteristic
70C2×C8×C127—42no—
71C4⋊C4×C127—41no—
72C4⋊C4×C127—41no—
73C4⋊C4×C127—42no—
74C4⋊C4×C127—42no—
75C4⋊C4×C127—44no—
76C4⋊C4×C127—42no—
77C4⋊C4×C127—44no—
78C22⋊C4×C127—24no—
79C22⋊C4×C127—41no—
80C22⋊C4×C127—41no—
81C22⋊C4×C127—42no—
82C22⋊C4×C127—42no—
83C22⋊C4×C127—42no—
84C42×C127—21no—
85C42×C127—21no—
86C42×C127—21no—
87C42×C127—21no—
88C42×C127—44no—
89C23×C127—21no—
90C23×C127—21no—
91C23×C127—41no—
92C23×C127—41no—
93C23×C127C2×C22⋊C411yescharacteristic
94C23×C127C2×D811yescharacteristic
95C23×C127C2×SD1611yescharacteristic
96Q8×C127—82no—
97Q8×C127—82no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
101C2×C4×C127—22no—
102C2×C4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—44no—
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—82no—
120C2×C4×C127—82no—
121C2×C4×C127#9911yescharacteristic
122C2×C4×C127#9911yescharacteristic
123C2×C4×C127#10212yes—
124C2×C4×C127#10212yes—
125C2×C4×C127#10412yes—
126C8×C127—82no—
127C22×C127—21no—
128C22×C127—21no—
129C22×C127—21no—
130C22×C127—21no—
131C22×C127—41no—
132C22×C127—41no—
133C22×C127—41no—
134C22×C127—41no—
135C22×C127—81no—
136C22×C127C2×#9911yescharacteristic
137C22×C127C2×#10211yescharacteristic
138C22×C127C2×#10411yescharacteristic
139C4×C127—22no—
140C4×C127—41no—
141C4×C127—41no—
142C4×C127—41no—
143C4×C127—41no—
144C4×C127—82no—
145C4×C127—84no—
146C4×C127—82no—
147C4×C127#32712yes—
148C4×C127#32712yes—
149#11694C12711yesmaximal, characteristic, direct product factor, Sylow
150C2×C127—21no—
151C2×C127—21no—
152C2×C127—41no—
153C2×C127—41no—
154C2×C127C2×#32711yescharacteristic
155C2×C127#212311yescharacteristic
156C2×C127#212911yescharacteristic
157#2621C2×C12711yescharacteristic
158#1740C2×C12711yescharacteristic
159#1458C2×C12711yescharacteristic
160#1344C2×C12714yes—
161C127#1169411yescharacteristic, direct product factor, Sylow
162C2×C4⋊Q8C4×C12711yescharacteristic, complemented
163C2×C22⋊Q8—21no—
164C2×#117—21no—
165C2×#117C22×C12711yescharacteristic
166#431—21no—
167#398—21no—
168#398C4×C12712yescomplemented
169#397—21no—
170#397C22×C12712yes—
171#391—21no—
172#382C4×C12711yescharacteristic, complemented
173#342—21no—
174#342C22×C12712yes—
175#324—21no—
176#323C22×C12712yes—
177C22×Q8—21no—
178C23×C4D4×C12711yescharacteristic
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183#117—24no—
184#117—41no—
185#117—41no—
186#117—42no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4C2×C4×C12712yes—
192C2×C4⋊C4C2×C4×C12711yescharacteristic
193C2×C4⋊C4D4×C12711yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4D4×C12711yescharacteristic
197C2×C22⋊C4C23×C12711yescharacteristic
198C2×C42—41no—
199C2×C42C2×C4×C12711yescharacteristic
200C2×C42D4×C12711yescharacteristic
201C4⋊C8—41no—
202C22⋊C8—22no—
203C4×C8—41no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95C2×C4×C12712yes—
208C24C2×D4×C12711yescharacteristic
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4C22⋊C4×C12711yescharacteristic
220C22×C4C22⋊C4×C12711yescharacteristic
221C22×C4C22⋊C4×C12711yescharacteristic
222C22×C4C22⋊C4×C12711yescharacteristic
223C22×C4D8×C12712yes—
224C22×C4SD16×C12712yes—
225C22×C4C22×C4×C12711yescharacteristic
226C22×C4C2×D4×C12711yescharacteristic
227C2×C8—42no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C22⋊C4—24no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—44no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23C2×C22⋊C4×C12711yescharacteristic
251C23C2×D8×C12711yescharacteristic
252C23C2×SD16×C12711yescharacteristic
253Q8—82no—
254Q8—82no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#99×C12711yescharacteristic
279C2×C4#99×C12711yescharacteristic
280C2×C4#102×C12712yes—
281C2×C4#102×C12712yes—
282C2×C4#104×C12712yes—
283C8—82no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22C2×#99×C12711yescharacteristic
294C22C2×#102×C12711yescharacteristic
295C22C2×#104×C12711yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—82no—
302C4—84no—
303C4—82no—
304C4#327×C12712yes—
305C4#327×C12712yes—
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#327×C12711yescharacteristic
311C2C127×#212311yescharacteristic
312C2C127×#212911yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC24×C127C2×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×#117×C127 ◁ G

Complements

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

NH ≅ G/Ntype
5. C2×C4⋊Q8×C127C4semidirect
11. #398×C127C4semidirect
15. #382×C127C4semidirect
149. #11694C127direct (N × H)
161. C127#11694direct (N × H)
162. C2×C4⋊Q8C4×C127semidirect
168. #398C4×C127semidirect
172. #382C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#26212
C127×#17402
C127×#14582
C127×#13442
#11694127