Finite Group Databasebeta

← C127×#11693 #44575366

Subgroups — C127×#11693 #44575366

Order 32512.

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

normal subgroups only

This group has 568 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×#2619C211yesmaximal, characteristic
2C127×#1740C211yesmaximal, characteristic, complemented
3C127×#1458C211yesmaximal, characteristic, complemented
4C127×#1343C214yesmaximal, complemented
5C2×#127×C127C411yescharacteristic, complemented
6C2×#121×C127—21no—
7C2×#117×C127—21no—
8C2×#117×C127C2211yescharacteristic
9#431×C127—21no—
10#394×C127—21no—
11#394×C127C412yescomplemented
12#392×C127—21no—
13#392×C127C2212yes—
14#390×C127—21no—
15#382×C127C411yescharacteristic, complemented
16#342×C127—21no—
17#342×C127C2212yes—
18#324×C127—21no—
19#323×C127C2212yes—
20C22×D4×C127—22no—
21C22×D4×C127—21no—
22C22×D4×C127—21no—
23C22×D4×C127C2×C412yes—
24C23×C4×C127D411yescharacteristic
25#127×C127—41no—
26#127×C127—41no—
27#121×C127—42no—
28#121×C127—42no—
29#117×C127—24no—
30#117×C127—41no—
31#117×C127—41no—
32#117×C127—42no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—41no—
35C2×C4⋊C4×C127C2×C411yescharacteristic
36C2×C4⋊C4×C127D411yescharacteristic
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127D411yescharacteristic
42C2×C22⋊C4×C127C2311yescharacteristic
43C2×C42×C127—41no—
44C2×C42×C127C2×C411yescharacteristic
45C2×C42×C127D411yescharacteristic
46C4⋊C8×C127—41no—
47C22⋊C8×C127—22no—
48C4×C8×C127—41no—
49#95×C127—21no—
50#95×C127C2×C412yes—
51C24×C127—22no—
52C24×C127C2×D411yescharacteristic
53C2×D4×C127—42no—
54C2×D4×C127—42no—
55C2×D4×C127—42no—
56C2×D4×C127—42no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C2×D4×C127—42no—
60C2×D4×C127—42no—
61C2×D4×C127—82no—
62C2×D4×C127—82no—
63C2×D4×C127—82no—
64C2×D4×C127—82no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127—42no—
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22⋊C411yescharacteristic
72C22×C4×C127C22⋊C411yescharacteristic
73C22×C4×C127C22⋊C411yescharacteristic
74C22×C4×C127D812yes—
75C22×C4×C127SD1612yes—
76C22×C4×C127C22×C411yescharacteristic
77C22×C4×C127C2×D411yescharacteristic
78C2×C8×C127—42no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127—41no—
81C4⋊C4×C127—44no—
82C4⋊C4×C127—42no—
83C22⋊C4×C127—24no—
84C22⋊C4×C127—41no—
85C22⋊C4×C127—41no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—44no—
88C22⋊C4×C127—82no—
89C22⋊C4×C127—82no—
90C42×C127—21no—
91C42×C127—21no—
92C42×C127—21no—
93C42×C127—21no—
94C42×C127—44no—
95C23×C127—21no—
96C23×C127—21no—
97C23×C127—41no—
98C23×C127—41no—
99C23×C127—42no—
100C23×C127—42no—
101C23×C127—42no—
102C23×C127—42no—
103C23×C127—42no—
104C23×C127—42no—
105C23×C127—42no—
106C23×C127C2×C22⋊C411yescharacteristic
107C23×C127C2×D811yescharacteristic
108C23×C127C2×SD1611yescharacteristic
109D4×C127—84no—
110D4×C127—84no—
111D4×C127—84no—
112D4×C127—84no—
113D4×C127—82no—
114D4×C127—82no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—22no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—44no—
130C2×C4×C127—42no—
131C2×C4×C127—42no—
132C2×C4×C127—42no—
133C2×C4×C127—82no—
134C2×C4×C127—82no—
135C2×C4×C127#9911yescharacteristic
136C2×C4×C127#9911yescharacteristic
137C2×C4×C127#10212yes—
138C2×C4×C127#10212yes—
139C2×C4×C127#10412yes—
140C8×C127—82no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—21no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—81no—
150C22×C127—82no—
151C22×C127—82no—
152C22×C127—82no—
153C22×C127—82no—
154C22×C127—82no—
155C22×C127—82no—
156C22×C127—82no—
157C22×C127C2×#9911yescharacteristic
158C22×C127C2×#10211yescharacteristic
159C22×C127C2×#10411yescharacteristic
160C4×C127—22no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—41no—
164C4×C127—41no—
165C4×C127—82no—
166C4×C127—84no—
167C4×C127#32712yes—
168C4×C127#32712yes—
169#11693C12711yesmaximal, characteristic, direct product factor, Sylow
170C2×C127—21no—
171C2×C127—21no—
172C2×C127—41no—
173C2×C127—41no—
174C2×C127—162no—
175C2×C127C2×#32711yescharacteristic
176C2×C127#212211yescharacteristic
177C2×C127#212811yescharacteristic
178#2619C2×C12711yescharacteristic
179#1740C2×C12711yescharacteristic, complemented
180#1458C2×C12711yescharacteristic, complemented
181#1343C2×C12714yescomplemented
182C127#1169311yescharacteristic, direct product factor, Sylow
183C2×#127C4×C12711yescharacteristic, complemented
184C2×#121—21no—
185C2×#117—21no—
186C2×#117C22×C12711yescharacteristic
187#431—21no—
188#394—21no—
189#394C4×C12712yescomplemented
190#392—21no—
191#392C22×C12712yes—
192#390—21no—
193#382C4×C12711yescharacteristic, complemented
194#342—21no—
195#342C22×C12712yes—
196#324—21no—
197#323C22×C12712yes—
198C22×D4—22no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4C2×C4×C12712yes—
202C23×C4D4×C12711yescharacteristic
203#127—41no—
204#127—41no—
205#121—42no—
206#121—42no—
207#117—24no—
208#117—41no—
209#117—41no—
210#117—42no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4C2×C4×C12711yescharacteristic
214C2×C4⋊C4D4×C12711yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4D4×C12711yescharacteristic
220C2×C22⋊C4C23×C12711yescharacteristic
221C2×C42—41no—
222C2×C42C2×C4×C12711yescharacteristic
223C2×C42D4×C12711yescharacteristic
224C4⋊C8—41no—
225C22⋊C8—22no—
226C4×C8—41no—
227#95—21no—
228#95C2×C4×C12712yes—
229C24—22no—
230C24C2×D4×C12711yescharacteristic
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—82no—
240C2×D4—82no—
241C2×D4—82no—
242C2×D4—82no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—42no—
248C22×C4C22⋊C4×C12711yescharacteristic
249C22×C4C22⋊C4×C12711yescharacteristic
250C22×C4C22⋊C4×C12711yescharacteristic
251C22×C4C22⋊C4×C12711yescharacteristic
252C22×C4D8×C12712yes—
253C22×C4SD16×C12712yes—
254C22×C4C22×C4×C12711yescharacteristic
255C22×C4C2×D4×C12711yescharacteristic
256C2×C8—42no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C22⋊C4—24no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—44no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23C2×C22⋊C4×C12711yescharacteristic
285C23C2×D8×C12711yescharacteristic
286C23C2×SD16×C12711yescharacteristic
287D4—84no—
288D4—84no—
289D4—84no—
290D4—84no—
291D4—82no—
292D4—82no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4#99×C12711yescharacteristic
314C2×C4#99×C12711yescharacteristic
315C2×C4#102×C12712yes—
316C2×C4#102×C12712yes—
317C2×C4#104×C12712yes—
318C8—82no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22C2×#99×C12711yescharacteristic
336C22C2×#102×C12711yescharacteristic
337C22C2×#104×C12711yescharacteristic
338C4—22no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—82no—
344C4—84no—
345C4#327×C12712yes—
346C4#327×C12712yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—41no—
351C2—162no—
352C2C2×#327×C12711yescharacteristic
353C2C127×#212211yescharacteristic
354C2C127×#212811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2619C2
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
2. C127×#1740C2semidirect
3. C127×#1458C2semidirect
4. C127×#1343C2semidirect
5. C2×#127×C127C4semidirect
11. #394×C127C4semidirect
15. #382×C127C4semidirect
169. #11693C127direct (N × H)
179. #1740C2×C127semidirect
180. #1458C2×C127semidirect
181. #1343C2×C127semidirect
182. C127#11693direct (N × H)
183. C2×#127C4×C127semidirect
189. #394C4×C127semidirect
193. #382C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#26192
C127×#17402
C127×#14582
C127×#13432
#11693127