Finite Group Databasebeta

← C127×#8524 #44572197

Subgroups — C127×#8524 #44572197

Order 32512.

910 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#536×C127C211yesmaximal, characteristic
2C127×#1558C212yesmaximal, complemented
3C127×#1533C212yesmaximal, complemented
4C127×#1464C212yesmaximal, complemented
5C22×#41×C127C411yescharacteristic
6C23×Q8×C127C411yescharacteristic
7#536×C127—24no—
8#536×C127—24no—
9C2×C4⋊Q8×C127C411yescharacteristic
10C2×C4⋊Q8×C127C2211yescharacteristic
11C2×#124×C127C411yescharacteristic
12C2×#124×C127C2211yescharacteristic
13C2×#123×C127—22no—
14C2×C22⋊Q8×C127—22no—
15C2×#117×C127C2211yescharacteristic
16#432×C127C2214yes—
17#333×C127—24no—
18#329×C127—24no—
19#324×C127—22no—
20#323×C127—22no—
21C2×#41×C127—22no—
22C2×#41×C127—22no—
23C2×#41×C127—42no—
24C2×#41×C127—42no—
25C22×Q8×C127—22no—
26C22×Q8×C127—22no—
27C22×Q8×C127—21no—
28C22×Q8×C127—42no—
29C22×Q8×C127—42no—
30C22×Q8×C127C2×C411yescharacteristic
31C22×Q8×C127C2×C411yescharacteristic
32C22×Q8×C127C2×C411yescharacteristic
33C22×D4×C127—21no—
34C22×D4×C127C2×C411yescharacteristic
35C23×C4×C127—21no—
36C23×C4×C127—21no—
37C23×C4×C127C2×C411yescharacteristic
38C4⋊Q8×C127—22no—
39C4⋊Q8×C127—22no—
40C4⋊Q8×C127—22no—
41C4⋊Q8×C127D414yes—
42#124×C127—22no—
43#124×C127—22no—
44#124×C127—22no—
45#124×C127D414yes—
46#123×C127—216no—
47C22⋊Q8×C127—216no—
48#117×C127—22no—
49#117×C127—22no—
50C2×C4⋊C4×C127—22no—
51C2×C4⋊C4×C127—22no—
52C2×C4⋊C4×C127D412yes—
53C2×C22⋊C4×C127—22no—
54C2×C22⋊C4×C127—22no—
55C2×C22⋊C4×C127D412yes—
56C2×C42×C127C2×C411yescharacteristic
57C2×C42×C127C2311yescharacteristic
58C4⋊C8×C127—24no—
59C22⋊C8×C127—24no—
60#97×C127—24no—
61C24×C127—21no—
62C24×C127C22⋊C411yescharacteristic
63#41×C127—48no—
64#41×C127—48no—
65C2×Q8×C127—24no—
66C2×Q8×C127—21no—
67C2×Q8×C127—21no—
68C2×Q8×C127—21no—
69C2×Q8×C127—48no—
70C2×Q8×C127—48no—
71C2×Q8×C127—42no—
72C2×Q8×C127—41no—
73C2×Q8×C127—41no—
74C2×Q8×C127—41no—
75C2×Q8×C127—41no—
76C2×Q8×C127—41no—
77C2×Q8×C127—41no—
78C2×Q8×C127—42no—
79C2×Q8×C127C22⋊C412yes—
80C2×Q8×C127C22⋊C412yes—
81C2×Q8×C127C22⋊C412yes—
82C2×D4×C127—24no—
83C2×D4×C127—21no—
84C2×D4×C127—41no—
85C2×D4×C127—41no—
86C2×D4×C127—42no—
87C2×D4×C127—42no—
88C2×D4×C127C22⋊C412yes—
89C22×C4×C127—22no—
90C22×C4×C127—22no—
91C22×C4×C127—21no—
92C22×C4×C127—21no—
93C22×C4×C127—24no—
94C22×C4×C127—24no—
95C22×C4×C127—21no—
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127—42no—
99C22×C4×C127—42no—
100C22×C4×C127—42no—
101C22×C4×C127—42no—
102C22×C4×C127C22⋊C411yescharacteristic
103C22×C4×C127C22⋊C412yes—
104C22×C4×C127C22×C411yescharacteristic
105C22×C4×C127C2×D412yes—
106C2×C8×C127—44no—
107C4⋊C4×C127—28no—
108C4⋊C4×C127—28no—
109C4⋊C4×C127—24no—
110C22⋊C4×C127—28no—
111C22⋊C4×C127—28no—
112C22⋊C4×C127—24no—
113C42×C127—21no—
114C42×C127—21no—
115C42×C127C2×D412yes—
116C42×C127C2×D412yes—
117C23×C127—21no—
118C23×C127—21no—
119C23×C127—22no—
120C23×C127—42no—
121C23×C127—44no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127#10012yes—
125C23×C127C2×C22⋊C411yescharacteristic
126Q8×C127—42no—
127Q8×C127—42no—
128Q8×C127—42no—
129Q8×C127—48no—
130Q8×C127—42no—
131Q8×C127—42no—
132Q8×C127—42no—
133D4×C127—42no—
134D4×C127—42no—
135D4×C127—48no—
136C2×C4×C127—24no—
137C2×C4×C127—24no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—22no—
143C2×C4×C127—22no—
144C2×C4×C127—22no—
145C2×C4×C127—22no—
146C2×C4×C127—42no—
147C2×C4×C127—48no—
148C2×C4×C127—48no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—41no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—42no—
156C2×C4×C127#10012yes—
157C2×C4×C127C2×C22⋊C411yescharacteristic
158C2×C4×C127C2×C22⋊C411yescharacteristic
159C2×C4×C127#12014yes—
160C8×C127—84no—
161C22×C127—22no—
162C22×C127—22no—
163C22×C127—42no—
164C22×C127—42no—
165C22×C127—42no—
166C22×C127—44no—
167C22×C127—42no—
168C22×C127—82no—
169C22×C127—82no—
170C22×C127#37911yescharacteristic
171C22×C127C2×#10011yescharacteristic
172C22×C127C2×#10011yescharacteristic
173C4×C127—24no—
174C4×C127—22no—
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—42no—
178C4×C127—42no—
179C4×C127—42no—
180C4×C127—42no—
181#8524C12711yesmaximal, characteristic, direct product factor, Sylow
182C2×C127—21no—
183C2×C127—21no—
184C2×C127—42no—
185C2×C127—82no—
186C2×C127#178311yescharacteristic
187C2×C127#179612yes—
188C2×#536C2×C12711yescharacteristic
189#1558C2×C12712yescomplemented
190#1533C2×C12712yescomplemented
191#1464C2×C12712yescomplemented
192C127#852411yescharacteristic, direct product factor, Sylow
193C22×#41C4×C12711yescharacteristic
194C23×Q8C4×C12711yescharacteristic
195#536—24no—
196#536—24no—
197C2×C4⋊Q8C4×C12711yescharacteristic
198C2×C4⋊Q8C22×C12711yescharacteristic
199C2×#124C4×C12711yescharacteristic
200C2×#124C22×C12711yescharacteristic
201C2×#123—22no—
202C2×C22⋊Q8—22no—
203C2×#117C22×C12711yescharacteristic
204#432C22×C12714yes—
205#333—24no—
206#329—24no—
207#324—22no—
208#323—22no—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—42no—
212C2×#41—42no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—21no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8C2×C4×C12711yescharacteristic
219C22×Q8C2×C4×C12711yescharacteristic
220C22×Q8C2×C4×C12711yescharacteristic
221C22×D4—21no—
222C22×D4C2×C4×C12711yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C23×C4C2×C4×C12711yescharacteristic
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8D4×C12714yes—
230#124—22no—
231#124—22no—
232#124—22no—
233#124D4×C12714yes—
234#123—216no—
235C22⋊Q8—216no—
236#117—22no—
237#117—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4D4×C12712yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4D4×C12712yes—
244C2×C42C2×C4×C12711yescharacteristic
245C2×C42C23×C12711yescharacteristic
246C4⋊C8—24no—
247C22⋊C8—24no—
248#97—24no—
249C24—21no—
250C24C22⋊C4×C12711yescharacteristic
251#41—48no—
252#41—48no—
253C2×Q8—24no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—48no—
258C2×Q8—48no—
259C2×Q8—42no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8C22⋊C4×C12712yes—
268C2×Q8C22⋊C4×C12712yes—
269C2×Q8C22⋊C4×C12712yes—
270C2×D4—24no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4C22⋊C4×C12712yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C22⋊C4×C12711yescharacteristic
291C22×C4C22⋊C4×C12712yes—
292C22×C4C22×C4×C12711yescharacteristic
293C22×C4C2×D4×C12712yes—
294C2×C8—44no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C22⋊C4—28no—
299C22⋊C4—28no—
300C22⋊C4—24no—
301C42—21no—
302C42—21no—
303C42C2×D4×C12712yes—
304C42C2×D4×C12712yes—
305C23—21no—
306C23—21no—
307C23—22no—
308C23—42no—
309C23—44no—
310C23—41no—
311C23—41no—
312C23#100×C12712yes—
313C23C2×C22⋊C4×C12711yescharacteristic
314Q8—42no—
315Q8—42no—
316Q8—42no—
317Q8—48no—
318Q8—42no—
319Q8—42no—
320Q8—42no—
321D4—42no—
322D4—42no—
323D4—48no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4#100×C12712yes—
345C2×C4C2×C22⋊C4×C12711yescharacteristic
346C2×C4C2×C22⋊C4×C12711yescharacteristic
347C2×C4#120×C12714yes—
348C8—84no—
349C22—22no—
350C22—22no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—44no—
356C22—82no—
357C22—82no—
358C22#379×C12711yescharacteristic
359C22C2×#100×C12711yescharacteristic
360C22C2×#100×C12711yescharacteristic
361C4—22no—
362C4—21no—
363C4—21no—
364C4—24no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4—42no—
369C2—21no—
370C2—21no—
371C2—42no—
372C2—82no—
373C2C127×#178311yescharacteristic
374C2C127×#179612yes—
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC22×#41×C127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1558C2semidirect
3. C127×#1533C2semidirect
4. C127×#1464C2semidirect
181. #8524C127direct (N × H)
189. #1558C2×C127semidirect
190. #1533C2×C127semidirect
191. #1464C2×C127semidirect
192. C127#8524direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#536×C1272
C127×#15582
C127×#15332
C127×#14642
#8524127