Finite Group Databasebeta

← C127×#8522 #44572195

Subgroups — C127×#8522 #44572195

Order 32512.

1102 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1102 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×#535×C127C211yesmaximal, characteristic
2C127×#1555C212yesmaximal, complemented
3C127×#1533C212yesmaximal, complemented
4C127×#1464C212yesmaximal, complemented
5C22×#41×C127C411yescharacteristic
6C23×D4×C127C411yescharacteristic
7#535×C127—24no—
8#535×C127—24no—
9C2×#127×C127C411yescharacteristic
10C2×#127×C127C2211yescharacteristic
11C2×#124×C127C411yescharacteristic
12C2×#124×C127C2211yescharacteristic
13C2×#121×C127—22no—
14C2×#120×C127—22no—
15C2×#117×C127C2211yescharacteristic
16#432×C127C2214yescomplemented
17#331×C127—24no—
18#329×C127—24no—
19#324×C127—22no—
20#323×C127—22no—
21C25×C127—21no—
22C2×#41×C127—22no—
23C2×#41×C127—22no—
24C2×#41×C127—42no—
25C2×#41×C127—42no—
26C22×Q8×C127C2×C411yescharacteristic
27C22×D4×C127—21no—
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127—21no—
31C22×D4×C127—22no—
32C22×D4×C127—22no—
33C22×D4×C127—42no—
34C22×D4×C127—42no—
35C22×D4×C127C2×C411yescharacteristic
36C22×D4×C127C2×C411yescharacteristic
37C22×D4×C127C2×C411yescharacteristic
38C23×C4×C127—21no—
39C23×C4×C127C2×C411yescharacteristic
40#127×C127—22no—
41#127×C127—22no—
42#127×C127—22no—
43#127×C127D414yes—
44#124×C127—22no—
45#124×C127—22no—
46#124×C127—22no—
47#124×C127D414yes—
48#121×C127—216no—
49#120×C127—216no—
50#117×C127—22no—
51#117×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—24no—
62C24×C127—21no—
63C24×C127—21no—
64C24×C127—21no—
65C24×C127—42no—
66C24×C127—42no—
67C24×C127—84no—
68C24×C127C22⋊C411yescharacteristic
69#41×C127—48no—
70#41×C127—48no—
71C2×Q8×C127—21no—
72C2×Q8×C127—41no—
73C2×Q8×C127—41no—
74C2×Q8×C127C22⋊C412yes—
75C2×D4×C127—24no—
76C2×D4×C127—24no—
77C2×D4×C127—28no—
78C2×D4×C127—21no—
79C2×D4×C127—21no—
80C2×D4×C127—21no—
81C2×D4×C127—28no—
82C2×D4×C127—41no—
83C2×D4×C127—41no—
84C2×D4×C127—41no—
85C2×D4×C127—41no—
86C2×D4×C127—41no—
87C2×D4×C127—41no—
88C2×D4×C127—48no—
89C2×D4×C127—48no—
90C2×D4×C127—42no—
91C2×D4×C127—42no—
92C2×D4×C127—42no—
93C2×D4×C127—42no—
94C2×D4×C127—84no—
95C2×D4×C127—84no—
96C2×D4×C127C22⋊C412yes—
97C2×D4×C127C22⋊C412yes—
98C2×D4×C127C22⋊C412yes—
99C22×C4×C127—22no—
100C22×C4×C127—22no—
101C22×C4×C127—21no—
102C22×C4×C127—21no—
103C22×C4×C127—24no—
104C22×C4×C127—21no—
105C22×C4×C127—42no—
106C22×C4×C127—42no—
107C22×C4×C127C22⋊C411yescharacteristic
108C22×C4×C127C22⋊C412yes—
109C22×C4×C127C22×C411yescharacteristic
110C22×C4×C127C2×D412yes—
111C2×C8×C127—44no—
112C4⋊C4×C127—24no—
113C22⋊C4×C127—28no—
114C22⋊C4×C127—28no—
115C22⋊C4×C127—24no—
116C42×C127—21no—
117C42×C127—21no—
118C42×C127C2×D412yes—
119C42×C127C2×D412yes—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—22no—
123C23×C127—22no—
124C23×C127—22no—
125C23×C127—42no—
126C23×C127—48no—
127C23×C127—44no—
128C23×C127—44no—
129C23×C127—44no—
130C23×C127—41no—
131C23×C127—41no—
132C23×C127—41no—
133C23×C127—41no—
134C23×C127—41no—
135C23×C127—41no—
136C23×C127—88no—
137C23×C127—88no—
138C23×C127—88no—
139C23×C127#10012yes—
140C23×C127C2×C22⋊C411yescharacteristic
141Q8×C127—42no—
142Q8×C127—42no—
143D4×C127—42no—
144D4×C127—42no—
145D4×C127—42no—
146D4×C127—48no—
147D4×C127—42no—
148D4×C127—42no—
149D4×C127—42no—
150D4×C127—48no—
151D4×C127—88no—
152D4×C127—88no—
153C2×C4×C127—24no—
154C2×C4×C127—24no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—22no—
160C2×C4×C127—22no—
161C2×C4×C127—42no—
162C2×C4×C127—48no—
163C2×C4×C127—41no—
164C2×C4×C127—41no—
165C2×C4×C127—42no—
166C2×C4×C127#10012yes—
167C2×C4×C127C2×C22⋊C411yescharacteristic
168C2×C4×C127C2×C22⋊C411yescharacteristic
169C2×C4×C127#12014yes—
170C8×C127—84no—
171C22×C127—22no—
172C22×C127—22no—
173C22×C127—42no—
174C22×C127—42no—
175C22×C127—42no—
176C22×C127—44no—
177C22×C127—44no—
178C22×C127—44no—
179C22×C127—42no—
180C22×C127—42no—
181C22×C127—42no—
182C22×C127—82no—
183C22×C127—82no—
184C22×C127—82no—
185C22×C127—82no—
186C22×C127—82no—
187C22×C127—82no—
188C22×C127—816no—
189C22×C127#37911yescharacteristic
190C22×C127C2×#10011yescharacteristic
191C22×C127C2×#10011yescharacteristic
192C4×C127—24no—
193C4×C127—22no—
194C4×C127—21no—
195C4×C127—21no—
196C4×C127—42no—
197C4×C127—42no—
198#8522C12711yesmaximal, characteristic, direct product factor, Sylow
199C2×C127—21no—
200C2×C127—21no—
201C2×C127—42no—
202C2×C127—82no—
203C2×C127—82no—
204C2×C127—82no—
205C2×C127#178311yescharacteristic
206C2×C127#179512yes—
207C2×#535C2×C12711yescharacteristic
208#1555C2×C12712yescomplemented
209#1533C2×C12712yescomplemented
210#1464C2×C12712yescomplemented
211C127#852211yescharacteristic, direct product factor, Sylow
212C22×#41C4×C12711yescharacteristic
213C23×D4C4×C12711yescharacteristic
214#535—24no—
215#535—24no—
216C2×#127C4×C12711yescharacteristic
217C2×#127C22×C12711yescharacteristic
218C2×#124C4×C12711yescharacteristic
219C2×#124C22×C12711yescharacteristic
220C2×#121—22no—
221C2×#120—22no—
222C2×#117C22×C12711yescharacteristic
223#432C22×C12714yescomplemented
224#331—24no—
225#329—24no—
226#324—22no—
227#323—22no—
228C25—21no—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—42no—
232C2×#41—42no—
233C22×Q8C2×C4×C12711yescharacteristic
234C22×D4—21no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—42no—
241C22×D4—42no—
242C22×D4C2×C4×C12711yescharacteristic
243C22×D4C2×C4×C12711yescharacteristic
244C22×D4C2×C4×C12711yescharacteristic
245C23×C4—21no—
246C23×C4C2×C4×C12711yescharacteristic
247#127—22no—
248#127—22no—
249#127—22no—
250#127D4×C12714yes—
251#124—22no—
252#124—22no—
253#124—22no—
254#124D4×C12714yes—
255#121—216no—
256#120—216no—
257#117—22no—
258#117—22no—
259C2×C4⋊C4D4×C12712yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4D4×C12712yes—
263C2×C42C2×C4×C12711yescharacteristic
264C2×C42C23×C12711yescharacteristic
265C4⋊C8—24no—
266C22⋊C8—24no—
267#97—24no—
268C24—24no—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—42no—
273C24—42no—
274C24—84no—
275C24C22⋊C4×C12711yescharacteristic
276#41—48no—
277#41—48no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8C22⋊C4×C12712yes—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—28no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—28no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4C22⋊C4×C12712yes—
304C2×D4C22⋊C4×C12712yes—
305C2×D4C22⋊C4×C12712yes—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4C22⋊C4×C12711yescharacteristic
315C22×C4C22⋊C4×C12712yes—
316C22×C4C22×C4×C12711yescharacteristic
317C22×C4C2×D4×C12712yes—
318C2×C8—44no—
319C4⋊C4—24no—
320C22⋊C4—28no—
321C22⋊C4—28no—
322C22⋊C4—24no—
323C42—21no—
324C42—21no—
325C42C2×D4×C12712yes—
326C42C2×D4×C12712yes—
327C23—21no—
328C23—21no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—42no—
333C23—48no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—88no—
344C23—88no—
345C23—88no—
346C23#100×C12712yes—
347C23C2×C22⋊C4×C12711yescharacteristic
348Q8—42no—
349Q8—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—48no—
354D4—42no—
355D4—42no—
356D4—42no—
357D4—48no—
358D4—88no—
359D4—88no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—48no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4#100×C12712yes—
374C2×C4C2×C22⋊C4×C12711yescharacteristic
375C2×C4C2×C22⋊C4×C12711yescharacteristic
376C2×C4#120×C12714yes—
377C8—84no—
378C22—22no—
379C22—22no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—82no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—816no—
396C22#379×C12711yescharacteristic
397C22C2×#100×C12711yescharacteristic
398C22C2×#100×C12711yescharacteristic
399C4—22no—
400C4—21no—
401C4—21no—
402C4—24no—
403C4—42no—
404C4—42no—
405C2—21no—
406C2—21no—
407C2—42no—
408C2—82no—
409C2—82no—
410C2—82no—
411C2C127×#178311yescharacteristic
412C2C127×#179512yes—
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#535×C127C2
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×#1555C2semidirect
3. C127×#1533C2semidirect
4. C127×#1464C2semidirect
16. #432×C127C22semidirect
198. #8522C127direct (N × H)
208. #1555C2×C127semidirect
209. #1533C2×C127semidirect
210. #1464C2×C127semidirect
211. C127#8522direct (N × H)
223. #432C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#535×C1272
C127×#15552
C127×#15332
C127×#14642
#8522127