Finite Group Databasebeta

← C127×#10341 #44574014

Subgroups — C127×#10341 #44574014

Order 32512.

684 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#2659C211yesmaximal, characteristic
2C127×#1787C212yesmaximal
3C127×#1446C211yesmaximal, characteristic
4C127×#1294C212yesmaximal
5C127×#1280C211yesmaximal, characteristic
6C23×Q8×C127C2211yescharacteristic
7C2×C22⋊Q8×C127—21no—
8C2×#103×C127—22no—
9C2×#103×C127—22no—
10C2×C22⋊C8×C127C2212yes—
11#398×C127—22no—
12#395×C127—22no—
13#380×C127—21no—
14#380×C127C2212yes—
15C2×#95×C127—21no—
16C2×#95×C127—21no—
17C2×#95×C127C2211yescharacteristic
18C2×#95×C127C2211yescharacteristic
19C22×Q8×C127—21no—
20C22×Q8×C127—21no—
21C22×Q8×C127—22no—
22C22×Q8×C127—22no—
23C22×Q8×C127—42no—
24C22×Q8×C127—42no—
25C22×Q8×C127Q812yescomplemented
26C23×C4×C127—21no—
27C23×C4×C127—21no—
28C23×C4×C127D411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C23×C4×C127C2311yescharacteristic
31C22×C8×C127—22no—
32C22⋊Q8×C127—42no—
33C22⋊Q8×C127—42no—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127D412yes—
39C2×C4⋊C4×C127D412yes—
40C2×C22⋊C4×C127—22no—
41#103×C127—44no—
42#103×C127—44no—
43C22⋊C8×C127—22no—
44C22⋊C8×C127—22no—
45#95×C127—22no—
46#95×C127—22no—
47#95×C127—22no—
48#95×C127—22no—
49#95×C127—42no—
50#95×C127—42no—
51C24×C127C2×D411yescharacteristic
52C2×Q8×C127—24no—
53C2×Q8×C127—24no—
54C2×Q8×C127—24no—
55C2×Q8×C127—44no—
56C2×Q8×C127—42no—
57C2×Q8×C127—42no—
58C2×Q8×C127—44no—
59C2×Q8×C127—42no—
60C2×Q8×C127—48no—
61C2×Q8×C127—42no—
62C2×Q8×C127—81no—
63C2×Q8×C127—81no—
64C22×C4×C127—22no—
65C22×C4×C127—22no—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—22no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—41no—
75C22×C4×C127—41no—
76C22×C4×C127—41no—
77C22×C4×C127—41no—
78C22×C4×C127—41no—
79C22×C4×C127—41no—
80C22×C4×C127—41no—
81C22×C4×C127—41no—
82C22×C4×C127—41no—
83C22×C4×C127—41no—
84C22×C4×C127—41no—
85C22×C4×C127—41no—
86C22×C4×C127—41no—
87C22×C4×C127—41no—
88C22×C4×C127—41no—
89C22×C4×C127—41no—
90C22×C4×C127SD1612yes—
91C22×C4×C127Q1612yes—
92C22×C4×C127C2×D412yes—
93C22×C4×C127C2×Q811yescharacteristic
94C22×C4×C127#4112yes—
95C22×C4×C127#4111yescharacteristic
96C2×C8×C127—42no—
97C2×C8×C127—42no—
98C2×C8×C127—42no—
99C4⋊C4×C127—42no—
100C4⋊C4×C127—42no—
101C4⋊C4×C127—42no—
102C4⋊C4×C127—44no—
103C4⋊C4×C127—81no—
104C4⋊C4×C127—81no—
105C22⋊C4×C127—42no—
106C22⋊C4×C127—42no—
107C23×C127—21no—
108C23×C127—21no—
109C23×C127—21no—
110C23×C127—21no—
111C23×C127#12011yescharacteristic
112C23×C127C22⋊Q811yescharacteristic
113C23×C127#12411yescharacteristic
114C23×C127C2×SD1611yescharacteristic
115C23×C127C2×Q1611yescharacteristic
116C23×C127AGL(1,ℤ/8)11yescharacteristic
117C23×C127#13711yescharacteristic
118Q8×C127—28no—
119Q8×C127—44no—
120Q8×C127—44no—
121Q8×C127—44no—
122Q8×C127—84no—
123C2×C4×C127—22no—
124C2×C4×C127—22no—
125C2×C4×C127—24no—
126C2×C4×C127—22no—
127C2×C4×C127—22no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—42no—
131C2×C4×C127—42no—
132C2×C4×C127—42no—
133C2×C4×C127—42no—
134C2×C4×C127—42no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—42no—
139C2×C4×C127—42no—
140C2×C4×C127—41no—
141C2×C4×C127—42no—
142C2×C4×C127—42no—
143C2×C4×C127—42no—
144C2×C4×C127—42no—
145C2×C4×C127—42no—
146C2×C4×C127—42no—
147C2×C4×C127—42no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127—42no—
151C2×C4×C127—42no—
152C2×C4×C127—81no—
153C2×C4×C127—81no—
154C2×C4×C127—81no—
155C2×C4×C127—81no—
156C2×C4×C127—81no—
157C2×C4×C127—81no—
158C2×C4×C127C22⋊Q812yes—
159C2×C4×C127#12412yes—
160C8×C127—82no—
161C22×C127—21no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—21no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127#39311yescharacteristic
174C22×C127#45011yescharacteristic
175C22×C127#45111yescharacteristic
176C22×C127#45511yescharacteristic
177C22×C127#45611yescharacteristic
178C22×C127B(4,𝔽2)11yescharacteristic
179C22×C127B(4,𝔽2)11yescharacteristic
180C22×C127#47511yescharacteristic
181C22×C127#47711yescharacteristic
182C22×C127#48711yescharacteristic
183C22×C127#48911yescharacteristic
184C4×C127—22no—
185C4×C127—22no—
186C4×C127—42no—
187C4×C127—42no—
188C4×C127—44no—
189C4×C127—82no—
190C4×C127—82no—
191C4×C127—82no—
192#10341C12711yesmaximal, characteristic, direct product factor, Sylow
193C2×C127—21no—
194C2×C127—21no—
195C2×C127—21no—
196C2×C127—21no—
197C2×C127#160111yescharacteristic
198C2×C127#160211yescharacteristic
199C2×C127#160411yescharacteristic
200C2×C127#160511yescharacteristic
201C2×C127#202411yescharacteristic
202C2×C127#202811yescharacteristic
203C2×C127#202911yescharacteristic
204#2659C2×C12711yescharacteristic
205#1787C2×C12712yes—
206#1446C2×C12711yescharacteristic
207#1294C2×C12712yes—
208#1280C2×C12711yescharacteristic
209C127#1034111yescharacteristic, direct product factor, Sylow
210C23×Q8C22×C12711yescharacteristic
211C2×C22⋊Q8—21no—
212C2×#103—22no—
213C2×#103—22no—
214C2×C22⋊C8C22×C12712yes—
215#398—22no—
216#395—22no—
217#380—21no—
218#380C22×C12712yes—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95C22×C12711yescharacteristic
222C2×#95C22×C12711yescharacteristic
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8Q8×C12712yescomplemented
230C23×C4—21no—
231C23×C4—21no—
232C23×C4D4×C12711yescharacteristic
233C23×C4D4×C12711yescharacteristic
234C23×C4C23×C12711yescharacteristic
235C22×C8—22no—
236C22⋊Q8—42no—
237C22⋊Q8—42no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4D4×C12712yes—
243C2×C4⋊C4D4×C12712yes—
244C2×C22⋊C4—22no—
245#103—44no—
246#103—44no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—42no—
254#95—42no—
255C24C2×D4×C12711yescharacteristic
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—42no—
264C2×Q8—48no—
265C2×Q8—42no—
266C2×Q8—81no—
267C2×Q8—81no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4SD16×C12712yes—
295C22×C4Q16×C12712yes—
296C22×C4C2×D4×C12712yes—
297C22×C4C2×Q8×C12711yescharacteristic
298C22×C4#41×C12712yes—
299C22×C4#41×C12711yescharacteristic
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23#120×C12711yescharacteristic
316C23C22⋊Q8×C12711yescharacteristic
317C23#124×C12711yescharacteristic
318C23C2×SD16×C12711yescharacteristic
319C23C2×Q16×C12711yescharacteristic
320C23AGL(1,ℤ/8)×C12711yescharacteristic
321C23#137×C12711yescharacteristic
322Q8—28no—
323Q8—44no—
324Q8—44no—
325Q8—44no—
326Q8—84no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4C22⋊Q8×C12712yes—
363C2×C4#124×C12712yes—
364C8—82no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22#393×C12711yescharacteristic
378C22#450×C12711yescharacteristic
379C22#451×C12711yescharacteristic
380C22#455×C12711yescharacteristic
381C22#456×C12711yescharacteristic
382C22B(4,𝔽2)×C12711yescharacteristic
383C22B(4,𝔽2)×C12711yescharacteristic
384C22#475×C12711yescharacteristic
385C22#477×C12711yescharacteristic
386C22#487×C12711yescharacteristic
387C22#489×C12711yescharacteristic
388C4—22no—
389C4—22no—
390C4—42no—
391C4—42no—
392C4—44no—
393C4—82no—
394C4—82no—
395C4—82no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2C127×#160111yescharacteristic
401C2C127×#160211yescharacteristic
402C2C127×#160411yescharacteristic
403C2C127×#160511yescharacteristic
404C2C127×#202411yescharacteristic
405C2C127×#202811yescharacteristic
406C2C127×#202911yescharacteristic
subgroupG/H
center, socleC23×C127#120
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC24×C127C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
25. C22×Q8×C127Q8semidirect
192. #10341C127direct (N × H)
209. C127#10341direct (N × H)
229. C22×Q8Q8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#26592
C127×#17872
C127×#14462
C127×#12942
C127×#12802
#10341127