Finite Group Databasebeta

← C127×#10344 #44574017

Subgroups — C127×#10344 #44574017

Order 32512.

856 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2646C211yesmaximal, characteristic, complemented
2C127×#1786C212yesmaximal
3C127×#1438C211yesmaximal, characteristic
4C127×#1294C212yesmaximal, complemented
5C127×#1280C211yesmaximal, characteristic, complemented
6C23×D4×C127C2211yescharacteristic
7C22×C22⋊C4×C127—22no—
8C2×#102×C127—22no—
9C2×#102×C127—22no—
10C2×C22⋊C8×C127C2212yes—
11#400×C127—22no—
12#398×C127—22no—
13#385×C127—22no—
14#380×C127C2212yes—
15C2×#95×C127C2211yescharacteristic
16C2×#95×C127C2211yescharacteristic
17C25×C127—21no—
18C22×D4×C127—22no—
19C22×D4×C127—22no—
20C22×D4×C127—21no—
21C22×D4×C127—44no—
22C22×D4×C127Q812yes—
23C23×C4×C127—21no—
24C23×C4×C127D412yes—
25C23×C4×C127C2311yescharacteristic
26C22×C8×C127—22no—
27C2×C4⋊C4×C127—22no—
28C2×C4⋊C4×C127—22no—
29C2×C4⋊C4×C127D412yes—
30C2×C4⋊C4×C127D412yes—
31C2×C22⋊C4×C127—22no—
32C2×C22⋊C4×C127—44no—
33C2×C22⋊C4×C127—44no—
34C2×C22⋊C4×C127—44no—
35#102×C127—44no—
36#102×C127—44no—
37C22⋊C8×C127—22no—
38C22⋊C8×C127—22no—
39#95×C127—22no—
40#95×C127—22no—
41#95×C127—22no—
42#95×C127—22no—
43#95×C127—22no—
44C24×C127—24no—
45C24×C127—22no—
46C24×C127—42no—
47C24×C127—42no—
48C24×C127—42no—
49C24×C127—41no—
50C24×C127—41no—
51C24×C127—42no—
52C24×C127—42no—
53C24×C127C2×D411yescharacteristic
54C2×D4×C127—24no—
55C2×D4×C127—24no—
56C2×D4×C127—24no—
57C2×D4×C127—42no—
58C2×D4×C127—42no—
59C2×D4×C127—44no—
60C2×D4×C127—48no—
61C2×D4×C127—44no—
62C2×D4×C127—42no—
63C22×C4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—22no—
66C22×C4×C127—22no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127—42no—
70C22×C4×C127—42no—
71C22×C4×C127—42no—
72C22×C4×C127—42no—
73C22×C4×C127—41no—
74C22×C4×C127—41no—
75C22×C4×C127—42no—
76C22×C4×C127C2×D412yes—
77C22×C4×C127C2×Q811yescharacteristic
78C22×C4×C127#4112yes—
79C22×C4×C127#4111yescharacteristic
80C2×C8×C127—42no—
81C2×C8×C127—42no—
82C2×C8×C127—42no—
83C4⋊C4×C127—42no—
84C4⋊C4×C127—42no—
85C4⋊C4×C127—82no—
86C4⋊C4×C127—82no—
87C22⋊C4×C127—416no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—42no—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—22no—
93C23×C127—24no—
94C23×C127—44no—
95C23×C127—44no—
96C23×C127—42no—
97C23×C127—44no—
98C23×C127—44no—
99C23×C127—44no—
100C23×C127—44no—
101C23×C127—42no—
102C23×C127—44no—
103C23×C127—44no—
104C23×C127—82no—
105C23×C127—82no—
106C23×C127—82no—
107C23×C127—82no—
108C23×C127—82no—
109C23×C127—82no—
110C23×C127—81no—
111C23×C127—81no—
112C23×C127—81no—
113C23×C127—81no—
114C23×C127#12011yescharacteristic
115C23×C127C22⋊Q811yescharacteristic
116C23×C127#12411yescharacteristic
117C23×C127#13512yes—
118C23×C127AGL(1,ℤ/8)12yes—
119D4×C127—28no—
120D4×C127—44no—
121D4×C127—44no—
122D4×C127—44no—
123C2×C4×C127—22no—
124C2×C4×C127—22no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—42no—
128C2×C4×C127—44no—
129C2×C4×C127—44no—
130C2×C4×C127—42no—
131C2×C4×C127—42no—
132C2×C4×C127—42no—
133C2×C4×C127—42no—
134C2×C4×C127—42no—
135C2×C4×C127—42no—
136C2×C4×C127—42no—
137C2×C4×C127—42no—
138C2×C4×C127—42no—
139C2×C4×C127—42no—
140C2×C4×C127—82no—
141C2×C4×C127—81no—
142C2×C4×C127—81no—
143C2×C4×C127—82no—
144C2×C4×C127C22⋊Q812yes—
145C2×C4×C127#12412yes—
146C8×C127—82no—
147C22×C127—22no—
148C22×C127—22no—
149C22×C127—22no—
150C22×C127—22no—
151C22×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—48no—
156C22×C127—44no—
157C22×C127—84no—
158C22×C127—84no—
159C22×C127—84no—
160C22×C127—84no—
161C22×C127—84no—
162C22×C127—82no—
163C22×C127—82no—
164C22×C127#39311yescharacteristic
165C22×C127#44912yes—
166C22×C127#45311yescharacteristic
167C22×C127#45411yescharacteristic
168C22×C127B(4,𝔽2)12yes—
169C22×C127#47812yes—
170C22×C127#48811yescharacteristic
171C22×C127#49011yescharacteristic
172C4×C127—22no—
173C4×C127—22no—
174C4×C127—82no—
175C4×C127—82no—
176C4×C127—82no—
177C4×C127—82no—
178#10344C12711yesmaximal, characteristic, direct product factor, Sylow
179C2×C127—21no—
180C2×C127—22no—
181C2×C127—21no—
182C2×C127—84no—
183C2×C127#159712yes—
184C2×C127#160012yes—
185C2×C127#202511yescharacteristic
186C2×C127#202711yescharacteristic
187C2×C127#202911yescharacteristic
188#2646C2×C12711yescharacteristic, complemented
189#1786C2×C12712yes—
190#1438C2×C12711yescharacteristic
191#1294C2×C12712yescomplemented
192#1280C2×C12711yescharacteristic, complemented
193C127#1034411yescharacteristic, direct product factor, Sylow
194C23×D4C22×C12711yescharacteristic
195C22×C22⋊C4—22no—
196C2×#102—22no—
197C2×#102—22no—
198C2×C22⋊C8C22×C12712yes—
199#400—22no—
200#398—22no—
201#385—22no—
202#380C22×C12712yes—
203C2×#95C22×C12711yescharacteristic
204C2×#95C22×C12711yescharacteristic
205C25—21no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—44no—
210C22×D4Q8×C12712yes—
211C23×C4—21no—
212C23×C4D4×C12712yes—
213C23×C4C23×C12711yescharacteristic
214C22×C8—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4D4×C12712yes—
218C2×C4⋊C4D4×C12712yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223#102—44no—
224#102—44no—
225C22⋊C8—22no—
226C22⋊C8—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24—24no—
233C24—22no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—41no—
238C24—41no—
239C24—42no—
240C24—42no—
241C24C2×D4×C12711yescharacteristic
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—48no—
249C2×D4—44no—
250C2×D4—42no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4C2×D4×C12712yes—
265C22×C4C2×Q8×C12711yescharacteristic
266C22×C4#41×C12712yes—
267C22×C4#41×C12711yescharacteristic
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—82no—
274C4⋊C4—82no—
275C22⋊C4—416no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—24no—
282C23—44no—
283C23—44no—
284C23—42no—
285C23—44no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—42no—
290C23—44no—
291C23—44no—
292C23—82no—
293C23—82no—
294C23—82no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23—81no—
299C23—81no—
300C23—81no—
301C23—81no—
302C23#120×C12711yescharacteristic
303C23C22⋊Q8×C12711yescharacteristic
304C23#124×C12711yescharacteristic
305C23#135×C12712yes—
306C23AGL(1,ℤ/8)×C12712yes—
307D4—28no—
308D4—44no—
309D4—44no—
310D4—44no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4C22⋊Q8×C12712yes—
333C2×C4#124×C12712yes—
334C8—82no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—48no—
344C22—44no—
345C22—84no—
346C22—84no—
347C22—84no—
348C22—84no—
349C22—84no—
350C22—82no—
351C22—82no—
352C22#393×C12711yescharacteristic
353C22#449×C12712yes—
354C22#453×C12711yescharacteristic
355C22#454×C12711yescharacteristic
356C22B(4,𝔽2)×C12712yes—
357C22#478×C12712yes—
358C22#488×C12711yescharacteristic
359C22#490×C12711yescharacteristic
360C4—22no—
361C4—22no—
362C4—82no—
363C4—82no—
364C4—82no—
365C4—82no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2—84no—
370C2C127×#159712yes—
371C2C127×#160012yes—
372C2C127×#202511yescharacteristic
373C2C127×#202711yescharacteristic
374C2C127×#202911yescharacteristic
subgroupG/H
center, socleC23×C127#120
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC23×D4×C127C22
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
1. C127×#2646C2semidirect
4. C127×#1294C2semidirect
5. C127×#1280C2semidirect
178. #10344C127direct (N × H)
188. #2646C2×C127semidirect
191. #1294C2×C127semidirect
192. #1280C2×C127semidirect
193. C127#10344direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#26462
C127×#17862
C127×#14382
C127×#12942
C127×#12802
#10344127