Finite Group Databasebeta

← C127×#12229 #44575902

Subgroups — C127×#12229 #44575902

Order 32512.

380 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2517C211yesmaximal, characteristic
2C127×#2062C211yesmaximal, characteristic
3C127×#1614C211yesmaximal, characteristic
4C127×#1608C211yesmaximal, characteristic
5C127×#1352C211yesmaximal, characteristic
6C127×#1344C211yesmaximal, characteristic
7C127×#1326C211yesmaximal, characteristic
8#543×C127C2211yescharacteristic
9#484×C127—21no—
10#483×C127—21no—
11#399×C127C2211yescharacteristic
12#397×C127—21no—
13#397×C127C2211yescharacteristic
14#387×C127—21no—
15#387×C127—21no—
16#387×C127—21no—
17#385×C127—21no—
18#385×C127C2211yescharacteristic
19#382×C127—21no—
20C4×C22⋊C4×C127—21no—
21#342×C127—21no—
22#342×C127C2211yescharacteristic
23#328×C127—21no—
24#328×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C23×C4×C127D411yescharacteristic
28C22⋊Q8×C127—21no—
29C22⋊Q8×C127—21no—
30C22⋊Q8×C127—41no—
31C22⋊Q8×C127—41no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127—41no—
36C2×C4⋊C4×C127D411yescharacteristic
37C2×C4⋊C4×C127D411yescharacteristic
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—21no—
41C2×C22⋊C4×C127—21no—
42C2×C22⋊C4×C127D411yescharacteristic
43C2×C22⋊C4×C127C2311yescharacteristic
44C2×C42×C127—21no—
45C2×C42×C127—21no—
46#107×C127—41no—
47#106×C127—41no—
48#103×C127—41no—
49#103×C127—41no—
50C22⋊C8×C127—21no—
51C22⋊C8×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—21no—
59#95×C127—41no—
60#95×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C24×C127C2×D411yescharacteristic
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—42no—
74C22×C4×C127—41no—
75C22×C4×C127—41no—
76C22×C4×C127C2×D411yescharacteristic
77C22×C4×C127C2×D411yescharacteristic
78C2×C8×C127—41no—
79C2×C8×C127—41no—
80C4⋊C4×C127—21no—
81C4⋊C4×C127—21no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—42no—
85C4⋊C4×C127—42no—
86C4⋊C4×C127—42no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—82no—
90C22⋊C4×C127—22no—
91C22⋊C4×C127—21no—
92C22⋊C4×C127—21no—
93C22⋊C4×C127—41no—
94C22⋊C4×C127—42no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C42×C127—42no—
101C42×C127—81no—
102C23×C127—21no—
103C23×C127—21no—
104C23×C127—41no—
105C23×C127—41no—
106C23×C127#12011yescharacteristic
107C23×C127#13511yescharacteristic
108C23×C127#13711yescharacteristic
109Q8×C127—81no—
110Q8×C127—81no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—42no—
122C2×C4×C127—41no—
123C2×C4×C127—42no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—81no—
138C2×C4×C127—81no—
139C2×C4×C127—81no—
140C2×C4×C127—81no—
141C2×C4×C127—81no—
142C8×C127—81no—
143C8×C127—81no—
144C22×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127—41no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—41no—
152C22×C127—81no—
153C22×C127#45211yescharacteristic
154C22×C127#45611yescharacteristic
155C22×C127B(4,𝔽2)11yescharacteristic
156C4×C127—42no—
157C4×C127—41no—
158C4×C127—42no—
159C4×C127—41no—
160C4×C127—81no—
161C4×C127—81no—
162C4×C127—82no—
163C4×C127—81no—
164C4×C127—81no—
165C4×C127—81no—
166C4×C127—81no—
167#12229C12711yesmaximal, characteristic, direct product factor, Sylow
168C2×C127—21no—
169C2×C127—21no—
170C2×C127—41no—
171C2×C127—41no—
172C2×C127#160311yescharacteristic
173C2×C127#219711yescharacteristic
174C2×C127#219811yescharacteristic
175#2517C2×C12711yescharacteristic
176#2062C2×C12711yescharacteristic
177#1614C2×C12711yescharacteristic
178#1608C2×C12711yescharacteristic
179#1352C2×C12711yescharacteristic
180#1344C2×C12711yescharacteristic
181#1326C2×C12711yescharacteristic
182C127#1222911yescharacteristic, direct product factor, Sylow
183#543C22×C12711yescharacteristic
184#484—21no—
185#483—21no—
186#399C22×C12711yescharacteristic
187#397—21no—
188#397C22×C12711yescharacteristic
189#387—21no—
190#387—21no—
191#387—21no—
192#385—21no—
193#385C22×C12711yescharacteristic
194#382—21no—
195C4×C22⋊C4—21no—
196#342—21no—
197#342C22×C12711yescharacteristic
198#328—21no—
199#328—21no—
200#323C22×C12711yescharacteristic
201#323C22×C12711yescharacteristic
202C23×C4D4×C12711yescharacteristic
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4D4×C12711yescharacteristic
212C2×C4⋊C4D4×C12711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4D4×C12711yescharacteristic
218C2×C22⋊C4C23×C12711yescharacteristic
219C2×C42—21no—
220C2×C42—21no—
221#107—41no—
222#106—41no—
223#103—41no—
224#103—41no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—41no—
235#95D4×C12711yescharacteristic
236#95D4×C12711yescharacteristic
237C24C2×D4×C12711yescharacteristic
238C2×Q8—21no—
239C2×Q8—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C2×D4×C12711yescharacteristic
252C22×C4C2×D4×C12711yescharacteristic
253C2×C8—41no—
254C2×C8—41no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—82no—
265C22⋊C4—22no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42—42no—
276C42—81no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23#120×C12711yescharacteristic
282C23#135×C12711yescharacteristic
283C23#137×C12711yescharacteristic
284Q8—81no—
285Q8—81no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C8—81no—
318C8—81no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22#452×C12711yescharacteristic
329C22#456×C12711yescharacteristic
330C22B(4,𝔽2)×C12711yescharacteristic
331C4—42no—
332C4—41no—
333C4—42no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—82no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2C127×#160311yescharacteristic
347C2C127×#219711yescharacteristic
348C2C127×#219811yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC24×C127C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
167. #12229C127direct (N × H)
182. C127#12229direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#25172
C127×#20622
C127×#16142
C127×#16082
C127×#13522
C127×#13442
C127×#13262
#12229127