Finite Group Databasebeta

← C127×#12238 #44575911

Subgroups — C127×#12238 #44575911

Order 32512.

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

normal subgroups only

This group has 368 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×#2795C211yesmaximal, characteristic
2C127×#2060C211yesmaximal, characteristic, complemented
3C127×#1617C211yesmaximal, characteristic, complemented
4C127×#1599C211yesmaximal, characteristic
5C127×#1353C211yesmaximal, characteristic, complemented
6C127×#1345C211yesmaximal, characteristic
7C127×#1325C211yesmaximal, characteristic, complemented
8#544×C127C2211yescharacteristic
9#484×C127—21no—
10#450×C127—21no—
11#428×C127—21no—
12#400×C127C2211yescharacteristic
13#396×C127—21no—
14#394×C127—21no—
15#394×C127C2211yescharacteristic
16#388×C127—21no—
17#388×C127—21no—
18#388×C127—21no—
19#387×C127—21no—
20#384×C127C2211yescharacteristic
21#343×C127C2211yescharacteristic
22#339×C127—21no—
23#328×C127—21no—
24#328×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C22×D4×C127D411yescharacteristic
28C2×SD16×C127—41no—
29C2×M16×C127—21no—
30C4⋊Q8×C127—41no—
31#125×C127—41no—
32C22⋊Q8×C127—21no—
33C22⋊Q8×C127—21no—
34#117×C127—21no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127D411yescharacteristic
39C2×C4⋊C4×C127D411yescharacteristic
40C2×C4⋊C4×C127C2311yescharacteristic
41C2×C22⋊C4×C127—21no—
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—21no—
47C2×C42×C127—21no—
48C2×C42×C127D411yescharacteristic
49#107×C127—41no—
50#107×C127—41no—
51#106×C127—41no—
52#103×C127—41no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C22⋊C8×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—21no—
59#95×C127—21no—
60#95×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C24×C127—21no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×D4×C127—41no—
66C2×D4×C127—81no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127C2×D411yescharacteristic
73C22×C4×C127C2×D411yescharacteristic
74C22×C4×C127C2×D411yescharacteristic
75SD16×C127—82no—
76M16×C127—22no—
77M16×C127—41no—
78C2×C8×C127—41no—
79C2×C8×C127—41no—
80C2×C8×C127—41no—
81C4⋊C4×C127—21no—
82C4⋊C4×C127—21no—
83C4⋊C4×C127—21no—
84C4⋊C4×C127—21no—
85C4⋊C4×C127—41no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127—42no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—81no—
92C4⋊C4×C127—81no—
93C22⋊C4×C127—42no—
94C22⋊C4×C127—42no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—81no—
99C22⋊C4×C127—81no—
100C22⋊C4×C127—81no—
101C42×C127—21no—
102C42×C127—21no—
103C42×C127—41no—
104C42×C127—81no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127—41no—
109C23×C127—41no—
110C23×C127—41no—
111C23×C127—41no—
112C23×C127#12011yescharacteristic
113Q8×C127—81no—
114Q8×C127—81no—
115D4×C127—42no—
116D4×C127—81no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—22no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
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—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127#13511yescharacteristic
141C2×C4×C127#13711yescharacteristic
142C8×C127—42no—
143C8×C127—81no—
144C8×C127—81no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—81no—
148C22×C127—81no—
149C22×C127—81no—
150C22×C127—81no—
151C22×C127—81no—
152C22×C127—81no—
153C22×C127—81no—
154C22×C127#45211yescharacteristic
155C22×C127#45611yescharacteristic
156C22×C127B(4,𝔽2)11yescharacteristic
157C4×C127—21no—
158C4×C127—21no—
159C4×C127—42no—
160C4×C127—41no—
161C4×C127—41no—
162C4×C127—81no—
163C4×C127—81no—
164C4×C127—81no—
165C4×C127—81no—
166C4×C127—81no—
167C4×C127—81no—
168#12238C12711yesmaximal, characteristic, direct product factor, Sylow
169C2×C127—21no—
170C2×C127—21no—
171C2×C127—161no—
172C2×C127#160311yescharacteristic
173C2×C127#220011yescharacteristic
174C2×C127#220411yescharacteristic
175#2795C2×C12711yescharacteristic
176#2060C2×C12711yescharacteristic, complemented
177#1617C2×C12711yescharacteristic, complemented
178#1599C2×C12711yescharacteristic
179#1353C2×C12711yescharacteristic, complemented
180#1345C2×C12711yescharacteristic
181#1325C2×C12711yescharacteristic, complemented
182C127#1223811yescharacteristic, direct product factor, Sylow
183#544C22×C12711yescharacteristic
184#484—21no—
185#450—21no—
186#428—21no—
187#400C22×C12711yescharacteristic
188#396—21no—
189#394—21no—
190#394C22×C12711yescharacteristic
191#388—21no—
192#388—21no—
193#388—21no—
194#387—21no—
195#384C22×C12711yescharacteristic
196#343C22×C12711yescharacteristic
197#339—21no—
198#328—21no—
199#328—21no—
200#324C22×C12711yescharacteristic
201#324C22×C12711yescharacteristic
202C22×D4D4×C12711yescharacteristic
203C2×SD16—41no—
204C2×M16—21no—
205C4⋊Q8—41no—
206#125—41no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209#117—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4D4×C12711yescharacteristic
214C2×C4⋊C4D4×C12711yescharacteristic
215C2×C4⋊C4C23×C12711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42D4×C12711yescharacteristic
224#107—41no—
225#107—41no—
226#106—41no—
227#103—41no—
228#102—41no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95D4×C12711yescharacteristic
236#95D4×C12711yescharacteristic
237C24—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×D4—41no—
241C2×D4—81no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4C2×D4×C12711yescharacteristic
248C22×C4C2×D4×C12711yescharacteristic
249C22×C4C2×D4×C12711yescharacteristic
250SD16—82no—
251M16—22no—
252M16—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—81no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—21no—
277C42—21no—
278C42—41no—
279C42—81no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23#120×C12711yescharacteristic
288Q8—81no—
289Q8—81no—
290D4—42no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
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—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4#135×C12711yescharacteristic
316C2×C4#137×C12711yescharacteristic
317C8—42no—
318C8—81no—
319C8—81no—
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#452×C12711yescharacteristic
330C22#456×C12711yescharacteristic
331C22B(4,𝔽2)×C12711yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C2—21no—
344C2—21no—
345C2—161no—
346C2C127×#160311yescharacteristic
347C2C127×#220011yescharacteristic
348C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC22×D4×C127D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2060C2semidirect
3. C127×#1617C2semidirect
5. C127×#1353C2semidirect
7. C127×#1325C2semidirect
168. #12238C127direct (N × H)
176. #2060C2×C127semidirect
177. #1617C2×C127semidirect
179. #1353C2×C127semidirect
181. #1325C2×C127semidirect
182. C127#12238direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#27952
C127×#20602
C127×#16172
C127×#15992
C127×#13532
C127×#13452
C127×#13252
#12238127