Finite Group Databasebeta

← C127×#12182 #44575855

Subgroups — C127×#12182 #44575855

Order 32512.

382 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2612C211yesmaximal, characteristic, complemented
2C127×#2040C211yesmaximal, characteristic
3C127×#1616C211yesmaximal, characteristic
4C127×#1604C211yesmaximal, characteristic
5C127×#1354C211yesmaximal, characteristic, complemented
6C127×#1353C211yesmaximal, characteristic, complemented
7C127×#1324C211yesmaximal, characteristic, complemented
8#537×C127C2211yescharacteristic
9#481×C127—21no—
10#448×C127—21no—
11C2×#104×C127—21no—
12#417×C127—21no—
13#400×C127C2211yescharacteristic
14#398×C127—21no—
15#395×C127C2211yescharacteristic
16#391×C127—21no—
17#389×C127—21no—
18#387×C127—21no—
19#387×C127—21no—
20#382×C127C2211yescharacteristic
21#376×C127—21no—
22#343×C127C2211yescharacteristic
23#327×C127—21no—
24#327×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C22×Q8×C127D411yescharacteristic
29C2×SD16×C127—41no—
30C2×M16×C127—21no—
31C22⋊Q8×C127—41no—
32#121×C127—21no—
33#121×C127—21no—
34#121×C127—41no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127D411yescharacteristic
40C2×C4⋊C4×C127D411yescharacteristic
41C2×C4⋊C4×C127C2311yescharacteristic
42C2×C42×C127—21no—
43C2×C42×C127—21no—
44C2×C42×C127D411yescharacteristic
45#106×C127—41no—
46#104×C127—42no—
47#103×C127—41no—
48#103×C127—41no—
49#102×C127—41no—
50#102×C127—41no—
51C22⋊C8×C127—21no—
52C22⋊C8×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—21no—
60#95×C127—21no—
61#95×C127D411yescharacteristic
62#95×C127D411yescharacteristic
63#41×C127—41no—
64#41×C127—41no—
65#41×C127—41no—
66#41×C127—41no—
67C2×Q8×C127—21no—
68C2×Q8×C127—42no—
69C2×Q8×C127—41no—
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—41no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127—41no—
81C22×C4×C127C2×D411yescharacteristic
82C22×C4×C127C2×D411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84SD16×C127—82no—
85M16×C127—41no—
86M16×C127—41no—
87C2×C8×C127—41no—
88C2×C8×C127—41no—
89C2×C8×C127—41no—
90C4⋊C4×C127—21no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—42no—
96C4⋊C4×C127—42no—
97C4⋊C4×C127—42no—
98C4⋊C4×C127—81no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C42×C127—21no—
102C42×C127—42no—
103C42×C127—42no—
104C42×C127SD1612yescomplemented
105C23×C127—41no—
106C23×C127#12011yescharacteristic
107Q8×C127—41no—
108Q8×C127—41no—
109Q8×C127—42no—
110Q8×C127—81no—
111D4×C127—41no—
112D4×C127—41no—
113D4×C127—81no—
114D4×C127—81no—
115D4×C127—81no—
116D4×C127—81no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—22no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
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—41no—
141C2×C4×C127—41no—
142C2×C4×C127—81no—
143C2×C4×C127—81no—
144C2×C4×C127C2×SD1611yescharacteristic
145C2×C4×C127#13711yescharacteristic
146C8×C127—81no—
147C8×C127—81no—
148C8×C127—81no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—81no—
152C22×C127—81no—
153C22×C127—81no—
154C22×C127#44811yescharacteristic
155C22×C127#45411yescharacteristic
156C22×C127B(4,𝔽2)11yescharacteristic
157C4×C127—21no—
158C4×C127—21no—
159C4×C127—42no—
160C4×C127—42no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—42no—
164C4×C127—81no—
165C4×C127—81no—
166C4×C127—81no—
167C4×C127—81no—
168#12182C12711yesmaximal, characteristic, direct product factor, Sylow
169C2×C127—21no—
170C2×C127—21no—
171C2×C127—161no—
172C2×C127#159911yescharacteristic
173C2×C127#219711yescharacteristic
174C2×C127#220411yescharacteristic
175#2612C2×C12711yescharacteristic, complemented
176#2040C2×C12711yescharacteristic
177#1616C2×C12711yescharacteristic
178#1604C2×C12711yescharacteristic
179#1354C2×C12711yescharacteristic, complemented
180#1353C2×C12711yescharacteristic, complemented
181#1324C2×C12711yescharacteristic, complemented
182C127#1218211yescharacteristic, direct product factor, Sylow
183#537C22×C12711yescharacteristic
184#481—21no—
185#448—21no—
186C2×#104—21no—
187#417—21no—
188#400C22×C12711yescharacteristic
189#398—21no—
190#395C22×C12711yescharacteristic
191#391—21no—
192#389—21no—
193#387—21no—
194#387—21no—
195#382C22×C12711yescharacteristic
196#376—21no—
197#343C22×C12711yescharacteristic
198#327—21no—
199#327—21no—
200#324C22×C12711yescharacteristic
201#324C22×C12711yescharacteristic
202C2×#41—21no—
203C22×Q8D4×C12711yescharacteristic
204C2×SD16—41no—
205C2×M16—21no—
206C22⋊Q8—41no—
207#121—21no—
208#121—21no—
209#121—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4D4×C12711yescharacteristic
215C2×C4⋊C4D4×C12711yescharacteristic
216C2×C4⋊C4C23×C12711yescharacteristic
217C2×C42—21no—
218C2×C42—21no—
219C2×C42D4×C12711yescharacteristic
220#106—41no—
221#104—42no—
222#103—41no—
223#103—41no—
224#102—41no—
225#102—41no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95D4×C12711yescharacteristic
237#95D4×C12711yescharacteristic
238#41—41no—
239#41—41no—
240#41—41no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4C2×D4×C12711yescharacteristic
257C22×C4C2×D4×C12711yescharacteristic
258C22×C4C2×D4×C12711yescharacteristic
259SD16—82no—
260M16—41no—
261M16—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—81no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—21no—
277C42—42no—
278C42—42no—
279C42SD16×C12712yescomplemented
280C23—41no—
281C23#120×C12711yescharacteristic
282Q8—41no—
283Q8—41no—
284Q8—42no—
285Q8—81no—
286D4—41no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—81no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
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—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4C2×SD16×C12711yescharacteristic
320C2×C4#137×C12711yescharacteristic
321C8—81no—
322C8—81no—
323C8—81no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#448×C12711yescharacteristic
330C22#454×C12711yescharacteristic
331C22B(4,𝔽2)×C12711yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—42no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C2—21no—
344C2—21no—
345C2—161no—
346C2C127×#159911yescharacteristic
347C2C127×#219711yescharacteristic
348C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radical#537×C127C22
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
1. C127×#2612C2semidirect
5. C127×#1354C2semidirect
6. C127×#1353C2semidirect
7. C127×#1324C2semidirect
104. C42×C127SD16semidirect
168. #12182C127direct (N × H)
175. #2612C2×C127semidirect
179. #1354C2×C127semidirect
180. #1353C2×C127semidirect
181. #1324C2×C127semidirect
182. C127#12182direct (N × H)
279. C42SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#26122
C127×#20402
C127×#16162
C127×#16042
C127×#13542
C127×#13532
C127×#13242
#12182127