Finite Group Databasebeta

← C127×#12172 #44575845

Subgroups — C127×#12172 #44575845

Order 32512.

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

normal subgroups only

This group has 344 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×#3263C211yesmaximal, characteristic, complemented
2C127×#2191C211yesmaximal, characteristic, complemented
3C127×#2190C211yesmaximal, characteristic
4C127×#2053C211yesmaximal, characteristic, complemented
5C127×#1340C211yesmaximal, characteristic, complemented
6C127×#1339C211yesmaximal, characteristic, complemented
7C127×#1328C211yesmaximal, characteristic, complemented
8#575×C127C2211yescharacteristic
9#542×C127—21no—
10#510×C127—21no—
11#509×C127—21no—
12#476×C127—21no—
13#475×C127—21no—
14#470×C127C2211yescharacteristic
15#469×C127C2211yescharacteristic
16#465×C127—21no—
17#464×C127—21no—
18#463×C127—21no—
19#424×C127C2211yescharacteristic
20#416×C127—21no—
21#410×C127—21no—
22#360×C127—21no—
23#360×C127—21no—
24#349×C127C2211yescharacteristic
25#349×C127C2211yescharacteristic
26H(3,ℤ/4)×C127C2211yescharacteristic
27C2×#41×C127—21no—
28#137×C127—41no—
29AGL(1,ℤ/8)×C127—41no—
30#135×C127—21no—
31#135×C127—21no—
32C2×Q16×C127D411yescharacteristic
33C2×SD16×C127—21no—
34C2×D8×C127D411yescharacteristic
35C2×M16×C127C2311yescharacteristic
36C22×C8×C127D411yescharacteristic
37C4⋊Q8×C127—21no—
38#126×C127—41no—
39#124×C127—21no—
40#124×C127—41no—
41#123×C127—41no—
42C22⋊Q8×C127—21no—
43C22⋊Q8×C127—41no—
44#121×C127—21no—
45C4×Q8×C127—21no—
46C4×D4×C127—21no—
47#117×C127—21no—
48#117×C127D411yescharacteristic
49Q32×C127—41no—
50SD32×C127—41no—
51SD32×C127—41no—
52D16×C127—41no—
53M32×C127—21no—
54M32×C127—21no—
55#106×C127—21no—
56#106×C127D411yescharacteristic
57#106×C127D411yescharacteristic
58C4⋊C8×C127—21no—
59C4⋊C8×C127—21no—
60#103×C127—21no—
61#103×C127—21no—
62#103×C127—41no—
63#102×C127—21no—
64#102×C127—21no—
65#102×C127—41no—
66#99×C127—41no—
67#99×C127—41no—
68#41×C127—41no—
69#41×C127—41no—
70#41×C127—41no—
71#41×C127—41no—
72C2×Q8×C127—21no—
73C2×Q8×C127—21no—
74C2×Q8×C127—41no—
75C2×D4×C127—21no—
76C2×D4×C127—21no—
77C2×D4×C127—41no—
78C2×D4×C127—41no—
79C22×C4×C127—41no—
80C22×C4×C127—41no—
81C22×C4×C127C2×D411yescharacteristic
82Q16×C127—21no—
83Q16×C127—21no—
84Q16×C127—41no—
85SD16×C127—41no—
86SD16×C127—41no—
87SD16×C127—81no—
88D8×C127—21no—
89D8×C127—21no—
90D8×C127—41no—
91M16×C127—41no—
92C2×C8×C127—21no—
93C2×C8×C127—41no—
94C2×C8×C127C2×D411yescharacteristic
95C2×C8×C127C2×D411yescharacteristic
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C22⋊C4×C127—41no—
104C22⋊C4×C127—41no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127D811yescharacteristic
110C22⋊C4×C127D811yescharacteristic
111C42×C127—21no—
112C42×C127—41no—
113C42×C127—41no—
114C16×C127—41no—
115C16×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127C2×D811yescharacteristic
119Q8×C127—41no—
120Q8×C127—41no—
121Q8×C127—41no—
122Q8×C127—41no—
123Q8×C127—161no—
124D4×C127—41no—
125D4×C127—41no—
126D4×C127—41no—
127D4×C127—41no—
128D4×C127—81no—
129D4×C127—81no—
130D4×C127—161no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—81no—
140C2×C4×C127—81no—
141C2×C4×C127—81no—
142C2×C4×C127—161no—
143C2×C4×C127#12011yescharacteristic
144C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
145C8×C127—21no—
146C8×C127—21no—
147C8×C127—41no—
148C22×C127—21no—
149C22×C127—41no—
150C22×C127—81no—
151C22×C127—81no—
152C22×C127—81no—
153C22×C127—81no—
154C22×C127—161no—
155C22×C127#44711yescharacteristic
156C4×C127—21no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127—81no—
160C4×C127—81no—
161C4×C127—81no—
162C4×C127—81no—
163C4×C127—161no—
164C4×C127#45411yescharacteristic
165C4×C127B(4,𝔽2)11yescharacteristic
166#12172C12711yesmaximal, characteristic, direct product factor, Sylow
167C2×C127—21no—
168C2×C127—41no—
169C2×C127—161no—
170C2×C127—161no—
171C2×C127#159811yescharacteristic
172#3263C2×C12711yescharacteristic, complemented
173#2191C2×C12711yescharacteristic, complemented
174#2190C2×C12711yescharacteristic
175#2053C2×C12711yescharacteristic, complemented
176#1340C2×C12711yescharacteristic, complemented
177#1339C2×C12711yescharacteristic, complemented
178#1328C2×C12711yescharacteristic, complemented
179C127#1217211yescharacteristic, direct product factor, Sylow
180#575C22×C12711yescharacteristic
181#542—21no—
182#510—21no—
183#509—21no—
184#476—21no—
185#475—21no—
186#470C22×C12711yescharacteristic
187#469C22×C12711yescharacteristic
188#465—21no—
189#464—21no—
190#463—21no—
191#424C22×C12711yescharacteristic
192#416—21no—
193#410—21no—
194#360—21no—
195#360—21no—
196#349C22×C12711yescharacteristic
197#349C22×C12711yescharacteristic
198H(3,ℤ/4)C22×C12711yescharacteristic
199C2×#41—21no—
200#137—41no—
201AGL(1,ℤ/8)—41no—
202#135—21no—
203#135—21no—
204C2×Q16D4×C12711yescharacteristic
205C2×SD16—21no—
206C2×D8D4×C12711yescharacteristic
207C2×M16C23×C12711yescharacteristic
208C22×C8D4×C12711yescharacteristic
209C4⋊Q8—21no—
210#126—41no—
211#124—21no—
212#124—41no—
213#123—41no—
214C22⋊Q8—21no—
215C22⋊Q8—41no—
216#121—21no—
217C4×Q8—21no—
218C4×D4—21no—
219#117—21no—
220#117D4×C12711yescharacteristic
221Q32—41no—
222SD32—41no—
223SD32—41no—
224D16—41no—
225M32—21no—
226M32—21no—
227#106—21no—
228#106D4×C12711yescharacteristic
229#106D4×C12711yescharacteristic
230C4⋊C8—21no—
231C4⋊C8—21no—
232#103—21no—
233#103—21no—
234#103—41no—
235#102—21no—
236#102—21no—
237#102—41no—
238#99—41no—
239#99—41no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C2×D4×C12711yescharacteristic
254Q16—21no—
255Q16—21no—
256Q16—41no—
257SD16—41no—
258SD16—41no—
259SD16—81no—
260D8—21no—
261D8—21no—
262D8—41no—
263M16—41no—
264C2×C8—21no—
265C2×C8—41no—
266C2×C8C2×D4×C12711yescharacteristic
267C2×C8C2×D4×C12711yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4D8×C12711yescharacteristic
282C22⋊C4D8×C12711yescharacteristic
283C42—21no—
284C42—41no—
285C42—41no—
286C16—41no—
287C16—41no—
288C23—41no—
289C23—41no—
290C23C2×D8×C12711yescharacteristic
291Q8—41no—
292Q8—41no—
293Q8—41no—
294Q8—41no—
295Q8—161no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—81no—
301D4—81no—
302D4—161no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—161no—
315C2×C4#120×C12711yescharacteristic
316C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
317C8—21no—
318C8—21no—
319C8—41no—
320C22—21no—
321C22—41no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—161no—
327C22#447×C12711yescharacteristic
328C4—21no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—161no—
336C4#454×C12711yescharacteristic
337C4B(4,𝔽2)×C12711yescharacteristic
338C2—21no—
339C2—41no—
340C2—161no—
341C2—161no—
342C2C127×#159811yescharacteristic
subgroupG/H
center, socleC2×C127#1598
commutator, Frattini, cosocleC2×M16C23×C127
prime radicalC127×#2190C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3263C2semidirect
2. C127×#2191C2semidirect
4. C127×#2053C2semidirect
5. C127×#1340C2semidirect
6. C127×#1339C2semidirect
7. C127×#1328C2semidirect
166. #12172C127direct (N × H)
172. #3263C2×C127semidirect
173. #2191C2×C127semidirect
175. #2053C2×C127semidirect
176. #1340C2×C127semidirect
177. #1339C2×C127semidirect
178. #1328C2×C127semidirect
179. C127#12172direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32632
C127×#21912
C127×#21902
C127×#20532
C127×#13402
C127×#13392
C127×#13282
#12172127