Finite Group Databasebeta

← C127×#12326 #44575999

Subgroups — C127×#12326 #44575999

Order 32512.

390 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3273C211yesmaximal, characteristic
2C127×#2251C211yesmaximal, characteristic, complemented
3C127×#2208C211yesmaximal, characteristic, complemented
4C127×#1689C211yesmaximal, characteristic, complemented
5C127×#1360C211yesmaximal, characteristic, complemented
6C127×#1333C211yesmaximal, characteristic, complemented
7C127×#1330C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127—21no—
9#546×C127—21no—
10C2×Q32×C127—21no—
11C2×SD32×C127—21no—
12C2×SD32×C127—21no—
13#495×C127C2211yescharacteristic
14#494×C127C2211yescharacteristic, complemented
15#474×C127—21no—
16#468×C127—21no—
17#463×C127—21no—
18#452×C127—21no—
19#447×C127—21no—
20C4×D8×C127C2211yescharacteristic
21#436×C127C2211yescharacteristic
22C4⋊C16×C127C2211yescharacteristic, complemented
23#358×C127—21no—
24#346×C127C2211yescharacteristic, complemented
25#334×C127C2211yescharacteristic, complemented
26#332×C127—21no—
27C2×#41×C127—21no—
28C22×D4×C127—21no—
29AGL(1,ℤ/8)×C127—42no—
30AGL(1,ℤ/8)×C127—42no—
31C2×Q16×C127—21no—
32C2×Q16×C127—21no—
33C2×Q16×C127D411yescharacteristic
34C2×SD16×C127—21no—
35C2×SD16×C127—21no—
36C2×D8×C127—21no—
37C2×D8×C127D411yescharacteristic
38C2×D8×C127D411yescharacteristic
39C2×M16×C127—21no—
40C4⋊Q8×C127—21no—
41#124×C127—21no—
42#123×C127—41no—
43C22⋊Q8×C127—21no—
44#121×C127—21no—
45#121×C127—41no—
46#120×C127—41no—
47C4×D4×C127—21no—
48C4×D4×C127D411yescharacteristic
49C2×C22⋊C4×C127—41no—
50Q32×C127—42no—
51SD32×C127—42no—
52SD32×C127—42no—
53C2×C16×C127—21no—
54C2×C16×C127—21no—
55#107×C127—41no—
56#107×C127D411yescharacteristic
57C4⋊C8×C127—21no—
58C4⋊C8×C127D411yescharacteristic
59#103×C127—21no—
60#102×C127—21no—
61#102×C127—21no—
62#102×C127—41no—
63C22⋊C8×C127—21no—
64C4×C8×C127C2311yescharacteristic
65C24×C127—41no—
66#41×C127—42no—
67#41×C127—81no—
68C2×Q8×C127—21no—
69C2×Q8×C127—41no—
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—41no—
73C2×D4×C127—42no—
74C2×D4×C127—41no—
75C2×D4×C127—41no—
76C2×D4×C127—81no—
77C2×D4×C127D811yescharacteristic
78C22×C4×C127—21no—
79C22×C4×C127—41no—
80C22×C4×C127—41no—
81Q16×C127—41no—
82Q16×C127—41no—
83Q16×C127—41no—
84Q16×C127—41no—
85Q16×C127—41no—
86SD16×C127—42no—
87SD16×C127—41no—
88SD16×C127—41no—
89D8×C127—21no—
90D8×C127—41no—
91D8×C127—42no—
92D8×C127D812yes—
93M16×C127—41no—
94M16×C127—41no—
95C2×C8×C127—21no—
96C2×C8×C127—41no—
97C2×C8×C127C2×D411yescharacteristic
98C2×C8×C127C2×D411yescharacteristic
99C4⋊C4×C127—21no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127D811yescharacteristic
104C22⋊C4×C127—21no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—41no—
110C42×C127C2×D411yescharacteristic
111C16×C127—22no—
112C16×C127—41no—
113C23×C127—21no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—82no—
117C23×C127—82no—
118C23×C127—161no—
119Q8×C127—41no—
120Q8×C127—41no—
121Q8×C127—81no—
122Q8×C127—81no—
123D4×C127—22no—
124D4×C127—41no—
125D4×C127—41no—
126D4×C127—41no—
127D4×C127—41no—
128D4×C127—82no—
129D4×C127—82no—
130D4×C127—161no—
131C2×C4×C127—21no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—82no—
137C2×C4×C127—81no—
138C2×C4×C127—161no—
139C2×C4×C127#12011yescharacteristic
140C2×C4×C127C2×D811yescharacteristic
141C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
142C8×C127—21no—
143C8×C127—41no—
144C8×C127—81no—
145C8×C127C2×D811yescharacteristic
146C8×C127AGL(1,ℤ/8)11yescharacteristic
147C22×C127—41no—
148C22×C127—42no—
149C22×C127—41no—
150C22×C127—41no—
151C22×C127—81no—
152C22×C127—81no—
153C22×C127—81no—
154C22×C127—81no—
155C22×C127—81no—
156C22×C127—164no—
157C22×C127#44711yescharacteristic
158C4×C127—21no—
159C4×C127—21no—
160C4×C127—41no—
161C4×C127—81no—
162C4×C127—81no—
163C4×C127—161no—
164C4×C127#44711yescharacteristic
165C4×C127#45311yescharacteristic
166#12326C12711yesmaximal, characteristic, direct product factor, Sylow
167C2×C127—81no—
168C2×C127—82no—
169C2×C127—161no—
170C2×C127#161911yescharacteristic
171C2×C127#219111yescharacteristic
172C2×C127#219411yescharacteristic
173#3273C2×C12711yescharacteristic
174#2251C2×C12711yescharacteristic, complemented
175#2208C2×C12711yescharacteristic, complemented
176#1689C2×C12711yescharacteristic, complemented
177#1360C2×C12711yescharacteristic, complemented
178#1333C2×C12711yescharacteristic, complemented
179#1330C2×C12711yescharacteristic, complemented
180C127#1232611yescharacteristic, direct product factor, Sylow
181C2×AGL(1,ℤ/8)—21no—
182#546—21no—
183C2×Q32—21no—
184C2×SD32—21no—
185C2×SD32—21no—
186#495C22×C12711yescharacteristic
187#494C22×C12711yescharacteristic, complemented
188#474—21no—
189#468—21no—
190#463—21no—
191#452—21no—
192#447—21no—
193C4×D8C22×C12711yescharacteristic
194#436C22×C12711yescharacteristic
195C4⋊C16C22×C12711yescharacteristic, complemented
196#358—21no—
197#346C22×C12711yescharacteristic, complemented
198#334C22×C12711yescharacteristic, complemented
199#332—21no—
200C2×#41—21no—
201C22×D4—21no—
202AGL(1,ℤ/8)—42no—
203AGL(1,ℤ/8)—42no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16D4×C12711yescharacteristic
207C2×SD16—21no—
208C2×SD16—21no—
209C2×D8—21no—
210C2×D8D4×C12711yescharacteristic
211C2×D8D4×C12711yescharacteristic
212C2×M16—21no—
213C4⋊Q8—21no—
214#124—21no—
215#123—41no—
216C22⋊Q8—21no—
217#121—21no—
218#121—41no—
219#120—41no—
220C4×D4—21no—
221C4×D4D4×C12711yescharacteristic
222C2×C22⋊C4—41no—
223Q32—42no—
224SD32—42no—
225SD32—42no—
226C2×C16—21no—
227C2×C16—21no—
228#107—41no—
229#107D4×C12711yescharacteristic
230C4⋊C8—21no—
231C4⋊C8D4×C12711yescharacteristic
232#103—21no—
233#102—21no—
234#102—21no—
235#102—41no—
236C22⋊C8—21no—
237C4×C8C23×C12711yescharacteristic
238C24—41no—
239#41—42no—
240#41—81no—
241C2×Q8—21no—
242C2×Q8—41no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—81no—
250C2×D4D8×C12711yescharacteristic
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254Q16—41no—
255Q16—41no—
256Q16—41no—
257Q16—41no—
258Q16—41no—
259SD16—42no—
260SD16—41no—
261SD16—41no—
262D8—21no—
263D8—41no—
264D8—42no—
265D8D8×C12712yes—
266M16—41no—
267M16—41no—
268C2×C8—21no—
269C2×C8—41no—
270C2×C8C2×D4×C12711yescharacteristic
271C2×C8C2×D4×C12711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4D8×C12711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C42C2×D4×C12711yescharacteristic
284C16—22no—
285C16—41no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—82no—
290C23—82no—
291C23—161no—
292Q8—41no—
293Q8—41no—
294Q8—81no—
295Q8—81no—
296D4—22no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—82no—
302D4—82no—
303D4—161no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—161no—
312C2×C4#120×C12711yescharacteristic
313C2×C4C2×D8×C12711yescharacteristic
314C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
315C8—21no—
316C8—41no—
317C8—81no—
318C8C2×D8×C12711yescharacteristic
319C8AGL(1,ℤ/8)×C12711yescharacteristic
320C22—41no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—164no—
330C22#447×C12711yescharacteristic
331C4—21no—
332C4—21no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—161no—
337C4#447×C12711yescharacteristic
338C4#453×C12711yescharacteristic
339C2—81no—
340C2—82no—
341C2—161no—
342C2C127×#161911yescharacteristic
343C2C127×#219111yescharacteristic
344C2C127×#219411yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3273C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ #436×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2251C2semidirect
3. C127×#2208C2semidirect
4. C127×#1689C2semidirect
5. C127×#1360C2semidirect
6. C127×#1333C2semidirect
7. C127×#1330C2semidirect
14. #494×C127C22semidirect
22. C4⋊C16×C127C22semidirect
24. #346×C127C22semidirect
25. #334×C127C22semidirect
166. #12326C127direct (N × H)
174. #2251C2×C127semidirect
175. #2208C2×C127semidirect
176. #1689C2×C127semidirect
177. #1360C2×C127semidirect
178. #1333C2×C127semidirect
179. #1330C2×C127semidirect
180. C127#12326direct (N × H)
187. #494C22×C127semidirect
195. C4⋊C16C22×C127semidirect
197. #346C22×C127semidirect
198. #334C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32732
C127×#22512
C127×#22082
C127×#16892
C127×#13602
C127×#13332
C127×#13302
#12326127