Finite Group Databasebeta

← C127×#12175 #44575848

Subgroups — C127×#12175 #44575848

Order 32512.

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

normal subgroups only

This group has 346 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×#3256C211yesmaximal, characteristic, complemented
2C127×#2191C211yesmaximal, characteristic, complemented
3C127×#2190C211yesmaximal, characteristic
4C127×#2050C211yesmaximal, characteristic, complemented
5C127×#1342C211yesmaximal, characteristic, complemented
6C127×#1341C211yesmaximal, characteristic, complemented
7C127×#1327C211yesmaximal, characteristic, complemented
8#575×C127C2211yescharacteristic
9#529×C127—21no—
10#510×C127—21no—
11#509×C127—21no—
12#479×C127—21no—
13#478×C127—21no—
14#470×C127C2211yescharacteristic
15#469×C127C2211yescharacteristic
16#468×C127—21no—
17#464×C127—21no—
18#463×C127—21no—
19#423×C127C2211yescharacteristic
20C2×#104×C127—21no—
21#413×C127—21no—
22#360×C127—21no—
23#360×C127—21no—
24#349×C127C2211yescharacteristic
25#349×C127C2211yescharacteristic
26#338×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×C127—21no—
36C2×M16×C127D411yescharacteristic
37C2×M16×C127C2311yescharacteristic
38#126×C127—41no—
39#125×C127—21no—
40#124×C127—21no—
41#123×C127—41no—
42C22⋊Q8×C127—21no—
43#121×C127—21no—
44C4×Q8×C127—21no—
45C4×D4×C127—21no—
46C4×D4×C127—41no—
47#117×C127—41no—
48C2×C42×C127D411yescharacteristic
49Q32×C127—41no—
50SD32×C127—41no—
51SD32×C127—41no—
52D16×C127—41no—
53M32×C127—21no—
54M32×C127—21no—
55#107×C127—21no—
56#106×C127D411yescharacteristic
57#106×C127D411yescharacteristic
58C4⋊C8×C127—21no—
59C4⋊C8×C127—21no—
60#104×C127—41no—
61#104×C127—41no—
62#103×C127—21no—
63#103×C127—21no—
64#102×C127—21no—
65#102×C127—21no—
66#101×C127—41no—
67#100×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×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127—41no—
77C2×D4×C127—41no—
78C22×C4×C127—21no—
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—
92M16×C127—41no—
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—21no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127—41no—
104C4⋊C4×C127—41no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—41no—
110C42×C127—21no—
111C42×C127—41no—
112C42×C127D811yescharacteristic
113C42×C127D811yescharacteristic
114C16×C127—41no—
115C16×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127AGL(1,ℤ/8)11yescharacteristic
119Q8×C127—41no—
120Q8×C127—41no—
121Q8×C127—41no—
122Q8×C127—41no—
123D4×C127—41no—
124D4×C127—41no—
125D4×C127—41no—
126D4×C127—41no—
127D4×C127—81no—
128D4×C127—82no—
129D4×C127—81no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—21no—
133C2×C4×C127—21no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—81no—
139C2×C4×C127—81no—
140C2×C4×C127—82no—
141C2×C4×C127—81no—
142C2×C4×C127#12011yescharacteristic
143C2×C4×C127C2×D811yescharacteristic
144C8×C127—21no—
145C8×C127—21no—
146C8×C127—41no—
147C8×C127—81no—
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#45411yescharacteristic
164C4×C127B(4,𝔽2)11yescharacteristic
165#12175C12711yesmaximal, characteristic, direct product factor, Sylow
166C2×C127—21no—
167C2×C127—41no—
168C2×C127—161no—
169C2×C127—161no—
170C2×C127#159811yescharacteristic
171#3256C2×C12711yescharacteristic, complemented
172#2191C2×C12711yescharacteristic, complemented
173#2190C2×C12711yescharacteristic
174#2050C2×C12711yescharacteristic, complemented
175#1342C2×C12711yescharacteristic, complemented
176#1341C2×C12711yescharacteristic, complemented
177#1327C2×C12711yescharacteristic, complemented
178C127#1217511yescharacteristic, direct product factor, Sylow
179#575C22×C12711yescharacteristic
180#529—21no—
181#510—21no—
182#509—21no—
183#479—21no—
184#478—21no—
185#470C22×C12711yescharacteristic
186#469C22×C12711yescharacteristic
187#468—21no—
188#464—21no—
189#463—21no—
190#423C22×C12711yescharacteristic
191C2×#104—21no—
192#413—21no—
193#360—21no—
194#360—21no—
195#349C22×C12711yescharacteristic
196#349C22×C12711yescharacteristic
197#338C22×C12711yescharacteristic
198C2×#41—21no—
199#137—41no—
200AGL(1,ℤ/8)—41no—
201#135—21no—
202#135—21no—
203C2×Q16D4×C12711yescharacteristic
204C2×SD16—21no—
205C2×D8D4×C12711yescharacteristic
206C2×M16—21no—
207C2×M16D4×C12711yescharacteristic
208C2×M16C23×C12711yescharacteristic
209#126—41no—
210#125—21no—
211#124—21no—
212#123—41no—
213C22⋊Q8—21no—
214#121—21no—
215C4×Q8—21no—
216C4×D4—21no—
217C4×D4—41no—
218#117—41no—
219C2×C42D4×C12711yescharacteristic
220Q32—41no—
221SD32—41no—
222SD32—41no—
223D16—41no—
224M32—21no—
225M32—21no—
226#107—21no—
227#106D4×C12711yescharacteristic
228#106D4×C12711yescharacteristic
229C4⋊C8—21no—
230C4⋊C8—21no—
231#104—41no—
232#104—41no—
233#103—21no—
234#103—21no—
235#102—21no—
236#102—21no—
237#101—41no—
238#100—41no—
239#41—41no—
240#41—41no—
241#41—41no—
242#41—41no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4C2×D4×C12711yescharacteristic
253Q16—21no—
254Q16—21no—
255Q16—41no—
256SD16—41no—
257SD16—41no—
258SD16—81no—
259D8—21no—
260D8—21no—
261D8—41no—
262M16—41no—
263M16—41no—
264M16—41no—
265M16—41no—
266C2×C8—21no—
267C2×C8—41no—
268C2×C8C2×D4×C12711yescharacteristic
269C2×C8C2×D4×C12711yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C42—21no—
282C42—41no—
283C42D8×C12711yescharacteristic
284C42D8×C12711yescharacteristic
285C16—41no—
286C16—41no—
287C23—41no—
288C23—41no—
289C23AGL(1,ℤ/8)×C12711yescharacteristic
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—81no—
299D4—82no—
300D4—81no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4#120×C12711yescharacteristic
314C2×C4C2×D8×C12711yescharacteristic
315C8—21no—
316C8—21no—
317C8—41no—
318C8—81no—
319C22—21no—
320C22—41no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—161no—
326C22#447×C12711yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4#454×C12711yescharacteristic
335C4B(4,𝔽2)×C12711yescharacteristic
336C2—21no—
337C2—41no—
338C2—161no—
339C2—161no—
340C2C127×#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×#3256C2semidirect
2. C127×#2191C2semidirect
4. C127×#2050C2semidirect
5. C127×#1342C2semidirect
6. C127×#1341C2semidirect
7. C127×#1327C2semidirect
165. #12175C127direct (N × H)
171. #3256C2×C127semidirect
172. #2191C2×C127semidirect
174. #2050C2×C127semidirect
175. #1342C2×C127semidirect
176. #1341C2×C127semidirect
177. #1327C2×C127semidirect
178. C127#12175direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32562
C127×#21912
C127×#21902
C127×#20502
C127×#13422
C127×#13412
C127×#13272
#12175127