Finite Group Databasebeta

← C127×#13216 #44576889

Subgroups — C127×#13216 #44576889

Order 32512.

434 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3156C211yesmaximal, characteristic
2C2×#362×C127C211yesmaximal, characteristic, complemented
3C127×#2145C211yesmaximal, characteristic
4C127×#2143C211yesmaximal, characteristic
5C127×#2141C211yesmaximal, characteristic, complemented
6C127×#2110C211yesmaximal, characteristic, complemented
7C127×#1833C211yesmaximal, characteristic, complemented
8C2×#137×C127—21no—
9C2×#135×C127C2211yescharacteristic
10C22×Q16×C127C2211yescharacteristic
11#536×C127—21no—
12C2×M32×C127C2211yescharacteristic
13C2×M32×C127C2211yescharacteristic
14#497×C127—22no—
15#495×C127C412yescomplemented
16#494×C127C412yescomplemented
17C2×#108×C127C2211yescharacteristic
18#427×C127C2211yescharacteristic
19#405×C127C2211yescharacteristic
20#362×C127—22no—
21#360×C127—22no—
22#359×C127—22no—
23#358×C127—22no—
24#347×C127—22no—
25#344×C127—21no—
26H(3,ℤ/4)×C127—21no—
27C2×#41×C127—21no—
28C22×Q8×C127—21no—
29#137×C127—42no—
30#137×C127—42no—
31#135×C127—21no—
32#135×C127—21no—
33#135×C127—21no—
34#135×C127—21no—
35C2×Q16×C127—22no—
36C2×Q16×C127—21no—
37C2×Q16×C127—21no—
38C2×Q16×C127—21no—
39C2×Q16×C127—21no—
40C2×Q16×C127—21no—
41C2×Q16×C127C2×C412yes—
42C2×Q16×C127C2×C411yescharacteristic
43C2×SD16×C127—22no—
44C2×SD16×C127—21no—
45C2×D8×C127C2×C411yescharacteristic
46C2×M16×C127—21no—
47C2×M16×C127D411yescharacteristic
48C22×C8×C127C2311yescharacteristic
49C4⋊Q8×C127—22no—
50#124×C127—22no—
51#123×C127—42no—
52C22⋊Q8×C127—42no—
53#117×C127—21no—
54#117×C127D411yescharacteristic
55M32×C127—21no—
56M32×C127—21no—
57M32×C127—21no—
58M32×C127D412yes—
59C2×C16×C127—21no—
60C2×C16×C127D412yes—
61#108×C127—21no—
62#108×C127—21no—
63#107×C127—21no—
64#106×C127—21no—
65#97×C127D412yes—
66C4×C8×C127C2×C412yes—
67#41×C127—41no—
68#41×C127—41no—
69#41×C127—41no—
70#41×C127—41no—
71C2×Q8×C127—22no—
72C2×Q8×C127—21no—
73C2×Q8×C127—41no—
74C2×Q8×C127—41no—
75C2×Q8×C127—41no—
76C2×Q8×C127—41no—
77C2×Q8×C127—41no—
78C2×D4×C127—21no—
79C2×D4×C127—41no—
80C22×C4×C127—41no—
81C22×C4×C127—41no—
82C22×C4×C127C2×D411yescharacteristic
83Q16×C127—22no—
84Q16×C127—22no—
85Q16×C127—21no—
86Q16×C127—21no—
87Q16×C127—44no—
88Q16×C127—41no—
89Q16×C127—41no—
90SD16×C127—44no—
91SD16×C127—41no—
92SD16×C127—41no—
93D8×C127—21no—
94D8×C127—21no—
95M16×C127—21no—
96M16×C127—21no—
97M16×C127—41no—
98M16×C127—41no—
99C2×C8×C127—41no—
100C2×C8×C127C22⋊C412yes—
101C2×C8×C127C22×C411yescharacteristic
102C2×C8×C127C2×D411yescharacteristic
103C2×C8×C127C2×D411yescharacteristic
104C2×C8×C127C2×D411yescharacteristic
105C2×C8×C127#4111yescharacteristic
106C2×C8×C127#4111yescharacteristic
107C4⋊C4×C127—21no—
108C4⋊C4×C127—44no—
109C4⋊C4×C127—41no—
110C22⋊C4×C127—21no—
111C22⋊C4×C127—44no—
112C22⋊C4×C127—41no—
113C42×C127—41no—
114C42×C127C22⋊C412yes—
115C16×C127—22no—
116C16×C127—22no—
117C23×C127—41no—
118C23×C127AGL(1,ℤ/8)11yescharacteristic
119Q8×C127—41no—
120Q8×C127—42no—
121Q8×C127—42no—
122Q8×C127—41no—
123Q8×C127—81no—
124Q8×C127—81no—
125D4×C127—41no—
126D4×C127—41no—
127D4×C127—81no—
128D4×C127—81no—
129C2×C4×C127—22no—
130C2×C4×C127—42no—
131C2×C4×C127—41no—
132C2×C4×C127—81no—
133C2×C4×C127—81no—
134C2×C4×C127—81no—
135C2×C4×C127—81no—
136C2×C4×C127—81no—
137C2×C4×C127C2×C22⋊C411yescharacteristic
138C2×C4×C127#12411yescharacteristic
139C2×C4×C127#12711yescharacteristic
140C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
143C8×C127—22no—
144C8×C127—81no—
145C8×C127C4×D411yescharacteristic
146C8×C127C4×D411yescharacteristic
147C8×C127#12111yescharacteristic
148C8×C127#12111yescharacteristic
149C22×C127—41no—
150C22×C127—82no—
151C22×C127—81no—
152C22×C127#41811yescharacteristic
153C22×C127#49011yescharacteristic
154C22×C127#49611yescharacteristic
155C4×C127—42no—
156C4×C127—81no—
157C4×C127—82no—
158C4×C127—161no—
159C4×C127#39011yescharacteristic
160C4×C127#41811yescharacteristic
161C4×C127#49011yescharacteristic
162C4×C127#49611yescharacteristic
163#13216C12711yesmaximal, characteristic, direct product factor, Sylow
164C2×C127—21no—
165C2×C127—21no—
166C2×C127—21no—
167C2×C127—161no—
168C2×C127#196111yescharacteristic
169#3156C2×C12711yescharacteristic
170C2×#362C2×C12711yescharacteristic, complemented
171#2145C2×C12711yescharacteristic
172#2143C2×C12711yescharacteristic
173#2141C2×C12711yescharacteristic, complemented
174#2110C2×C12711yescharacteristic, complemented
175#1833C2×C12711yescharacteristic, complemented
176C127#1321611yescharacteristic, direct product factor, Sylow
177C2×#137—21no—
178C2×#135C22×C12711yescharacteristic
179C22×Q16C22×C12711yescharacteristic
180#536—21no—
181C2×M32C22×C12711yescharacteristic
182C2×M32C22×C12711yescharacteristic
183#497—22no—
184#495C4×C12712yescomplemented
185#494C4×C12712yescomplemented
186C2×#108C22×C12711yescharacteristic
187#427C22×C12711yescharacteristic
188#405C22×C12711yescharacteristic
189#362—22no—
190#360—22no—
191#359—22no—
192#358—22no—
193#347—22no—
194#344—21no—
195H(3,ℤ/4)—21no—
196C2×#41—21no—
197C22×Q8—21no—
198#137—42no—
199#137—42no—
200#135—21no—
201#135—21no—
202#135—21no—
203#135—21no—
204C2×Q16—22no—
205C2×Q16—21no—
206C2×Q16—21no—
207C2×Q16—21no—
208C2×Q16—21no—
209C2×Q16—21no—
210C2×Q16C2×C4×C12712yes—
211C2×Q16C2×C4×C12711yescharacteristic
212C2×SD16—22no—
213C2×SD16—21no—
214C2×D8C2×C4×C12711yescharacteristic
215C2×M16—21no—
216C2×M16D4×C12711yescharacteristic
217C22×C8C23×C12711yescharacteristic
218C4⋊Q8—22no—
219#124—22no—
220#123—42no—
221C22⋊Q8—42no—
222#117—21no—
223#117D4×C12711yescharacteristic
224M32—21no—
225M32—21no—
226M32—21no—
227M32D4×C12712yes—
228C2×C16—21no—
229C2×C16D4×C12712yes—
230#108—21no—
231#108—21no—
232#107—21no—
233#106—21no—
234#97D4×C12712yes—
235C4×C8C2×C4×C12712yes—
236#41—41no—
237#41—41no—
238#41—41no—
239#41—41no—
240C2×Q8—22no—
241C2×Q8—21no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4C2×D4×C12711yescharacteristic
252Q16—22no—
253Q16—22no—
254Q16—21no—
255Q16—21no—
256Q16—44no—
257Q16—41no—
258Q16—41no—
259SD16—44no—
260SD16—41no—
261SD16—41no—
262D8—21no—
263D8—21no—
264M16—21no—
265M16—21no—
266M16—41no—
267M16—41no—
268C2×C8—41no—
269C2×C8C22⋊C4×C12712yes—
270C2×C8C22×C4×C12711yescharacteristic
271C2×C8C2×D4×C12711yescharacteristic
272C2×C8C2×D4×C12711yescharacteristic
273C2×C8C2×D4×C12711yescharacteristic
274C2×C8#41×C12711yescharacteristic
275C2×C8#41×C12711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C22⋊C4—21no—
280C22⋊C4—44no—
281C22⋊C4—41no—
282C42—41no—
283C42C22⋊C4×C12712yes—
284C16—22no—
285C16—22no—
286C23—41no—
287C23AGL(1,ℤ/8)×C12711yescharacteristic
288Q8—41no—
289Q8—42no—
290Q8—42no—
291Q8—41no—
292Q8—81no—
293Q8—81no—
294D4—41no—
295D4—41no—
296D4—81no—
297D4—81no—
298C2×C4—22no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4C2×C22⋊C4×C12711yescharacteristic
307C2×C4#124×C12711yescharacteristic
308C2×C4#127×C12711yescharacteristic
309C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
310C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
311C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
312C8—22no—
313C8—81no—
314C8C4×D4×C12711yescharacteristic
315C8C4×D4×C12711yescharacteristic
316C8#121×C12711yescharacteristic
317C8#121×C12711yescharacteristic
318C22—41no—
319C22—82no—
320C22—81no—
321C22#418×C12711yescharacteristic
322C22#490×C12711yescharacteristic
323C22#496×C12711yescharacteristic
324C4—42no—
325C4—81no—
326C4—82no—
327C4—161no—
328C4#390×C12711yescharacteristic
329C4#418×C12711yescharacteristic
330C4#490×C12711yescharacteristic
331C4#496×C12711yescharacteristic
332C2—21no—
333C2—21no—
334C2—21no—
335C2—161no—
336C2C127×#196111yescharacteristic
subgroupG/H
center, socleC2×C127#1961
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC2×#135×C127C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#362×C127C2semidirect
5. C127×#2141C2semidirect
6. C127×#2110C2semidirect
7. C127×#1833C2semidirect
15. #495×C127C4semidirect
16. #494×C127C4semidirect
163. #13216C127direct (N × H)
170. C2×#362C2×C127semidirect
173. #2141C2×C127semidirect
174. #2110C2×C127semidirect
175. #1833C2×C127semidirect
176. C127#13216direct (N × H)
184. #495C4×C127semidirect
185. #494C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31562
C2×#362×C1272
C127×#21452
C127×#21432
C127×#21412
C127×#21102
C127×#18332
#13216127