Finite Group Databasebeta

← C127×#13215 #44576888

Subgroups — C127×#13215 #44576888

Order 32512.

466 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3155C211yesmaximal, characteristic
2C2×#361×C127C211yesmaximal, characteristic, complemented
3C127×#2145C211yesmaximal, characteristic, complemented
4C127×#2143C211yesmaximal, characteristic, complemented
5C127×#2140C211yesmaximal, characteristic, complemented
6C127×#2110C211yesmaximal, characteristic, complemented
7C127×#1833C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127—21no—
9C2×#135×C127C2211yescharacteristic
10C22×D8×C127C2211yescharacteristic
11#535×C127—21no—
12C2×M32×C127C2211yescharacteristic
13C2×M32×C127C2211yescharacteristic
14#496×C127—22no—
15#495×C127C412yescomplemented
16#493×C127C412yescomplemented
17C2×#108×C127C2211yescharacteristic
18#427×C127C2211yescharacteristic
19#405×C127C2211yescharacteristic
20#361×C127—22no—
21#360×C127—22no—
22#359×C127—22no—
23#357×C127—22no—
24#347×C127—22no—
25#344×C127—21no—
26H(3,ℤ/4)×C127—21no—
27C2×#41×C127—21no—
28C22×D4×C127—21no—
29AGL(1,ℤ/8)×C127—42no—
30AGL(1,ℤ/8)×C127—42no—
31#135×C127—21no—
32#135×C127—21no—
33#135×C127—21no—
34#135×C127—21no—
35C2×Q16×C127C2×C411yescharacteristic
36C2×SD16×C127—22no—
37C2×SD16×C127—21no—
38C2×D8×C127—22no—
39C2×D8×C127—21no—
40C2×D8×C127—21no—
41C2×D8×C127—21no—
42C2×D8×C127—21no—
43C2×D8×C127—21no—
44C2×D8×C127C2×C412yes—
45C2×D8×C127C2×C411yescharacteristic
46C2×M16×C127—21no—
47C2×M16×C127D411yescharacteristic
48C22×C8×C127C2311yescharacteristic
49#127×C127—22no—
50#124×C127—22no—
51#121×C127—42no—
52#120×C127—42no—
53#117×C127—21no—
54#117×C127D411yescharacteristic
55M32×C127—21no—
56M32×C127—21no—
57M32×C127—21no—
58M32×C127D412yescomplemented
59C2×C16×C127—21no—
60C2×C16×C127D412yescomplemented
61#108×C127—21no—
62#108×C127—21no—
63#107×C127—21no—
64#106×C127—21no—
65#97×C127D412yes—
66C4×C8×C127C2×C412yes—
67C24×C127—41no—
68#41×C127—41no—
69#41×C127—41no—
70#41×C127—41no—
71#41×C127—41no—
72C2×Q8×C127—21no—
73C2×D4×C127—22no—
74C2×D4×C127—21no—
75C2×D4×C127—44no—
76C2×D4×C127—41no—
77C2×D4×C127—41no—
78C2×D4×C127—41no—
79C2×D4×C127—41no—
80C2×D4×C127—41no—
81C2×D4×C127—41no—
82C22×C4×C127—41no—
83C22×C4×C127C2×D411yescharacteristic
84Q16×C127—21no—
85Q16×C127—21no—
86SD16×C127—44no—
87SD16×C127—41no—
88SD16×C127—41no—
89D8×C127—22no—
90D8×C127—22no—
91D8×C127—21no—
92D8×C127—21no—
93D8×C127—44no—
94D8×C127—41no—
95D8×C127—41no—
96M16×C127—21no—
97M16×C127—21no—
98M16×C127—41no—
99M16×C127—41no—
100C2×C8×C127—41no—
101C2×C8×C127C22⋊C412yes—
102C2×C8×C127C22×C411yescharacteristic
103C2×C8×C127C2×D411yescharacteristic
104C2×C8×C127C2×D411yescharacteristic
105C2×C8×C127C2×D411yescharacteristic
106C2×C8×C127#4111yescharacteristic
107C2×C8×C127#4111yescharacteristic
108C4⋊C4×C127—21no—
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—42no—
118C23×C127—41no—
119C23×C127—81no—
120C23×C127—81no—
121C23×C127—162no—
122C23×C127AGL(1,ℤ/8)11yescharacteristic
123Q8×C127—41no—
124Q8×C127—41no—
125D4×C127—42no—
126D4×C127—41no—
127D4×C127—41no—
128D4×C127—42no—
129D4×C127—81no—
130D4×C127—81no—
131D4×C127—81no—
132D4×C127—81no—
133D4×C127—164no—
134C2×C4×C127—22no—
135C2×C4×C127—41no—
136C2×C4×C127—81no—
137C2×C4×C127—81no—
138C2×C4×C127—81no—
139C2×C4×C127C2×C22⋊C411yescharacteristic
140C2×C4×C127#12411yescharacteristic
141C2×C4×C127#12711yescharacteristic
142C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
145C8×C127—22no—
146C8×C127—81no—
147C8×C127C4×D411yescharacteristic
148C8×C127C4×D411yescharacteristic
149C8×C127#12111yescharacteristic
150C8×C127#12111yescharacteristic
151C22×C127—41no—
152C22×C127—82no—
153C22×C127—84no—
154C22×C127—81no—
155C22×C127—82no—
156C22×C127—162no—
157C22×C127—162no—
158C22×C127#41811yescharacteristic
159C22×C127#49011yescharacteristic
160C22×C127#49611yescharacteristic
161C4×C127—42no—
162C4×C127—81no—
163C4×C127—161no—
164C4×C127#39011yescharacteristic
165C4×C127#41811yescharacteristic
166C4×C127#49011yescharacteristic
167C4×C127#49611yescharacteristic
168#13215C12711yesmaximal, characteristic, direct product factor, Sylow
169C2×C127—21no—
170C2×C127—21no—
171C2×C127—21no—
172C2×C127—161no—
173C2×C127—162no—
174C2×C127#196111yescharacteristic
175#3155C2×C12711yescharacteristic
176C2×#361C2×C12711yescharacteristic, complemented
177#2145C2×C12711yescharacteristic, complemented
178#2143C2×C12711yescharacteristic, complemented
179#2140C2×C12711yescharacteristic, complemented
180#2110C2×C12711yescharacteristic, complemented
181#1833C2×C12711yescharacteristic, complemented
182C127#1321511yescharacteristic, direct product factor, Sylow
183C2×AGL(1,ℤ/8)—21no—
184C2×#135C22×C12711yescharacteristic
185C22×D8C22×C12711yescharacteristic
186#535—21no—
187C2×M32C22×C12711yescharacteristic
188C2×M32C22×C12711yescharacteristic
189#496—22no—
190#495C4×C12712yescomplemented
191#493C4×C12712yescomplemented
192C2×#108C22×C12711yescharacteristic
193#427C22×C12711yescharacteristic
194#405C22×C12711yescharacteristic
195#361—22no—
196#360—22no—
197#359—22no—
198#357—22no—
199#347—22no—
200#344—21no—
201H(3,ℤ/4)—21no—
202C2×#41—21no—
203C22×D4—21no—
204AGL(1,ℤ/8)—42no—
205AGL(1,ℤ/8)—42no—
206#135—21no—
207#135—21no—
208#135—21no—
209#135—21no—
210C2×Q16C2×C4×C12711yescharacteristic
211C2×SD16—22no—
212C2×SD16—21no—
213C2×D8—22no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8C2×C4×C12712yes—
220C2×D8C2×C4×C12711yescharacteristic
221C2×M16—21no—
222C2×M16D4×C12711yescharacteristic
223C22×C8C23×C12711yescharacteristic
224#127—22no—
225#124—22no—
226#121—42no—
227#120—42no—
228#117—21no—
229#117D4×C12711yescharacteristic
230M32—21no—
231M32—21no—
232M32—21no—
233M32D4×C12712yescomplemented
234C2×C16—21no—
235C2×C16D4×C12712yescomplemented
236#108—21no—
237#108—21no—
238#107—21no—
239#106—21no—
240#97D4×C12712yes—
241C4×C8C2×C4×C12712yes—
242C24—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246#41—41no—
247C2×Q8—21no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C22×C4—41no—
258C22×C4C2×D4×C12711yescharacteristic
259Q16—21no—
260Q16—21no—
261SD16—44no—
262SD16—41no—
263SD16—41no—
264D8—22no—
265D8—22no—
266D8—21no—
267D8—21no—
268D8—44no—
269D8—41no—
270D8—41no—
271M16—21no—
272M16—21no—
273M16—41no—
274M16—41no—
275C2×C8—41no—
276C2×C8C22⋊C4×C12712yes—
277C2×C8C22×C4×C12711yescharacteristic
278C2×C8C2×D4×C12711yescharacteristic
279C2×C8C2×D4×C12711yescharacteristic
280C2×C8C2×D4×C12711yescharacteristic
281C2×C8#41×C12711yescharacteristic
282C2×C8#41×C12711yescharacteristic
283C4⋊C4—21no—
284C4⋊C4—41no—
285C22⋊C4—21no—
286C22⋊C4—44no—
287C22⋊C4—41no—
288C42—41no—
289C42C22⋊C4×C12712yes—
290C16—22no—
291C16—22no—
292C23—42no—
293C23—41no—
294C23—81no—
295C23—81no—
296C23—162no—
297C23AGL(1,ℤ/8)×C12711yescharacteristic
298Q8—41no—
299Q8—41no—
300D4—42no—
301D4—41no—
302D4—41no—
303D4—42no—
304D4—81no—
305D4—81no—
306D4—81no—
307D4—81no—
308D4—164no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4C2×C22⋊C4×C12711yescharacteristic
315C2×C4#124×C12711yescharacteristic
316C2×C4#127×C12711yescharacteristic
317C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
318C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
319C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
320C8—22no—
321C8—81no—
322C8C4×D4×C12711yescharacteristic
323C8C4×D4×C12711yescharacteristic
324C8#121×C12711yescharacteristic
325C8#121×C12711yescharacteristic
326C22—41no—
327C22—82no—
328C22—84no—
329C22—81no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22#418×C12711yescharacteristic
334C22#490×C12711yescharacteristic
335C22#496×C12711yescharacteristic
336C4—42no—
337C4—81no—
338C4—161no—
339C4#390×C12711yescharacteristic
340C4#418×C12711yescharacteristic
341C4#490×C12711yescharacteristic
342C4#496×C12711yescharacteristic
343C2—21no—
344C2—21no—
345C2—21no—
346C2—162no—
347C2—161no—
348C2C127×#196111yescharacteristic
subgroupG/H
center, socleC2×C127#1961
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC127×#3155C2
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×#361×C127C2semidirect
3. C127×#2145C2semidirect
4. C127×#2143C2semidirect
5. C127×#2140C2semidirect
6. C127×#2110C2semidirect
7. C127×#1833C2semidirect
15. #495×C127C4semidirect
16. #493×C127C4semidirect
58. M32×C127D4semidirect
60. C2×C16×C127D4semidirect
168. #13215C127direct (N × H)
176. C2×#361C2×C127semidirect
177. #2145C2×C127semidirect
178. #2143C2×C127semidirect
179. #2140C2×C127semidirect
180. #2110C2×C127semidirect
181. #1833C2×C127semidirect
182. C127#13215direct (N × H)
190. #495C4×C127semidirect
191. #493C4×C127semidirect
233. M32D4×C127semidirect
235. C2×C16D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31552
C2×#361×C1272
C127×#21452
C127×#21432
C127×#21402
C127×#21102
C127×#18332
#13215127