Finite Group Databasebeta

← C127×#8456 #44572129

Subgroups — C127×#8456 #44572129

Order 32512.

1026 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 154 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×#3496C211yesmaximal, characteristic
2C127×#1955C211yesmaximal, characteristic, complemented
3C127×#1551C212yesmaximal, complemented
4C127×#1530C212yesmaximal, complemented
5C127×#1466C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C127C412yes—
7#547×C127—22no—
8#546×C127—24no—
9D42×C127—22no—
10#537×C127C412yes—
11#534×C127—22no—
12#532×C127C2212yes—
13C2×#121×C127—22no—
14C2×C4×D4×C127C2211yescharacteristic
15#432×C127C2212yescomplemented
16C2×C4⋊C8×C127C2211yescharacteristic, complemented
17C2×C22⋊C8×C127—22no—
18C2×#97×C127C2211yescharacteristic, complemented
19#333×C127—22no—
20#331×C127—22no—
21#329×C127—24no—
22#323×C127—22no—
23H(5,𝔽2)×C127—48no—
24C2×#41×C127—24no—
25C2×#41×C127C2×C414yes—
26C22×D4×C127—24no—
27C22×D4×C127—22no—
28C22×D4×C127—24no—
29C22×D4×C127C2×C411yescharacteristic
30C23×C4×C127D412yes—
31C22×C8×C127—21no—
32C22×C8×C127—21no—
33C4⋊Q8×C127D412yes—
34#127×C127D412yes—
35#124×C127D414yes—
36#123×C127—28no—
37C22⋊Q8×C127—24no—
38#121×C127—24no—
39#121×C127—24no—
40#121×C127—24no—
41#121×C127—24no—
42#121×C127—22no—
43#121×C127—22no—
44#120×C127—28no—
45C4×D4×C127—22no—
46C4×D4×C127—22no—
47C4×D4×C127—22no—
48C4×D4×C127—22no—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127C2×C411yescharacteristic
51C2×C22⋊C4×C127—24no—
52C2×C22⋊C4×C127D412yes—
53C2×C42×C127C2311yescharacteristic
54C4⋊C8×C127—22no—
55C4⋊C8×C127—22no—
56C22⋊C8×C127—22no—
57C22⋊C8×C127—44no—
58#97×C127—22no—
59#97×C127—22no—
60C24×C127—24no—
61C24×C127C22⋊C412yes—
62#41×C127—48no—
63#41×C127—416no—
64C2×Q8×C127C22⋊C414yes—
65C2×D4×C127—24no—
66C2×D4×C127—28no—
67C2×D4×C127—24no—
68C2×D4×C127—24no—
69C2×D4×C127—22no—
70C2×D4×C127—22no—
71C2×D4×C127—22no—
72C2×D4×C127—416no—
73C2×D4×C127—48no—
74C2×D4×C127—48no—
75C2×D4×C127—82no—
76C2×D4×C127—82no—
77C2×D4×C127C22⋊C414yes—
78C22×C4×C127—24no—
79C22×C4×C127—24no—
80C22×C4×C127—22no—
81C22×C4×C127—22no—
82C22×C4×C127—42no—
83C22×C4×C127—42no—
84C22×C4×C127C22⋊C412yes—
85C22×C4×C127C22×C411yescharacteristic
86C22×C4×C127C2×D411yescharacteristic
87C22×C4×C127C2×D411yescharacteristic
88C2×C8×C127—22no—
89C2×C8×C127—41no—
90C2×C8×C127—41no—
91C2×C8×C127—42no—
92C2×C8×C127—42no—
93C2×C8×C127—41no—
94C4⋊C4×C127—24no—
95C4⋊C4×C127—22no—
96C4⋊C4×C127—22no—
97C4⋊C4×C127—22no—
98C22⋊C4×C127—28no—
99C22⋊C4×C127—24no—
100C22⋊C4×C127—24no—
101C22⋊C4×C127—24no—
102C42×C127C2×D412yes—
103C42×C127C2×D412yes—
104C23×C127—22no—
105C23×C127—22no—
106C23×C127—24no—
107C23×C127—42no—
108C23×C127—42no—
109C23×C127—48no—
110C23×C127—84no—
111C23×C127—84no—
112C23×C127#10014yes—
113C23×C127C2×C22⋊C411yescharacteristic
114Q8×C127—44no—
115D4×C127—28no—
116D4×C127—44no—
117D4×C127—416no—
118D4×C127—48no—
119D4×C127—42no—
120D4×C127—42no—
121D4×C127—82no—
122D4×C127—84no—
123C2×C4×C127—22no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—22no—
127C2×C4×C127—24no—
128C2×C4×C127—41no—
129C2×C4×C127—48no—
130C2×C4×C127—48no—
131C2×C4×C127—42no—
132C2×C4×C127—42no—
133C2×C4×C127—42no—
134C2×C4×C127—82no—
135C2×C4×C127C2×C22⋊C412yes—
136C2×C4×C127#12012yes—
137C2×C4×C127#12011yescharacteristic
138C2×C4×C127#12011yescharacteristic
139C8×C127—42no—
140C8×C127—82no—
141C8×C127—81no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—24no—
145C22×C127—44no—
146C22×C127—42no—
147C22×C127—42no—
148C22×C127—44no—
149C22×C127—44no—
150C22×C127—44no—
151C22×C127—82no—
152C22×C127—168no—
153C22×C127#37911yescharacteristic
154C22×C127C2×#10012yes—
155C4×C127—21no—
156C4×C127—22no—
157C4×C127—22no—
158C4×C127—42no—
159C4×C127—44no—
160C4×C127#45311yescharacteristic
161C4×C127#45511yescharacteristic
162#8456C12711yesmaximal, characteristic, direct product factor, Sylow
163C2×C127—22no—
164C2×C127—44no—
165C2×C127—84no—
166C2×C127#178311yescharacteristic
167C2×C127#179211yescharacteristic
168C2×C127#179411yescharacteristic
169#3496C2×C12711yescharacteristic
170#1955C2×C12711yescharacteristic, complemented
171#1551C2×C12712yescomplemented
172#1530C2×C12712yescomplemented
173#1466C2×C12711yescharacteristic, complemented
174C127#845611yescharacteristic, direct product factor, Sylow
175C2×H(5,𝔽2)C4×C12712yes—
176#547—22no—
177#546—24no—
178D42—22no—
179#537C4×C12712yes—
180#534—22no—
181#532C22×C12712yes—
182C2×#121—22no—
183C2×C4×D4C22×C12711yescharacteristic
184#432C22×C12712yescomplemented
185C2×C4⋊C8C22×C12711yescharacteristic, complemented
186C2×C22⋊C8—22no—
187C2×#97C22×C12711yescharacteristic, complemented
188#333—22no—
189#331—22no—
190#329—24no—
191#323—22no—
192H(5,𝔽2)—48no—
193C2×#41—24no—
194C2×#41C2×C4×C12714yes—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4C2×C4×C12711yescharacteristic
199C23×C4D4×C12712yes—
200C22×C8—21no—
201C22×C8—21no—
202C4⋊Q8D4×C12712yes—
203#127D4×C12712yes—
204#124D4×C12714yes—
205#123—28no—
206C22⋊Q8—24no—
207#121—24no—
208#121—24no—
209#121—24no—
210#121—24no—
211#121—22no—
212#121—22no—
213#120—28no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C2×C4×C12711yescharacteristic
220C2×C22⋊C4—24no—
221C2×C22⋊C4D4×C12712yes—
222C2×C42C23×C12711yescharacteristic
223C4⋊C8—22no—
224C4⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—44no—
227#97—22no—
228#97—22no—
229C24—24no—
230C24C22⋊C4×C12712yes—
231#41—48no—
232#41—416no—
233C2×Q8C22⋊C4×C12714yes—
234C2×D4—24no—
235C2×D4—28no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—416no—
242C2×D4—48no—
243C2×D4—48no—
244C2×D4—82no—
245C2×D4—82no—
246C2×D4C22⋊C4×C12714yes—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4C22⋊C4×C12712yes—
254C22×C4C22×C4×C12711yescharacteristic
255C22×C4C2×D4×C12711yescharacteristic
256C22×C4C2×D4×C12711yescharacteristic
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C22⋊C4—28no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42C2×D4×C12712yes—
272C42C2×D4×C12712yes—
273C23—22no—
274C23—22no—
275C23—24no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—84no—
280C23—84no—
281C23#100×C12714yes—
282C23C2×C22⋊C4×C12711yescharacteristic
283Q8—44no—
284D4—28no—
285D4—44no—
286D4—416no—
287D4—48no—
288D4—42no—
289D4—42no—
290D4—82no—
291D4—84no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4C2×C22⋊C4×C12712yes—
305C2×C4#120×C12712yes—
306C2×C4#120×C12711yescharacteristic
307C2×C4#120×C12711yescharacteristic
308C8—42no—
309C8—82no—
310C8—81no—
311C22—21no—
312C22—21no—
313C22—24no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—44no—
319C22—44no—
320C22—82no—
321C22—168no—
322C22#379×C12711yescharacteristic
323C22C2×#100×C12712yes—
324C4—22no—
325C4—21no—
326C4—22no—
327C4—44no—
328C4—42no—
329C4#453×C12711yescharacteristic
330C4#455×C12711yescharacteristic
331C2—22no—
332C2—44no—
333C2—84no—
334C2C127×#178311yescharacteristic
335C2C127×#179211yescharacteristic
336C2C127×#179411yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1955C2semidirect
3. C127×#1551C2semidirect
4. C127×#1530C2semidirect
5. C127×#1466C2semidirect
15. #432×C127C22semidirect
16. C2×C4⋊C8×C127C22semidirect
18. C2×#97×C127C22semidirect
162. #8456C127direct (N × H)
170. #1955C2×C127semidirect
171. #1551C2×C127semidirect
172. #1530C2×C127semidirect
173. #1466C2×C127semidirect
174. C127#8456direct (N × H)
184. #432C22×C127semidirect
185. C2×C4⋊C8C22×C127semidirect
187. C2×#97C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34962
C127×#19552
C127×#15512
C127×#15302
C127×#14662
#8456127