Finite Group Databasebeta

← C127×#8354 #44572027

Subgroups — C127×#8354 #44572027

Order 32512.

950 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3497C211yesmaximal, characteristic
2C127×#1984C211yesmaximal, characteristic, complemented
3C127×#1801C211yesmaximal, characteristic, complemented
4C127×#1547C212yesmaximal, complemented
5C127×#1492C212yesmaximal, complemented
6C2×2−1+4×C127—21no—
7#550×C127—22no—
8D4×Q8×C127—22no—
9#548×C127—24no—
10#537×C127—21no—
11#537×C127C414yes—
12#532×C127—22no—
13#532×C127C2212yes—
14C2×C4×Q8×C127C2211yescharacteristic
15#423×C127C2212yes—
16C2×C4⋊C8×C127C2211yescharacteristic
17C2×#101×C127—21no—
18C2×C4×C8×C127C2211yescharacteristic
19#343×C127—21no—
20#336×C127—22no—
21#332×C127—22no—
22#331×C127—22no—
23D4⋊C8×C127—24no—
242−1+4×C127—44no—
252−1+4×C127—42no—
262−1+4×C127—42no—
27C2×#41×C127—24no—
28C2×#41×C127—24no—
29C2×#41×C127—22no—
30C22×Q8×C127—24no—
31C22×Q8×C127D411yescharacteristic
32C2×M16×C127—22no—
33C22×C8×C127—21no—
34C22×C8×C127—21no—
35C4⋊Q8×C127—22no—
36C4⋊Q8×C127D412yes—
37#127×C127—22no—
38#127×C127D412yes—
39#124×C127—24no—
40#124×C127—22no—
41C22⋊Q8×C127—24no—
42C22⋊Q8×C127—28no—
43#121×C127—24no—
44#121×C127—28no—
45#121×C127—24no—
46#121×C127—22no—
47#121×C127C2×C414yes—
48C4×Q8×C127—22no—
49C4×Q8×C127—22no—
50C4×D4×C127—24no—
51C4×D4×C127—24no—
52C4×D4×C127—22no—
53C4×D4×C127D414yes—
54C2×C4⋊C4×C127C2×C412yes—
55C2×C4⋊C4×C127D411yescharacteristic
56C2×C42×C127D412yes—
57C2×C42×C127C2311yescharacteristic
58C4⋊C8×C127—22no—
59C4⋊C8×C127—22no—
60C4⋊C8×C127—22no—
61#101×C127—42no—
62C4×C8×C127—22no—
63#41×C127—48no—
64#41×C127—48no—
65#41×C127—48no—
66#41×C127—48no—
67#41×C127—42no—
68#41×C127—42no—
69#41×C127—42no—
70#41×C127—42no—
71C2×Q8×C127—24no—
72C2×Q8×C127—22no—
73C2×Q8×C127—22no—
74C2×Q8×C127—22no—
75C2×Q8×C127—48no—
76C2×Q8×C127—48no—
77C2×Q8×C127D812yes—
78C2×Q8×C127SD1612yes—
79C2×D4×C127—24no—
80C2×D4×C127—24no—
81C2×D4×C127—24no—
82C2×D4×C127—22no—
83C2×D4×C127—22no—
84C2×D4×C127C22⋊C414yes—
85C22×C4×C127—28no—
86C22×C4×C127—24no—
87C22×C4×C127C22⋊C412yes—
88C22×C4×C127C22⋊C412yes—
89C22×C4×C127C22×C411yescharacteristic
90C22×C4×C127C2×D411yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92M16×C127—42no—
93M16×C127—42no—
94C2×C8×C127—22no—
95C2×C8×C127—41no—
96C2×C8×C127—41no—
97C2×C8×C127—42no—
98C2×C8×C127—41no—
99C2×C8×C127—42no—
100C4⋊C4×C127—24no—
101C4⋊C4×C127—24no—
102C4⋊C4×C127—22no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127C22⋊C414yes—
105C22⋊C4×C127—28no—
106C22⋊C4×C127—24no—
107C22⋊C4×C127—24no—
108C42×C127—22no—
109C42×C127—22no—
110C42×C127C2×D412yes—
111C42×C127C2×D412yes—
112C23×C127—24no—
113C23×C127C2×C22⋊C411yescharacteristic
114Q8×C127—24no—
115Q8×C127—24no—
116Q8×C127—48no—
117Q8×C127—42no—
118Q8×C127—42no—
119D4×C127—44no—
120D4×C127—48no—
121D4×C127—44no—
122D4×C127—44no—
123D4×C127—42no—
124D4×C127—42no—
125D4×C127—42no—
126D4×C127—42no—
127D4×C127—84no—
128C2×C4×C127—22no—
129C2×C4×C127—22no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—22no—
133C2×C4×C127—24no—
134C2×C4×C127—41no—
135C2×C4×C127—416no—
136C2×C4×C127—44no—
137C2×C4×C127—44no—
138C2×C4×C127—42no—
139C2×C4×C127#10214yes—
140C2×C4×C127C2×C22⋊C412yes—
141C2×C4×C127#12011yescharacteristic
142C2×C4×C127#12012yes—
143C2×C4×C127#12011yescharacteristic
144C2×C4×C127C2×D811yescharacteristic
145C2×C4×C127C2×SD1611yescharacteristic
146C2×C4×C127AGL(1,ℤ/8)12yes—
147C8×C127—42no—
148C8×C127—82no—
149C8×C127—81no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127—44no—
153C22×C127—44no—
154C22×C127—44no—
155C22×C127#37911yescharacteristic
156C22×C127C2×#10211yescharacteristic
157C22×C127#41811yescharacteristic
158C4×C127—21no—
159C4×C127—24no—
160C4×C127—22no—
161C4×C127—44no—
162C4×C127—42no—
163C4×C127#44712yes—
164C4×C127#45012yes—
165C4×C127#45411yescharacteristic
166C4×C127#45611yescharacteristic
167#8354C12711yesmaximal, characteristic, direct product factor, Sylow
168C2×C127—21no—
169C2×C127—21no—
170C2×C127—84no—
171C2×C127#178611yescharacteristic
172C2×C127#179111yescharacteristic
173C2×C127#179311yescharacteristic
174#3497C2×C12711yescharacteristic
175#1984C2×C12711yescharacteristic, complemented
176#1801C2×C12711yescharacteristic, complemented
177#1547C2×C12712yescomplemented
178#1492C2×C12712yescomplemented
179C127#835411yescharacteristic, direct product factor, Sylow
180C2×2−1+4—21no—
181#550—22no—
182D4×Q8—22no—
183#548—24no—
184#537—21no—
185#537C4×C12714yes—
186#532—22no—
187#532C22×C12712yes—
188C2×C4×Q8C22×C12711yescharacteristic
189#423C22×C12712yes—
190C2×C4⋊C8C22×C12711yescharacteristic
191C2×#101—21no—
192C2×C4×C8C22×C12711yescharacteristic
193#343—21no—
194#336—22no—
195#332—22no—
196#331—22no—
197D4⋊C8—24no—
1982−1+4—44no—
1992−1+4—42no—
2002−1+4—42no—
201C2×#41—24no—
202C2×#41—24no—
203C2×#41—22no—
204C22×Q8—24no—
205C22×Q8D4×C12711yescharacteristic
206C2×M16—22no—
207C22×C8—21no—
208C22×C8—21no—
209C4⋊Q8—22no—
210C4⋊Q8D4×C12712yes—
211#127—22no—
212#127D4×C12712yes—
213#124—24no—
214#124—22no—
215C22⋊Q8—24no—
216C22⋊Q8—28no—
217#121—24no—
218#121—28no—
219#121—24no—
220#121—22no—
221#121C2×C4×C12714yes—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—22no—
227C4×D4D4×C12714yes—
228C2×C4⋊C4C2×C4×C12712yes—
229C2×C4⋊C4D4×C12711yescharacteristic
230C2×C42D4×C12712yes—
231C2×C42C23×C12711yescharacteristic
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235#101—42no—
236C4×C8—22no—
237#41—48no—
238#41—48no—
239#41—48no—
240#41—48no—
241#41—42no—
242#41—42no—
243#41—42no—
244#41—42no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8D8×C12712yes—
252C2×Q8SD16×C12712yes—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4C22⋊C4×C12714yes—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4C22⋊C4×C12712yes—
262C22×C4C22⋊C4×C12712yes—
263C22×C4C22×C4×C12711yescharacteristic
264C22×C4C2×D4×C12711yescharacteristic
265C22×C4C2×D4×C12711yescharacteristic
266M16—42no—
267M16—42no—
268C2×C8—22no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—42no—
272C2×C8—41no—
273C2×C8—42no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4C22⋊C4×C12714yes—
279C22⋊C4—28no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C42—22no—
283C42—22no—
284C42C2×D4×C12712yes—
285C42C2×D4×C12712yes—
286C23—24no—
287C23C2×C22⋊C4×C12711yescharacteristic
288Q8—24no—
289Q8—24no—
290Q8—48no—
291Q8—42no—
292Q8—42no—
293D4—44no—
294D4—48no—
295D4—44no—
296D4—44no—
297D4—42no—
298D4—42no—
299D4—42no—
300D4—42no—
301D4—84no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—41no—
309C2×C4—416no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4#102×C12714yes—
314C2×C4C2×C22⋊C4×C12712yes—
315C2×C4#120×C12711yescharacteristic
316C2×C4#120×C12712yes—
317C2×C4#120×C12711yescharacteristic
318C2×C4C2×D8×C12711yescharacteristic
319C2×C4C2×SD16×C12711yescharacteristic
320C2×C4AGL(1,ℤ/8)×C12712yes—
321C8—42no—
322C8—82no—
323C8—81no—
324C22—21no—
325C22—21no—
326C22—44no—
327C22—44no—
328C22—44no—
329C22#379×C12711yescharacteristic
330C22C2×#102×C12711yescharacteristic
331C22#418×C12711yescharacteristic
332C4—24no—
333C4—22no—
334C4—21no—
335C4—44no—
336C4—42no—
337C4#447×C12712yes—
338C4#450×C12712yes—
339C4#454×C12711yescharacteristic
340C4#456×C12711yescharacteristic
341C2—21no—
342C2—21no—
343C2—84no—
344C2C127×#178611yescharacteristic
345C2C127×#179111yescharacteristic
346C2C127×#179311yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3497C2
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×#1984C2semidirect
3. C127×#1801C2semidirect
4. C127×#1547C2semidirect
5. C127×#1492C2semidirect
167. #8354C127direct (N × H)
175. #1984C2×C127semidirect
176. #1801C2×C127semidirect
177. #1547C2×C127semidirect
178. #1492C2×C127semidirect
179. C127#8354direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34972
C127×#19842
C127×#18012
C127×#15472
C127×#14922
#8354127