Finite Group Databasebeta

← C127×#8397 #44572070

Subgroups — C127×#8397 #44572070

Order 32512.

802 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555×C127C211yesmaximal, characteristic
2C127×#1800C211yesmaximal, characteristic, complemented
3C127×#1549C212yesmaximal, complemented
4C127×#1486C212yesmaximal, complemented
5C127×#1460C211yesmaximal, characteristic, complemented
6#555×C127—24no—
7#555×C127—24no—
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×#124×C127—22no—
10C2×#121×C127—21no—
11C2×#121×C127C412yes—
12C2×C4×D4×C127C2211yescharacteristic
13C2×C4×D4×C127C2211yescharacteristic
14C2×#117×C127—21no—
15C2×#117×C127C412yes—
16#431×C127C2212yes—
17#423×C127C2212yes—
18C2×#100×C127—21no—
19#343×C127—21no—
20#332×C127—24no—
21D4⋊C8×C127—24no—
22#323×C127—22no—
23C22×Q8×C127—21no—
24C22×D4×C127—22no—
25C22×D4×C127C2×C411yescharacteristic
26C22×D4×C127D411yescharacteristic
27C23×C4×C127—21no—
28C23×C4×C127C2×C412yes—
29C2×M16×C127—22no—
30C4⋊Q8×C127—22no—
31C4⋊Q8×C127D414yes—
32#124×C127—216no—
33#121×C127—28no—
34#121×C127—24no—
35#121×C127—24no—
36C4×D4×C127—22no—
37C4×D4×C127—22no—
38C4×D4×C127—22no—
39C4×D4×C127D414yes—
40#117×C127—28no—
41#117×C127—24no—
42#117×C127—24no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127C2×C411yescharacteristic
45C2×C4⋊C4×C127C2×C412yes—
46C2×C4⋊C4×C127D411yescharacteristic
47C2×C22⋊C4×C127—22no—
48C2×C22⋊C4×C127—22no—
49C2×C22⋊C4×C127—21no—
50C2×C22⋊C4×C127D412yes—
51C2×C42×C127—22no—
52C2×C42×C127C2311yescharacteristic
53C4⋊C8×C127—22no—
54C4⋊C8×C127—22no—
55C4⋊C8×C127—22no—
56#100×C127—42no—
57C22⋊C8×C127—22no—
58C4×C8×C127—22no—
59C24×C127—21no—
60C24×C127C22⋊C412yes—
61C2×Q8×C127—24no—
62C2×Q8×C127—82no—
63C2×D4×C127—28no—
64C2×D4×C127—21no—
65C2×D4×C127—22no—
66C2×D4×C127—22no—
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—42no—
70C2×D4×C127—84no—
71C2×D4×C127C22⋊C412yes—
72C22×C4×C127—21no—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—24no—
77C22×C4×C127—44no—
78C22×C4×C127—42no—
79C22×C4×C127—41no—
80C22×C4×C127—41no—
81C22×C4×C127C22⋊C412yes—
82C22×C4×C127C22×C411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84C22×C4×C127C2×D411yescharacteristic
85M16×C127—42no—
86M16×C127—42no—
87C2×C8×C127—42no—
88C2×C8×C127—42no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—22no—
91C4⋊C4×C127—24no—
92C4⋊C4×C127—22no—
93C4⋊C4×C127C22⋊C412yes—
94C4⋊C4×C127C22⋊C414yes—
95C22⋊C4×C127—28no—
96C22⋊C4×C127—28no—
97C22⋊C4×C127—24no—
98C22⋊C4×C127—24no—
99C42×C127—28no—
100C42×C127C2×D412yes—
101C42×C127C2×D412yes—
102C23×C127—22no—
103C23×C127—22no—
104C23×C127—22no—
105C23×C127—22no—
106C23×C127—42no—
107C23×C127—42no—
108C23×C127—42no—
109C23×C127—42no—
110C23×C127—41no—
111C23×C127—41no—
112C23×C127C2×C22⋊C411yescharacteristic
113Q8×C127—84no—
114D4×C127—24no—
115D4×C127—42no—
116D4×C127—42no—
117D4×C127—42no—
118D4×C127—88no—
119C2×C4×C127—22no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—24no—
123C2×C4×C127—41no—
124C2×C4×C127—44no—
125C2×C4×C127—44no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—44no—
129C2×C4×C127—44no—
130C2×C4×C127—44no—
131C2×C4×C127—42no—
132C2×C4×C127C2×C22⋊C411yescharacteristic
133C2×C4×C127C2×C22⋊C411yescharacteristic
134C2×C4×C127#12012yes—
135C2×C4×C127#12012yes—
136C2×C4×C127#13512yes—
137C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C127#13711yescharacteristic
139C8×C127—82no—
140C8×C127—82no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—42no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127—42no—
147C22×C127—42no—
148C22×C127—42no—
149C22×C127—82no—
150C22×C127—82no—
151C22×C127—82no—
152C22×C127—82no—
153C22×C127#37911yescharacteristic
154C22×C127#41611yescharacteristic
155C22×C127#41711yescharacteristic
156C4×C127—22no—
157C4×C127—21no—
158C4×C127—44no—
159C4×C127—42no—
160C4×C127—42no—
161C4×C127#41312yes—
162C4×C127#44912yes—
163C4×C127#45212yes—
164#8397C12711yesmaximal, characteristic, direct product factor, Sylow
165C2×C127—21no—
166C2×C127—21no—
167C2×C127—82no—
168C2×C127—82no—
169C2×C127#178511yescharacteristic
170C2×C127#178811yescharacteristic
171C2×C127#179511yescharacteristic
172C2×#555C2×C12711yescharacteristic
173#1800C2×C12711yescharacteristic, complemented
174#1549C2×C12712yescomplemented
175#1486C2×C12712yescomplemented
176#1460C2×C12711yescharacteristic, complemented
177C127#839711yescharacteristic, direct product factor, Sylow
178#555—24no—
179#555—24no—
180C2×C4⋊Q8C22×C12711yescharacteristic
181C2×#124—22no—
182C2×#121—21no—
183C2×#121C4×C12712yes—
184C2×C4×D4C22×C12711yescharacteristic
185C2×C4×D4C22×C12711yescharacteristic
186C2×#117—21no—
187C2×#117C4×C12712yes—
188#431C22×C12712yes—
189#423C22×C12712yes—
190C2×#100—21no—
191#343—21no—
192#332—24no—
193D4⋊C8—24no—
194#323—22no—
195C22×Q8—21no—
196C22×D4—22no—
197C22×D4C2×C4×C12711yescharacteristic
198C22×D4D4×C12711yescharacteristic
199C23×C4—21no—
200C23×C4C2×C4×C12712yes—
201C2×M16—22no—
202C4⋊Q8—22no—
203C4⋊Q8D4×C12714yes—
204#124—216no—
205#121—28no—
206#121—24no—
207#121—24no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4D4×C12714yes—
212#117—28no—
213#117—24no—
214#117—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2×C4×C12711yescharacteristic
217C2×C4⋊C4C2×C4×C12712yes—
218C2×C4⋊C4D4×C12711yescharacteristic
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4D4×C12712yes—
223C2×C42—22no—
224C2×C42C23×C12711yescharacteristic
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228#100—42no—
229C22⋊C8—22no—
230C4×C8—22no—
231C24—21no—
232C24C22⋊C4×C12712yes—
233C2×Q8—24no—
234C2×Q8—82no—
235C2×D4—28no—
236C2×D4—21no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—42no—
242C2×D4—84no—
243C2×D4C22⋊C4×C12712yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4C22⋊C4×C12712yes—
254C22×C4C22×C4×C12711yescharacteristic
255C22×C4C2×D4×C12711yescharacteristic
256C22×C4C2×D4×C12711yescharacteristic
257M16—42no—
258M16—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4C22⋊C4×C12712yes—
266C4⋊C4C22⋊C4×C12714yes—
267C22⋊C4—28no—
268C22⋊C4—28no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C42—28no—
272C42C2×D4×C12712yes—
273C42C2×D4×C12712yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊C4×C12711yescharacteristic
285Q8—84no—
286D4—24no—
287D4—42no—
288D4—42no—
289D4—42no—
290D4—88no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4C2×C22⋊C4×C12711yescharacteristic
305C2×C4C2×C22⋊C4×C12711yescharacteristic
306C2×C4#120×C12712yes—
307C2×C4#120×C12712yes—
308C2×C4#135×C12712yes—
309C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
310C2×C4#137×C12711yescharacteristic
311C8—82no—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22#379×C12711yescharacteristic
326C22#416×C12711yescharacteristic
327C22#417×C12711yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—42no—
332C4—44no—
333C4#413×C12712yes—
334C4#449×C12712yes—
335C4#452×C12712yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—82no—
340C2C127×#178511yescharacteristic
341C2C127×#178811yescharacteristic
342C2C127×#179511yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#555×C127C2
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×#1800C2semidirect
3. C127×#1549C2semidirect
4. C127×#1486C2semidirect
5. C127×#1460C2semidirect
164. #8397C127direct (N × H)
173. #1800C2×C127semidirect
174. #1549C2×C127semidirect
175. #1486C2×C127semidirect
176. #1460C2×C127semidirect
177. C127#8397direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#555×C1272
C127×#18002
C127×#15492
C127×#14862
C127×#14602
#8397127