Finite Group Databasebeta

← C127×#10318 #44573991

Subgroups — C127×#10318 #44573991

Order 32512.

602 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#498×C127C211yesmaximal, characteristic
2C2×#494×C127C211yesmaximal, characteristic
3C2×#487×C127C211yesmaximal, characteristic
4C127×#1990C211yesmaximal, characteristic
5C127×#1990C211yesmaximal, characteristic
6C127×#1990C211yesmaximal, characteristic
7C127×#1273C211yesmaximal, characteristic
8C22×Q16×C127—22no—
9C2×C4⋊Q8×C127C2211yescharacteristic
10C2×C4⋊Q8×C127C2211yescharacteristic
11C2×C4⋊Q8×C127C2211yescharacteristic
12#498×C127—22no—
13#498×C127—22no—
14#494×C127—22no—
15#494×C127—22no—
16#487×C127—22no—
17#487×C127—22no—
18C2×#106×C127—22no—
19C2×#103×C127—22no—
20C2×C4×C8×C127C2211yescharacteristic
21C2×C4×C8×C127C2211yescharacteristic
22C2×C4×C8×C127C2211yescharacteristic
23#384×C127C2211yescharacteristic
24#340×C127—22no—
25#340×C127—22no—
26#340×C127—22no—
27C22×Q8×C127—21no—
28C22×Q8×C127—21no—
29C22×Q8×C127—21no—
30C2×Q16×C127—28no—
31C2×Q16×C127—44no—
32C2×Q16×C127—44no—
33C22×C8×C127D412yes—
34C22×C8×C127D412yes—
35C22×C8×C127D412yes—
36C4⋊Q8×C127—21no—
37C4⋊Q8×C127—21no—
38C4⋊Q8×C127—21no—
39C4⋊Q8×C127—21no—
40C4⋊Q8×C127—21no—
41C4⋊Q8×C127—21no—
42C4⋊Q8×C127—21no—
43C4⋊Q8×C127—21no—
44C4⋊Q8×C127—21no—
45C4⋊Q8×C127D412yes—
46C4⋊Q8×C127D412yes—
47C4⋊Q8×C127D412yes—
48C2×C4⋊C4×C127—21no—
49C2×C4⋊C4×C127—21no—
50C2×C4⋊C4×C127—21no—
51C2×C4⋊C4×C127—21no—
52C2×C4⋊C4×C127—21no—
53C2×C4⋊C4×C127—21no—
54C2×C4⋊C4×C127—21no—
55C2×C4⋊C4×C127—21no—
56C2×C4⋊C4×C127—21no—
57C2×C42×C127C2311yescharacteristic
58#106×C127—28no—
59#103×C127—28no—
60C4×C8×C127—21no—
61C4×C8×C127—21no—
62C4×C8×C127—21no—
63C4×C8×C127—21no—
64C4×C8×C127—21no—
65C4×C8×C127—21no—
66C2×Q8×C127—22no—
67C2×Q8×C127—22no—
68C2×Q8×C127—22no—
69C2×Q8×C127—41no—
70C2×Q8×C127—41no—
71C2×Q8×C127—41no—
72C2×Q8×C127—42no—
73C2×Q8×C127—42no—
74C2×Q8×C127—42no—
75C2×Q8×C127—42no—
76C2×Q8×C127—42no—
77C2×Q8×C127—42no—
78C22×C4×C127—41no—
79C22×C4×C127—41no—
80C22×C4×C127—41no—
81C22×C4×C127—41no—
82C22×C4×C127C2×D411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84C22×C4×C127C2×D411yescharacteristic
85Q16×C127—416no—
86C2×C8×C127—22no—
87C2×C8×C127—22no—
88C2×C8×C127—22no—
89C2×C8×C127—22no—
90C2×C8×C127—22no—
91C2×C8×C127—22no—
92C2×C8×C127—22no—
93C2×C8×C127SD1614yes—
94C2×C8×C127Q1614yescomplemented
95C4⋊C4×C127—22no—
96C4⋊C4×C127—22no—
97C4⋊C4×C127—22no—
98C4⋊C4×C127—22no—
99C4⋊C4×C127—22no—
100C4⋊C4×C127—22no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127—41no—
104C4⋊C4×C127—41no—
105C4⋊C4×C127—41no—
106C4⋊C4×C127—41no—
107C4⋊C4×C127—42no—
108C4⋊C4×C127—42no—
109C4⋊C4×C127—42no—
110C42×C127C2×D411yescharacteristic
111C42×C127C2×D411yescharacteristic
112C42×C127C2×D411yescharacteristic
113C42×C127#4111yescharacteristic
114C23×C127#12711yescharacteristic
115Q8×C127—44no—
116Q8×C127—44no—
117Q8×C127—44no—
118Q8×C127—82no—
119Q8×C127—82no—
120Q8×C127—82no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—42no—
124C2×C4×C127—81no—
125C2×C4×C127—81no—
126C2×C4×C127—81no—
127C2×C4×C127—81no—
128C2×C4×C127—81no—
129C2×C4×C127—81no—
130C2×C4×C127—81no—
131C2×C4×C127—81no—
132C2×C4×C127—81no—
133C2×C4×C127#12011yescharacteristic
134C2×C4×C127#12011yescharacteristic
135C2×C4×C127#12011yescharacteristic
136C2×C4×C127#12111yescharacteristic
137C2×C4×C127#12111yescharacteristic
138C2×C4×C127#12111yescharacteristic
139C2×C4×C127C2×SD1612yes—
140C2×C4×C127C2×Q1612yes—
141C2×C4×C127#13512yes—
142C2×C4×C127AGL(1,ℤ/8)12yes—
143C2×C4×C127#13712yes—
144C2×C4×C127#13712yes—
145C8×C127—24no—
146C8×C127—24no—
147C8×C127—42no—
148C22×C127#39211yescharacteristic
149C22×C127#49211yescharacteristic
150C22×C127#49411yescharacteristic
151C22×C127#49511yescharacteristic
152C22×C127#49611yescharacteristic
153C22×C127#49711yescharacteristic
154C22×C127#49711yescharacteristic
155C4×C127—82no—
156C4×C127—82no—
157C4×C127—82no—
158C4×C127—161no—
159C4×C127#45012yes—
160C4×C127#45112yes—
161C4×C127#45212yes—
162C4×C127#46012yes—
163C4×C127#46212yes—
164C4×C127#46412yes—
165#10318C12711yesmaximal, characteristic, direct product factor, Sylow
166C2×C127#170711yescharacteristic
167C2×C127#171611yescharacteristic
168C2×C127#172011yescharacteristic
169C2×C127#172311yescharacteristic
170C2×C127#201411yescharacteristic
171C2×C127#201511yescharacteristic
172C2×C127#201611yescharacteristic
173C2×#498C2×C12711yescharacteristic
174C2×#494C2×C12711yescharacteristic
175C2×#487C2×C12711yescharacteristic
176#1990C2×C12711yescharacteristic
177#1990C2×C12711yescharacteristic
178#1990C2×C12711yescharacteristic
179#1273C2×C12711yescharacteristic
180C127#1031811yescharacteristic, direct product factor, Sylow
181C22×Q16—22no—
182C2×C4⋊Q8C22×C12711yescharacteristic
183C2×C4⋊Q8C22×C12711yescharacteristic
184C2×C4⋊Q8C22×C12711yescharacteristic
185#498—22no—
186#498—22no—
187#494—22no—
188#494—22no—
189#487—22no—
190#487—22no—
191C2×#106—22no—
192C2×#103—22no—
193C2×C4×C8C22×C12711yescharacteristic
194C2×C4×C8C22×C12711yescharacteristic
195C2×C4×C8C22×C12711yescharacteristic
196#384C22×C12711yescharacteristic
197#340—22no—
198#340—22no—
199#340—22no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C2×Q16—28no—
204C2×Q16—44no—
205C2×Q16—44no—
206C22×C8D4×C12712yes—
207C22×C8D4×C12712yes—
208C22×C8D4×C12712yes—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8D4×C12712yes—
219C4⋊Q8D4×C12712yes—
220C4⋊Q8D4×C12712yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C42C23×C12711yescharacteristic
231#106—28no—
232#103—28no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×D4×C12711yescharacteristic
256C22×C4C2×D4×C12711yescharacteristic
257C22×C4C2×D4×C12711yescharacteristic
258Q16—416no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8SD16×C12714yes—
267C2×C8Q16×C12714yescomplemented
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C42C2×D4×C12711yescharacteristic
284C42C2×D4×C12711yescharacteristic
285C42C2×D4×C12711yescharacteristic
286C42#41×C12711yescharacteristic
287C23#127×C12711yescharacteristic
288Q8—44no—
289Q8—44no—
290Q8—44no—
291Q8—82no—
292Q8—82no—
293Q8—82no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4#120×C12711yescharacteristic
307C2×C4#120×C12711yescharacteristic
308C2×C4#120×C12711yescharacteristic
309C2×C4#121×C12711yescharacteristic
310C2×C4#121×C12711yescharacteristic
311C2×C4#121×C12711yescharacteristic
312C2×C4C2×SD16×C12712yes—
313C2×C4C2×Q16×C12712yes—
314C2×C4#135×C12712yes—
315C2×C4AGL(1,ℤ/8)×C12712yes—
316C2×C4#137×C12712yes—
317C2×C4#137×C12712yes—
318C8—24no—
319C8—24no—
320C8—42no—
321C22#392×C12711yescharacteristic
322C22#492×C12711yescharacteristic
323C22#494×C12711yescharacteristic
324C22#495×C12711yescharacteristic
325C22#496×C12711yescharacteristic
326C22#497×C12711yescharacteristic
327C22#497×C12711yescharacteristic
328C4—82no—
329C4—82no—
330C4—82no—
331C4—161no—
332C4#450×C12712yes—
333C4#451×C12712yes—
334C4#452×C12712yes—
335C4#460×C12712yes—
336C4#462×C12712yes—
337C4#464×C12712yes—
338C2C127×#170711yescharacteristic
339C2C127×#171611yescharacteristic
340C2C127×#172011yescharacteristic
341C2C127×#172311yescharacteristic
342C2C127×#201411yescharacteristic
343C2C127×#201511yescharacteristic
344C2C127×#201611yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C127#127
commutator, Frattini, cosocleC2×C42C23×C127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
94. C2×C8×C127Q16semidirect
165. #10318C127direct (N × H)
180. C127#10318direct (N × H)
267. C2×C8Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#498×C1272
C2×#494×C1272
C2×#487×C1272
C127×#19902
C127×#12732
#10318127