Finite Group Databasebeta

← C127×#12698 #44576371

Subgroups — C127×#12698 #44576371

Order 32512.

990 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C2×#536×C127C211yesmaximal, characteristic
2C2×#354×C127C214yesmaximal, complemented
3C127×#1782C212yesmaximal
4C22×#41×C127C2211yescharacteristic
5C23×Q8×C127C411yescharacteristic, complemented
6#536×C127—28no—
7C2×C4⋊Q8×C127C412yescomplemented
8C2×#124×C127C2212yes—
9C2×#123×C127—22no—
10C2×C22⋊Q8×C127—22no—
11C2×#117×C127C411yescharacteristic, complemented
12C2×#99×C127—22no—
13C2×#99×C127C2214yes—
14#381×C127—22no—
15#354×C127—28no—
16C2×#41×C127—24no—
17C2×#41×C127—42no—
18C2×#41×C127—42no—
19C22×Q8×C127—24no—
20C22×Q8×C127—21no—
21C22×Q8×C127—42no—
22C22×Q8×C127—42no—
23C22×Q8×C127C2×C412yes—
24C22×Q8×C127C2×C411yescharacteristic
25C22×D4×C127D412yes—
26C22×D4×C127C2311yescharacteristic
27C23×C4×C127—21no—
28C23×C4×C127C2×C411yescharacteristic
29C23×C4×C127D412yes—
30C4⋊Q8×C127—24no—
31C4⋊Q8×C127—24no—
32#124×C127—24no—
33#124×C127D418yes—
34#123×C127—216no—
35C22⋊Q8×C127—216no—
36#117×C127—24no—
37C2×C4⋊C4×C127—24no—
38C2×C4⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—24no—
41C2×C22⋊C4×C127—24no—
42C2×C42×C127C2×C412yes—
43#99×C127—28no—
44#99×C127—44no—
45#95×C127—44no—
46C24×C127C22⋊C411yescharacteristic
47C24×C127C2×D412yes—
48#41×C127—416no—
49C2×Q8×C127—24no—
50C2×Q8×C127—22no—
51C2×Q8×C127—22no—
52C2×Q8×C127—41no—
53C2×Q8×C127—41no—
54C2×Q8×C127—416no—
55C2×Q8×C127—42no—
56C2×Q8×C127—42no—
57C2×Q8×C127—42no—
58C2×Q8×C127—42no—
59C2×Q8×C127C22⋊C414yes—
60C2×D4×C127—24no—
61C2×D4×C127—24no—
62C2×D4×C127—42no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127C2×D414yes—
66C22×C4×C127—22no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127—24no—
70C22×C4×C127—22no—
71C22×C4×C127—22no—
72C22×C4×C127—24no—
73C22×C4×C127—24no—
74C22×C4×C127—42no—
75C22×C4×C127—42no—
76C22×C4×C127—42no—
77C22×C4×C127—44no—
78C22×C4×C127C22⋊C412yes—
79C22×C4×C127C22⋊C411yescharacteristic
80C22×C4×C127C22×C411yescharacteristic
81C4⋊C4×C127—24no—
82C4⋊C4×C127—216no—
83C22⋊C4×C127—24no—
84C22⋊C4×C127—216no—
85C22⋊C4×C127—44no—
86C22⋊C4×C127—44no—
87C42×C127—22no—
88C42×C127C22⋊C414yes—
89C23×C127—21no—
90C23×C127—21no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—42no—
94C23×C127—44no—
95C23×C127#9912yes—
96C23×C127C2×C22⋊C411yescharacteristic
97C23×C127#12014yes—
98Q8×C127—44no—
99Q8×C127—48no—
100Q8×C127—44no—
101Q8×C127—44no—
102D4×C127—44no—
103D4×C127—48no—
104C2×C4×C127—28no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—24no—
110C2×C4×C127—24no—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—48no—
115C2×C4×C127—44no—
116C2×C4×C127—48no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—84no—
120C2×C4×C127—88no—
121C2×C4×C127#9912yes—
122C2×C4×C127C2×C22⋊C412yes—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—24no—
126C22×C127—42no—
127C22×C127—44no—
128C22×C127—42no—
129C22×C127—44no—
130C22×C127—42no—
131C22×C127—82no—
132C22×C127#37911yescharacteristic
133C22×C127C2×#9911yescharacteristic
134C22×C127C2×#9911yescharacteristic
135C4×C127—22no—
136C4×C127—22no—
137C4×C127—44no—
138C4×C127—42no—
139C4×C127—44no—
140C4×C127—164no—
141#12698C12711yesmaximal, characteristic, direct product factor, Sylow
142C2×C127—21no—
143C2×C127—21no—
144C2×C127—44no—
145C2×C127—42no—
146C2×C127#178111yescharacteristic
147C2×C127#212912yes—
148C2×#536C2×C12711yescharacteristic
149C2×#354C2×C12714yescomplemented
150#1782C2×C12712yes—
151C127#1269811yescharacteristic, direct product factor, Sylow
152C22×#41C22×C12711yescharacteristic
153C23×Q8C4×C12711yescharacteristic, complemented
154#536—28no—
155C2×C4⋊Q8C4×C12712yescomplemented
156C2×#124C22×C12712yes—
157C2×#123—22no—
158C2×C22⋊Q8—22no—
159C2×#117C4×C12711yescharacteristic, complemented
160C2×#99—22no—
161C2×#99C22×C12714yes—
162#381—22no—
163#354—28no—
164C2×#41—24no—
165C2×#41—42no—
166C2×#41—42no—
167C22×Q8—24no—
168C22×Q8—21no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8C2×C4×C12712yes—
172C22×Q8C2×C4×C12711yescharacteristic
173C22×D4D4×C12712yes—
174C22×D4C23×C12711yescharacteristic
175C23×C4—21no—
176C23×C4C2×C4×C12711yescharacteristic
177C23×C4D4×C12712yes—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180#124—24no—
181#124D4×C12718yes—
182#123—216no—
183C22⋊Q8—216no—
184#117—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C42C2×C4×C12712yes—
191#99—28no—
192#99—44no—
193#95—44no—
194C24C22⋊C4×C12711yescharacteristic
195C24C2×D4×C12712yes—
196#41—416no—
197C2×Q8—24no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—416no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8C22⋊C4×C12714yes—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4C2×D4×C12714yes—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4C22⋊C4×C12712yes—
227C22×C4C22⋊C4×C12711yescharacteristic
228C22×C4C22×C4×C12711yescharacteristic
229C4⋊C4—24no—
230C4⋊C4—216no—
231C22⋊C4—24no—
232C22⋊C4—216no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C42—22no—
236C42C22⋊C4×C12714yes—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—22no—
241C23—42no—
242C23—44no—
243C23#99×C12712yes—
244C23C2×C22⋊C4×C12711yescharacteristic
245C23#120×C12714yes—
246Q8—44no—
247Q8—48no—
248Q8—44no—
249Q8—44no—
250D4—44no—
251D4—48no—
252C2×C4—28no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—48no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—84no—
268C2×C4—88no—
269C2×C4#99×C12712yes—
270C2×C4C2×C22⋊C4×C12712yes—
271C22—22no—
272C22—22no—
273C22—24no—
274C22—42no—
275C22—44no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—82no—
280C22#379×C12711yescharacteristic
281C22C2×#99×C12711yescharacteristic
282C22C2×#99×C12711yescharacteristic
283C4—22no—
284C4—22no—
285C4—44no—
286C4—42no—
287C4—44no—
288C4—164no—
289C2—21no—
290C2—21no—
291C2—44no—
292C2—42no—
293C2C127×#178111yescharacteristic
294C2C127×#212912yes—
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC22×#41×C127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#354×C127C2semidirect
5. C23×Q8×C127C4semidirect
7. C2×C4⋊Q8×C127C4semidirect
11. C2×#117×C127C4semidirect
141. #12698C127direct (N × H)
149. C2×#354C2×C127semidirect
151. C127#12698direct (N × H)
153. C23×Q8C4×C127semidirect
155. C2×C4⋊Q8C4×C127semidirect
159. C2×#117C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#536×C1272
C2×#354×C1272
C127×#17822
#12698127