Finite Group Databasebeta

← C127×#11815 #44575488

Subgroups — C127×#11815 #44575488

Order 32512.

586 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 106 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×#395×C127C211yesmaximal, characteristic
2C127×#1821C211yesmaximal, characteristic
3C127×#1463C211yesmaximal, characteristic
4C127×#1354C214yesmaximal, complemented
5C23×Q8×C127C411yescharacteristic
6C22×C4⋊C4×C127—21no—
7C22×C4⋊C4×C127C2211yescharacteristic
8#407×C127—21no—
9#407×C127—21no—
10#395×C127—21no—
11#395×C127—21no—
12#395×C127C412yes—
13#395×C127C2212yes—
14C2×#95×C127—21no—
15C2×#95×C127C411yescharacteristic
16#343×C127—21no—
17#343×C127C2212yes—
18#324×C127—21no—
19#324×C127C2212yes—
20C22×Q8×C127—21no—
21C22×Q8×C127—44no—
22C22×Q8×C127—41no—
23C22×Q8×C127—41no—
24C22×Q8×C127C2×C412yes—
25C23×C4×C127—21no—
26C23×C4×C127—21no—
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127D411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C2×M16×C127—22no—
31C2×M16×C127—41no—
32C2×C4⋊C4×C127—22no—
33C2×C4⋊C4×C127—22no—
34C2×C4⋊C4×C127—41no—
35C2×C4⋊C4×C127—41no—
36C2×C4⋊C4×C127—44no—
37C2×C4⋊C4×C127C2×C411yescharacteristic
38C2×C4⋊C4×C127D411yescharacteristic
39C2×C4⋊C4×C127D411yescharacteristic
40C2×C4⋊C4×C127C2311yescharacteristic
41C22⋊C8×C127—22no—
42C22⋊C8×C127—41no—
43C22⋊C8×C127—41no—
44C22⋊C8×C127—41no—
45#95×C127—22no—
46#95×C127—21no—
47#95×C127—22no—
48#95×C127C2×C412yes—
49C24×C127C22⋊C411yescharacteristic
50C2×Q8×C127—44no—
51C2×Q8×C127—44no—
52C2×Q8×C127—42no—
53C2×Q8×C127—42no—
54C2×Q8×C127—48no—
55C2×Q8×C127—44no—
56C2×Q8×C127—44no—
57C22×C4×C127—24no—
58C22×C4×C127—24no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127—21no—
64C22×C4×C127—22no—
65C22×C4×C127—22no—
66C22×C4×C127—21no—
67C22×C4×C127—22no—
68C22×C4×C127—41no—
69C22×C4×C127—42no—
70C22×C4×C127—42no—
71C22×C4×C127—42no—
72C22×C4×C127—41no—
73C22×C4×C127—41no—
74C22×C4×C127—42no—
75C22×C4×C127—42no—
76C22×C4×C127C22⋊C411yescharacteristic
77C22×C4×C127C22⋊C411yescharacteristic
78C22×C4×C127C22⋊C411yescharacteristic
79C22×C4×C127C22×C411yescharacteristic
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82M16×C127—42no—
83M16×C127—42no—
84M16×C127—82no—
85C2×C8×C127—42no—
86C2×C8×C127—42no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—44no—
92C4⋊C4×C127—44no—
93C23×C127—22no—
94C23×C127—22no—
95C23×C127—21no—
96C23×C127—21no—
97C23×C127#9911yescharacteristic
98C23×C127#9911yescharacteristic
99C23×C127C2×C22⋊C411yescharacteristic
100Q8×C127—44no—
101Q8×C127—44no—
102Q8×C127—48no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—24no—
106C2×C4×C127—24no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127—48no—
118C2×C4×C127—42no—
119C2×C4×C127—44no—
120C2×C4×C127—42no—
121C2×C4×C127—42no—
122C2×C4×C127—44no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127—42no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—81no—
131C2×C4×C127#13712yes—
132C8×C127—82no—
133C8×C127—82no—
134C22×C127—24no—
135C22×C127—24no—
136C22×C127—21no—
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127AΣL(1,𝔽16)12yes—
143C22×C127#35612yes—
144C22×C127C2×#9911yescharacteristic
145C22×C127#41911yescharacteristic
146C22×C127#42111yescharacteristic
147C4×C127—22no—
148C4×C127—44no—
149C4×C127—41no—
150C4×C127—44no—
151C4×C127—41no—
152C4×C127—41no—
153C4×C127—41no—
154C4×C127—41no—
155#11815C12711yesmaximal, characteristic, direct product factor, Sylow
156C2×C127—21no—
157C2×C127—22no—
158C2×C127—22no—
159C2×C127—21no—
160C2×C127#151111yescharacteristic
161C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
162C2×C127C2×#35611yescharacteristic
163C2×#395C2×C12711yescharacteristic
164#1821C2×C12711yescharacteristic
165#1463C2×C12711yescharacteristic
166#1354C2×C12714yescomplemented
167C127#1181511yescharacteristic, direct product factor, Sylow
168C23×Q8C4×C12711yescharacteristic
169C22×C4⋊C4—21no—
170C22×C4⋊C4C22×C12711yescharacteristic
171#407—21no—
172#407—21no—
173#395—21no—
174#395—21no—
175#395C4×C12712yes—
176#395C22×C12712yes—
177C2×#95—21no—
178C2×#95C4×C12711yescharacteristic
179#343—21no—
180#343C22×C12712yes—
181#324—21no—
182#324C22×C12712yes—
183C22×Q8—21no—
184C22×Q8—44no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8C2×C4×C12712yes—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C2×C4×C12711yescharacteristic
191C23×C4D4×C12711yescharacteristic
192C23×C4D4×C12711yescharacteristic
193C2×M16—22no—
194C2×M16—41no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4C2×C4×C12711yescharacteristic
201C2×C4⋊C4D4×C12711yescharacteristic
202C2×C4⋊C4D4×C12711yescharacteristic
203C2×C4⋊C4C23×C12711yescharacteristic
204C22⋊C8—22no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95C2×C4×C12712yes—
212C24C22⋊C4×C12711yescharacteristic
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—48no—
218C2×Q8—44no—
219C2×Q8—44no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊C4×C12711yescharacteristic
240C22×C4C22⋊C4×C12711yescharacteristic
241C22×C4C22⋊C4×C12711yescharacteristic
242C22×C4C22×C4×C12711yescharacteristic
243C22×C4C2×D4×C12711yescharacteristic
244C22×C4C2×D4×C12711yescharacteristic
245M16—42no—
246M16—42no—
247M16—82no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—21no—
260C23#99×C12711yescharacteristic
261C23#99×C12711yescharacteristic
262C23C2×C22⋊C4×C12711yescharacteristic
263Q8—44no—
264Q8—44no—
265Q8—48no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4#137×C12712yes—
295C8—82no—
296C8—82no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22AΣL(1,𝔽16)×C12712yes—
306C22#356×C12712yes—
307C22C2×#99×C12711yescharacteristic
308C22#419×C12711yescharacteristic
309C22#421×C12711yescharacteristic
310C4—22no—
311C4—44no—
312C4—41no—
313C4—44no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C2—21no—
319C2—22no—
320C2—22no—
321C2—21no—
322C2C127×#151111yescharacteristic
323C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
324C2C2×#356×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C22×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1354C2semidirect
155. #11815C127direct (N × H)
166. #1354C2×C127semidirect
167. C127#11815direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#395×C1272
C127×#18212
C127×#14632
C127×#13542
#11815127