Finite Group Databasebeta

← C127×#7236 #44570909

Subgroups — C127×#7236 #44570909

Order 32512.

372 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C127×#1869C211yesmaximal, characteristic
2C127×#1800C211yesmaximal, characteristic
3C127×#1799C211yesmaximal, characteristic
4C23×C8×C127C411yescharacteristic, complemented
5C2×C4×D4×C127C2211yescharacteristic
6#423×C127—21no—
7C2×#100×C127—21no—
8C2×#99×C127—21no—
9C2×C22⋊C8×C127—21no—
10C2×C22⋊C8×C127C411yescharacteristic, complemented
11#380×C127—21no—
12#343×C127C412yescomplemented
13#342×C127C412yes—
14#336×C127—21no—
15C22×D4×C127D411yescharacteristic
16C23×C4×C127C2×C411yescharacteristic
17C23×C4×C127D411yescharacteristic
18C2×M16×C127—22no—
19C22×C8×C127—21no—
20C22×C8×C127—21no—
21C22×C8×C127—42no—
22C22×C8×C127—41no—
23C4×D4×C127—41no—
24C4×D4×C127—41no—
25C2×C4⋊C4×C127—41no—
26C2×C4⋊C4×C127C2×C411yescharacteristic
27C2×C22⋊C4×C127—22no—
28C2×C22⋊C4×C127C2×C411yescharacteristic
29C2×C22⋊C4×C127Q811yescharacteristic
30C2×C42×C127D411yescharacteristic
31C4⋊C8×C127—41no—
32C4⋊C8×C127—41no—
33#100×C127—41no—
34#100×C127—41no—
35#99×C127—41no—
36#99×C127—41no—
37C22⋊C8×C127—42no—
38C22⋊C8×C127—41no—
39#95×C127—41no—
40C24×C127C22⋊C411yescharacteristic
41C24×C127C4⋊C411yescharacteristic
42C2×D4×C127—41no—
43C2×D4×C127—41no—
44C2×D4×C127D811yescharacteristic
45C2×D4×C127SD1611yescharacteristic
46C2×D4×C127SD1611yescharacteristic
47C2×D4×C127Q1611yescharacteristic
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127—21no—
51C22×C4×C127—21no—
52C22×C4×C127—41no—
53C22×C4×C127—41no—
54C22×C4×C127—41no—
55C22×C4×C127—41no—
56C22×C4×C127—42no—
57C22×C4×C127C4211yescharacteristic
58C22×C4×C127C22⋊C411yescharacteristic
59C22×C4×C127C22⋊C411yescharacteristic
60C22×C4×C127C4⋊C411yescharacteristic
61C22×C4×C127C4⋊C411yescharacteristic
62M16×C127—42no—
63M16×C127—42no—
64C2×C8×C127—22no—
65C2×C8×C127—41no—
66C2×C8×C127—42no—
67C2×C8×C127—41no—
68C2×C8×C127—42no—
69C2×C8×C127—41no—
70C2×C8×C127—42no—
71C2×C8×C127—41no—
72C2×C8×C127—41no—
73C4⋊C4×C127—41no—
74C4⋊C4×C127—81no—
75C4⋊C4×C127—81no—
76C22⋊C4×C127—41no—
77C22⋊C4×C127—41no—
78C22⋊C4×C127—42no—
79C22⋊C4×C127—42no—
80C42×C127—21no—
81C42×C127—21no—
82C23×C127—21no—
83C23×C127—21no—
84C23×C127—21no—
85C23×C127—21no—
86C23×C127—41no—
87C23×C127—41no—
88C23×C127—41no—
89C23×C127—41no—
90C23×C127#9511yescharacteristic
91C23×C127#10211yescharacteristic
92C23×C127#10311yescharacteristic
93C23×C127#10611yescharacteristic
94C23×C127#10711yescharacteristic
95D4×C127—41no—
96D4×C127—41no—
97D4×C127—41no—
98D4×C127—41no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—42no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127—81no—
115C2×C4×C127—82no—
116C2×C4×C127—82no—
117C2×C4×C127—82no—
118C2×C4×C127#9911yescharacteristic
119C2×C4×C127#9911yescharacteristic
120C2×C4×C127#10211yescharacteristic
121C2×C4×C127#10311yescharacteristic
122C8×C127—42no—
123C8×C127—81no—
124C8×C127—82no—
125C22×C127—21no—
126C22×C127—21no—
127C22×C127—21no—
128C22×C127—21no—
129C22×C127—21no—
130C22×C127—21no—
131C22×C127—41no—
132C22×C127—41no—
133C22×C127—41no—
134C22×C127—41no—
135C22×C127—42no—
136C22×C127—41no—
137C22×C127—41no—
138C22×C127—41no—
139C22×C127—41no—
140C22×C127—81no—
141C22×C127H(3,ℤ/4)11yescharacteristic
142C22×C127#34011yescharacteristic
143C22×C127#34211yescharacteristic
144C4×C127—21no—
145C4×C127—21no—
146C4×C127—21no—
147C4×C127—41no—
148C4×C127—41no—
149C4×C127—41no—
150C4×C127—162no—
151C4×C127AΣL(1,𝔽16)11yescharacteristic
152C4×C127#35211yescharacteristic
153#7236C12711yesmaximal, characteristic, direct product factor, Sylow
154C2×C127—21no—
155C2×C127—21no—
156C2×C127—41no—
157C2×C127—41no—
158C2×C127—41no—
159C2×C127—41no—
160C2×C127#128911yescharacteristic
161C2×C127#139011yescharacteristic
162C2×C127#139511yescharacteristic
163#1869C2×C12711yescharacteristic
164#1800C2×C12711yescharacteristic
165#1799C2×C12711yescharacteristic
166C127#723611yescharacteristic, direct product factor, Sylow
167C23×C8C4×C12711yescharacteristic, complemented
168C2×C4×D4C22×C12711yescharacteristic
169#423—21no—
170C2×#100—21no—
171C2×#99—21no—
172C2×C22⋊C8—21no—
173C2×C22⋊C8C4×C12711yescharacteristic, complemented
174#380—21no—
175#343C4×C12712yescomplemented
176#342C4×C12712yes—
177#336—21no—
178C22×D4D4×C12711yescharacteristic
179C23×C4C2×C4×C12711yescharacteristic
180C23×C4D4×C12711yescharacteristic
181C2×M16—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—42no—
185C22×C8—41no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×C4×C12711yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4C2×C4×C12711yescharacteristic
192C2×C22⋊C4Q8×C12711yescharacteristic
193C2×C42D4×C12711yescharacteristic
194C4⋊C8—41no—
195C4⋊C8—41no—
196#100—41no—
197#100—41no—
198#99—41no—
199#99—41no—
200C22⋊C8—42no—
201C22⋊C8—41no—
202#95—41no—
203C24C22⋊C4×C12711yescharacteristic
204C24C4⋊C4×C12711yescharacteristic
205C2×D4—41no—
206C2×D4—41no—
207C2×D4D8×C12711yescharacteristic
208C2×D4SD16×C12711yescharacteristic
209C2×D4SD16×C12711yescharacteristic
210C2×D4Q16×C12711yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4C42×C12711yescharacteristic
221C22×C4C22⋊C4×C12711yescharacteristic
222C22×C4C22⋊C4×C12711yescharacteristic
223C22×C4C4⋊C4×C12711yescharacteristic
224C22×C4C4⋊C4×C12711yescharacteristic
225M16—42no—
226M16—42no—
227C2×C8—22no—
228C2×C8—41no—
229C2×C8—42no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8—41no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—21no—
244C42—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#95×C12711yescharacteristic
254C23#102×C12711yescharacteristic
255C23#103×C12711yescharacteristic
256C23#106×C12711yescharacteristic
257C23#107×C12711yescharacteristic
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#99×C12711yescharacteristic
282C2×C4#99×C12711yescharacteristic
283C2×C4#102×C12711yescharacteristic
284C2×C4#103×C12711yescharacteristic
285C8—42no—
286C8—81no—
287C8—82no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22H(3,ℤ/4)×C12711yescharacteristic
305C22#340×C12711yescharacteristic
306C22#342×C12711yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—162no—
314C4AΣL(1,𝔽16)×C12711yescharacteristic
315C4#352×C12711yescharacteristic
316C2—21no—
317C2—21no—
318C2—41no—
319C2—41no—
320C2—41no—
321C2—41no—
322C2C127×#128911yescharacteristic
323C2C127×#139011yescharacteristic
324C2C127×#139511yescharacteristic
subgroupG/H
center, socleC22×C127#342
commutatorC22×C4C42×C127
Frattini, cosocleC2×C4×D4C22×C127
prime radicalC22×D4×C127D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×C8×C127C4semidirect
10. C2×C22⋊C8×C127C4semidirect
12. #343×C127C4semidirect
153. #7236C127direct (N × H)
166. C127#7236direct (N × H)
167. C23×C8C4×C127semidirect
173. C2×C22⋊C8C4×C127semidirect
175. #343C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#18692
C127×#18002
C127×#17992
#7236127