Finite Group Databasebeta

← C127×#12124 #44575797

Subgroups — C127×#12124 #44575797

Order 32512.

542 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2842C211yesmaximal, characteristic, complemented
2C127×#2021C211yesmaximal, characteristic
3C127×#1613C212yesmaximal, complemented
4C127×#1352C212yesmaximal, complemented
5C127×#1316C211yesmaximal, characteristic, complemented
6#534×C127C2211yescharacteristic
7C2×#120×C127—21no—
8#481×C127—22no—
9C2×#99×C127—21no—
10#399×C127—21no—
11#399×C127C2212yes—
12#396×C127—22no—
13#396×C127—22no—
14#379×C127—21no—
15#379×C127C2211yescharacteristic
16#342×C127C2211yescharacteristic
17#327×C127—22no—
18#323×C127C2212yes—
19C25×C127D411yescharacteristic
20C22×D4×C127—21no—
21C22×D4×C127—41no—
22#123×C127—41no—
23#121×C127—22no—
24#120×C127—42no—
25#120×C127—41no—
26#120×C127—42no—
27C2×C4⋊C4×C127—22no—
28C2×C4⋊C4×C127—21no—
29C2×C4⋊C4×C127—22no—
30C2×C4⋊C4×C127D412yescomplemented
31C2×C22⋊C4×C127—21no—
32C2×C22⋊C4×C127—21no—
33C2×C22⋊C4×C127—22no—
34C2×C22⋊C4×C127—21no—
35C2×C22⋊C4×C127—22no—
36C2×C22⋊C4×C127—22no—
37C2×C22⋊C4×C127—41no—
38C2×C22⋊C4×C127D411yescharacteristic
39C2×C22⋊C4×C127C2311yescharacteristic
40#106×C127—42no—
41#102×C127—42no—
42#99×C127—42no—
43C22⋊C8×C127—22no—
44#95×C127—21no—
45#95×C127D412yescomplemented
46C24×C127—21no—
47C24×C127—21no—
48C24×C127—21no—
49C24×C127—41no—
50C24×C127—41no—
51C24×C127—82no—
52C24×C127SD1612yes—
53C24×C127C2×D411yescharacteristic
54C2×D4×C127—22no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127—41no—
58C2×D4×C127—44no—
59C2×D4×C127—41no—
60C2×D4×C127—41no—
61C2×D4×C127—41no—
62C2×D4×C127—84no—
63C22×C4×C127—21no—
64C22×C4×C127—22no—
65C22×C4×C127—21no—
66C22×C4×C127—22no—
67C22×C4×C127—41no—
68C22×C4×C127C2×D412yes—
69C2×C8×C127—42no—
70C4⋊C4×C127—22no—
71C4⋊C4×C127—42no—
72C4⋊C4×C127—82no—
73C4⋊C4×C127—82no—
74C4⋊C4×C127—81no—
75C22⋊C4×C127—22no—
76C22⋊C4×C127—41no—
77C22⋊C4×C127—42no—
78C22⋊C4×C127—44no—
79C22⋊C4×C127—41no—
80C22⋊C4×C127—41no—
81C22⋊C4×C127—82no—
82C22⋊C4×C127—82no—
83C22⋊C4×C127—82no—
84C22⋊C4×C127—81no—
85C22⋊C4×C127—81no—
86C23×C127—21no—
87C23×C127—21no—
88C23×C127—22no—
89C23×C127—42no—
90C23×C127—42no—
91C23×C127—42no—
92C23×C127—42no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127—44no—
96C23×C127—41no—
97C23×C127—41no—
98C23×C127—41no—
99C23×C127—81no—
100C23×C127—82no—
101C23×C127—82no—
102C23×C127—82no—
103C23×C127—84no—
104C23×C127—82no—
105C23×C127#12011yescharacteristic
106C23×C127C2×SD1611yescharacteristic
107C23×C127#13711yescharacteristic
108D4×C127—82no—
109D4×C127—81no—
110D4×C127—81no—
111D4×C127—88no—
112D4×C127—81no—
113D4×C127—81no—
114C2×C4×C127—22no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127—41no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127—42no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—42no—
126C2×C4×C127—41no—
127C2×C4×C127—42no—
128C2×C4×C127—82no—
129C8×C127—82no—
130C22×C127—22no—
131C22×C127—21no—
132C22×C127—21no—
133C22×C127—41no—
134C22×C127—41no—
135C22×C127—42no—
136C22×C127—42no—
137C22×C127—42no—
138C22×C127—44no—
139C22×C127—42no—
140C22×C127—81no—
141C22×C127—82no—
142C22×C127—82no—
143C22×C127—84no—
144C22×C127—82no—
145C22×C127—82no—
146C22×C127—81no—
147C22×C127—82no—
148C22×C127—161no—
149C22×C127#44811yescharacteristic
150C22×C127#45311yescharacteristic
151C22×C127B(4,𝔽2)11yescharacteristic
152C4×C127—42no—
153C4×C127—81no—
154C4×C127—82no—
155C4×C127—82no—
156C4×C127—81no—
157C4×C127—82no—
158#12124C12711yesmaximal, characteristic, direct product factor, Sylow
159C2×C127—21no—
160C2×C127—21no—
161C2×C127—41no—
162C2×C127—41no—
163C2×C127—82no—
164C2×C127—161no—
165C2×C127#159611yescharacteristic
166C2×C127#220012yes—
167#2842C2×C12711yescharacteristic, complemented
168#2021C2×C12711yescharacteristic
169#1613C2×C12712yescomplemented
170#1352C2×C12712yescomplemented
171#1316C2×C12711yescharacteristic, complemented
172C127#1212411yescharacteristic, direct product factor, Sylow
173#534C22×C12711yescharacteristic
174C2×#120—21no—
175#481—22no—
176C2×#99—21no—
177#399—21no—
178#399C22×C12712yes—
179#396—22no—
180#396—22no—
181#379—21no—
182#379C22×C12711yescharacteristic
183#342C22×C12711yescharacteristic
184#327—22no—
185#323C22×C12712yes—
186C25D4×C12711yescharacteristic
187C22×D4—21no—
188C22×D4—41no—
189#123—41no—
190#121—22no—
191#120—42no—
192#120—41no—
193#120—42no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4D4×C12712yescomplemented
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4D4×C12711yescharacteristic
206C2×C22⋊C4C23×C12711yescharacteristic
207#106—42no—
208#102—42no—
209#99—42no—
210C22⋊C8—22no—
211#95—21no—
212#95D4×C12712yescomplemented
213C24—21no—
214C24—21no—
215C24—21no—
216C24—41no—
217C24—41no—
218C24—82no—
219C24SD16×C12712yes—
220C24C2×D4×C12711yescharacteristic
221C2×D4—22no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—44no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—84no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4C2×D4×C12712yes—
236C2×C8—42no—
237C4⋊C4—22no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—81no—
242C22⋊C4—22no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—82no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C22⋊C4—81no—
252C22⋊C4—81no—
253C23—21no—
254C23—21no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—44no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—81no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23—84no—
271C23—82no—
272C23#120×C12711yescharacteristic
273C23C2×SD16×C12711yescharacteristic
274C23#137×C12711yescharacteristic
275D4—82no—
276D4—81no—
277D4—81no—
278D4—88no—
279D4—81no—
280D4—81no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—82no—
296C8—82no—
297C22—22no—
298C22—21no—
299C22—21no—
300C22—41no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—44no—
306C22—42no—
307C22—81no—
308C22—82no—
309C22—82no—
310C22—84no—
311C22—82no—
312C22—82no—
313C22—81no—
314C22—82no—
315C22—161no—
316C22#448×C12711yescharacteristic
317C22#453×C12711yescharacteristic
318C22B(4,𝔽2)×C12711yescharacteristic
319C4—42no—
320C4—81no—
321C4—82no—
322C4—82no—
323C4—81no—
324C4—82no—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2—82no—
330C2—161no—
331C2C127×#159611yescharacteristic
332C2C127×#220012yes—
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2021C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2842C2semidirect
3. C127×#1613C2semidirect
4. C127×#1352C2semidirect
5. C127×#1316C2semidirect
30. C2×C4⋊C4×C127D4semidirect
45. #95×C127D4semidirect
158. #12124C127direct (N × H)
167. #2842C2×C127semidirect
169. #1613C2×C127semidirect
170. #1352C2×C127semidirect
171. #1316C2×C127semidirect
172. C127#12124direct (N × H)
197. C2×C4⋊C4D4×C127semidirect
212. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#28422
C127×#20212
C127×#16132
C127×#13522
C127×#13162
#12124127