Finite Group Databasebeta

← C127×#9970 #44573643

Subgroups — C127×#9970 #44573643

Order 32512.

772 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 138 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×#480×C127C212yesmaximal, complemented
2C127×#2656C211yesmaximal, characteristic
3C2×#327×C127C212yesmaximal, complemented
4C127×#1445C211yesmaximal, characteristic, complemented
5C127×#1280C211yesmaximal, characteristic, complemented
6C2×#121×C127—22no—
7C2×#121×C127C2212yes—
8C22×C4⋊C4×C127C2211yescharacteristic
9#480×C127—24no—
10#480×C127—24no—
11C2×#107×C127—22no—
12C2×#102×C127—22no—
13C2×C22⋊C8×C127C2212yes—
14#399×C127—22no—
15#392×C127—22no—
16C2×#95×C127—22no—
17C2×#95×C127C2211yescharacteristic
18C2×#95×C127C2211yescharacteristic
19#327×C127—24no—
20C22×D4×C127—22no—
21C22×D4×C127—22no—
22C22×D4×C127D412yes—
23C23×C4×C127—21no—
24C23×C4×C127—21no—
25C23×C4×C127D412yes—
26C23×C4×C127C2311yescharacteristic
27C22×C8×C127—22no—
28#121×C127—22no—
29#121×C127—22no—
30#121×C127—22no—
31#121×C127—22no—
32#121×C127—44no—
33#121×C127—44no—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—22no—
36C2×C4⋊C4×C127—22no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127D412yes—
39C2×C4⋊C4×C127D412yescomplemented
40C2×C22⋊C4×C127—24no—
41C2×C22⋊C4×C127—22no—
42#107×C127—28no—
43#102×C127—28no—
44C22⋊C8×C127—22no—
45C22⋊C8×C127D414yescomplemented
46#95×C127—22no—
47#95×C127—22no—
48#95×C127—44no—
49C24×C127—22no—
50C24×C127C2×D411yescharacteristic
51C2×D4×C127—22no—
52C2×D4×C127—22no—
53C2×D4×C127—44no—
54C2×D4×C127—44no—
55C2×D4×C127—42no—
56C2×D4×C127—42no—
57C2×D4×C127—44no—
58C2×D4×C127—44no—
59C2×D4×C127—82no—
60C2×D4×C127—82no—
61C22×C4×C127—22no—
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—22no—
66C22×C4×C127—42no—
67C22×C4×C127—42no—
68C22×C4×C127—42no—
69C22×C4×C127—41no—
70C22×C4×C127—42no—
71C22×C4×C127—41no—
72C22×C4×C127—41no—
73C22×C4×C127—41no—
74C22×C4×C127—42no—
75C22×C4×C127—42no—
76C22×C4×C127—42no—
77C22×C4×C127D814yes—
78C22×C4×C127C2×D412yes—
79C22×C4×C127C2×D411yescharacteristic
80C22×C4×C127C2×D412yes—
81C22×C4×C127#4111yescharacteristic
82C2×C8×C127—24no—
83C2×C8×C127—44no—
84C4⋊C4×C127—24no—
85C4⋊C4×C127—22no—
86C4⋊C4×C127—22no—
87C4⋊C4×C127—44no—
88C4⋊C4×C127—42no—
89C22⋊C4×C127—48no—
90C22⋊C4×C127—42no—
91C22⋊C4×C127—42no—
92C23×C127—21no—
93C23×C127—21no—
94C23×C127—22no—
95C23×C127—42no—
96C23×C127—44no—
97C23×C127—42no—
98C23×C127—42no—
99C23×C127—44no—
100C23×C127#12011yescharacteristic
101C23×C127#12111yescharacteristic
102C23×C127#12711yescharacteristic
103C23×C127C2×D812yes—
104C23×C127AGL(1,ℤ/8)12yes—
105D4×C127—44no—
106D4×C127—82no—
107D4×C127—88no—
108D4×C127—84no—
109D4×C127—82no—
110D4×C127—82no—
111C2×C4×C127—24no—
112C2×C4×C127—24no—
113C2×C4×C127—41no—
114C2×C4×C127—42no—
115C2×C4×C127—41no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—44no—
119C2×C4×C127—44no—
120C2×C4×C127—42no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—44no—
124C2×C4×C127—44no—
125C2×C4×C127—82no—
126C2×C4×C127—82no—
127C2×C4×C127—81no—
128C2×C4×C127—82no—
129C2×C4×C127—81no—
130C2×C4×C127—81no—
131C2×C4×C127—81no—
132C2×C4×C127#12012yes—
133C2×C4×C127#12112yes—
134C8×C127—44no—
135C22×C127—22no—
136C22×C127—22no—
137C22×C127—22no—
138C22×C127—22no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—84no—
144C22×C127—84no—
145C22×C127—82no—
146C22×C127—82no—
147C22×C127—82no—
148C22×C127#39211yescharacteristic
149C22×C127#44712yes—
150C22×C127#45511yescharacteristic
151C22×C127#45611yescharacteristic
152C22×C127#45812yes—
153C22×C127#45912yes—
154C22×C127#49311yescharacteristic
155C22×C127#49611yescharacteristic
156C4×C127—22no—
157C4×C127—22no—
158C4×C127—44no—
159C4×C127—82no—
160C4×C127—82no—
161C4×C127—82no—
162#9970C12711yesmaximal, characteristic, direct product factor, Sylow
163C2×C127—21no—
164C2×C127—22no—
165C2×C127—21no—
166C2×C127—162no—
167C2×C127#160712yes—
168C2×C127#161012yes—
169C2×C127#200711yescharacteristic
170C2×C127#201111yescharacteristic
171C2×C127#202211yescharacteristic
172C2×#480C2×C12712yescomplemented
173#2656C2×C12711yescharacteristic
174C2×#327C2×C12712yescomplemented
175#1445C2×C12711yescharacteristic, complemented
176#1280C2×C12711yescharacteristic, complemented
177C127#997011yescharacteristic, direct product factor, Sylow
178C2×#121—22no—
179C2×#121C22×C12712yes—
180C22×C4⋊C4C22×C12711yescharacteristic
181#480—24no—
182#480—24no—
183C2×#107—22no—
184C2×#102—22no—
185C2×C22⋊C8C22×C12712yes—
186#399—22no—
187#392—22no—
188C2×#95—22no—
189C2×#95C22×C12711yescharacteristic
190C2×#95C22×C12711yescharacteristic
191#327—24no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4D4×C12712yes—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4D4×C12712yes—
198C23×C4C23×C12711yescharacteristic
199C22×C8—22no—
200#121—22no—
201#121—22no—
202#121—22no—
203#121—22no—
204#121—44no—
205#121—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4D4×C12712yes—
211C2×C4⋊C4D4×C12712yescomplemented
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214#107—28no—
215#102—28no—
216C22⋊C8—22no—
217C22⋊C8D4×C12714yescomplemented
218#95—22no—
219#95—22no—
220#95—44no—
221C24—22no—
222C24C2×D4×C12711yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—82no—
232C2×D4—82no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4D8×C12714yes—
250C22×C4C2×D4×C12712yes—
251C22×C4C2×D4×C12711yescharacteristic
252C22×C4C2×D4×C12712yes—
253C22×C4#41×C12711yescharacteristic
254C2×C8—24no—
255C2×C8—44no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C22⋊C4—48no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—42no—
268C23—44no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23#120×C12711yescharacteristic
273C23#121×C12711yescharacteristic
274C23#127×C12711yescharacteristic
275C23C2×D8×C12712yes—
276C23AGL(1,ℤ/8)×C12712yes—
277D4—44no—
278D4—82no—
279D4—88no—
280D4—82no—
281D4—84no—
282D4—82no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4#120×C12712yes—
305C2×C4#121×C12712yes—
306C8—44no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—84no—
316C22—84no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22#392×C12711yescharacteristic
321C22#447×C12712yes—
322C22#455×C12711yescharacteristic
323C22#456×C12711yescharacteristic
324C22#458×C12712yes—
325C22#459×C12712yes—
326C22#493×C12711yescharacteristic
327C22#496×C12711yescharacteristic
328C4—22no—
329C4—22no—
330C4—44no—
331C4—82no—
332C4—82no—
333C4—82no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—162no—
338C2C127×#160712yes—
339C2C127×#161012yes—
340C2C127×#200711yescharacteristic
341C2C127×#201111yescharacteristic
342C2C127×#202211yescharacteristic
subgroupG/H
center, socleC23×C127#120
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC127×#2656C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#480×C127C2semidirect
3. C2×#327×C127C2semidirect
4. C127×#1445C2semidirect
5. C127×#1280C2semidirect
39. C2×C4⋊C4×C127D4semidirect
45. C22⋊C8×C127D4semidirect
162. #9970C127direct (N × H)
172. C2×#480C2×C127semidirect
174. C2×#327C2×C127semidirect
175. #1445C2×C127semidirect
176. #1280C2×C127semidirect
177. C127#9970direct (N × H)
211. C2×C4⋊C4D4×C127semidirect
217. C22⋊C8D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#480×C1272
C127×#26562
C2×#327×C1272
C127×#14452
C127×#12802
#9970127