Finite Group Databasebeta

← C127×#10564 #44574237

Subgroups — C127×#10564 #44574237

Order 32512.

574 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3186C211yesmaximal, characteristic
2C127×#1964C212yesmaximal, complemented
3C127×#1893C212yesmaximal, complemented
4C127×#1850C211yesmaximal, characteristic, complemented
5C127×#1289C211yesmaximal, characteristic, complemented
6C23×C8×C127C2211yescharacteristic
7C2×#121×C127C2212yes—
8#480×C127—22no—
9#466×C127—21no—
10#466×C127—21no—
11#447×C127—22no—
12C2×C4⋊C8×C127—21no—
13C2×#102×C127—22no—
14C2×#99×C127—22no—
15C2×#99×C127—22no—
16C2×C22⋊C8×C127C2211yescharacteristic
17#380×C127C2212yes—
18C4×C22⋊C4×C127C2211yescharacteristic
19#340×C127—22no—
20#336×C127—21no—
21C22×D4×C127—22no—
22C22×D4×C127D412yes—
23C23×C4×C127C2311yescharacteristic
24C2×D8×C127—21no—
25C2×D8×C127—21no—
26C22×C8×C127—21no—
27C22×C8×C127—22no—
28C22×C8×C127—42no—
29C22×C8×C127D411yescharacteristic
30C22×C8×C127D411yescharacteristic
31#121×C127—22no—
32#121×C127—22no—
33#121×C127—22no—
34#121×C127—22no—
35C2×C4⋊C4×C127—22no—
36C2×C4⋊C4×C127D412yes—
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127D411yescharacteristic
40C2×C22⋊C4×C127D411yescharacteristic
41C2×C42×C127—21no—
42#107×C127—21no—
43#107×C127—21no—
44C4⋊C8×C127—42no—
45#102×C127—22no—
46#102×C127—22no—
47#102×C127—44no—
48#99×C127—44no—
49#99×C127—44no—
50C22⋊C8×C127—22no—
51#95×C127—21no—
52#95×C127—22no—
53C24×C127—22no—
54C24×C127C2×D411yescharacteristic
55C2×D4×C127—22no—
56C2×D4×C127—22no—
57C2×D4×C127—42no—
58C2×D4×C127—42no—
59C2×D4×C127—42no—
60C2×D4×C127—42no—
61C2×D4×C127—42no—
62C2×D4×C127—42no—
63C2×D4×C127—42no—
64C2×D4×C127—42no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—22no—
68C22×C4×C127—41no—
69C22×C4×C127—41no—
70C22×C4×C127—42no—
71C22×C4×C127C2×D412yes—
72C22×C4×C127C2×D411yescharacteristic
73C22×C4×C127#4111yescharacteristic
74C22×C4×C127#4112yes—
75D8×C127—81no—
76D8×C127—81no—
77C2×C8×C127—24no—
78C2×C8×C127—22no—
79C2×C8×C127—21no—
80C2×C8×C127—21no—
81C2×C8×C127—41no—
82C2×C8×C127—42no—
83C2×C8×C127—41no—
84C2×C8×C127—41no—
85C2×C8×C127—41no—
86C4⋊C4×C127—22no—
87C4⋊C4×C127—22no—
88C4⋊C4×C127—42no—
89C4⋊C4×C127—42no—
90C22⋊C4×C127—21no—
91C22⋊C4×C127—22no—
92C22⋊C4×C127—42no—
93C22⋊C4×C127—42no—
94C22⋊C4×C127—42no—
95C22⋊C4×C127—42no—
96C22⋊C4×C127D812yescomplemented
97C42×C127—41no—
98C42×C127—41no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—42no—
102C23×C127—42no—
103C23×C127—42no—
104C23×C127—82no—
105C23×C127—82no—
106C23×C127#12112yes—
107C23×C127#12111yescharacteristic
108C23×C127C2×D811yescharacteristic
109C23×C127#13511yescharacteristic
110C23×C127AGL(1,ℤ/8)12yes—
111D4×C127—44no—
112D4×C127—42no—
113D4×C127—42no—
114D4×C127—82no—
115D4×C127—82no—
116D4×C127—82no—
117D4×C127—82no—
118C2×C4×C127—21no—
119C2×C4×C127—22no—
120C2×C4×C127—21no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—41no—
124C2×C4×C127—42no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—42no—
130C2×C4×C127—42no—
131C2×C4×C127—42no—
132C2×C4×C127—41no—
133C2×C4×C127—82no—
134C2×C4×C127—82no—
135C2×C4×C127—82no—
136C2×C4×C127#12011yescharacteristic
137C2×C4×C127#12312yes—
138C2×C4×C127#12411yescharacteristic
139C8×C127—42no—
140C8×C127—41no—
141C8×C127—41no—
142C22×C127—21no—
143C22×C127—22no—
144C22×C127—21no—
145C22×C127—22no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—42no—
149C22×C127—82no—
150C22×C127—82no—
151C22×C127—82no—
152C22×C127—162no—
153C22×C127—162no—
154C22×C127#39411yescharacteristic
155C22×C127#45912yes—
156C22×C127#46312yes—
157C22×C127#46611yescharacteristic
158C22×C127#46911yescharacteristic
159C4×C127—22no—
160C4×C127—21no—
161C4×C127—42no—
162C4×C127—81no—
163C4×C127—82no—
164C4×C127—162no—
165C4×C127B(4,𝔽2)11yescharacteristic
166C4×C127#45811yescharacteristic
167#10564C12711yesmaximal, characteristic, direct product factor, Sylow
168C2×C127—21no—
169C2×C127—21no—
170C2×C127—22no—
171C2×C127—22no—
172C2×C127—162no—
173C2×C127#204411yescharacteristic
174C2×C127#205311yescharacteristic
175C2×C127#205411yescharacteristic
176#3186C2×C12711yescharacteristic
177#1964C2×C12712yescomplemented
178#1893C2×C12712yescomplemented
179#1850C2×C12711yescharacteristic, complemented
180#1289C2×C12711yescharacteristic, complemented
181C127#1056411yescharacteristic, direct product factor, Sylow
182C23×C8C22×C12711yescharacteristic
183C2×#121C22×C12712yes—
184#480—22no—
185#466—21no—
186#466—21no—
187#447—22no—
188C2×C4⋊C8—21no—
189C2×#102—22no—
190C2×#99—22no—
191C2×#99—22no—
192C2×C22⋊C8C22×C12711yescharacteristic
193#380C22×C12712yes—
194C4×C22⋊C4C22×C12711yescharacteristic
195#340—22no—
196#336—21no—
197C22×D4—22no—
198C22×D4D4×C12712yes—
199C23×C4C23×C12711yescharacteristic
200C2×D8—21no—
201C2×D8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—42no—
205C22×C8D4×C12711yescharacteristic
206C22×C8D4×C12711yescharacteristic
207#121—22no—
208#121—22no—
209#121—22no—
210#121—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4D4×C12712yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4D4×C12711yescharacteristic
216C2×C22⋊C4D4×C12711yescharacteristic
217C2×C42—21no—
218#107—21no—
219#107—21no—
220C4⋊C8—42no—
221#102—22no—
222#102—22no—
223#102—44no—
224#99—44no—
225#99—44no—
226C22⋊C8—22no—
227#95—21no—
228#95—22no—
229C24—22no—
230C24C2×D4×C12711yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4C2×D4×C12712yes—
248C22×C4C2×D4×C12711yescharacteristic
249C22×C4#41×C12711yescharacteristic
250C22×C4#41×C12712yes—
251D8—81no—
252D8—81no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C22⋊C4—21no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4D8×C12712yescomplemented
273C42—41no—
274C42—41no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—82no—
281C23—82no—
282C23#121×C12712yes—
283C23#121×C12711yescharacteristic
284C23C2×D8×C12711yescharacteristic
285C23#135×C12711yescharacteristic
286C23AGL(1,ℤ/8)×C12712yes—
287D4—44no—
288D4—42no—
289D4—42no—
290D4—82no—
291D4—82no—
292D4—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4#120×C12711yescharacteristic
313C2×C4#123×C12712yes—
314C2×C4#124×C12711yescharacteristic
315C8—42no—
316C8—41no—
317C8—41no—
318C22—21no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—162no—
329C22—162no—
330C22#394×C12711yescharacteristic
331C22#459×C12712yes—
332C22#463×C12712yes—
333C22#466×C12711yescharacteristic
334C22#469×C12711yescharacteristic
335C4—22no—
336C4—21no—
337C4—42no—
338C4—81no—
339C4—82no—
340C4—162no—
341C4B(4,𝔽2)×C12711yescharacteristic
342C4#458×C12711yescharacteristic
343C2—21no—
344C2—21no—
345C2—22no—
346C2—22no—
347C2—162no—
348C2C127×#204411yescharacteristic
349C2C127×#205311yescharacteristic
350C2C127×#205411yescharacteristic
subgroupG/H
center, socleC22×C127#394
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC127×#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1964C2semidirect
3. C127×#1893C2semidirect
4. C127×#1850C2semidirect
5. C127×#1289C2semidirect
96. C22⋊C4×C127D8semidirect
167. #10564C127direct (N × H)
177. #1964C2×C127semidirect
178. #1893C2×C127semidirect
179. #1850C2×C127semidirect
180. #1289C2×C127semidirect
181. C127#10564direct (N × H)
272. C22⋊C4D8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31862
C127×#19642
C127×#18932
C127×#18502
C127×#12892
#10564127