Finite Group Databasebeta

← C127×#21174 #44584847

Subgroups — C127×#21174 #44584847

Order 32512.

990 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#428×C127C211yesmaximal, characteristic
2C22×#106×C127C211yesmaximal, characteristic
3C2×#424×C127C211yesmaximal, characteristic
4C127×#2317C211yesmaximal, characteristic
5C127×#2274C211yesmaximal, characteristic
6C127×#1949C214yesmaximal, complemented
7C127×#1948C214yesmaximal, complemented
8C127×#1736C211yesmaximal, characteristic
9C127×#1735C211yesmaximal, characteristic
10C22×M16×C127C2211yescharacteristic
11C23×C8×C127C2211yescharacteristic
12C2×#117×C127C2211yescharacteristic
13C2×#117×C127C2211yescharacteristic
14C2×#117×C127C2211yescharacteristic
15C22×C4⋊C4×C127C2211yescharacteristic
16C22×C4⋊C4×C127C2211yescharacteristic
17#428×C127—24no—
18C2×#107×C127C2212yes—
19C2×#106×C127—24no—
20C2×#106×C127C2214yes—
21C2×#106×C127C2212yes—
22#424×C127C418yescomplemented
23C2×C4⋊C8×C127C2214yes—
24#389×C127—22no—
25#385×C127—22no—
26#384×C127C2212yes—
27#382×C127C2212yes—
28C4×C4⋊C4×C127—22no—
29C4×C4⋊C4×C127C2214yes—
30C4×C22⋊C4×C127—22no—
31#340×C127C2212yes—
32#340×C127C2214yes—
33#340×C127C2212yes—
34C23×C4×C127—21no—
35C23×C4×C127—21no—
36C23×C4×C127C2311yescharacteristic
37C2×M16×C127—22no—
38C2×M16×C127—22no—
39C2×M16×C127C2×C414yes—
40C22×C8×C127—22no—
41C22×C8×C127—22no—
42C22×C8×C127C2×C414yes—
43C22×C8×C127C2312yes—
44C22×C8×C127C2312yes—
45#117×C127—22no—
46#117×C127—22no—
47#117×C127—22no—
48#117×C127C2×C414yes—
49#117×C127Q814yes—
50#117×C127Q814yes—
51C2×C4⋊C4×C127—21no—
52C2×C4⋊C4×C127—22no—
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127—22no—
55C2×C4⋊C4×C127—21no—
56C2×C4⋊C4×C127—24no—
57C2×C4⋊C4×C127D414yes—
58C2×C4⋊C4×C127D412yes—
59C2×C4⋊C4×C127C2312yes—
60C2×C4⋊C4×C127C2311yescharacteristic
61C2×C4⋊C4×C127C2311yescharacteristic
62C2×C4⋊C4×C127C2312yes—
63C2×C4⋊C4×C127C2312yes—
64C2×C22⋊C4×C127—21no—
65C2×C22⋊C4×C127—21no—
66C2×C22⋊C4×C127D412yes—
67C2×C42×C127—21no—
68C2×C42×C127—21no—
69C2×C42×C127—24no—
70C2×C42×C127C2312yes—
71#107×C127—24no—
72#106×C127—28no—
73#106×C127—24no—
74C4⋊C8×C127C2×C4116yescomplemented
75#95×C127—24no—
76#95×C127—24no—
77C24×C127C2×D411yescharacteristic
78C22×C4×C127—22no—
79C22×C4×C127—22no—
80C22×C4×C127—24no—
81C22×C4×C127—21no—
82C22×C4×C127—22no—
83C22×C4×C127—21no—
84C22×C4×C127—42no—
85C22×C4×C127—82no—
86C22×C4×C127—82no—
87C22×C4×C127C22×C412yes—
88C22×C4×C127C2×D414yes—
89C22×C4×C127C2×D411yescharacteristic
90C22×C4×C127C2×D412yes—
91C22×C4×C127C2×Q812yes—
92C22×C4×C127C2×Q812yes—
93C22×C4×C127#4112yes—
94C22×C4×C127#4112yes—
95C22×C4×C127C2411yescharacteristic
96M16×C127—28no—
97C2×C8×C127—28no—
98C2×C8×C127—22no—
99C2×C8×C127—22no—
100C2×C8×C127—22no—
101C2×C8×C127—22no—
102C2×C8×C127C22×C414yes—
103C2×C8×C127C22×C414yes—
104C4⋊C4×C127—24no—
105C4⋊C4×C127—22no—
106C4⋊C4×C127—22no—
107C4⋊C4×C127—22no—
108C4⋊C4×C127—21no—
109C4⋊C4×C127—21no—
110C4⋊C4×C127—22no—
111C4⋊C4×C127—21no—
112C4⋊C4×C127—21no—
113C4⋊C4×C127—42no—
114C4⋊C4×C127—84no—
115C4⋊C4×C127C4⋊C418yes—
116C4⋊C4×C127C2×D414yes—
117C4⋊C4×C127C2×Q814yes—
118C22⋊C4×C127—24no—
119C22⋊C4×C127—42no—
120C22⋊C4×C127C4⋊C418yes—
121C42×C127—24no—
122C42×C127—42no—
123C42×C127—84no—
124C42×C127C22×C414yes—
125C42×C127C2×Q814yes—
126C23×C127—22no—
127C23×C127—22no—
128C23×C127C2×C4⋊C412yes—
129C23×C127#12112yes—
130C23×C127C22⋊Q812yes—
131C23×C127C22×D411yescharacteristic
132C2×C4×C127—24no—
133C2×C4×C127—24no—
134C2×C4×C127—24no—
135C2×C4×C127—24no—
136C2×C4×C127—42no—
137C2×C4×C127—44no—
138C2×C4×C127—42no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—42no—
142C2×C4×C127—42no—
143C2×C4×C127—44no—
144C2×C4×C127—42no—
145C2×C4×C127—84no—
146C2×C4×C127—84no—
147C2×C4×C127C2×C4⋊C418yes—
148C2×C4×C127C2×C4⋊C412yes—
149C2×C4×C127C4×D414yes—
150C2×C4×C127C4×Q814yes—
151C2×C4×C127#12112yes—
152C2×C4×C127C22⋊Q812yes—
153C2×C4×C127#12514yes—
154C2×C4×C127C4⋊Q814yes—
155C2×C4×C127C23×C411yescharacteristic
156C2×C4×C127C22×D411yescharacteristic
157C2×C4×C127C22×Q811yescharacteristic
158C2×C4×C127C22×Q811yescharacteristic
159C2×C4×C127C2×#4111yescharacteristic
160C2×C4×C127C2×#4111yescharacteristic
161C8×C127—24no—
162C8×C127—24no—
163C22×C127—24no—
164C22×C127—24no—
165C22×C127—22no—
166C22×C127—22no—
167C22×C127#38914yes—
168C22×C127C22×C4⋊C411yescharacteristic
169C22×C127C2×#12111yescharacteristic
170C22×C127C2×C22⋊Q811yescharacteristic
171C22×C127#57712yes—
172C22×C127#57712yes—
173C4×C127—28no—
174C4×C127—44no—
175C4×C127—44no—
176C4×C127—81no—
177C4×C127—81no—
178C4×C127—82no—
179C4×C127#38914yes—
180C4×C127C2×C4×D411yescharacteristic
181C4×C127C2×C4×Q811yescharacteristic
182C4×C127C2×#12511yescharacteristic
183C4×C127C2×C4⋊Q811yescharacteristic
184#21174C12711yesmaximal, characteristic, direct product factor, Sylow
185C2×C127—24no—
186C2×C127C2×#38911yescharacteristic
187C2×C127#291212yes—
188C2×C127#304211yescharacteristic
189C2×C127#304311yescharacteristic
190C2×C127#308412yes—
191C2×#428C2×C12711yescharacteristic
192C22×#106C2×C12711yescharacteristic
193C2×#424C2×C12711yescharacteristic
194#2317C2×C12711yescharacteristic
195#2274C2×C12711yescharacteristic
196#1949C2×C12714yescomplemented
197#1948C2×C12714yescomplemented
198#1736C2×C12711yescharacteristic
199#1735C2×C12711yescharacteristic
200C127#2117411yescharacteristic, direct product factor, Sylow
201C22×M16C22×C12711yescharacteristic
202C23×C8C22×C12711yescharacteristic
203C2×#117C22×C12711yescharacteristic
204C2×#117C22×C12711yescharacteristic
205C2×#117C22×C12711yescharacteristic
206C22×C4⋊C4C22×C12711yescharacteristic
207C22×C4⋊C4C22×C12711yescharacteristic
208#428—24no—
209C2×#107C22×C12712yes—
210C2×#106—24no—
211C2×#106C22×C12714yes—
212C2×#106C22×C12712yes—
213#424C4×C12718yescomplemented
214C2×C4⋊C8C22×C12714yes—
215#389—22no—
216#385—22no—
217#384C22×C12712yes—
218#382C22×C12712yes—
219C4×C4⋊C4—22no—
220C4×C4⋊C4C22×C12714yes—
221C4×C22⋊C4—22no—
222#340C22×C12712yes—
223#340C22×C12714yes—
224#340C22×C12712yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4C23×C12711yescharacteristic
228C2×M16—22no—
229C2×M16—22no—
230C2×M16C2×C4×C12714yes—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8C2×C4×C12714yes—
234C22×C8C23×C12712yes—
235C22×C8C23×C12712yes—
236#117—22no—
237#117—22no—
238#117—22no—
239#117C2×C4×C12714yes—
240#117Q8×C12714yes—
241#117Q8×C12714yes—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4D4×C12714yes—
249C2×C4⋊C4D4×C12712yes—
250C2×C4⋊C4C23×C12712yes—
251C2×C4⋊C4C23×C12711yescharacteristic
252C2×C4⋊C4C23×C12711yescharacteristic
253C2×C4⋊C4C23×C12712yes—
254C2×C4⋊C4C23×C12712yes—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4D4×C12712yes—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—24no—
261C2×C42C23×C12712yes—
262#107—24no—
263#106—28no—
264#106—24no—
265C4⋊C8C2×C4×C127116yescomplemented
266#95—24no—
267#95—24no—
268C24C2×D4×C12711yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4C22×C4×C12712yes—
279C22×C4C2×D4×C12714yes—
280C22×C4C2×D4×C12711yescharacteristic
281C22×C4C2×D4×C12712yes—
282C22×C4C2×Q8×C12712yes—
283C22×C4C2×Q8×C12712yes—
284C22×C4#41×C12712yes—
285C22×C4#41×C12712yes—
286C22×C4C24×C12711yescharacteristic
287M16—28no—
288C2×C8—28no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8C22×C4×C12714yes—
294C2×C8C22×C4×C12714yes—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—84no—
306C4⋊C4C4⋊C4×C12718yes—
307C4⋊C4C2×D4×C12714yes—
308C4⋊C4C2×Q8×C12714yes—
309C22⋊C4—24no—
310C22⋊C4—42no—
311C22⋊C4C4⋊C4×C12718yes—
312C42—24no—
313C42—42no—
314C42—84no—
315C42C22×C4×C12714yes—
316C42C2×Q8×C12714yes—
317C23—22no—
318C23—22no—
319C23C2×C4⋊C4×C12712yes—
320C23#121×C12712yes—
321C23C22⋊Q8×C12712yes—
322C23C22×D4×C12711yescharacteristic
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4C2×C4⋊C4×C12718yes—
339C2×C4C2×C4⋊C4×C12712yes—
340C2×C4C4×D4×C12714yes—
341C2×C4C4×Q8×C12714yes—
342C2×C4#121×C12712yes—
343C2×C4C22⋊Q8×C12712yes—
344C2×C4#125×C12714yes—
345C2×C4C4⋊Q8×C12714yes—
346C2×C4C23×C4×C12711yescharacteristic
347C2×C4C22×D4×C12711yescharacteristic
348C2×C4C22×Q8×C12711yescharacteristic
349C2×C4C22×Q8×C12711yescharacteristic
350C2×C4C2×#41×C12711yescharacteristic
351C2×C4C2×#41×C12711yescharacteristic
352C8—24no—
353C8—24no—
354C22—24no—
355C22—24no—
356C22—22no—
357C22—22no—
358C22#389×C12714yes—
359C22C22×C4⋊C4×C12711yescharacteristic
360C22C2×#121×C12711yescharacteristic
361C22C2×C22⋊Q8×C12711yescharacteristic
362C22#577×C12712yes—
363C22#577×C12712yes—
364C4—28no—
365C4—44no—
366C4—44no—
367C4—81no—
368C4—82no—
369C4—81no—
370C4#389×C12714yes—
371C4C2×C4×D4×C12711yescharacteristic
372C4C2×C4×Q8×C12711yescharacteristic
373C4C2×#125×C12711yescharacteristic
374C4C2×C4⋊Q8×C12711yescharacteristic
375C2—24no—
376C2C2×#389×C12711yescharacteristic
377C2C127×#291212yes—
378C2C127×#304211yescharacteristic
379C2C127×#304311yescharacteristic
380C2C127×#308412yes—
subgroupG/H
center, socleC23×C127C22×D4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC22×C4C24×C127
prime radicalC24×C127C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×#117×C127 ◁ G

Complements

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

NH ≅ G/Ntype
6. C127×#1949C2semidirect
7. C127×#1948C2semidirect
22. #424×C127C4semidirect
74. C4⋊C8×C127C2×C4semidirect
184. #21174C127direct (N × H)
196. #1949C2×C127semidirect
197. #1948C2×C127semidirect
200. C127#21174direct (N × H)
213. #424C4×C127semidirect
265. C4⋊C8C2×C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#428×C1272
C22×#106×C1272
C2×#424×C1272
C127×#23172
C127×#22742
C127×#19492
C127×#19482
C127×#17362
C127×#17352
#21174127