Finite Group Databasebeta

← C127×#12098 #44575771

Subgroups — C127×#12098 #44575771

Order 32512.

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

normal subgroups only

This group has 476 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×#2728C211yesmaximal, characteristic, complemented
2C127×#2007C211yesmaximal, characteristic, complemented
3C127×#1610C211yesmaximal, characteristic, complemented
4C127×#1598C211yesmaximal, characteristic
5C127×#1346C211yesmaximal, characteristic, complemented
6C127×#1345C211yesmaximal, characteristic, complemented
7C127×#1319C211yesmaximal, characteristic, complemented
8C2×#137×C127—21no—
9#537×C127C2211yescharacteristic
10C2×C22⋊Q8×C127—21no—
11#480×C127—21no—
12#447×C127—21no—
13C2×#104×C127—21no—
14#400×C127—21no—
15#398×C127C2211yescharacteristic
16#397×C127—21no—
17#394×C127C2211yescharacteristic
18#391×C127C2211yescharacteristic
19#388×C127—21no—
20#386×C127—21no—
21#385×C127—21no—
22#343×C127C2211yescharacteristic
23#327×C127—21no—
24#327×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C22×Q8×C127D411yescharacteristic
29C22×D4×C127D411yescharacteristic
30C23×C4×C127—21no—
31#137×C127—42no—
32#137×C127—42no—
33C2×Q16×C127—41no—
34C2×SD16×C127—41no—
35C2×D8×C127—41no—
36C2×M16×C127—21no—
37C22⋊Q8×C127—41no—
38C22⋊Q8×C127—42no—
39C22⋊Q8×C127—42no—
40#121×C127—21no—
41#121×C127—21no—
42#121×C127—41no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127—21no—
45C2×C4⋊C4×C127—21no—
46C2×C4⋊C4×C127D411yescharacteristic
47C2×C4⋊C4×C127C2311yescharacteristic
48C2×C22⋊C4×C127—21no—
49C2×C22⋊C4×C127—21no—
50C2×C22⋊C4×C127—21no—
51C2×C22⋊C4×C127—21no—
52C2×C22⋊C4×C127—21no—
53C2×C42×C127D411yescharacteristic
54#107×C127—41no—
55#104×C127—42no—
56#102×C127—41no—
57#102×C127—41no—
58C22⋊C8×C127—21no—
59C22⋊C8×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127—21no—
63#95×C127—21no—
64#95×C127—21no—
65#95×C127D411yescharacteristic
66#95×C127D411yescharacteristic
67C24×C127—21no—
68#41×C127—41no—
69#41×C127—41no—
70#41×C127—41no—
71#41×C127—41no—
72C2×Q8×C127—21no—
73C2×Q8×C127—21no—
74C2×Q8×C127—41no—
75C2×Q8×C127—41no—
76C2×Q8×C127D812yescomplemented
77C2×D4×C127—21no—
78C2×D4×C127—21no—
79C2×D4×C127—21no—
80C2×D4×C127—41no—
81C2×D4×C127—42no—
82C2×D4×C127—41no—
83C22×C4×C127—21no—
84C22×C4×C127—21no—
85C22×C4×C127—21no—
86C22×C4×C127—21no—
87C22×C4×C127—21no—
88C22×C4×C127—41no—
89C22×C4×C127—42no—
90C22×C4×C127—81no—
91C22×C4×C127—81no—
92C22×C4×C127C2×D411yescharacteristic
93C22×C4×C127C2×D411yescharacteristic
94C22×C4×C127C2×D411yescharacteristic
95Q16×C127—44no—
96SD16×C127—44no—
97D8×C127—82no—
98M16×C127—22no—
99M16×C127—41no—
100C2×C8×C127—41no—
101C2×C8×C127—41no—
102C2×C8×C127—41no—
103C4⋊C4×C127—21no—
104C4⋊C4×C127—21no—
105C4⋊C4×C127—41no—
106C4⋊C4×C127—41no—
107C4⋊C4×C127—42no—
108C4⋊C4×C127—42no—
109C4⋊C4×C127—41no—
110C4⋊C4×C127—41no—
111C22⋊C4×C127—42no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—41no—
114C22⋊C4×C127—42no—
115C22⋊C4×C127—42no—
116C22⋊C4×C127—41no—
117C22⋊C4×C127—41no—
118C22⋊C4×C127—81no—
119C42×C127—41no—
120C23×C127—22no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—41no—
126C23×C127—41no—
127C23×C127—41no—
128C23×C127#12011yescharacteristic
129Q8×C127—42no—
130Q8×C127—41no—
131Q8×C127—41no—
132Q8×C127—81no—
133D4×C127—41no—
134D4×C127—41no—
135D4×C127—42no—
136D4×C127—81no—
137D4×C127—81no—
138D4×C127—81no—
139D4×C127—81no—
140D4×C127—81no—
141C2×C4×C127—21no—
142C2×C4×C127—21no—
143C2×C4×C127—22no—
144C2×C4×C127—21no—
145C2×C4×C127—21no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—41no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—41no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—41no—
156C2×C4×C127—41no—
157C2×C4×C127—41no—
158C2×C4×C127—41no—
159C2×C4×C127—41no—
160C2×C4×C127—41no—
161C2×C4×C127—41no—
162C2×C4×C127—41no—
163C2×C4×C127—81no—
164C2×C4×C127—81no—
165C2×C4×C127—84no—
166C2×C4×C127C2×D811yescharacteristic
167C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
168C8×C127—42no—
169C8×C127—81no—
170C8×C127—81no—
171C22×C127—21no—
172C22×C127—21no—
173C22×C127—42no—
174C22×C127—42no—
175C22×C127—42no—
176C22×C127—81no—
177C22×C127—81no—
178C22×C127—81no—
179C22×C127—81no—
180C22×C127—81no—
181C22×C127—81no—
182C22×C127—81no—
183C22×C127#44711yescharacteristic
184C22×C127#45311yescharacteristic
185C22×C127B(4,𝔽2)11yescharacteristic
186C4×C127—21no—
187C4×C127—21no—
188C4×C127—42no—
189C4×C127—41no—
190C4×C127—41no—
191C4×C127—81no—
192C4×C127—81no—
193C4×C127—81no—
194C4×C127—81no—
195C4×C127—81no—
196#12098C12711yesmaximal, characteristic, direct product factor, Sylow
197C2×C127—21no—
198C2×C127—21no—
199C2×C127—82no—
200C2×C127—161no—
201C2×C127#159511yescharacteristic
202C2×C127#219711yescharacteristic
203C2×C127#220311yescharacteristic
204#2728C2×C12711yescharacteristic, complemented
205#2007C2×C12711yescharacteristic, complemented
206#1610C2×C12711yescharacteristic, complemented
207#1598C2×C12711yescharacteristic
208#1346C2×C12711yescharacteristic, complemented
209#1345C2×C12711yescharacteristic, complemented
210#1319C2×C12711yescharacteristic, complemented
211C127#1209811yescharacteristic, direct product factor, Sylow
212C2×#137—21no—
213#537C22×C12711yescharacteristic
214C2×C22⋊Q8—21no—
215#480—21no—
216#447—21no—
217C2×#104—21no—
218#400—21no—
219#398C22×C12711yescharacteristic
220#397—21no—
221#394C22×C12711yescharacteristic
222#391C22×C12711yescharacteristic
223#388—21no—
224#386—21no—
225#385—21no—
226#343C22×C12711yescharacteristic
227#327—21no—
228#327—21no—
229#324C22×C12711yescharacteristic
230#324C22×C12711yescharacteristic
231C2×#41—21no—
232C22×Q8D4×C12711yescharacteristic
233C22×D4D4×C12711yescharacteristic
234C23×C4—21no—
235#137—42no—
236#137—42no—
237C2×Q16—41no—
238C2×SD16—41no—
239C2×D8—41no—
240C2×M16—21no—
241C22⋊Q8—41no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244#121—21no—
245#121—21no—
246#121—41no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4D4×C12711yescharacteristic
251C2×C4⋊C4C23×C12711yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42D4×C12711yescharacteristic
258#107—41no—
259#104—42no—
260#102—41no—
261#102—41no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95D4×C12711yescharacteristic
270#95D4×C12711yescharacteristic
271C24—21no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8D8×C12712yescomplemented
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—41no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4C2×D4×C12711yescharacteristic
297C22×C4C2×D4×C12711yescharacteristic
298C22×C4C2×D4×C12711yescharacteristic
299Q16—44no—
300SD16—44no—
301D8—82no—
302M16—22no—
303M16—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—81no—
323C42—41no—
324C23—22no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23#120×C12711yescharacteristic
333Q8—42no—
334Q8—41no—
335Q8—41no—
336Q8—81no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—84no—
370C2×C4C2×D8×C12711yescharacteristic
371C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
372C8—42no—
373C8—81no—
374C8—81no—
375C22—21no—
376C22—21no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22#447×C12711yescharacteristic
388C22#453×C12711yescharacteristic
389C22B(4,𝔽2)×C12711yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C2—21no—
401C2—21no—
402C2—82no—
403C2—161no—
404C2C127×#159511yescharacteristic
405C2C127×#219711yescharacteristic
406C2C127×#220311yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC127×#1598C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2728C2semidirect
2. C127×#2007C2semidirect
3. C127×#1610C2semidirect
5. C127×#1346C2semidirect
6. C127×#1345C2semidirect
7. C127×#1319C2semidirect
76. C2×Q8×C127D8semidirect
196. #12098C127direct (N × H)
204. #2728C2×C127semidirect
205. #2007C2×C127semidirect
206. #1610C2×C127semidirect
208. #1346C2×C127semidirect
209. #1345C2×C127semidirect
210. #1319C2×C127semidirect
211. C127#12098direct (N × H)
280. C2×Q8D8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#27282
C127×#20072
C127×#16102
C127×#15982
C127×#13462
C127×#13452
C127×#13192
#12098127