Finite Group Databasebeta

← C127×#10330 #44574003

Subgroups — C127×#10330 #44574003

Order 32512.

780 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3193C211yesmaximal, characteristic
2C127×#2436C211yesmaximal, characteristic
3C127×#2434C211yesmaximal, characteristic, complemented
4C127×#1784C211yesmaximal, characteristic, complemented
5C127×#1508C211yesmaximal, characteristic, complemented
6C127×#1507C211yesmaximal, characteristic
7C127×#1297C211yesmaximal, characteristic, complemented
8C23×Q8×C127C2211yescharacteristic
9C2×C4⋊Q8×C127—21no—
10C2×#124×C127—21no—
11C2×C22⋊Q8×C127C2211yescharacteristic
12C2×#121×C127C2211yescharacteristic
13#484×C127—21no—
14#480×C127—21no—
15#468×C127—22no—
16#450×C127—21no—
17#448×C127—21no—
18C2×#101×C127—22no—
19#407×C127C2211yescharacteristic
20#407×C127C2211yescharacteristic
21#395×C127—22no—
22#393×C127—22no—
23#391×C127—21no—
24#391×C127—21no—
25#386×C127—22no—
26#385×C127—22no—
27C4×C22⋊C4×C127C2211yescharacteristic
28C4×C22⋊C4×C127C2211yescharacteristic
29#328×C127—22no—
30#327×C127—22no—
31C22×Q8×C127—21no—
32C22×Q8×C127—21no—
33C22×Q8×C127—21no—
34C22×Q8×C127—21no—
35C22×Q8×C127—21no—
36C22×Q8×C127—21no—
37C22×Q8×C127—44no—
38C22×Q8×C127D411yescharacteristic
39C22×D4×C127—21no—
40C22×D4×C127D411yescharacteristic
41C23×C4×C127D412yes—
42C23×C4×C127C2311yescharacteristic
43C2×SD16×C127—22no—
44C2×M16×C127—21no—
45C2×M16×C127D412yes—
46C4⋊Q8×C127—44no—
47#124×C127—44no—
48C22⋊Q8×C127—21no—
49C22⋊Q8×C127—21no—
50C22⋊Q8×C127—21no—
51C22⋊Q8×C127—21no—
52#121×C127—21no—
53#121×C127—21no—
54#121×C127—21no—
55#121×C127—21no—
56C2×C4⋊C4×C127—24no—
57C2×C4⋊C4×C127—21no—
58C2×C4⋊C4×C127D411yescharacteristic
59C2×C4⋊C4×C127D411yescharacteristic
60C2×C22⋊C4×C127—21no—
61C2×C22⋊C4×C127—24no—
62C2×C22⋊C4×C127—21no—
63C2×C22⋊C4×C127—21no—
64C2×C22⋊C4×C127—21no—
65C2×C42×C127—21no—
66C2×C42×C127—21no—
67#107×C127—22no—
68#103×C127—22no—
69#102×C127—22no—
70#101×C127—44no—
71C22⋊C8×C127—21no—
72C22⋊C8×C127—21no—
73C22⋊C8×C127—21no—
74C22⋊C8×C127—21no—
75#95×C127—24no—
76#95×C127—24no—
77#95×C127D412yes—
78#95×C127D412yes—
79C24×C127—21no—
80C24×C127C2×D411yescharacteristic
81C2×Q8×C127—24no—
82C2×Q8×C127—24no—
83C2×Q8×C127—24no—
84C2×Q8×C127—21no—
85C2×Q8×C127—21no—
86C2×Q8×C127—42no—
87C2×Q8×C127—42no—
88C2×Q8×C127—42no—
89C2×Q8×C127—44no—
90C2×Q8×C127—44no—
91C2×Q8×C127—44no—
92C2×Q8×C127—42no—
93C2×Q8×C127—41no—
94C2×Q8×C127—41no—
95C2×Q8×C127—41no—
96C2×Q8×C127—41no—
97C2×D4×C127—21no—
98C2×D4×C127—21no—
99C2×D4×C127—42no—
100C2×D4×C127—44no—
101C2×D4×C127—41no—
102C2×D4×C127—41no—
103C22×C4×C127—22no—
104C22×C4×C127—22no—
105C22×C4×C127—22no—
106C22×C4×C127—22no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127—22no—
110C22×C4×C127—21no—
111C22×C4×C127—44no—
112C22×C4×C127—42no—
113C22×C4×C127C2×D411yescharacteristic
114C22×C4×C127C2×D411yescharacteristic
115C22×C4×C127C2×D411yescharacteristic
116C22×C4×C127C2×D411yescharacteristic
117C22×C4×C127C2×D411yescharacteristic
118C22×C4×C127#4111yescharacteristic
119SD16×C127—82no—
120M16×C127—42no—
121M16×C127—42no—
122C2×C8×C127—22no—
123C2×C8×C127—41no—
124C4⋊C4×C127—21no—
125C4⋊C4×C127—21no—
126C4⋊C4×C127—21no—
127C4⋊C4×C127—21no—
128C4⋊C4×C127—48no—
129C4⋊C4×C127—41no—
130C4⋊C4×C127—41no—
131C22⋊C4×C127—24no—
132C22⋊C4×C127—48no—
133C22⋊C4×C127—42no—
134C22⋊C4×C127—41no—
135C22⋊C4×C127—41no—
136C22⋊C4×C127—41no—
137C22⋊C4×C127—41no—
138C42×C127—42no—
139C42×C127—42no—
140C23×C127—22no—
141C23×C127—22no—
142C23×C127—41no—
143C23×C127—42no—
144C23×C127—42no—
145C23×C127—41no—
146C23×C127—42no—
147C23×C127#12011yescharacteristic
148C23×C127#12011yescharacteristic
149C23×C127#12111yescharacteristic
150Q8×C127—44no—
151Q8×C127—44no—
152Q8×C127—44no—
153Q8×C127—44no—
154Q8×C127—42no—
155Q8×C127—81no—
156D4×C127—42no—
157D4×C127—81no—
158D4×C127—82no—
159D4×C127—82no—
160C2×C4×C127—22no—
161C2×C4×C127—22no—
162C2×C4×C127—22no—
163C2×C4×C127—24no—
164C2×C4×C127—24no—
165C2×C4×C127—24no—
166C2×C4×C127—24no—
167C2×C4×C127—41no—
168C2×C4×C127—41no—
169C2×C4×C127—41no—
170C2×C4×C127—42no—
171C2×C4×C127—42no—
172C2×C4×C127—42no—
173C2×C4×C127—42no—
174C2×C4×C127—42no—
175C2×C4×C127—42no—
176C2×C4×C127—42no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—42no—
180C2×C4×C127—41no—
181C2×C4×C127—41no—
182C2×C4×C127—41no—
183C2×C4×C127—42no—
184C2×C4×C127—42no—
185C2×C4×C127#12011yescharacteristic
186C2×C4×C127#12111yescharacteristic
187C2×C4×C127#12111yescharacteristic
188C2×C4×C127#12711yescharacteristic
189C8×C127—42no—
190C8×C127—81no—
191C22×C127—21no—
192C22×C127—21no—
193C22×C127—21no—
194C22×C127—21no—
195C22×C127—21no—
196C22×C127—21no—
197C22×C127—21no—
198C22×C127—44no—
199C22×C127—84no—
200C22×C127—81no—
201C22×C127—81no—
202C22×C127—81no—
203C22×C127#39211yescharacteristic
204C22×C127#45412yes—
205C22×C127#45612yes—
206C4×C127—21no—
207C4×C127—21no—
208C4×C127—21no—
209C4×C127—44no—
210C4×C127—44no—
211C4×C127—82no—
212C4×C127—81no—
213C4×C127—81no—
214C4×C127—81no—
215C4×C127B(4,𝔽2)12yes—
216#10330C12711yesmaximal, characteristic, direct product factor, Sylow
217C2×C127—22no—
218C2×C127—22no—
219C2×C127—41no—
220C2×C127—161no—
221C2×C127#200311yescharacteristic
222C2×C127#200511yescharacteristic
223C2×C127#202111yescharacteristic
224#3193C2×C12711yescharacteristic
225#2436C2×C12711yescharacteristic
226#2434C2×C12711yescharacteristic, complemented
227#1784C2×C12711yescharacteristic, complemented
228#1508C2×C12711yescharacteristic, complemented
229#1507C2×C12711yescharacteristic
230#1297C2×C12711yescharacteristic, complemented
231C127#1033011yescharacteristic, direct product factor, Sylow
232C23×Q8C22×C12711yescharacteristic
233C2×C4⋊Q8—21no—
234C2×#124—21no—
235C2×C22⋊Q8C22×C12711yescharacteristic
236C2×#121C22×C12711yescharacteristic
237#484—21no—
238#480—21no—
239#468—22no—
240#450—21no—
241#448—21no—
242C2×#101—22no—
243#407C22×C12711yescharacteristic
244#407C22×C12711yescharacteristic
245#395—22no—
246#393—22no—
247#391—21no—
248#391—21no—
249#386—22no—
250#385—22no—
251C4×C22⋊C4C22×C12711yescharacteristic
252C4×C22⋊C4C22×C12711yescharacteristic
253#328—22no—
254#327—22no—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—44no—
262C22×Q8D4×C12711yescharacteristic
263C22×D4—21no—
264C22×D4D4×C12711yescharacteristic
265C23×C4D4×C12712yes—
266C23×C4C23×C12711yescharacteristic
267C2×SD16—22no—
268C2×M16—21no—
269C2×M16D4×C12712yes—
270C4⋊Q8—44no—
271#124—44no—
272C22⋊Q8—21no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276#121—21no—
277#121—21no—
278#121—21no—
279#121—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4D4×C12711yescharacteristic
283C2×C4⋊C4D4×C12711yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C42—21no—
290C2×C42—21no—
291#107—22no—
292#103—22no—
293#102—22no—
294#101—44no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299#95—24no—
300#95—24no—
301#95D4×C12712yes—
302#95D4×C12712yes—
303C24—21no—
304C24C2×D4×C12711yescharacteristic
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—42no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—41no—
326C2×D4—41no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4C2×D4×C12711yescharacteristic
338C22×C4C2×D4×C12711yescharacteristic
339C22×C4C2×D4×C12711yescharacteristic
340C22×C4C2×D4×C12711yescharacteristic
341C22×C4C2×D4×C12711yescharacteristic
342C22×C4#41×C12711yescharacteristic
343SD16—82no—
344M16—42no—
345M16—42no—
346C2×C8—22no—
347C2×C8—41no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—48no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C22⋊C4—24no—
356C22⋊C4—48no—
357C22⋊C4—42no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C42—42no—
363C42—42no—
364C23—22no—
365C23—22no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—41no—
370C23—42no—
371C23#120×C12711yescharacteristic
372C23#120×C12711yescharacteristic
373C23#121×C12711yescharacteristic
374Q8—44no—
375Q8—44no—
376Q8—44no—
377Q8—44no—
378Q8—42no—
379Q8—81no—
380D4—42no—
381D4—81no—
382D4—82no—
383D4—82no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4#120×C12711yescharacteristic
410C2×C4#121×C12711yescharacteristic
411C2×C4#121×C12711yescharacteristic
412C2×C4#127×C12711yescharacteristic
413C8—42no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—44no—
423C22—84no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22#392×C12711yescharacteristic
428C22#454×C12712yes—
429C22#456×C12712yes—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—44no—
434C4—44no—
435C4—82no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4B(4,𝔽2)×C12712yes—
440C2—22no—
441C2—22no—
442C2—41no—
443C2—161no—
444C2C127×#200311yescharacteristic
445C2C127×#200511yescharacteristic
446C2C127×#202111yescharacteristic
subgroupG/H
center, socleC22×C127#392
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC2×#121×C127C22
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
3. C127×#2434C2semidirect
4. C127×#1784C2semidirect
5. C127×#1508C2semidirect
7. C127×#1297C2semidirect
216. #10330C127direct (N × H)
226. #2434C2×C127semidirect
227. #1784C2×C127semidirect
228. #1508C2×C127semidirect
230. #1297C2×C127semidirect
231. C127#10330direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31932
C127×#24362
C127×#24342
C127×#17842
C127×#15082
C127×#15072
C127×#12972
#10330127