Finite Group Databasebeta

← C127×#10029 #44573702

Subgroups — C127×#10029 #44573702

Order 32512.

490 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3246C211yesmaximal, characteristic
2C127×#3113C211yesmaximal, characteristic, complemented
3C127×#3112C211yesmaximal, characteristic
4C127×#1985C211yesmaximal, characteristic, complemented
5C127×#1531C211yesmaximal, characteristic, complemented
6C127×#1530C211yesmaximal, characteristic
7C127×#1290C211yesmaximal, characteristic, complemented
8#533×C127C2211yescharacteristic
9#532×C127C2211yescharacteristic
10C2×C4×Q8×C127C2211yescharacteristic
11#483×C127—21no—
12#480×C127—21no—
13#477×C127—21no—
14#476×C127—21no—
15#475×C127—21no—
16#475×C127—21no—
17#465×C127—22no—
18#462×C127—21no—
19#461×C127—21no—
20#460×C127—21no—
21#460×C127—21no—
22#451×C127—21no—
23#448×C127—21no—
24#441×C127—22no—
25#440×C127—22no—
26#432×C127C2211yescharacteristic
27#432×C127C2211yescharacteristic
28C2×C4⋊C8×C127C2211yescharacteristic
29C2×#97×C127C2211yescharacteristic
30#336×C127—22no—
31#330×C127—22no—
32#329×C127—22no—
33C2×#41×C127—21no—
34C22×Q8×C127D411yescharacteristic
35C2×Q16×C127—22no—
36C2×SD16×C127—22no—
37C2×SD16×C127—22no—
38C22×C8×C127—21no—
39C22×C8×C127D412yes—
40C4⋊Q8×C127D411yescharacteristic
41C4⋊Q8×C127D411yescharacteristic
42C4⋊Q8×C127D411yescharacteristic
43#127×C127D411yescharacteristic
44#125×C127—21no—
45#124×C127—21no—
46C22⋊Q8×C127—21no—
47C22⋊Q8×C127—21no—
48#121×C127—21no—
49#121×C127—21no—
50C4×Q8×C127—21no—
51C4×Q8×C127—21no—
52C4×Q8×C127—21no—
53C4×Q8×C127—21no—
54C4×Q8×C127—21no—
55C4×Q8×C127—21no—
56C4×D4×C127—21no—
57C4×D4×C127—21no—
58#117×C127—21no—
59C2×C4⋊C4×C127—21no—
60C2×C4⋊C4×C127D411yescharacteristic
61C2×C42×C127—21no—
62C2×C42×C127C2311yescharacteristic
63#107×C127—22no—
64#106×C127—22no—
65#106×C127—22no—
66C4⋊C8×C127—21no—
67C4⋊C8×C127—21no—
68C4⋊C8×C127—21no—
69C4⋊C8×C127—21no—
70#103×C127—22no—
71#103×C127—22no—
72#103×C127—22no—
73#103×C127—22no—
74#102×C127—22no—
75#102×C127—22no—
76C22⋊C8×C127—21no—
77C22⋊C8×C127—21no—
78#97×C127—21no—
79#97×C127—21no—
80#97×C127D412yes—
81#97×C127D412yes—
82#41×C127—82no—
83C2×Q8×C127—21no—
84C2×Q8×C127—21no—
85C2×Q8×C127—21no—
86C2×Q8×C127—21no—
87C2×Q8×C127—21no—
88C2×Q8×C127—42no—
89C2×D4×C127—21no—
90C2×D4×C127—21no—
91C2×D4×C127—21no—
92C2×D4×C127—41no—
93C22×C4×C127—21no—
94C22×C4×C127—21no—
95C22×C4×C127—41no—
96C22×C4×C127C2×D411yescharacteristic
97C22×C4×C127C2×D411yescharacteristic
98C22×C4×C127#4111yescharacteristic
99Q16×C127—44no—
100SD16×C127—44no—
101SD16×C127—82no—
102C2×C8×C127—22no—
103C2×C8×C127—22no—
104C2×C8×C127—22no—
105C2×C8×C127—22no—
106C2×C8×C127—41no—
107C2×C8×C127—41no—
108C2×C8×C127—42no—
109C4⋊C4×C127—21no—
110C4⋊C4×C127—21no—
111C4⋊C4×C127—21no—
112C4⋊C4×C127—21no—
113C4⋊C4×C127—21no—
114C4⋊C4×C127—21no—
115C4⋊C4×C127—21no—
116C4⋊C4×C127—21no—
117C4⋊C4×C127—41no—
118C4⋊C4×C127—41no—
119C4⋊C4×C127—41no—
120C4⋊C4×C127—41no—
121C22⋊C4×C127—41no—
122C22⋊C4×C127—41no—
123C22⋊C4×C127—41no—
124C42×C127—41no—
125C42×C127—41no—
126C42×C127—41no—
127C42×C127C2×D411yescharacteristic
128C42×C127C2×D411yescharacteristic
129C42×C127C2×D411yescharacteristic
130C42×C127C2×D411yescharacteristic
131C23×C127—41no—
132C23×C127#12111yescharacteristic
133Q8×C127—22no—
134Q8×C127—22no—
135Q8×C127—41no—
136Q8×C127—41no—
137Q8×C127—42no—
138Q8×C127—41no—
139Q8×C127—41no—
140Q8×C127—81no—
141D4×C127—42no—
142D4×C127—41no—
143D4×C127—41no—
144D4×C127—81no—
145D4×C127—82no—
146C2×C4×C127—22no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—22no—
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—42no—
158C2×C4×C127—82no—
159C2×C4×C127#12011yescharacteristic
160C2×C4×C127#12011yescharacteristic
161C2×C4×C127#12011yescharacteristic
162C2×C4×C127#12111yescharacteristic
163C2×C4×C127#12111yescharacteristic
164C2×C4×C127#12711yescharacteristic
165C8×C127—42no—
166C8×C127—42no—
167C8×C127—42no—
168C8×C127—42no—
169C22×C127—21no—
170C22×C127—21no—
171C22×C127—81no—
172C22×C127—81no—
173C22×C127—81no—
174C22×C127#39211yescharacteristic
175C22×C127#47112yes—
176C4×C127—21no—
177C4×C127—21no—
178C4×C127—21no—
179C4×C127—21no—
180C4×C127—21no—
181C4×C127—42no—
182C4×C127—81no—
183C4×C127—81no—
184C4×C127—81no—
185C4×C127—81no—
186C4×C127#45511yescharacteristic
187C4×C127#45611yescharacteristic
188#10029C12711yesmaximal, characteristic, direct product factor, Sylow
189C2×C127—22no—
190C2×C127—161no—
191C2×C127#200711yescharacteristic
192C2×C127#201011yescharacteristic
193C2×C127#201911yescharacteristic
194#3246C2×C12711yescharacteristic
195#3113C2×C12711yescharacteristic, complemented
196#3112C2×C12711yescharacteristic
197#1985C2×C12711yescharacteristic, complemented
198#1531C2×C12711yescharacteristic, complemented
199#1530C2×C12711yescharacteristic
200#1290C2×C12711yescharacteristic, complemented
201C127#1002911yescharacteristic, direct product factor, Sylow
202#533C22×C12711yescharacteristic
203#532C22×C12711yescharacteristic
204C2×C4×Q8C22×C12711yescharacteristic
205#483—21no—
206#480—21no—
207#477—21no—
208#476—21no—
209#475—21no—
210#475—21no—
211#465—22no—
212#462—21no—
213#461—21no—
214#460—21no—
215#460—21no—
216#451—21no—
217#448—21no—
218#441—22no—
219#440—22no—
220#432C22×C12711yescharacteristic
221#432C22×C12711yescharacteristic
222C2×C4⋊C8C22×C12711yescharacteristic
223C2×#97C22×C12711yescharacteristic
224#336—22no—
225#330—22no—
226#329—22no—
227C2×#41—21no—
228C22×Q8D4×C12711yescharacteristic
229C2×Q16—22no—
230C2×SD16—22no—
231C2×SD16—22no—
232C22×C8—21no—
233C22×C8D4×C12712yes—
234C4⋊Q8D4×C12711yescharacteristic
235C4⋊Q8D4×C12711yescharacteristic
236C4⋊Q8D4×C12711yescharacteristic
237#127D4×C12711yescharacteristic
238#125—21no—
239#124—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242#121—21no—
243#121—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×D4—21no—
251C4×D4—21no—
252#117—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4D4×C12711yescharacteristic
255C2×C42—21no—
256C2×C42C23×C12711yescharacteristic
257#107—22no—
258#106—22no—
259#106—22no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264#103—22no—
265#103—22no—
266#103—22no—
267#103—22no—
268#102—22no—
269#102—22no—
270C22⋊C8—21no—
271C22⋊C8—21no—
272#97—21no—
273#97—21no—
274#97D4×C12712yes—
275#97D4×C12712yes—
276#41—82no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—42no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—41no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×D4×C12711yescharacteristic
291C22×C4C2×D4×C12711yescharacteristic
292C22×C4#41×C12711yescharacteristic
293Q16—44no—
294SD16—44no—
295SD16—82no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42C2×D4×C12711yescharacteristic
322C42C2×D4×C12711yescharacteristic
323C42C2×D4×C12711yescharacteristic
324C42C2×D4×C12711yescharacteristic
325C23—41no—
326C23#121×C12711yescharacteristic
327Q8—22no—
328Q8—22no—
329Q8—41no—
330Q8—41no—
331Q8—42no—
332Q8—41no—
333Q8—41no—
334Q8—81no—
335D4—42no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—82no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4#120×C12711yescharacteristic
354C2×C4#120×C12711yescharacteristic
355C2×C4#120×C12711yescharacteristic
356C2×C4#121×C12711yescharacteristic
357C2×C4#121×C12711yescharacteristic
358C2×C4#127×C12711yescharacteristic
359C8—42no—
360C8—42no—
361C8—42no—
362C8—42no—
363C22—21no—
364C22—21no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#392×C12711yescharacteristic
369C22#471×C12712yes—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4#455×C12711yescharacteristic
381C4#456×C12711yescharacteristic
382C2—22no—
383C2—161no—
384C2C127×#200711yescharacteristic
385C2C127×#201011yescharacteristic
386C2C127×#201911yescharacteristic
subgroupG/H
center, socleC22×C127#392
commutator, Frattini, cosocleC2×C42C23×C127
prime radical#532×C127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#3113C2semidirect
4. C127×#1985C2semidirect
5. C127×#1531C2semidirect
7. C127×#1290C2semidirect
188. #10029C127direct (N × H)
195. #3113C2×C127semidirect
197. #1985C2×C127semidirect
198. #1531C2×C127semidirect
200. #1290C2×C127semidirect
201. C127#10029direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32462
C127×#31132
C127×#31122
C127×#19852
C127×#15312
C127×#15302
C127×#12902
#10029127