Finite Group Databasebeta

← C127×#10510 #44574183

Subgroups — C127×#10510 #44574183

Order 32512.

392 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3215C211yesmaximal, characteristic
2C127×#1983C211yesmaximal, characteristic
3C127×#1573C211yesmaximal, characteristic, complemented
4C127×#1552C211yesmaximal, characteristic, complemented
5C127×#1503C211yesmaximal, characteristic, complemented
6C127×#1469C211yesmaximal, characteristic
7C127×#1285C211yesmaximal, characteristic, complemented
8#533×C127C2211yescharacteristic
9C2×#124×C127C2211yescharacteristic
10#497×C127—21no—
11#489×C127—21no—
12#464×C127—21no—
13#464×C127—21no—
14#451×C127—21no—
15#450×C127—21no—
16#432×C127C2211yescharacteristic
17#432×C127C2211yescharacteristic
18#431×C127C2211yescharacteristic
19C2×C4⋊C8×C127C2211yescharacteristic
20C2×#103×C127—21no—
21C2×#102×C127—21no—
22C4×C4⋊C4×C127C2211yescharacteristic
23#335×C127—21no—
24#334×C127—21no—
25#332×C127—21no—
26#332×C127—21no—
27#328×C127—22no—
28#323×C127—22no—
29C22×Q8×C127D411yescharacteristic
30C22×D4×C127D411yescharacteristic
31C2×Q16×C127—21no—
32C2×Q16×C127—21no—
33C2×SD16×C127—21no—
34C2×SD16×C127—21no—
35C22×C8×C127—21no—
36C4⋊Q8×C127D411yescharacteristic
37C4⋊Q8×C127D411yescharacteristic
38#125×C127—21no—
39#124×C127—21no—
40#124×C127—21no—
41#124×C127—21no—
42#124×C127—21no—
43C22⋊Q8×C127—21no—
44C22⋊Q8×C127—21no—
45C4×Q8×C127—21no—
46C4×Q8×C127—21no—
47#117×C127—21no—
48C2×C4⋊C4×C127—21no—
49C2×C22⋊C4×C127—21no—
50C2×C22⋊C4×C127—21no—
51C2×C42×C127—21no—
52C2×C42×C127C2311yescharacteristic
53C4⋊C8×C127—21no—
54C4⋊C8×C127—21no—
55C4⋊C8×C127—21no—
56C4⋊C8×C127—21no—
57C4⋊C8×C127D411yescharacteristic
58C4⋊C8×C127D411yescharacteristic
59#103×C127—21no—
60#103×C127—21no—
61#103×C127—42no—
62#102×C127—21no—
63#102×C127—21no—
64#102×C127—42no—
65C22⋊C8×C127—21no—
66C22⋊C8×C127—21no—
67C22⋊C8×C127D412yes—
68#97×C127—21no—
69#97×C127—21no—
70C4×C8×C127—21no—
71#95×C127—21no—
72C24×C127—21no—
73C2×Q8×C127—21no—
74C2×Q8×C127—21no—
75C2×Q8×C127—21no—
76C2×Q8×C127—21no—
77C2×Q8×C127—41no—
78C2×Q8×C127—41no—
79C2×D4×C127—21no—
80C2×D4×C127—21no—
81C2×D4×C127—41no—
82C2×D4×C127—41no—
83C22×C4×C127—21no—
84C22×C4×C127—41no—
85C22×C4×C127C2×D411yescharacteristic
86C22×C4×C127C2×D411yescharacteristic
87C22×C4×C127#4111yescharacteristic
88Q16×C127—81no—
89Q16×C127—81no—
90SD16×C127—81no—
91SD16×C127—81no—
92C2×C8×C127—21no—
93C2×C8×C127—21no—
94C2×C8×C127—22no—
95C2×C8×C127—41no—
96C2×C8×C127—41no—
97C2×C8×C127—41no—
98C2×C8×C127—41no—
99C4⋊C4×C127—22no—
100C4⋊C4×C127—21no—
101C4⋊C4×C127—21no—
102C4⋊C4×C127—21no—
103C4⋊C4×C127—21no—
104C4⋊C4×C127—41no—
105C4⋊C4×C127—41no—
106C4⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—41no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—41no—
112C42×C127—41no—
113C42×C127—81no—
114C42×C127C2×D411yescharacteristic
115C42×C127C2×D411yescharacteristic
116C42×C127#4111yescharacteristic
117C42×C127#4111yescharacteristic
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127—81no—
122C23×C127—81no—
123C23×C127#12111yescharacteristic
124Q8×C127—42no—
125Q8×C127—41no—
126Q8×C127—41no—
127Q8×C127—41no—
128Q8×C127—41no—
129Q8×C127—81no—
130D4×C127—42no—
131D4×C127—41no—
132D4×C127—41no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—41no—
138C2×C4×C127—42no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—81no—
143C2×C4×C127—81no—
144C2×C4×C127—81no—
145C2×C4×C127#12011yescharacteristic
146C2×C4×C127#12111yescharacteristic
147C2×C4×C127#12111yescharacteristic
148C2×C4×C127#12311yescharacteristic
149C2×C4×C127#12311yescharacteristic
150C2×C4×C127#12411yescharacteristic
151C8×C127—42no—
152C8×C127—41no—
153C8×C127—41no—
154C8×C127—81no—
155C8×C127—81no—
156C22×C127—21no—
157C22×C127—21no—
158C22×C127—81no—
159C22×C127—81no—
160C22×C127—81no—
161C22×C127—161no—
162C22×C127—161no—
163C22×C127#39411yescharacteristic
164C22×C127#47111yescharacteristic
165C22×C127#47311yescharacteristic
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—21no—
170C4×C127—21no—
171C4×C127—82no—
172C4×C127—81no—
173C4×C127—81no—
174C4×C127—81no—
175C4×C127#45311yescharacteristic
176C4×C127#45511yescharacteristic
177#10510C12711yesmaximal, characteristic, direct product factor, Sylow
178C2×C127—21no—
179C2×C127—21no—
180C2×C127—161no—
181C2×C127#203711yescharacteristic
182C2×C127#205211yescharacteristic
183C2×C127#205311yescharacteristic
184#3215C2×C12711yescharacteristic
185#1983C2×C12711yescharacteristic
186#1573C2×C12711yescharacteristic, complemented
187#1552C2×C12711yescharacteristic, complemented
188#1503C2×C12711yescharacteristic, complemented
189#1469C2×C12711yescharacteristic
190#1285C2×C12711yescharacteristic, complemented
191C127#1051011yescharacteristic, direct product factor, Sylow
192#533C22×C12711yescharacteristic
193C2×#124C22×C12711yescharacteristic
194#497—21no—
195#489—21no—
196#464—21no—
197#464—21no—
198#451—21no—
199#450—21no—
200#432C22×C12711yescharacteristic
201#432C22×C12711yescharacteristic
202#431C22×C12711yescharacteristic
203C2×C4⋊C8C22×C12711yescharacteristic
204C2×#103—21no—
205C2×#102—21no—
206C4×C4⋊C4C22×C12711yescharacteristic
207#335—21no—
208#334—21no—
209#332—21no—
210#332—21no—
211#328—22no—
212#323—22no—
213C22×Q8D4×C12711yescharacteristic
214C22×D4D4×C12711yescharacteristic
215C2×Q16—21no—
216C2×Q16—21no—
217C2×SD16—21no—
218C2×SD16—21no—
219C22×C8—21no—
220C4⋊Q8D4×C12711yescharacteristic
221C4⋊Q8D4×C12711yescharacteristic
222#125—21no—
223#124—21no—
224#124—21no—
225#124—21no—
226#124—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231#117—21no—
232C2×C4⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42—21no—
236C2×C42C23×C12711yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8D4×C12711yescharacteristic
242C4⋊C8D4×C12711yescharacteristic
243#103—21no—
244#103—21no—
245#103—42no—
246#102—21no—
247#102—21no—
248#102—42no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8D4×C12712yes—
252#97—21no—
253#97—21no—
254C4×C8—21no—
255#95—21no—
256C24—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4C2×D4×C12711yescharacteristic
270C22×C4C2×D4×C12711yescharacteristic
271C22×C4#41×C12711yescharacteristic
272Q16—81no—
273Q16—81no—
274SD16—81no—
275SD16—81no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C4⋊C4—22no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C42—41no—
297C42—81no—
298C42C2×D4×C12711yescharacteristic
299C42C2×D4×C12711yescharacteristic
300C42#41×C12711yescharacteristic
301C42#41×C12711yescharacteristic
302C23—41no—
303C23—41no—
304C23—41no—
305C23—81no—
306C23—81no—
307C23#121×C12711yescharacteristic
308Q8—42no—
309Q8—41no—
310Q8—41no—
311Q8—41no—
312Q8—41no—
313Q8—81no—
314D4—42no—
315D4—41no—
316D4—41no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4#120×C12711yescharacteristic
330C2×C4#121×C12711yescharacteristic
331C2×C4#121×C12711yescharacteristic
332C2×C4#123×C12711yescharacteristic
333C2×C4#123×C12711yescharacteristic
334C2×C4#124×C12711yescharacteristic
335C8—42no—
336C8—41no—
337C8—41no—
338C8—81no—
339C8—81no—
340C22—21no—
341C22—21no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—161no—
346C22—161no—
347C22#394×C12711yescharacteristic
348C22#471×C12711yescharacteristic
349C22#473×C12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—82no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4#453×C12711yescharacteristic
360C4#455×C12711yescharacteristic
361C2—21no—
362C2—21no—
363C2—161no—
364C2C127×#203711yescharacteristic
365C2C127×#205211yescharacteristic
366C2C127×#205311yescharacteristic
subgroupG/H
center, socleC22×C127#394
commutator, Frattini, cosocleC2×C42C23×C127
prime radicalC22×D4×C127D4
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
3. C127×#1573C2semidirect
4. C127×#1552C2semidirect
5. C127×#1503C2semidirect
7. C127×#1285C2semidirect
177. #10510C127direct (N × H)
186. #1573C2×C127semidirect
187. #1552C2×C127semidirect
188. #1503C2×C127semidirect
190. #1285C2×C127semidirect
191. C127#10510direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32152
C127×#19832
C127×#15732
C127×#15522
C127×#15032
C127×#14692
C127×#12852
#10510127