Finite Group Databasebeta

← C127×#12223 #44575896

Subgroups — C127×#12223 #44575896

Order 32512.

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

normal subgroups only

This group has 394 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×#2784C211yesmaximal, characteristic
2C127×#2027C211yesmaximal, characteristic
3C127×#1617C211yesmaximal, characteristic, complemented
4C127×#1605C211yesmaximal, characteristic, complemented
5C127×#1354C211yesmaximal, characteristic, complemented
6C127×#1353C211yesmaximal, characteristic, complemented
7C127×#1318C211yesmaximal, characteristic
8#544×C127C2211yescharacteristic
9#484×C127—21no—
10#451×C127—21no—
11#418×C127—21no—
12#400×C127C2211yescharacteristic
13#396×C127—21no—
14#395×C127C2211yescharacteristic
15#393×C127—21no—
16#390×C127C2211yescharacteristic
17#387×C127—21no—
18#386×C127—21no—
19#385×C127—21no—
20#380×C127—21no—
21#343×C127C2211yescharacteristic
22#339×C127—21no—
23#328×C127—21no—
24#328×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C22×Q8×C127D411yescharacteristic
28C22×D4×C127D411yescharacteristic
29C23×C4×C127—21no—
30C2×Q16×C127—41no—
31C2×M16×C127—21no—
32C4⋊Q8×C127—41no—
33#125×C127—41no—
34C22⋊Q8×C127—21no—
35C22⋊Q8×C127—21no—
36#117×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127D411yescharacteristic
41C2×C4⋊C4×C127C2311yescharacteristic
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—21no—
47C2×C42×C127D411yescharacteristic
48#107×C127—41no—
49#103×C127—41no—
50#103×C127—41no—
51#102×C127—41no—
52#102×C127—41no—
53C22⋊C8×C127—21no—
54C22⋊C8×C127—21no—
55#95×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—21no—
59#95×C127—21no—
60#95×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C24×C127—21no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—41no—
66C2×Q8×C127—81no—
67C2×D4×C127—21no—
68C2×D4×C127—41no—
69C2×D4×C127—41no—
70C2×D4×C127Q1612yescomplemented
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—42no—
77C22×C4×C127—81no—
78C22×C4×C127—81no—
79C22×C4×C127C2×D411yescharacteristic
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82Q16×C127—82no—
83M16×C127—41no—
84M16×C127—41no—
85C2×C8×C127—41no—
86C2×C8×C127—41no—
87C2×C8×C127—41no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—21no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—42no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—81no—
98C4⋊C4×C127—81no—
99C22⋊C4×C127—42no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—81no—
102C22⋊C4×C127—81no—
103C22⋊C4×C127—81no—
104C22⋊C4×C127—81no—
105C42×C127—41no—
106C42×C127—41no—
107C23×C127—22no—
108C23×C127—41no—
109C23×C127—41no—
110C23×C127—41no—
111C23×C127—41no—
112C23×C127—41no—
113C23×C127—41no—
114C23×C127#12011yescharacteristic
115Q8×C127—42no—
116Q8×C127—81no—
117Q8×C127—81no—
118Q8×C127—81no—
119D4×C127—42no—
120D4×C127—81no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—22no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—84no—
145C2×C4×C127C2×Q1611yescharacteristic
146C2×C4×C127#13711yescharacteristic
147C8×C127—81no—
148C8×C127—81no—
149C8×C127—81no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127—42no—
153C22×C127—42no—
154C22×C127—42no—
155C22×C127—81no—
156C22×C127—81no—
157C22×C127—81no—
158C22×C127—81no—
159C22×C127#45111yescharacteristic
160C22×C127#45511yescharacteristic
161C22×C127B(4,𝔽2)11yescharacteristic
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—42no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127—81no—
168C4×C127—81no—
169C4×C127—81no—
170C4×C127—81no—
171C4×C127—81no—
172C4×C127—81no—
173#12223C12711yesmaximal, characteristic, direct product factor, Sylow
174C2×C127—21no—
175C2×C127—21no—
176C2×C127—82no—
177C2×C127#160211yescharacteristic
178C2×C127#220011yescharacteristic
179C2×C127#220411yescharacteristic
180#2784C2×C12711yescharacteristic
181#2027C2×C12711yescharacteristic
182#1617C2×C12711yescharacteristic, complemented
183#1605C2×C12711yescharacteristic, complemented
184#1354C2×C12711yescharacteristic, complemented
185#1353C2×C12711yescharacteristic, complemented
186#1318C2×C12711yescharacteristic
187C127#1222311yescharacteristic, direct product factor, Sylow
188#544C22×C12711yescharacteristic
189#484—21no—
190#451—21no—
191#418—21no—
192#400C22×C12711yescharacteristic
193#396—21no—
194#395C22×C12711yescharacteristic
195#393—21no—
196#390C22×C12711yescharacteristic
197#387—21no—
198#386—21no—
199#385—21no—
200#380—21no—
201#343C22×C12711yescharacteristic
202#339—21no—
203#328—21no—
204#328—21no—
205#324C22×C12711yescharacteristic
206#324C22×C12711yescharacteristic
207C22×Q8D4×C12711yescharacteristic
208C22×D4D4×C12711yescharacteristic
209C23×C4—21no—
210C2×Q16—41no—
211C2×M16—21no—
212C4⋊Q8—41no—
213#125—41no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216#117—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4D4×C12711yescharacteristic
221C2×C4⋊C4C23×C12711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42D4×C12711yescharacteristic
228#107—41no—
229#103—41no—
230#103—41no—
231#102—41no—
232#102—41no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95D4×C12711yescharacteristic
241#95D4×C12711yescharacteristic
242C24—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—41no—
246C2×Q8—81no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4Q16×C12712yescomplemented
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4C2×D4×C12711yescharacteristic
260C22×C4C2×D4×C12711yescharacteristic
261C22×C4C2×D4×C12711yescharacteristic
262Q16—82no—
263M16—41no—
264M16—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—42no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C22⋊C4—42no—
280C22⋊C4—41no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C42—41no—
286C42—41no—
287C23—22no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#120×C12711yescharacteristic
295Q8—42no—
296Q8—81no—
297Q8—81no—
298Q8—81no—
299D4—42no—
300D4—81no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—84no—
325C2×C4C2×Q16×C12711yescharacteristic
326C2×C4#137×C12711yescharacteristic
327C8—81no—
328C8—81no—
329C8—81no—
330C22—21no—
331C22—21no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#451×C12711yescharacteristic
340C22#455×C12711yescharacteristic
341C22B(4,𝔽2)×C12711yescharacteristic
342C4—21no—
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C2—21no—
354C2—21no—
355C2—82no—
356C2C127×#160211yescharacteristic
357C2C127×#220011yescharacteristic
358C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC22×D4×C127D4
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
3. C127×#1617C2semidirect
4. C127×#1605C2semidirect
5. C127×#1354C2semidirect
6. C127×#1353C2semidirect
70. C2×D4×C127Q16semidirect
173. #12223C127direct (N × H)
182. #1617C2×C127semidirect
183. #1605C2×C127semidirect
184. #1354C2×C127semidirect
185. #1353C2×C127semidirect
187. C127#12223direct (N × H)
250. C2×D4Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#27842
C127×#20272
C127×#16172
C127×#16052
C127×#13542
C127×#13532
C127×#13182
#12223127