Finite Group Databasebeta

← C127×#11734 #44575407

Subgroups — C127×#11734 #44575407

Order 32512.

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

normal subgroups only

This group has 434 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×#2398C211yesmaximal, characteristic
2C4×#101×C127C211yesmaximal, characteristic, complemented
3C127×#1461C211yesmaximal, characteristic, complemented
4C127×#1353C212yesmaximal, complemented
5C127×#1345C212yesmaximal
6C2×C4×Q8×C127C2211yescharacteristic
7C2×C4×D4×C127C411yescharacteristic
8#431×C127—21no—
9C2×#101×C127—21no—
10C4×M16×C127—21no—
11#400×C127C411yescharacteristic
12#400×C127C2211yescharacteristic
13#397×C127—21no—
14#394×C127C411yescharacteristic
15#394×C127C2211yescharacteristic
16#393×C127—21no—
17C4×C22⋊C4×C127—21no—
18C4×C22⋊C4×C127—21no—
19#376×C127C411yescharacteristic
20#343×C127C2212yes—
21#324×C127—21no—
22#324×C127C2212yes—
23C22×Q8×C127D411yescharacteristic
24C22×D4×C127C2×C411yescharacteristic
25C23×C4×C127—21no—
26C2×M16×C127—22no—
27C4×Q8×C127—41no—
28C4×Q8×C127—41no—
29C4×D4×C127—41no—
30C4×D4×C127—41no—
31C2×C4⋊C4×C127C2×C411yescharacteristic
32C2×C4⋊C4×C127C2×C411yescharacteristic
33C2×C4⋊C4×C127D411yescharacteristic
34C2×C4⋊C4×C127C2311yescharacteristic
35C2×C22⋊C4×C127—21no—
36C2×C22⋊C4×C127—21no—
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C42×C127—21no—
41C2×C42×C127C2×C411yescharacteristic
42C2×C42×C127D411yescharacteristic
43C2×C42×C127D411yescharacteristic
44C4⋊C8×C127—41no—
45#101×C127—24no—
46C22⋊C8×C127—22no—
47#97×C127—41no—
48C4×C8×C127—41no—
49C4×C8×C127—41no—
50#95×C127—21no—
51#95×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127C2×C411yescharacteristic
56#95×C127C2×C411yescharacteristic
57C24×C127—21no—
58C2×Q8×C127—22no—
59C2×Q8×C127—22no—
60C2×Q8×C127—41no—
61C2×D4×C127—41no—
62C2×D4×C127—42no—
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127—42no—
67C22×C4×C127—42no—
68C22×C4×C127—41no—
69C22×C4×C127—41no—
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22⋊C411yescharacteristic
72C22×C4×C127C22⋊C411yescharacteristic
73C22×C4×C127C22⋊C411yescharacteristic
74C22×C4×C127C22×C411yescharacteristic
75C22×C4×C127C2×D411yescharacteristic
76C22×C4×C127C2×D411yescharacteristic
77M16×C127—24no—
78M16×C127—42no—
79C2×C8×C127—42no—
80C2×C8×C127—42no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—41no—
85C22⋊C4×C127—42no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—42no—
90C42×C127—21no—
91C42×C127—21no—
92C42×C127—21no—
93C42×C127—21no—
94C42×C127—21no—
95C42×C127—21no—
96C42×C127—42no—
97C23×C127—22no—
98C23×C127—41no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—41no—
102C23×C127—41no—
103C23×C127—41no—
104C23×C127C2×C22⋊C411yescharacteristic
105Q8×C127—44no—
106D4×C127—42no—
107D4×C127—42no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—82no—
132C2×C4×C127—82no—
133C2×C4×C127#9911yescharacteristic
134C2×C4×C127#9911yescharacteristic
135C2×C4×C127#10412yes—
136C2×C4×C127#13512yes—
137C8×C127—44no—
138C8×C127—82no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—42no—
142C22×C127—42no—
143C22×C127—42no—
144C22×C127—81no—
145C22×C127—81no—
146C22×C127—81no—
147C22×C127—81no—
148C22×C127C2×#9911yescharacteristic
149C22×C127#41611yescharacteristic
150C22×C127C2×#10411yescharacteristic
151C4×C127—21no—
152C4×C127—22no—
153C4×C127—22no—
154C4×C127—41no—
155C4×C127—41no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127—81no—
159C4×C127—81no—
160C4×C127—81no—
161C4×C127#35512yes—
162#11734C12711yesmaximal, characteristic, direct product factor, Sylow
163C2×C127—21no—
164C2×C127—21no—
165C2×C127—82no—
166C2×C127#150111yescharacteristic
167C2×C127#212011yescharacteristic
168C2×C127C2×#35511yescharacteristic
169#2398C2×C12711yescharacteristic
170C4×#101C2×C12711yescharacteristic, complemented
171#1461C2×C12711yescharacteristic, complemented
172#1353C2×C12712yescomplemented
173#1345C2×C12712yes—
174C127#1173411yescharacteristic, direct product factor, Sylow
175C2×C4×Q8C22×C12711yescharacteristic
176C2×C4×D4C4×C12711yescharacteristic
177#431—21no—
178C2×#101—21no—
179C4×M16—21no—
180#400C4×C12711yescharacteristic
181#400C22×C12711yescharacteristic
182#397—21no—
183#394C4×C12711yescharacteristic
184#394C22×C12711yescharacteristic
185#393—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188#376C4×C12711yescharacteristic
189#343C22×C12712yes—
190#324—21no—
191#324C22×C12712yes—
192C22×Q8D4×C12711yescharacteristic
193C22×D4C2×C4×C12711yescharacteristic
194C23×C4—21no—
195C2×M16—22no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4C2×C4×C12711yescharacteristic
201C2×C4⋊C4C2×C4×C12711yescharacteristic
202C2×C4⋊C4D4×C12711yescharacteristic
203C2×C4⋊C4C23×C12711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42C2×C4×C12711yescharacteristic
211C2×C42D4×C12711yescharacteristic
212C2×C42D4×C12711yescharacteristic
213C4⋊C8—41no—
214#101—24no—
215C22⋊C8—22no—
216#97—41no—
217C4×C8—41no—
218C4×C8—41no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95C2×C4×C12711yescharacteristic
225#95C2×C4×C12711yescharacteristic
226C24—21no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×D4—41no—
231C2×D4—42no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4C22⋊C4×C12711yescharacteristic
240C22×C4C22⋊C4×C12711yescharacteristic
241C22×C4C22⋊C4×C12711yescharacteristic
242C22×C4C22⋊C4×C12711yescharacteristic
243C22×C4C22×C4×C12711yescharacteristic
244C22×C4C2×D4×C12711yescharacteristic
245C22×C4C2×D4×C12711yescharacteristic
246M16—24no—
247M16—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—42no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23C2×C22⋊C4×C12711yescharacteristic
274Q8—44no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4#99×C12711yescharacteristic
303C2×C4#99×C12711yescharacteristic
304C2×C4#104×C12712yes—
305C2×C4#135×C12712yes—
306C8—44no—
307C8—82no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22C2×#99×C12711yescharacteristic
318C22#416×C12711yescharacteristic
319C22C2×#104×C12711yescharacteristic
320C4—21no—
321C4—22no—
322C4—22no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4#355×C12712yes—
331C2—21no—
332C2—21no—
333C2—82no—
334C2C127×#150111yescharacteristic
335C2C127×#212011yescharacteristic
336C2C2×#355×C12711yescharacteristic
subgroupG/H
centerC2×C4×C127#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
socleC22×C127C2×#99
prime radicalC22×D4×C127C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×C4×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#101×C127C2semidirect
3. C127×#1461C2semidirect
4. C127×#1353C2semidirect
162. #11734C127direct (N × H)
170. C4×#101C2×C127semidirect
171. #1461C2×C127semidirect
172. #1353C2×C127semidirect
174. C127#11734direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#23982
C4×#101×C1272
C127×#14612
C127×#13532
C127×#13452
#11734127