Finite Group Databasebeta

← C127×#9333 #44573006

Subgroups — C127×#9333 #44573006

Order 32512.

438 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3213C211yesmaximal, characteristic
2C127×#2980C211yesmaximal, characteristic, complemented
3C127×#1553C211yesmaximal, characteristic, complemented
4C127×#1533C211yesmaximal, characteristic, complemented
5C127×#1469C211yesmaximal, characteristic, complemented
6C127×#1308C211yesmaximal, characteristic, complemented
7C127×#1287C211yesmaximal, characteristic, complemented
8#532×C127C2211yescharacteristic
9C2×#124×C127C2211yescharacteristic
10#496×C127—21no—
11#489×C127—21no—
12#463×C127—22no—
13#448×C127C412yes—
14#447×C127C412yes—
15#435×C127—22no—
16#435×C127—22no—
17#432×C127C2211yescharacteristic
18#432×C127C2211yescharacteristic
19#432×C127C2211yescharacteristic
20#432×C127C2211yescharacteristic
21#431×C127C2211yescharacteristic
22#408×C127—22no—
23#332×C127—21no—
24#331×C127—21no—
25#329×C127—22no—
26#323×C127—22no—
27C2×#41×C127—21no—
28C22×Q8×C127C2×C411yescharacteristic
29C22×D4×C127C2×C411yescharacteristic
30C2×SD16×C127—22no—
31C2×D8×C127—22no—
32C2×M16×C127—22no—
33C22×C8×C127—22no—
34C4⋊Q8×C127D411yescharacteristic
35#127×C127D411yescharacteristic
36#124×C127—21no—
37#124×C127—21no—
38#124×C127—21no—
39#124×C127—21no—
40#124×C127—21no—
41#121×C127—21no—
42#121×C127C2×C412yes—
43C4×D4×C127—21no—
44C4×D4×C127D412yes—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—21no—
47C2×C42×C127C2311yescharacteristic
48C4⋊C8×C127—21no—
49C4⋊C8×C127—21no—
50C4⋊C8×C127—21no—
51C4⋊C8×C127—21no—
52C4⋊C8×C127—21no—
53#103×C127—22no—
54#102×C127—22no—
55C22⋊C8×C127—24no—
56C22⋊C8×C127—21no—
57C22⋊C8×C127—21no—
58C22⋊C8×C127—21no—
59C22⋊C8×C127C2×C412yes—
60C22⋊C8×C127D412yes—
61#97×C127—21no—
62#97×C127—21no—
63#97×C127—21no—
64#97×C127D412yes—
65C4×C8×C127—21no—
66C24×C127—21no—
67#41×C127—81no—
68#41×C127—81no—
69C2×Q8×C127—21no—
70C2×Q8×C127—21no—
71C2×Q8×C127—21no—
72C2×Q8×C127—81no—
73C2×D4×C127—22no—
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127—21no—
77C2×D4×C127—21no—
78C2×D4×C127—81no—
79C2×D4×C127C22⋊C412yes—
80C22×C4×C127—22no—
81C22×C4×C127—21no—
82C22×C4×C127C22×C411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84C22×C4×C127#4111yescharacteristic
85SD16×C127—82no—
86D8×C127—82no—
87M16×C127—82no—
88C2×C8×C127—22no—
89C2×C8×C127—22no—
90C2×C8×C127—22no—
91C2×C8×C127—41no—
92C2×C8×C127—41no—
93C2×C8×C127—82no—
94C4⋊C4×C127—21no—
95C4⋊C4×C127C22⋊C412yes—
96C22⋊C4×C127—22no—
97C22⋊C4×C127—22no—
98C22⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C42×C127C2×D411yescharacteristic
103C42×C127C2×D411yescharacteristic
104C42×C127C2×D411yescharacteristic
105C42×C127#4111yescharacteristic
106C23×C127—22no—
107C23×C127—42no—
108C23×C127—41no—
109C23×C127—81no—
110C23×C127—81no—
111C23×C127C4×D411yescharacteristic
112Q8×C127—42no—
113Q8×C127—81no—
114Q8×C127—81no—
115D4×C127—42no—
116D4×C127—42no—
117D4×C127—81no—
118D4×C127—81no—
119D4×C127—81no—
120D4×C127—82no—
121C2×C4×C127—22no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127—41no—
128C2×C4×C127—81no—
129C2×C4×C127C2×C22⋊C411yescharacteristic
130C2×C4×C127C4×D411yescharacteristic
131C2×C4×C127#12011yescharacteristic
132C2×C4×C127#12111yescharacteristic
133C2×C4×C127#12111yescharacteristic
134C2×C4×C127#12311yescharacteristic
135C8×C127—42no—
136C8×C127—42no—
137C8×C127—42no—
138C8×C127—81no—
139C8×C127—81no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—42no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127—81no—
147C22×C127—81no—
148C22×C127—81no—
149C22×C127—162no—
150C22×C127#38611yescharacteristic
151C4×C127—21no—
152C4×C127—21no—
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—42no—
158C4×C127—81no—
159#9333C12711yesmaximal, characteristic, direct product factor, Sylow
160C2×C127—41no—
161C2×C127—82no—
162C2×C127—161no—
163C2×C127#188111yescharacteristic
164C2×C127#190411yescharacteristic
165C2×C127#191411yescharacteristic
166#3213C2×C12711yescharacteristic
167#2980C2×C12711yescharacteristic, complemented
168#1553C2×C12711yescharacteristic, complemented
169#1533C2×C12711yescharacteristic, complemented
170#1469C2×C12711yescharacteristic, complemented
171#1308C2×C12711yescharacteristic, complemented
172#1287C2×C12711yescharacteristic, complemented
173C127#933311yescharacteristic, direct product factor, Sylow
174#532C22×C12711yescharacteristic
175C2×#124C22×C12711yescharacteristic
176#496—21no—
177#489—21no—
178#463—22no—
179#448C4×C12712yes—
180#447C4×C12712yes—
181#435—22no—
182#435—22no—
183#432C22×C12711yescharacteristic
184#432C22×C12711yescharacteristic
185#432C22×C12711yescharacteristic
186#432C22×C12711yescharacteristic
187#431C22×C12711yescharacteristic
188#408—22no—
189#332—21no—
190#331—21no—
191#329—22no—
192#323—22no—
193C2×#41—21no—
194C22×Q8C2×C4×C12711yescharacteristic
195C22×D4C2×C4×C12711yescharacteristic
196C2×SD16—22no—
197C2×D8—22no—
198C2×M16—22no—
199C22×C8—22no—
200C4⋊Q8D4×C12711yescharacteristic
201#127D4×C12711yescharacteristic
202#124—21no—
203#124—21no—
204#124—21no—
205#124—21no—
206#124—21no—
207#121—21no—
208#121C2×C4×C12712yes—
209C4×D4—21no—
210C4×D4D4×C12712yes—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C42C23×C12711yescharacteristic
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219#103—22no—
220#102—22no—
221C22⋊C8—24no—
222C22⋊C8—21no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8C2×C4×C12712yes—
226C22⋊C8D4×C12712yes—
227#97—21no—
228#97—21no—
229#97—21no—
230#97D4×C12712yes—
231C4×C8—21no—
232C24—21no—
233#41—81no—
234#41—81no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—81no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—81no—
245C2×D4C22⋊C4×C12712yes—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4C22×C4×C12711yescharacteristic
249C22×C4C2×D4×C12711yescharacteristic
250C22×C4#41×C12711yescharacteristic
251SD16—82no—
252D8—82no—
253M16—82no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—82no—
260C4⋊C4—21no—
261C4⋊C4C22⋊C4×C12712yes—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42C2×D4×C12711yescharacteristic
269C42C2×D4×C12711yescharacteristic
270C42C2×D4×C12711yescharacteristic
271C42#41×C12711yescharacteristic
272C23—22no—
273C23—42no—
274C23—41no—
275C23—81no—
276C23—81no—
277C23C4×D4×C12711yescharacteristic
278Q8—42no—
279Q8—81no—
280Q8—81no—
281D4—42no—
282D4—42no—
283D4—81no—
284D4—81no—
285D4—81no—
286D4—82no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4C2×C22⋊C4×C12711yescharacteristic
296C2×C4C4×D4×C12711yescharacteristic
297C2×C4#120×C12711yescharacteristic
298C2×C4#121×C12711yescharacteristic
299C2×C4#121×C12711yescharacteristic
300C2×C4#123×C12711yescharacteristic
301C8—42no—
302C8—42no—
303C8—42no—
304C8—81no—
305C8—81no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22—162no—
316C22#386×C12711yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—42no—
324C4—81no—
325C2—41no—
326C2—82no—
327C2—161no—
328C2C127×#188111yescharacteristic
329C2C127×#190411yescharacteristic
330C2C127×#191411yescharacteristic
subgroupG/H
center, socleC22×C127#386
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3213C2
LCSG ▷ C22×C4 ▷ 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×#2980C2semidirect
3. C127×#1553C2semidirect
4. C127×#1533C2semidirect
5. C127×#1469C2semidirect
6. C127×#1308C2semidirect
7. C127×#1287C2semidirect
159. #9333C127direct (N × H)
167. #2980C2×C127semidirect
168. #1553C2×C127semidirect
169. #1533C2×C127semidirect
170. #1469C2×C127semidirect
171. #1308C2×C127semidirect
172. #1287C2×C127semidirect
173. C127#9333direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32132
C127×#29802
C127×#15532
C127×#15332
C127×#14692
C127×#13082
C127×#12872
#9333127