Finite Group Databasebeta

← C127×#12328 #44576001

Subgroups — C127×#12328 #44576001

Order 32512.

378 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3391C211yesmaximal, characteristic, complemented
2C127×#2246C211yesmaximal, characteristic, complemented
3C127×#2206C211yesmaximal, characteristic, complemented
4C127×#1668C211yesmaximal, characteristic, complemented
5C127×#1361C211yesmaximal, characteristic, complemented
6C127×Q8⋊C16C211yesmaximal, characteristic, complemented
7C127×#1331C211yesmaximal, characteristic, complemented
8#550×C127—21no—
9C2×D16×C127—21no—
10C2×D16×C127—22no—
11#493×C127—21no—
12#493×C127C2211yescharacteristic
13#493×C127C2211yescharacteristic
14#460×C127—21no—
15#459×C127—22no—
16#459×C127—21no—
17C8×Q8×C127C2211yescharacteristic
18C4×D8×C127—21no—
19C4×D8×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic
21#357×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#331×C127—21no—
25C2×#41×C127—41no—
26C2×SD16×C127—41no—
27C2×D8×C127—22no—
28C2×D8×C127—21no—
29C2×D8×C127—21no—
30C2×D8×C127—21no—
31C2×D8×C127D411yescharacteristic
32C2×D8×C127D412yes—
33#127×C127—22no—
34#127×C127—21no—
35#127×C127—21no—
36#121×C127—42no—
37#121×C127—41no—
38C4×Q8×C127D411yescharacteristic
39C4×D4×C127—22no—
40C4×D4×C127—21no—
41D16×C127—44no—
42D16×C127—42no—
43C2×C16×C127—21no—
44C2×C16×C127—21no—
45#107×C127D411yescharacteristic
46C4⋊C8×C127—21no—
47C4⋊C8×C127—21no—
48C4⋊C8×C127D411yescharacteristic
49#102×C127—22no—
50#102×C127—21no—
51#102×C127—41no—
52C4×C8×C127—21no—
53C4×C8×C127C2311yescharacteristic
54#41×C127—82no—
55#41×C127—82no—
56C2×Q8×C127D811yescharacteristic
57C2×D4×C127—21no—
58C2×D4×C127—22no—
59C2×D4×C127—42no—
60C2×D4×C127—42no—
61C2×D4×C127—41no—
62C2×D4×C127—41no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C22×C4×C127—42no—
67C22×C4×C127—41no—
68SD16×C127—82no—
69D8×C127—21no—
70D8×C127—21no—
71D8×C127—42no—
72D8×C127—42no—
73D8×C127—41no—
74D8×C127—41no—
75D8×C127—41no—
76D8×C127—41no—
77D8×C127—82no—
78C2×C8×C127—21no—
79C2×C8×C127—41no—
80C2×C8×C127C2×D411yescharacteristic
81C2×C8×C127C2×D411yescharacteristic
82C4⋊C4×C127—21no—
83C4⋊C4×C127—21no—
84C4⋊C4×C127D811yescharacteristic
85C22⋊C4×C127—42no—
86C22⋊C4×C127—41no—
87C42×C127—21no—
88C42×C127C2×D411yescharacteristic
89C16×C127—22no—
90C16×C127—41no—
91C23×C127—42no—
92C23×C127—41no—
93C23×C127—81no—
94Q8×C127—21no—
95Q8×C127D1612yescomplemented
96D4×C127—41no—
97D4×C127—42no—
98D4×C127—42no—
99D4×C127—41no—
100D4×C127—84no—
101D4×C127—81no—
102D4×C127—81no—
103D4×C127—81no—
104D4×C127—81no—
105D4×C127—81no—
106D4×C127—81no—
107D4×C127—161no—
108D4×C127—161no—
109D4×C127—162no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—41no—
113C2×C4×C127—84no—
114C2×C4×C127—81no—
115C2×C4×C127—81no—
116C2×C4×C127#12011yescharacteristic
117C2×C4×C127C2×D811yescharacteristic
118C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
119C8×C127—21no—
120C8×C127—41no—
121C8×C127—81no—
122C8×C127#13511yescharacteristic
123C8×C127AGL(1,ℤ/8)11yescharacteristic
124C22×C127—81no—
125C22×C127—82no—
126C22×C127—82no—
127C22×C127—81no—
128C22×C127—82no—
129C22×C127—81no—
130C22×C127—161no—
131C22×C127—161no—
132C22×C127—161no—
133C22×C127#44711yescharacteristic
134C4×C127—22no—
135C4×C127—21no—
136C4×C127—41no—
137C4×C127—81no—
138C4×C127#44911yescharacteristic
139C4×C127#45311yescharacteristic
140C4×C127C2×D1611yescharacteristic
141C4×C127#50911yescharacteristic
142#12328C12711yesmaximal, characteristic, direct product factor, Sylow
143C2×C127—161no—
144C2×C127—162no—
145C2×C127—321no—
146C2×C127#162111yescharacteristic
147C2×C127#218411yescharacteristic
148C2×C127#219211yescharacteristic
149#3391C2×C12711yescharacteristic, complemented
150#2246C2×C12711yescharacteristic, complemented
151#2206C2×C12711yescharacteristic, complemented
152#1668C2×C12711yescharacteristic, complemented
153#1361C2×C12711yescharacteristic, complemented
154Q8⋊C16C2×C12711yescharacteristic, complemented
155#1331C2×C12711yescharacteristic, complemented
156C127#1232811yescharacteristic, direct product factor, Sylow
157#550—21no—
158C2×D16—21no—
159C2×D16—22no—
160#493—21no—
161#493C22×C12711yescharacteristic
162#493C22×C12711yescharacteristic
163#460—21no—
164#459—22no—
165#459—21no—
166C8×Q8C22×C12711yescharacteristic
167C4×D8—21no—
168C4×D8C22×C12711yescharacteristic
169C4⋊C16C22×C12711yescharacteristic
170#357—21no—
171C4×C16C22×C12711yescharacteristic
172#335C22×C12711yescharacteristic
173#331—21no—
174C2×#41—41no—
175C2×SD16—41no—
176C2×D8—22no—
177C2×D8—21no—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8D4×C12711yescharacteristic
181C2×D8D4×C12712yes—
182#127—22no—
183#127—21no—
184#127—21no—
185#121—42no—
186#121—41no—
187C4×Q8D4×C12711yescharacteristic
188C4×D4—22no—
189C4×D4—21no—
190D16—44no—
191D16—42no—
192C2×C16—21no—
193C2×C16—21no—
194#107D4×C12711yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8D4×C12711yescharacteristic
198#102—22no—
199#102—21no—
200#102—41no—
201C4×C8—21no—
202C4×C8C23×C12711yescharacteristic
203#41—82no—
204#41—82no—
205C2×Q8D8×C12711yescharacteristic
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—42no—
216C22×C4—41no—
217SD16—82no—
218D8—21no—
219D8—21no—
220D8—42no—
221D8—42no—
222D8—41no—
223D8—41no—
224D8—41no—
225D8—41no—
226D8—82no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8C2×D4×C12711yescharacteristic
230C2×C8C2×D4×C12711yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4D8×C12711yescharacteristic
234C22⋊C4—42no—
235C22⋊C4—41no—
236C42—21no—
237C42C2×D4×C12711yescharacteristic
238C16—22no—
239C16—41no—
240C23—42no—
241C23—41no—
242C23—81no—
243Q8—21no—
244Q8D16×C12712yescomplemented
245D4—41no—
246D4—42no—
247D4—42no—
248D4—41no—
249D4—84no—
250D4—81no—
251D4—81no—
252D4—81no—
253D4—81no—
254D4—81no—
255D4—81no—
256D4—161no—
257D4—161no—
258D4—162no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—84no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4#120×C12711yescharacteristic
266C2×C4C2×D8×C12711yescharacteristic
267C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
268C8—21no—
269C8—41no—
270C8—81no—
271C8#135×C12711yescharacteristic
272C8AGL(1,ℤ/8)×C12711yescharacteristic
273C22—81no—
274C22—82no—
275C22—82no—
276C22—81no—
277C22—82no—
278C22—81no—
279C22—161no—
280C22—161no—
281C22—161no—
282C22#447×C12711yescharacteristic
283C4—22no—
284C4—21no—
285C4—41no—
286C4—81no—
287C4#449×C12711yescharacteristic
288C4#453×C12711yescharacteristic
289C4C2×D16×C12711yescharacteristic
290C4#509×C12711yescharacteristic
291C2—161no—
292C2—162no—
293C2—321no—
294C2C127×#162111yescharacteristic
295C2C127×#218411yescharacteristic
296C2C127×#219211yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3391C2semidirect
2. C127×#2246C2semidirect
3. C127×#2206C2semidirect
4. C127×#1668C2semidirect
5. C127×#1361C2semidirect
6. C127×Q8⋊C16C2semidirect
7. C127×#1331C2semidirect
95. Q8×C127D16semidirect
142. #12328C127direct (N × H)
149. #3391C2×C127semidirect
150. #2246C2×C127semidirect
151. #2206C2×C127semidirect
152. #1668C2×C127semidirect
153. #1361C2×C127semidirect
154. Q8⋊C16C2×C127semidirect
155. #1331C2×C127semidirect
156. C127#12328direct (N × H)
244. Q8D16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#33912
C127×#22462
C127×#22062
C127×#16682
C127×#13612
C127×Q8⋊C162
C127×#13312
#12328127