Finite Group Databasebeta

← C127×#12318 #44575991

Subgroups — C127×#12318 #44575991

Order 32512.

384 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 90 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
1D4×Q16×C127C211yesmaximal, characteristic, complemented
2C127×#2248C211yesmaximal, characteristic, complemented
3C127×#2207C211yesmaximal, characteristic, complemented
4C127×#1667C211yesmaximal, characteristic
5C127×#1362C211yesmaximal, characteristic, complemented
6C127×#1332C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8C22×Q16×C127—21no—
9D4×Q8×C127—21no—
10C2×SD32×C127—21no—
11C2×SD32×C127—22no—
12#494×C127C2211yescharacteristic
13#493×C127C2211yescharacteristic
14#467×C127—21no—
15#462×C127—21no—
16#459×C127—21no—
17#451×C127—22no—
18C4×Q16×C127C2211yescharacteristic
19C8×D4×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic
21#357×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#331×C127—21no—
25C22×Q8×C127—22no—
26C2×Q16×C127—22no—
27C2×Q16×C127—21no—
28C2×Q16×C127—42no—
29C2×Q16×C127—42no—
30C2×Q16×C127D411yescharacteristic
31C2×Q16×C127D412yes—
32C2×D8×C127—21no—
33C2×D8×C127—21no—
34C2×D8×C127—41no—
35C22×C8×C127—21no—
36C4⋊Q8×C127—21no—
37C4⋊Q8×C127—41no—
38#127×C127—21no—
39C22⋊Q8×C127—22no—
40C22⋊Q8×C127—42no—
41C4×Q8×C127—21no—
42C4×D4×C127—41no—
43C4×D4×C127D411yescharacteristic
44SD32×C127—44no—
45SD32×C127—42no—
46C2×C16×C127—21no—
47C2×C16×C127—21no—
48#107×C127D411yescharacteristic
49C4⋊C8×C127—21no—
50C4⋊C8×C127D411yescharacteristic
51#103×C127—22no—
52#103×C127—21no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C4×C8×C127C2311yescharacteristic
56C2×Q8×C127—21no—
57C2×Q8×C127—22no—
58C2×Q8×C127—42no—
59C2×Q8×C127—44no—
60C2×Q8×C127—82no—
61C2×D4×C127—41no—
62C2×D4×C127—41no—
63C2×D4×C127—41no—
64C2×D4×C127D811yescharacteristic
65C22×C4×C127—21no—
66C22×C4×C127—42no—
67Q16×C127—21no—
68Q16×C127—42no—
69Q16×C127—44no—
70Q16×C127—41no—
71Q16×C127—41no—
72Q16×C127D812yescomplemented
73D8×C127—42no—
74D8×C127—41no—
75D8×C127—41no—
76D8×C127—82no—
77C2×C8×C127—21no—
78C2×C8×C127—41no—
79C2×C8×C127—41no—
80C2×C8×C127—41no—
81C2×C8×C127C2×D411yescharacteristic
82C2×C8×C127C2×D411yescharacteristic
83C4⋊C4×C127—21no—
84C4⋊C4×C127—42no—
85C4⋊C4×C127—41no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127D811yescharacteristic
89C22⋊C4×C127—21no—
90C22⋊C4×C127—42no—
91C42×C127—41no—
92C42×C127C2×D411yescharacteristic
93C16×C127—22no—
94C16×C127—41no—
95C23×C127—21no—
96C23×C127—81no—
97Q8×C127—22no—
98Q8×C127—41no—
99Q8×C127—42no—
100Q8×C127—42no—
101Q8×C127—84no—
102D4×C127—21no—
103D4×C127—81no—
104D4×C127—81no—
105D4×C127—81no—
106D4×C127—81no—
107D4×C127—161no—
108D4×C127SD3212yescomplemented
109C2×C4×C127—21no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127—42no—
115C2×C4×C127—84no—
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×C127C2×D811yescharacteristic
123C8×C127AGL(1,ℤ/8)11yescharacteristic
124C22×C127—22no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127—161no—
128C22×C127—161no—
129C22×C127—161no—
130C22×C127#44711yescharacteristic
131C4×C127—21no—
132C4×C127—41no—
133C4×C127—42no—
134C4×C127—82no—
135C4×C127—81no—
136C4×C127#44711yescharacteristic
137C4×C127#45311yescharacteristic
138C4×C127C2×SD3211yescharacteristic
139C4×C127#51011yescharacteristic
140#12318C12711yesmaximal, characteristic, direct product factor, Sylow
141C2×C127—42no—
142C2×C127—321no—
143C2×C127#161911yescharacteristic
144C2×C127#218511yescharacteristic
145C2×C127#219311yescharacteristic
146D4×Q16C2×C12711yescharacteristic, complemented
147#2248C2×C12711yescharacteristic, complemented
148#2207C2×C12711yescharacteristic, complemented
149#1667C2×C12711yescharacteristic
150#1362C2×C12711yescharacteristic, complemented
151#1332C2×C12711yescharacteristic, complemented
152D4⋊C16C2×C12711yescharacteristic, complemented
153C127#1231811yescharacteristic, direct product factor, Sylow
154C22×Q16—21no—
155D4×Q8—21no—
156C2×SD32—21no—
157C2×SD32—22no—
158#494C22×C12711yescharacteristic
159#493C22×C12711yescharacteristic
160#467—21no—
161#462—21no—
162#459—21no—
163#451—22no—
164C4×Q16C22×C12711yescharacteristic
165C8×D4C22×C12711yescharacteristic
166C4⋊C16C22×C12711yescharacteristic
167#357—21no—
168C4×C16C22×C12711yescharacteristic
169#335C22×C12711yescharacteristic
170#331—21no—
171C22×Q8—22no—
172C2×Q16—22no—
173C2×Q16—21no—
174C2×Q16—42no—
175C2×Q16—42no—
176C2×Q16D4×C12711yescharacteristic
177C2×Q16D4×C12712yes—
178C2×D8—21no—
179C2×D8—21no—
180C2×D8—41no—
181C22×C8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—41no—
184#127—21no—
185C22⋊Q8—22no—
186C22⋊Q8—42no—
187C4×Q8—21no—
188C4×D4—41no—
189C4×D4D4×C12711yescharacteristic
190SD32—44no—
191SD32—42no—
192C2×C16—21no—
193C2×C16—21no—
194#107D4×C12711yescharacteristic
195C4⋊C8—21no—
196C4⋊C8D4×C12711yescharacteristic
197#103—22no—
198#103—21no—
199#102—41no—
200C22⋊C8—21no—
201C4×C8C23×C12711yescharacteristic
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—42no—
205C2×Q8—44no—
206C2×Q8—82no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4D8×C12711yescharacteristic
211C22×C4—21no—
212C22×C4—42no—
213Q16—21no—
214Q16—42no—
215Q16—44no—
216Q16—41no—
217Q16—41no—
218Q16D8×C12712yescomplemented
219D8—42no—
220D8—41no—
221D8—41no—
222D8—82no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—41no—
227C2×C8C2×D4×C12711yescharacteristic
228C2×C8C2×D4×C12711yescharacteristic
229C4⋊C4—21no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4D8×C12711yescharacteristic
235C22⋊C4—21no—
236C22⋊C4—42no—
237C42—41no—
238C42C2×D4×C12711yescharacteristic
239C16—22no—
240C16—41no—
241C23—21no—
242C23—81no—
243Q8—22no—
244Q8—41no—
245Q8—42no—
246Q8—42no—
247Q8—84no—
248D4—21no—
249D4—81no—
250D4—81no—
251D4—81no—
252D4—81no—
253D4—161no—
254D4SD32×C12712yescomplemented
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—84no—
262C2×C4#120×C12711yescharacteristic
263C2×C4C2×D8×C12711yescharacteristic
264C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
265C8—21no—
266C8—41no—
267C8—81no—
268C8C2×D8×C12711yescharacteristic
269C8AGL(1,ℤ/8)×C12711yescharacteristic
270C22—22no—
271C22—41no—
272C22—41no—
273C22—161no—
274C22—161no—
275C22—161no—
276C22#447×C12711yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—82no—
281C4—81no—
282C4#447×C12711yescharacteristic
283C4#453×C12711yescharacteristic
284C4C2×SD32×C12711yescharacteristic
285C4#510×C12711yescharacteristic
286C2—42no—
287C2—321no—
288C2C127×#161911yescharacteristic
289C2C127×#218511yescharacteristic
290C2C127×#219311yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#1667C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. D4×Q16×C127C2semidirect
2. C127×#2248C2semidirect
3. C127×#2207C2semidirect
5. C127×#1362C2semidirect
6. C127×#1332C2semidirect
7. C127×D4⋊C16C2semidirect
72. Q16×C127D8semidirect
108. D4×C127SD32semidirect
140. #12318C127direct (N × H)
146. D4×Q16C2×C127semidirect
147. #2248C2×C127semidirect
148. #2207C2×C127semidirect
150. #1362C2×C127semidirect
151. #1332C2×C127semidirect
152. D4⋊C16C2×C127semidirect
153. C127#12318direct (N × H)
218. Q16D8×C127semidirect
254. D4SD32×C127semidirect

Maximal subgroups

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

subgroupindex
D4×Q16×C1272
C127×#22482
C127×#22072
C127×#16672
C127×#13622
C127×#13322
C127×D4⋊C162
#12318127