Finite Group Databasebeta

← C127×#12321 #44575994

Subgroups — C127×#12321 #44575994

Order 32512.

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

normal subgroups only

This group has 384 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×#3287C211yesmaximal, characteristic
2C127×#2248C211yesmaximal, characteristic, complemented
3C127×#2209C211yesmaximal, characteristic, complemented
4C127×#1669C211yesmaximal, characteristic, complemented
5C127×#1361C211yesmaximal, characteristic, complemented
6C127×#1332C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8C2×#135×C127—21no—
9#550×C127—21no—
10C2×Q32×C127—21no—
11C2×SD32×C127—22no—
12#494×C127C2211yescharacteristic
13#493×C127C2211yescharacteristic
14#474×C127—21no—
15#466×C127—21no—
16#460×C127—21no—
17#449×C127—22no—
18C4×Q16×C127C2211yescharacteristic
19C8×D4×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic
21#358×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#332×C127—21no—
25C2×#41×C127—22no—
26#135×C127—42no—
27#135×C127—42no—
28C2×Q16×C127—21no—
29C2×Q16×C127—21no—
30C2×Q16×C127D411yescharacteristic
31C2×SD16×C127—22no—
32C2×D8×C127—21no—
33C2×D8×C127D412yes—
34C22×C8×C127—21no—
35C4⋊Q8×C127—21no—
36#127×C127—21no—
37#127×C127—41no—
38#121×C127—22no—
39#121×C127—42no—
40C4×Q8×C127—21no—
41C4×D4×C127—41no—
42C4×D4×C127D411yescharacteristic
43Q32×C127—42no—
44SD32×C127—44no—
45C2×C16×C127—21no—
46C2×C16×C127—21no—
47#107×C127—41no—
48#107×C127D411yescharacteristic
49C4⋊C8×C127—21no—
50C4⋊C8×C127D411yescharacteristic
51#103×C127—21no—
52#102×C127—22no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C4×C8×C127C2311yescharacteristic
56#41×C127—44no—
57#41×C127—82no—
58C2×Q8×C127—21no—
59C2×Q8×C127—41no—
60C2×D4×C127—22no—
61C2×D4×C127—42no—
62C2×D4×C127—42no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127D811yescharacteristic
66C22×C4×C127—21no—
67C22×C4×C127—42no—
68Q16×C127—21no—
69Q16×C127—42no—
70Q16×C127—41no—
71Q16×C127—41no—
72Q16×C127D812yes—
73SD16×C127—44no—
74D8×C127—42no—
75D8×C127—41no—
76D8×C127—41no—
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—41no—
85C4⋊C4×C127—41no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127D811yescharacteristic
88C22⋊C4×C127—21no—
89C22⋊C4×C127—42no—
90C42×C127—41no—
91C42×C127C2×D411yescharacteristic
92C16×C127—22no—
93C16×C127—41no—
94C23×C127—21no—
95C23×C127—42no—
96Q8×C127—22no—
97Q8×C127—41no—
98Q8×C127—81no—
99Q8×C127—81no—
100D4×C127—21no—
101D4×C127—21no—
102D4×C127—42no—
103D4×C127—42no—
104D4×C127—84no—
105D4×C127—81no—
106D4×C127—81no—
107D4×C127—161no—
108D4×C127—162no—
109C2×C4×C127—21no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127—84no—
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×C127C2×D811yescharacteristic
123C8×C127AGL(1,ℤ/8)11yescharacteristic
124C22×C127—41no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127—82no—
128C22×C127—82no—
129C22×C127—82no—
130C22×C127#44711yescharacteristic
131C4×C127—21no—
132C4×C127—41no—
133C4×C127—42no—
134C4×C127—81no—
135C4×C127—161no—
136C4×C127#44711yescharacteristic
137C4×C127#45311yescharacteristic
138C4×C127#50811yescharacteristic
139C4×C127#51011yescharacteristic
140#12321C12711yesmaximal, characteristic, direct product factor, Sylow
141C2×C127—81no—
142C2×C127—162no—
143C2×C127#161911yescharacteristic
144C2×C127#218911yescharacteristic
145C2×C127#219411yescharacteristic
146#3287C2×C12711yescharacteristic
147#2248C2×C12711yescharacteristic, complemented
148#2209C2×C12711yescharacteristic, complemented
149#1669C2×C12711yescharacteristic, complemented
150#1361C2×C12711yescharacteristic, complemented
151#1332C2×C12711yescharacteristic, complemented
152D4⋊C16C2×C12711yescharacteristic, complemented
153C127#1232111yescharacteristic, direct product factor, Sylow
154C2×#135—21no—
155#550—21no—
156C2×Q32—21no—
157C2×SD32—22no—
158#494C22×C12711yescharacteristic
159#493C22×C12711yescharacteristic
160#474—21no—
161#466—21no—
162#460—21no—
163#449—22no—
164C4×Q16C22×C12711yescharacteristic
165C8×D4C22×C12711yescharacteristic
166C4⋊C16C22×C12711yescharacteristic
167#358—21no—
168C4×C16C22×C12711yescharacteristic
169#335C22×C12711yescharacteristic
170#332—21no—
171C2×#41—22no—
172#135—42no—
173#135—42no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16D4×C12711yescharacteristic
177C2×SD16—22no—
178C2×D8—21no—
179C2×D8D4×C12712yes—
180C22×C8—21no—
181C4⋊Q8—21no—
182#127—21no—
183#127—41no—
184#121—22no—
185#121—42no—
186C4×Q8—21no—
187C4×D4—41no—
188C4×D4D4×C12711yescharacteristic
189Q32—42no—
190SD32—44no—
191C2×C16—21no—
192C2×C16—21no—
193#107—41no—
194#107D4×C12711yescharacteristic
195C4⋊C8—21no—
196C4⋊C8D4×C12711yescharacteristic
197#103—21no—
198#102—22no—
199#102—41no—
200C22⋊C8—21no—
201C4×C8C23×C12711yescharacteristic
202#41—44no—
203#41—82no—
204C2×Q8—21no—
205C2×Q8—41no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4D8×C12711yescharacteristic
212C22×C4—21no—
213C22×C4—42no—
214Q16—21no—
215Q16—42no—
216Q16—41no—
217Q16—41no—
218Q16D8×C12712yes—
219SD16—44no—
220D8—42no—
221D8—41no—
222D8—41no—
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—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4D8×C12711yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—42no—
236C42—41no—
237C42C2×D4×C12711yescharacteristic
238C16—22no—
239C16—41no—
240C23—21no—
241C23—42no—
242Q8—22no—
243Q8—41no—
244Q8—81no—
245Q8—81no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—42no—
250D4—84no—
251D4—81no—
252D4—81no—
253D4—161no—
254D4—162no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—84no—
261C2×C4—81no—
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—41no—
271C22—41no—
272C22—41no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22#447×C12711yescharacteristic
277C4—21no—
278C4—41no—
279C4—42no—
280C4—81no—
281C4—161no—
282C4#447×C12711yescharacteristic
283C4#453×C12711yescharacteristic
284C4#508×C12711yescharacteristic
285C4#510×C12711yescharacteristic
286C2—81no—
287C2—162no—
288C2C127×#161911yescharacteristic
289C2C127×#218911yescharacteristic
290C2C127×#219411yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3287C2
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
2. C127×#2248C2semidirect
3. C127×#2209C2semidirect
4. C127×#1669C2semidirect
5. C127×#1361C2semidirect
6. C127×#1332C2semidirect
7. C127×D4⋊C16C2semidirect
140. #12321C127direct (N × H)
147. #2248C2×C127semidirect
148. #2209C2×C127semidirect
149. #1669C2×C127semidirect
150. #1361C2×C127semidirect
151. #1332C2×C127semidirect
152. D4⋊C16C2×C127semidirect
153. C127#12321direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32872
C127×#22482
C127×#22092
C127×#16692
C127×#13612
C127×#13322
C127×D4⋊C162
#12321127