Finite Group Databasebeta

← C127×#12106 #44575779

Subgroups — C127×#12106 #44575779

Order 32512.

424 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3189C211yesmaximal, characteristic
2C127×#2188C212yesmaximal, complemented
3C127×#2001C211yesmaximal, characteristic
4C127×#1348C212yesmaximal, complemented
5C127×#1314C211yesmaximal, characteristic
6C22×Q16×C127C2211yescharacteristic
7C2×C22⋊Q8×C127—21no—
8C2×C22⋊Q8×C127—21no—
9C2×Q32×C127—22no—
10#484×C127—22no—
11#467×C127—21no—
12#467×C127C2212yes—
13#451×C127—22no—
14C2×#103×C127—21no—
15C2×C22⋊C8×C127C2211yescharacteristic
16#358×C127—22no—
17C22⋊C16×C127C2212yes—
18#340×C127C2211yescharacteristic
19C22×Q8×C127—21no—
20C22×Q8×C127—21no—
21C23×C4×C127D411yescharacteristic
22C2×Q16×C127—21no—
23C2×Q16×C127—22no—
24C2×Q16×C127—42no—
25C2×Q16×C127D412yes—
26C22×C8×C127D411yescharacteristic
27C22×C8×C127C2311yescharacteristic
28C22⋊Q8×C127—22no—
29C22⋊Q8×C127—22no—
30C22⋊Q8×C127—42no—
31C22⋊Q8×C127—41no—
32C22⋊Q8×C127—42no—
33C22⋊Q8×C127—41no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—41no—
37C2×C4⋊C4×C127—41no—
38C2×C22⋊C4×C127—41no—
39C2×C22⋊C4×C127—41no—
40Q32×C127—44no—
41C2×C16×C127—22no—
42#107×C127—21no—
43#107×C127D412yes—
44#103×C127—22no—
45#103×C127—22no—
46#103×C127—42no—
47C22⋊C8×C127—22no—
48C24×C127D811yescharacteristic
49C2×Q8×C127—22no—
50C2×Q8×C127—22no—
51C2×Q8×C127—41no—
52C2×Q8×C127—41no—
53C2×Q8×C127—41no—
54C2×Q8×C127—41no—
55C2×Q8×C127—41no—
56C2×Q8×C127—41no—
57C2×Q8×C127—81no—
58C2×Q8×C127—81no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—41no—
62C22×C4×C127—41no—
63C22×C4×C127—41no—
64C22×C4×C127—41no—
65C22×C4×C127—41no—
66C22×C4×C127—41no—
67C22×C4×C127D811yescharacteristic
68C22×C4×C127C2×D411yescharacteristic
69Q16×C127—24no—
70Q16×C127—42no—
71Q16×C127—42no—
72Q16×C127—81no—
73C2×C8×C127—21no—
74C2×C8×C127—41no—
75C2×C8×C127—41no—
76C2×C8×C127C2×D412yes—
77C4⋊C4×C127—22no—
78C4⋊C4×C127—22no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127—41no—
81C4⋊C4×C127—42no—
82C4⋊C4×C127—44no—
83C4⋊C4×C127—42no—
84C22⋊C4×C127—42no—
85C22⋊C4×C127—44no—
86C22⋊C4×C127—42no—
87C16×C127—42no—
88C23×C127—21no—
89C23×C127—21no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127Q3212yescomplemented
93C23×C127C2×D811yescharacteristic
94Q8×C127—42no—
95Q8×C127—42no—
96Q8×C127—42no—
97Q8×C127—42no—
98Q8×C127—81no—
99Q8×C127—82no—
100Q8×C127—81no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—21no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—41no—
107C2×C4×C127—42no—
108C2×C4×C127—42no—
109C2×C4×C127—42no—
110C2×C4×C127—42no—
111C2×C4×C127—81no—
112C2×C4×C127—81no—
113C2×C4×C127—81no—
114C2×C4×C127—82no—
115C2×C4×C127—82no—
116C2×C4×C127—81no—
117C2×C4×C127—81no—
118C2×C4×C127#12011yescharacteristic
119C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
120C8×C127—22no—
121C8×C127—41no—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—22no—
125C22×C127—42no—
126C22×C127—41no—
127C22×C127—41no—
128C22×C127—81no—
129C22×C127#44711yescharacteristic
130C22×C127C2×Q3211yescharacteristic
131C22×C127#51011yescharacteristic
132C4×C127—21no—
133C4×C127—41no—
134C4×C127—82no—
135C4×C127—82no—
136C4×C127—82no—
137C4×C127—82no—
138C4×C127#45311yescharacteristic
139C4×C127B(4,𝔽2)11yescharacteristic
140#12106C12711yesmaximal, characteristic, direct product factor, Sylow
141C2×C127—21no—
142C2×C127—21no—
143C2×C127—42no—
144C2×C127#159511yescharacteristic
145C2×C127#218811yescharacteristic
146C2×C127#219111yescharacteristic
147#3189C2×C12711yescharacteristic
148#2188C2×C12712yescomplemented
149#2001C2×C12711yescharacteristic
150#1348C2×C12712yescomplemented
151#1314C2×C12711yescharacteristic
152C127#1210611yescharacteristic, direct product factor, Sylow
153C22×Q16C22×C12711yescharacteristic
154C2×C22⋊Q8—21no—
155C2×C22⋊Q8—21no—
156C2×Q32—22no—
157#484—22no—
158#467—21no—
159#467C22×C12712yes—
160#451—22no—
161C2×#103—21no—
162C2×C22⋊C8C22×C12711yescharacteristic
163#358—22no—
164C22⋊C16C22×C12712yes—
165#340C22×C12711yescharacteristic
166C22×Q8—21no—
167C22×Q8—21no—
168C23×C4D4×C12711yescharacteristic
169C2×Q16—21no—
170C2×Q16—22no—
171C2×Q16—42no—
172C2×Q16D4×C12712yes—
173C22×C8D4×C12711yescharacteristic
174C22×C8C23×C12711yescharacteristic
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—42no—
178C22⋊Q8—41no—
179C22⋊Q8—42no—
180C22⋊Q8—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187Q32—44no—
188C2×C16—22no—
189#107—21no—
190#107D4×C12712yes—
191#103—22no—
192#103—22no—
193#103—42no—
194C22⋊C8—22no—
195C24D8×C12711yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—81no—
205C2×Q8—81no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4D8×C12711yescharacteristic
215C22×C4C2×D4×C12711yescharacteristic
216Q16—24no—
217Q16—42no—
218Q16—42no—
219Q16—81no—
220C2×C8—21no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8C2×D4×C12712yes—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C16—42no—
235C23—21no—
236C23—21no—
237C23—41no—
238C23—41no—
239C23Q32×C12712yescomplemented
240C23C2×D8×C12711yescharacteristic
241Q8—42no—
242Q8—42no—
243Q8—42no—
244Q8—42no—
245Q8—81no—
246Q8—82no—
247Q8—81no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4#120×C12711yescharacteristic
266C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
267C8—22no—
268C8—41no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—81no—
276C22#447×C12711yescharacteristic
277C22C2×Q32×C12711yescharacteristic
278C22#510×C12711yescharacteristic
279C4—21no—
280C4—41no—
281C4—82no—
282C4—82no—
283C4—82no—
284C4—82no—
285C4#453×C12711yescharacteristic
286C4B(4,𝔽2)×C12711yescharacteristic
287C2—21no—
288C2—21no—
289C2—42no—
290C2C127×#159511yescharacteristic
291C2C127×#218811yescharacteristic
292C2C127×#219111yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC24×C127D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C22⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2188C2semidirect
4. C127×#1348C2semidirect
92. C23×C127Q32semidirect
140. #12106C127direct (N × H)
148. #2188C2×C127semidirect
150. #1348C2×C127semidirect
152. C127#12106direct (N × H)
239. C23Q32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31892
C127×#21882
C127×#20012
C127×#13482
C127×#13142
#12106127