Finite Group Databasebeta

← C127×#12217 #44575890

Subgroups — C127×#12217 #44575890

Order 32512.

484 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2840C211yesmaximal, characteristic
2C127×#2029C211yesmaximal, characteristic
3C127×#1602C212yesmaximal, complemented
4C127×#1351C212yesmaximal, complemented
5C127×#1316C211yesmaximal, characteristic
6#543×C127C2211yescharacteristic
7#451×C127—22no—
8#393×C127—22no—
9#393×C127—21no—
10#393×C127—22no—
11#393×C127C2212yes—
12#379×C127—21no—
13#379×C127—21no—
14#379×C127C2211yescharacteristic
15#342×C127—21no—
16#342×C127C2211yescharacteristic
17#328×C127—22no—
18#323×C127C2212yes—
19C25×C127D411yescharacteristic
20C22×Q8×C127—21no—
21C22×Q8×C127D412yes—
22C2×Q16×C127—42no—
23C22⋊Q8×C127—22no—
24C22⋊Q8×C127—42no—
25C2×C22⋊C4×C127—21no—
26C2×C22⋊C4×C127—21no—
27C2×C22⋊C4×C127—21no—
28C2×C22⋊C4×C127—22no—
29C2×C22⋊C4×C127—21no—
30C2×C22⋊C4×C127—22no—
31C2×C22⋊C4×C127—22no—
32C2×C22⋊C4×C127—42no—
33C2×C22⋊C4×C127D411yescharacteristic
34C2×C22⋊C4×C127C2311yescharacteristic
35#103×C127—42no—
36C22⋊C8×C127—22no—
37#95×C127—21no—
38#95×C127—22no—
39#95×C127—22no—
40#95×C127D412yes—
41C24×C127—21no—
42C24×C127—21no—
43C24×C127—41no—
44C24×C127—41no—
45C24×C127—82no—
46C24×C127Q1612yescomplemented
47C24×C127C2×D411yescharacteristic
48C2×Q8×C127—22no—
49C2×Q8×C127—42no—
50C2×Q8×C127—82no—
51C2×Q8×C127—82no—
52C2×Q8×C127—81no—
53C22×C4×C127—22no—
54C22×C4×C127—21no—
55C22×C4×C127—22no—
56C22×C4×C127—22no—
57C22×C4×C127—41no—
58C22×C4×C127C2×D412yes—
59Q16×C127—84no—
60C2×C8×C127—42no—
61C4⋊C4×C127—22no—
62C4⋊C4×C127—42no—
63C22⋊C4×C127—41no—
64C22⋊C4×C127—41no—
65C22⋊C4×C127—41no—
66C22⋊C4×C127—41no—
67C22⋊C4×C127—41no—
68C22⋊C4×C127—41no—
69C22⋊C4×C127—82no—
70C22⋊C4×C127—82no—
71C22⋊C4×C127—82no—
72C22⋊C4×C127—81no—
73C22⋊C4×C127—81no—
74C22⋊C4×C127—84no—
75C23×C127—21no—
76C23×C127—21no—
77C23×C127—22no—
78C23×C127—42no—
79C23×C127—42no—
80C23×C127—42no—
81C23×C127—42no—
82C23×C127—44no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127—81no—
86C23×C127—82no—
87C23×C127—82no—
88C23×C127—82no—
89C23×C127—84no—
90C23×C127#12011yescharacteristic
91C23×C127C2×Q1611yescharacteristic
92C23×C127#13711yescharacteristic
93Q8×C127—44no—
94Q8×C127—82no—
95Q8×C127—82no—
96Q8×C127—84no—
97C2×C4×C127—22no—
98C2×C4×C127—42no—
99C2×C4×C127—42no—
100C2×C4×C127—42no—
101C2×C4×C127—42no—
102C2×C4×C127—42no—
103C2×C4×C127—42no—
104C2×C4×C127—41no—
105C2×C4×C127—42no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127—82no—
109C2×C4×C127—81no—
110C2×C4×C127—82no—
111C8×C127—82no—
112C22×C127—22no—
113C22×C127—21no—
114C22×C127—21no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—44no—
121C22×C127—42no—
122C22×C127—81no—
123C22×C127—82no—
124C22×C127—82no—
125C22×C127—84no—
126C22×C127—82no—
127C22×C127#45111yescharacteristic
128C22×C127#45511yescharacteristic
129C22×C127B(4,𝔽2)11yescharacteristic
130C4×C127—42no—
131C4×C127—81no—
132C4×C127—82no—
133C4×C127—82no—
134C4×C127—82no—
135C4×C127—161no—
136#12217C12711yesmaximal, characteristic, direct product factor, Sylow
137C2×C127—21no—
138C2×C127—21no—
139C2×C127—41no—
140C2×C127—41no—
141C2×C127—82no—
142C2×C127#160211yescharacteristic
143C2×C127#219912yes—
144#2840C2×C12711yescharacteristic
145#2029C2×C12711yescharacteristic
146#1602C2×C12712yescomplemented
147#1351C2×C12712yescomplemented
148#1316C2×C12711yescharacteristic
149C127#1221711yescharacteristic, direct product factor, Sylow
150#543C22×C12711yescharacteristic
151#451—22no—
152#393—22no—
153#393—21no—
154#393—22no—
155#393C22×C12712yes—
156#379—21no—
157#379—21no—
158#379C22×C12711yescharacteristic
159#342—21no—
160#342C22×C12711yescharacteristic
161#328—22no—
162#323C22×C12712yes—
163C25D4×C12711yescharacteristic
164C22×Q8—21no—
165C22×Q8D4×C12712yes—
166C2×Q16—42no—
167C22⋊Q8—22no—
168C22⋊Q8—42no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4D4×C12711yescharacteristic
178C2×C22⋊C4C23×C12711yescharacteristic
179#103—42no—
180C22⋊C8—22no—
181#95—21no—
182#95—22no—
183#95—22no—
184#95D4×C12712yes—
185C24—21no—
186C24—21no—
187C24—41no—
188C24—41no—
189C24—82no—
190C24Q16×C12712yescomplemented
191C24C2×D4×C12711yescharacteristic
192C2×Q8—22no—
193C2×Q8—42no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8—81no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—41no—
202C22×C4C2×D4×C12712yes—
203Q16—84no—
204C2×C8—42no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C22⋊C4—84no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—44no—
227C23—41no—
228C23—41no—
229C23—81no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—84no—
234C23#120×C12711yescharacteristic
235C23C2×Q16×C12711yescharacteristic
236C23#137×C12711yescharacteristic
237Q8—44no—
238Q8—82no—
239Q8—82no—
240Q8—84no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4—82no—
255C8—82no—
256C22—22no—
257C22—21no—
258C22—21no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—81no—
267C22—82no—
268C22—82no—
269C22—84no—
270C22—82no—
271C22#451×C12711yescharacteristic
272C22#455×C12711yescharacteristic
273C22B(4,𝔽2)×C12711yescharacteristic
274C4—42no—
275C4—81no—
276C4—82no—
277C4—82no—
278C4—82no—
279C4—161no—
280C2—21no—
281C2—21no—
282C2—41no—
283C2—41no—
284C2—82no—
285C2C127×#160211yescharacteristic
286C2C127×#219912yes—
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC25×C127D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1602C2semidirect
4. C127×#1351C2semidirect
46. C24×C127Q16semidirect
136. #12217C127direct (N × H)
146. #1602C2×C127semidirect
147. #1351C2×C127semidirect
149. C127#12217direct (N × H)
190. C24Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#28402
C127×#20292
C127×#16022
C127×#13512
C127×#13162
#12217127