Finite Group Databasebeta

← C127×#12218 #44575891

Subgroups — C127×#12218 #44575891

Order 32512.

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

normal subgroups only

This group has 456 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×#2842C211yesmaximal, characteristic
2C127×#2029C211yesmaximal, characteristic
3C127×#1614C212yesmaximal, complemented
4C127×#1352C212yesmaximal, complemented
5C127×#1316C211yesmaximal, characteristic
6#543×C127C2211yescharacteristic
7#484×C127—22no—
8#399×C127—21no—
9#399×C127C2212yes—
10#396×C127—22no—
11#396×C127—22no—
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—22no—
21C22⋊Q8×C127—42no—
22C2×C4⋊C4×C127—22no—
23C2×C4⋊C4×C127—21no—
24C2×C4⋊C4×C127—22no—
25C2×C4⋊C4×C127D412yes—
26C2×C22⋊C4×C127—21no—
27C2×C22⋊C4×C127—21no—
28C2×C22⋊C4×C127—21no—
29C2×C22⋊C4×C127—22no—
30C2×C22⋊C4×C127—21no—
31C2×C22⋊C4×C127—22no—
32C2×C22⋊C4×C127—22no—
33C2×C22⋊C4×C127—42no—
34C2×C22⋊C4×C127D411yescharacteristic
35C2×C22⋊C4×C127C2311yescharacteristic
36#107×C127—42no—
37#103×C127—42no—
38C22⋊C8×C127—22no—
39#95×C127—21no—
40#95×C127D412yes—
41C24×C127—21no—
42C24×C127—21no—
43C24×C127—41no—
44C24×C127—41no—
45C24×C127—82no—
46C24×C127Q1612yes—
47C24×C127C2×D411yescharacteristic
48C2×Q8×C127—22no—
49C22×C4×C127—22no—
50C22×C4×C127—21no—
51C22×C4×C127—22no—
52C22×C4×C127—22no—
53C22×C4×C127—41no—
54C22×C4×C127C2×D412yes—
55C2×C8×C127—42no—
56C4⋊C4×C127—22no—
57C4⋊C4×C127—42no—
58C4⋊C4×C127—42no—
59C4⋊C4×C127—82no—
60C4⋊C4×C127—82no—
61C4⋊C4×C127—81no—
62C22⋊C4×C127—41no—
63C22⋊C4×C127—41no—
64C22⋊C4×C127—41no—
65C22⋊C4×C127—41no—
66C22⋊C4×C127—41no—
67C22⋊C4×C127—41no—
68C22⋊C4×C127—82no—
69C22⋊C4×C127—82no—
70C22⋊C4×C127—82no—
71C22⋊C4×C127—81no—
72C22⋊C4×C127—81no—
73C22⋊C4×C127—84no—
74C23×C127—21no—
75C23×C127—21no—
76C23×C127—22no—
77C23×C127—42no—
78C23×C127—42no—
79C23×C127—42no—
80C23×C127—42no—
81C23×C127—44no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—81no—
85C23×C127—82no—
86C23×C127—82no—
87C23×C127—82no—
88C23×C127—84no—
89C23×C127#12011yescharacteristic
90C23×C127C2×Q1611yescharacteristic
91C23×C127#13711yescharacteristic
92Q8×C127—82no—
93C2×C4×C127—22no—
94C2×C4×C127—42no—
95C2×C4×C127—42no—
96C2×C4×C127—42no—
97C2×C4×C127—42no—
98C2×C4×C127—42no—
99C2×C4×C127—42no—
100C2×C4×C127—41no—
101C2×C4×C127—42no—
102C2×C4×C127—42no—
103C2×C4×C127—42no—
104C2×C4×C127—82no—
105C2×C4×C127—81no—
106C2×C4×C127—82no—
107C8×C127—82no—
108C22×C127—22no—
109C22×C127—21no—
110C22×C127—21no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127—42no—
114C22×C127—42no—
115C22×C127—42no—
116C22×C127—44no—
117C22×C127—42no—
118C22×C127—81no—
119C22×C127—82no—
120C22×C127—82no—
121C22×C127—84no—
122C22×C127—82no—
123C22×C127#45111yescharacteristic
124C22×C127#45511yescharacteristic
125C22×C127B(4,𝔽2)11yescharacteristic
126C4×C127—42no—
127C4×C127—81no—
128C4×C127—82no—
129C4×C127—82no—
130C4×C127—82no—
131C4×C127—161no—
132#12218C12711yesmaximal, characteristic, direct product factor, Sylow
133C2×C127—21no—
134C2×C127—21no—
135C2×C127—41no—
136C2×C127—41no—
137C2×C127—82no—
138C2×C127#160211yescharacteristic
139C2×C127#220012yes—
140#2842C2×C12711yescharacteristic
141#2029C2×C12711yescharacteristic
142#1614C2×C12712yescomplemented
143#1352C2×C12712yescomplemented
144#1316C2×C12711yescharacteristic
145C127#1221811yescharacteristic, direct product factor, Sylow
146#543C22×C12711yescharacteristic
147#484—22no—
148#399—21no—
149#399C22×C12712yes—
150#396—22no—
151#396—22no—
152#379—21no—
153#379—21no—
154#379C22×C12711yescharacteristic
155#342—21no—
156#342C22×C12711yescharacteristic
157#328—22no—
158#323C22×C12712yes—
159C25D4×C12711yescharacteristic
160C22⋊Q8—22no—
161C22⋊Q8—42no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4D4×C12712yes—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4D4×C12711yescharacteristic
175C2×C22⋊C4C23×C12711yescharacteristic
176#107—42no—
177#103—42no—
178C22⋊C8—22no—
179#95—21no—
180#95D4×C12712yes—
181C24—21no—
182C24—21no—
183C24—41no—
184C24—41no—
185C24—82no—
186C24Q16×C12712yes—
187C24C2×D4×C12711yescharacteristic
188C2×Q8—22no—
189C22×C4—22no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4C2×D4×C12712yes—
195C2×C8—42no—
196C4⋊C4—22no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—81no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4—84no—
214C23—21no—
215C23—21no—
216C23—22no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—44no—
222C23—41no—
223C23—41no—
224C23—81no—
225C23—82no—
226C23—82no—
227C23—82no—
228C23—84no—
229C23#120×C12711yescharacteristic
230C23C2×Q16×C12711yescharacteristic
231C23#137×C12711yescharacteristic
232Q8—82no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—82no—
247C8—82no—
248C22—22no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—42no—
258C22—81no—
259C22—82no—
260C22—82no—
261C22—84no—
262C22—82no—
263C22#451×C12711yescharacteristic
264C22#455×C12711yescharacteristic
265C22B(4,𝔽2)×C12711yescharacteristic
266C4—42no—
267C4—81no—
268C4—82no—
269C4—82no—
270C4—82no—
271C4—161no—
272C2—21no—
273C2—21no—
274C2—41no—
275C2—41no—
276C2—82no—
277C2C127×#160211yescharacteristic
278C2C127×#220012yes—
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×#1614C2semidirect
4. C127×#1352C2semidirect
132. #12218C127direct (N × H)
142. #1614C2×C127semidirect
143. #1352C2×C127semidirect
145. C127#12218direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#28422
C127×#20292
C127×#16142
C127×#13522
C127×#13162
#12218127