Finite Group Databasebeta

← C127×#12221 #44575894

Subgroups — C127×#12221 #44575894

Order 32512.

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

normal subgroups only

This group has 392 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×#2804C211yesmaximal, characteristic
2C127×#2027C211yesmaximal, characteristic
3C127×#1617C212yesmaximal, complemented
4C127×#1353C212yesmaximal, complemented
5C127×#1318C211yesmaximal, characteristic
6#544×C127C2211yescharacteristic
7#484×C127—22no—
8#418×C127—21no—
9#400×C127C2212yes—
10#393×C127—22no—
11#391×C127—21no—
12#390×C127C2211yescharacteristic
13#386×C127—21no—
14#385×C127—22no—
15#343×C127C2211yescharacteristic
16#339×C127—21no—
17#328×C127—22no—
18#324×C127C2212yes—
19C22×Q8×C127—21no—
20C22×D4×C127D411yescharacteristic
21C23×C4×C127—21no—
22C2×M16×C127—21no—
23C4⋊Q8×C127—41no—
24#125×C127—41no—
25C22⋊Q8×C127—22no—
26#117×C127—21no—
27C2×C4⋊C4×C127—21no—
28C2×C4⋊C4×C127D412yes—
29C2×C4⋊C4×C127C2311yescharacteristic
30C2×C22⋊C4×C127—22no—
31C2×C22⋊C4×C127—22no—
32C2×C22⋊C4×C127—21no—
33C2×C42×C127D411yescharacteristic
34#107×C127—42no—
35#103×C127—42no—
36#102×C127—41no—
37#102×C127—41no—
38C22⋊C8×C127—22no—
39#95×C127—22no—
40#95×C127—22no—
41#95×C127—22no—
42#95×C127D412yes—
43C24×C127—21no—
44C2×Q8×C127—22no—
45C2×Q8×C127—42no—
46C2×Q8×C127—81no—
47C2×Q8×C127—81no—
48C2×D4×C127—21no—
49C2×D4×C127—41no—
50C2×D4×C127—41no—
51C2×D4×C127Q1612yes—
52C22×C4×C127—21no—
53C22×C4×C127—22no—
54C22×C4×C127—22no—
55C22×C4×C127—82no—
56C22×C4×C127—81no—
57C22×C4×C127C2×D411yescharacteristic
58C22×C4×C127C2×D412yes—
59M16×C127—41no—
60M16×C127—41no—
61C2×C8×C127—42no—
62C2×C8×C127—41no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127—21no—
65C4⋊C4×C127—22no—
66C4⋊C4×C127—41no—
67C4⋊C4×C127—42no—
68C4⋊C4×C127—42no—
69C4⋊C4×C127—81no—
70C22⋊C4×C127—44no—
71C22⋊C4×C127—41no—
72C22⋊C4×C127—82no—
73C22⋊C4×C127—81no—
74C42×C127—41no—
75C42×C127—41no—
76C23×C127—22no—
77C23×C127—42no—
78C23×C127—42no—
79C23×C127—41no—
80C23×C127—41no—
81C23×C127#12011yescharacteristic
82Q8×C127—82no—
83Q8×C127—84no—
84D4×C127—42no—
85D4×C127—81no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—22no—
89C2×C4×C127—22no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—42no—
93C2×C4×C127—42no—
94C2×C4×C127—42no—
95C2×C4×C127—42no—
96C2×C4×C127—42no—
97C2×C4×C127—42no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127—42no—
101C2×C4×C127—84no—
102C2×C4×C127C2×Q1611yescharacteristic
103C2×C4×C127#13711yescharacteristic
104C8×C127—81no—
105C8×C127—82no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127—44no—
109C22×C127—42no—
110C22×C127—82no—
111C22×C127—81no—
112C22×C127—81no—
113C22×C127#45111yescharacteristic
114C22×C127#45511yescharacteristic
115C22×C127B(4,𝔽2)11yescharacteristic
116C4×C127—21no—
117C4×C127—21no—
118C4×C127—42no—
119C4×C127—42no—
120C4×C127—82no—
121C4×C127—81no—
122C4×C127—81no—
123C4×C127—82no—
124#12221C12711yesmaximal, characteristic, direct product factor, Sylow
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—82no—
128C2×C127#160211yescharacteristic
129C2×C127#219811yescharacteristic
130C2×C127#220211yescharacteristic
131#2804C2×C12711yescharacteristic
132#2027C2×C12711yescharacteristic
133#1617C2×C12712yescomplemented
134#1353C2×C12712yescomplemented
135#1318C2×C12711yescharacteristic
136C127#1222111yescharacteristic, direct product factor, Sylow
137#544C22×C12711yescharacteristic
138#484—22no—
139#418—21no—
140#400C22×C12712yes—
141#393—22no—
142#391—21no—
143#390C22×C12711yescharacteristic
144#386—21no—
145#385—22no—
146#343C22×C12711yescharacteristic
147#339—21no—
148#328—22no—
149#324C22×C12712yes—
150C22×Q8—21no—
151C22×D4D4×C12711yescharacteristic
152C23×C4—21no—
153C2×M16—21no—
154C4⋊Q8—41no—
155#125—41no—
156C22⋊Q8—22no—
157#117—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4D4×C12712yes—
160C2×C4⋊C4C23×C12711yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C42D4×C12711yescharacteristic
165#107—42no—
166#103—42no—
167#102—41no—
168#102—41no—
169C22⋊C8—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95D4×C12712yes—
174C24—21no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—81no—
178C2×Q8—81no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4Q16×C12712yes—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—82no—
187C22×C4—81no—
188C22×C4C2×D4×C12711yescharacteristic
189C22×C4C2×D4×C12712yes—
190M16—41no—
191M16—41no—
192C2×C8—42no—
193C2×C8—41no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—22no—
197C4⋊C4—41no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C22⋊C4—44no—
202C22⋊C4—41no—
203C22⋊C4—82no—
204C22⋊C4—81no—
205C42—41no—
206C42—41no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23#120×C12711yescharacteristic
213Q8—82no—
214Q8—84no—
215D4—42no—
216D4—81no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—84no—
233C2×C4C2×Q16×C12711yescharacteristic
234C2×C4#137×C12711yescharacteristic
235C8—81no—
236C8—82no—
237C22—21no—
238C22—21no—
239C22—44no—
240C22—42no—
241C22—82no—
242C22—81no—
243C22—81no—
244C22#451×C12711yescharacteristic
245C22#455×C12711yescharacteristic
246C22B(4,𝔽2)×C12711yescharacteristic
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—82no—
252C4—81no—
253C4—82no—
254C4—81no—
255C2—21no—
256C2—21no—
257C2—82no—
258C2C127×#160211yescharacteristic
259C2C127×#219811yescharacteristic
260C2C127×#220211yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC22×D4×C127D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1617C2semidirect
4. C127×#1353C2semidirect
124. #12221C127direct (N × H)
133. #1617C2×C127semidirect
134. #1353C2×C127semidirect
136. C127#12221direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#28042
C127×#20272
C127×#16172
C127×#13532
C127×#13182
#12221127