Finite Group Databasebeta

← C127×#13604 #44577277

Subgroups — C127×#13604 #44577277

Order 32512.

322 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3142C211yesmaximal, characteristic
2C127×#2132C211yesmaximal, characteristic
3C127×#2126C211yesmaximal, characteristic
4C127×#1414C214yesmaximal, complemented
5#532×C127C2211yescharacteristic
6C2×#125×C127C411yescharacteristic, complemented
7#491×C127—21no—
8#490×C127—21no—
9#488×C127C412yescomplemented
10#488×C127C2212yes—
11#417×C127—21no—
12#417×C127—21no—
13#413×C127—21no—
14#410×C127—21no—
15C4×M16×C127C411yescharacteristic, complemented
16#356×C127—21no—
17#355×C127C2212yes—
18#354×C127C2212yes—
19#353×C127—21no—
20C2×#41×C127D411yescharacteristic
21C2×#41×C127D411yescharacteristic
22C2×M16×C127—21no—
23C2×M16×C127—41no—
24C4⋊Q8×C127D411yescharacteristic
25#127×C127D411yescharacteristic
26#125×C127—21no—
27#125×C127—41no—
28#125×C127C2×C412yes—
29#124×C127C2×C411yescharacteristic
30#124×C127C2311yescharacteristic
31#121×C127—41no—
32C4×D4×C127—41no—
33#117×C127—41no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127—41no—
36C2×C42×C127C2×C411yescharacteristic
37#103×C127—22no—
38#103×C127—22no—
39#102×C127—22no—
40#102×C127—22no—
41#101×C127—22no—
42#100×C127—41no—
43#99×C127—22no—
44#99×C127—41no—
45#97×C127—21no—
46C4×C8×C127C2×C412yes—
47#41×C127—41no—
48#41×C127—41no—
49#41×C127—41no—
50#41×C127—41no—
51C2×Q8×C127C22⋊C411yescharacteristic
52C2×Q8×C127C2×D411yescharacteristic
53C2×D4×C127—21no—
54C2×D4×C127—21no—
55C2×D4×C127—41no—
56C2×D4×C127C22⋊C411yescharacteristic
57C2×D4×C127C2×D411yescharacteristic
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—41no—
62C22×C4×C127C22⋊C411yescharacteristic
63M16×C127—41no—
64M16×C127—41no—
65M16×C127—42no—
66M16×C127—81no—
67C2×C8×C127—22no—
68C2×C8×C127—81no—
69C4⋊C4×C127—22no—
70C4⋊C4×C127—41no—
71C4⋊C4×C127—42no—
72C4⋊C4×C127—42no—
73C4⋊C4×C127—41no—
74C4⋊C4×C127—81no—
75C22⋊C4×C127—41no—
76C22⋊C4×C127—41no—
77C22⋊C4×C127—42no—
78C42×C127—21no—
79C42×C127—81no—
80C42×C127C22⋊C411yescharacteristic
81C42×C127C22×C411yescharacteristic
82C23×C127—21no—
83C23×C127—21no—
84C23×C127#9911yescharacteristic
85Q8×C127—41no—
86Q8×C127—41no—
87D4×C127—41no—
88D4×C127—41no—
89D4×C127—81no—
90D4×C127—81no—
91D4×C127—82no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—42no—
103C2×C4×C127—81no—
104C2×C4×C127—82no—
105C2×C4×C127—82no—
106C2×C4×C127#9911yescharacteristic
107C2×C4×C127C2×C22⋊C411yescharacteristic
108C8×C127—42no—
109C8×C127—82no—
110C22×C127—21no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127—41no—
114C22×C127—81no—
115C22×C127—81no—
116C22×C127C2×#9911yescharacteristic
117C4×C127—21no—
118C4×C127—41no—
119C4×C127—41no—
120C4×C127—41no—
121C4×C127—41no—
122C4×C127—82no—
123C4×C127—162no—
124C4×C127#35411yescharacteristic
125C4×C127#35411yescharacteristic
126#13604C12711yesmaximal, characteristic, direct product factor, Sylow
127C2×C127—21no—
128C2×C127—41no—
129C2×C127—81no—
130C2×C127—81no—
131C2×C127C2×#35411yescharacteristic
132#3142C2×C12711yescharacteristic
133#2132C2×C12711yescharacteristic
134#2126C2×C12711yescharacteristic
135#1414C2×C12714yescomplemented
136C127#1360411yescharacteristic, direct product factor, Sylow
137#532C22×C12711yescharacteristic
138C2×#125C4×C12711yescharacteristic, complemented
139#491—21no—
140#490—21no—
141#488C4×C12712yescomplemented
142#488C22×C12712yes—
143#417—21no—
144#417—21no—
145#413—21no—
146#410—21no—
147C4×M16C4×C12711yescharacteristic, complemented
148#356—21no—
149#355C22×C12712yes—
150#354C22×C12712yes—
151#353—21no—
152C2×#41D4×C12711yescharacteristic
153C2×#41D4×C12711yescharacteristic
154C2×M16—21no—
155C2×M16—41no—
156C4⋊Q8D4×C12711yescharacteristic
157#127D4×C12711yescharacteristic
158#125—21no—
159#125—41no—
160#125C2×C4×C12712yes—
161#124C2×C4×C12711yescharacteristic
162#124C23×C12711yescharacteristic
163#121—41no—
164C4×D4—41no—
165#117—41no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C42C2×C4×C12711yescharacteristic
169#103—22no—
170#103—22no—
171#102—22no—
172#102—22no—
173#101—22no—
174#100—41no—
175#99—22no—
176#99—41no—
177#97—21no—
178C4×C8C2×C4×C12712yes—
179#41—41no—
180#41—41no—
181#41—41no—
182#41—41no—
183C2×Q8C22⋊C4×C12711yescharacteristic
184C2×Q8C2×D4×C12711yescharacteristic
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—41no—
188C2×D4C22⋊C4×C12711yescharacteristic
189C2×D4C2×D4×C12711yescharacteristic
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4C22⋊C4×C12711yescharacteristic
195M16—41no—
196M16—41no—
197M16—42no—
198M16—81no—
199C2×C8—22no—
200C2×C8—81no—
201C4⋊C4—22no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—41no—
206C4⋊C4—81no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C42—21no—
211C42—81no—
212C42C22⋊C4×C12711yescharacteristic
213C42C22×C4×C12711yescharacteristic
214C23—21no—
215C23—21no—
216C23#99×C12711yescharacteristic
217Q8—41no—
218Q8—41no—
219D4—41no—
220D4—41no—
221D4—81no—
222D4—81no—
223D4—82no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4#99×C12711yescharacteristic
239C2×C4C2×C22⋊C4×C12711yescharacteristic
240C8—42no—
241C8—82no—
242C22—21no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22C2×#99×C12711yescharacteristic
249C4—21no—
250C4—41no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—82no—
255C4—162no—
256C4#354×C12711yescharacteristic
257C4#354×C12711yescharacteristic
258C2—21no—
259C2—41no—
260C2—81no—
261C2—81no—
262C2C2×#354×C12711yescharacteristic
subgroupG/H
center, socleC2×C127C2×#354
commutatorC42C22×C4×C127
Frattini, cosocle#124C23×C127
prime radical#532×C127C22
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ #532×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1414C2semidirect
6. C2×#125×C127C4semidirect
9. #488×C127C4semidirect
15. C4×M16×C127C4semidirect
126. #13604C127direct (N × H)
135. #1414C2×C127semidirect
136. C127#13604direct (N × H)
138. C2×#125C4×C127semidirect
141. #488C4×C127semidirect
147. C4×M16C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31422
C127×#21322
C127×#21262
C127×#14142
#13604127