Finite Group Databasebeta

← C127×#10307 #44573980

Subgroups — C127×#10307 #44573980

Order 32512.

754 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×#493×C127C211yesmaximal, characteristic
2C2×#486×C127C212yesmaximal, complemented
3C127×#1957C212yesmaximal, complemented
4C127×#1956C211yesmaximal, characteristic, complemented
5C127×#1273C211yesmaximal, characteristic, complemented
6C22×D8×C127—22no—
7C2×C4⋊Q8×C127C2211yescharacteristic
8C2×#127×C127C2212yes—
9#493×C127—24no—
10#486×C127—24no—
11#486×C127—24no—
12C2×#103×C127—22no—
13C2×#102×C127—24no—
14C2×#102×C127—24no—
15C2×C4×C8×C127C2212yes—
16C2×C4×C8×C127C2211yescharacteristic
17#384×C127C2211yescharacteristic
18C22×Q8×C127—21no—
19C22×D4×C127—22no—
20C22×D4×C127—22no—
21C22×D4×C127—22no—
22C2×D8×C127—28no—
23C2×D8×C127—44no—
24C2×D8×C127—44no—
25C22×C8×C127D414yes—
26C22×C8×C127D412yes—
27C4⋊Q8×C127—22no—
28C4⋊Q8×C127—21no—
29C4⋊Q8×C127D412yes—
30#127×C127—22no—
31#127×C127—22no—
32#127×C127—22no—
33#127×C127D414yes—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—22no—
36C2×C4⋊C4×C127—21no—
37C2×C42×C127C2311yescharacteristic
38#103×C127—44no—
39#102×C127—216no—
40#102×C127—48no—
41C4×C8×C127—22no—
42C4×C8×C127—22no—
43C4×C8×C127—22no—
44C24×C127—42no—
45C2×Q8×C127—22no—
46C2×Q8×C127—41no—
47C2×Q8×C127—42no—
48C2×Q8×C127—42no—
49C2×D4×C127—24no—
50C2×D4×C127—24no—
51C2×D4×C127—24no—
52C2×D4×C127—42no—
53C2×D4×C127—42no—
54C2×D4×C127—42no—
55C2×D4×C127—44no—
56C2×D4×C127—44no—
57C2×D4×C127—44no—
58C2×D4×C127—44no—
59C2×D4×C127—44no—
60C2×D4×C127—44no—
61C22×C4×C127—41no—
62C22×C4×C127—41no—
63C22×C4×C127C2×D411yescharacteristic
64C22×C4×C127C2×D412yes—
65D8×C127—416no—
66C2×C8×C127—22no—
67C2×C8×C127—22no—
68C2×C8×C127—24no—
69C2×C8×C127—24no—
70C2×C8×C127—24no—
71C2×C8×C127D814yes—
72C4⋊C4×C127—24no—
73C4⋊C4×C127—42no—
74C4⋊C4×C127—42no—
75C4⋊C4×C127—44no—
76C42×C127C2×D411yescharacteristic
77C42×C127C2×D412yes—
78C42×C127#4111yescharacteristic
79C23×C127—44no—
80C23×C127—82no—
81C23×C127—82no—
82C23×C127—82no—
83C23×C127—82no—
84C23×C127—82no—
85C23×C127—82no—
86C23×C127#12711yescharacteristic
87Q8×C127—44no—
88Q8×C127—82no—
89D4×C127—48no—
90D4×C127—48no—
91D4×C127—48no—
92D4×C127—84no—
93D4×C127—84no—
94D4×C127—84no—
95C2×C4×C127—42no—
96C2×C4×C127—81no—
97C2×C4×C127—81no—
98C2×C4×C127—81no—
99C2×C4×C127—82no—
100C2×C4×C127#12011yescharacteristic
101C2×C4×C127#12012yes—
102C2×C4×C127#12111yescharacteristic
103C2×C4×C127#12112yes—
104C2×C4×C127C2×D812yes—
105C2×C4×C127#13514yes—
106C2×C4×C127AGL(1,ℤ/8)12yes—
107C2×C4×C127AGL(1,ℤ/8)14yes—
108C8×C127—24no—
109C8×C127—44no—
110C22×C127—84no—
111C22×C127—84no—
112C22×C127—84no—
113C22×C127—162no—
114C22×C127—162no—
115C22×C127—162no—
116C22×C127—162no—
117C22×C127#39211yescharacteristic
118C22×C127#49311yescharacteristic
119C22×C127#49512yes—
120C22×C127#49611yescharacteristic
121C22×C127#49612yes—
122C4×C127—82no—
123C4×C127—161no—
124C4×C127#44712yes—
125C4×C127#44914yes—
126C4×C127#45912yes—
127C4×C127#46314yes—
128#10307C12711yesmaximal, characteristic, direct product factor, Sylow
129C2×C127—164no—
130C2×C127#170911yescharacteristic
131C2×C127#171712yes—
132C2×C127#172111yescharacteristic
133C2×C127#201111yescharacteristic
134C2×C127#201212yes—
135C2×#493C2×C12711yescharacteristic
136C2×#486C2×C12712yescomplemented
137#1957C2×C12712yescomplemented
138#1956C2×C12711yescharacteristic, complemented
139#1273C2×C12711yescharacteristic, complemented
140C127#1030711yescharacteristic, direct product factor, Sylow
141C22×D8—22no—
142C2×C4⋊Q8C22×C12711yescharacteristic
143C2×#127C22×C12712yes—
144#493—24no—
145#486—24no—
146#486—24no—
147C2×#103—22no—
148C2×#102—24no—
149C2×#102—24no—
150C2×C4×C8C22×C12712yes—
151C2×C4×C8C22×C12711yescharacteristic
152#384C22×C12711yescharacteristic
153C22×Q8—21no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C2×D8—28no—
158C2×D8—44no—
159C2×D8—44no—
160C22×C8D4×C12714yes—
161C22×C8D4×C12712yes—
162C4⋊Q8—22no—
163C4⋊Q8—21no—
164C4⋊Q8D4×C12712yes—
165#127—22no—
166#127—22no—
167#127—22no—
168#127D4×C12714yes—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C42C23×C12711yescharacteristic
173#103—44no—
174#102—216no—
175#102—48no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C24—42no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—44no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4C2×D4×C12711yescharacteristic
199C22×C4C2×D4×C12712yes—
200D8—416no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—24no—
204C2×C8—24no—
205C2×C8—24no—
206C2×C8D8×C12714yes—
207C4⋊C4—24no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C42C2×D4×C12711yescharacteristic
212C42C2×D4×C12712yes—
213C42#41×C12711yescharacteristic
214C23—44no—
215C23—82no—
216C23—82no—
217C23—82no—
218C23—82no—
219C23—82no—
220C23—82no—
221C23#127×C12711yescharacteristic
222Q8—44no—
223Q8—82no—
224D4—48no—
225D4—48no—
226D4—48no—
227D4—84no—
228D4—84no—
229D4—84no—
230C2×C4—42no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4#120×C12711yescharacteristic
236C2×C4#120×C12712yes—
237C2×C4#121×C12711yescharacteristic
238C2×C4#121×C12712yes—
239C2×C4C2×D8×C12712yes—
240C2×C4#135×C12714yes—
241C2×C4AGL(1,ℤ/8)×C12712yes—
242C2×C4AGL(1,ℤ/8)×C12714yes—
243C8—24no—
244C8—44no—
245C22—84no—
246C22—84no—
247C22—84no—
248C22—162no—
249C22—162no—
250C22—162no—
251C22—162no—
252C22#392×C12711yescharacteristic
253C22#493×C12711yescharacteristic
254C22#495×C12712yes—
255C22#496×C12711yescharacteristic
256C22#496×C12712yes—
257C4—82no—
258C4—161no—
259C4#447×C12712yes—
260C4#449×C12714yes—
261C4#459×C12712yes—
262C4#463×C12714yes—
263C2—164no—
264C2C127×#170911yescharacteristic
265C2C127×#171712yes—
266C2C127×#172111yescharacteristic
267C2C127×#201111yescharacteristic
268C2C127×#201212yes—
subgroupG/H
center, socleC23×C127#127
commutator, Frattini, cosocleC2×C42C23×C127
prime radicalC2×#493×C127C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#486×C127C2semidirect
3. C127×#1957C2semidirect
4. C127×#1956C2semidirect
5. C127×#1273C2semidirect
128. #10307C127direct (N × H)
136. C2×#486C2×C127semidirect
137. #1957C2×C127semidirect
138. #1956C2×C127semidirect
139. #1273C2×C127semidirect
140. C127#10307direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#493×C1272
C2×#486×C1272
C127×#19572
C127×#19562
C127×#12732
#10307127