Finite Group Databasebeta

← C127×#10312 #44573985

Subgroups — C127×#10312 #44573985

Order 32512.

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

normal subgroups only

This group has 638 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×#498×C127C211yesmaximal, characteristic, complemented
2C2×#486×C127C212yesmaximal
3C127×#1957C211yesmaximal, characteristic
4C127×#1956C212yesmaximal, complemented
5C127×#1273C211yesmaximal, characteristic, complemented
6C2×C4⋊Q8×C127C2212yes—
7C2×#127×C127C2211yescharacteristic
8#498×C127—24no—
9#486×C127—24no—
10#486×C127—24no—
11C2×#106×C127—22no—
12C2×#103×C127—24no—
13C2×#102×C127—24no—
14C2×#102×C127—22no—
15C2×C4×C8×C127C2212yes—
16C2×C4×C8×C127C2211yescharacteristic
17#384×C127C2211yescharacteristic
18C22×Q8×C127—22no—
19C22×D4×C127—21no—
20C22×D4×C127—22no—
21C22×C8×C127D414yes—
22C22×C8×C127D412yes—
23C4⋊Q8×C127—22no—
24C4⋊Q8×C127—22no—
25C4⋊Q8×C127—22no—
26C4⋊Q8×C127D414yes—
27#127×C127—22no—
28#127×C127—21no—
29#127×C127D412yes—
30C2×C4⋊C4×C127—22no—
31C2×C4⋊C4×C127—22no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—22no—
34C2×C42×C127C2311yescharacteristic
35#106×C127—28no—
36#103×C127—48no—
37#102×C127—216no—
38#102×C127—44no—
39C4×C8×C127—22no—
40C4×C8×C127—22no—
41C4×C8×C127—22no—
42C24×C127—41no—
43C2×Q8×C127—24no—
44C2×Q8×C127—42no—
45C2×Q8×C127—44no—
46C2×Q8×C127—44no—
47C2×D4×C127—24no—
48C2×D4×C127—22no—
49C2×D4×C127—41no—
50C2×D4×C127—42no—
51C2×D4×C127—42no—
52C2×D4×C127—44no—
53C2×D4×C127—44no—
54C2×D4×C127—42no—
55C22×C4×C127—42no—
56C22×C4×C127—41no—
57C22×C4×C127C2×D411yescharacteristic
58C22×C4×C127C2×D412yes—
59C2×C8×C127—22no—
60C2×C8×C127—22no—
61C2×C8×C127—24no—
62C2×C8×C127—24no—
63C2×C8×C127—24no—
64C2×C8×C127SD1614yes—
65C4⋊C4×C127—24no—
66C4⋊C4×C127—24no—
67C4⋊C4×C127—42no—
68C4⋊C4×C127—42no—
69C4⋊C4×C127—42no—
70C4⋊C4×C127—44no—
71C42×C127C2×D412yes—
72C42×C127C2×D411yescharacteristic
73C42×C127#4111yescharacteristic
74C23×C127—42no—
75C23×C127—81no—
76C23×C127—82no—
77C23×C127—82no—
78C23×C127—81no—
79C23×C127#12711yescharacteristic
80Q8×C127—48no—
81Q8×C127—84no—
82D4×C127—48no—
83D4×C127—44no—
84D4×C127—82no—
85D4×C127—84no—
86C2×C4×C127—44no—
87C2×C4×C127—82no—
88C2×C4×C127—82no—
89C2×C4×C127—81no—
90C2×C4×C127—82no—
91C2×C4×C127#12011yescharacteristic
92C2×C4×C127#12012yes—
93C2×C4×C127#12111yescharacteristic
94C2×C4×C127#12112yes—
95C2×C4×C127C2×SD1612yes—
96C2×C4×C127#13514yes—
97C2×C4×C127AGL(1,ℤ/8)14yes—
98C2×C4×C127#13712yes—
99C8×C127—24no—
100C8×C127—44no—
101C22×C127—84no—
102C22×C127—82no—
103C22×C127—161no—
104C22×C127—161no—
105C22×C127—162no—
106C22×C127#39211yescharacteristic
107C22×C127#49211yescharacteristic
108C22×C127#49512yes—
109C22×C127#49612yes—
110C22×C127#49711yescharacteristic
111C4×C127—84no—
112C4×C127—161no—
113C4×C127#44812yes—
114C4×C127#44914yes—
115C4×C127#46112yes—
116C4×C127#46314yes—
117#10312C12711yesmaximal, characteristic, direct product factor, Sylow
118C2×C127—162no—
119C2×C127#170811yescharacteristic
120C2×C127#171511yescharacteristic
121C2×C127#172012yes—
122C2×C127#201212yes—
123C2×C127#201311yescharacteristic
124C2×#498C2×C12711yescharacteristic, complemented
125C2×#486C2×C12712yes—
126#1957C2×C12711yescharacteristic
127#1956C2×C12712yescomplemented
128#1273C2×C12711yescharacteristic, complemented
129C127#1031211yescharacteristic, direct product factor, Sylow
130C2×C4⋊Q8C22×C12712yes—
131C2×#127C22×C12711yescharacteristic
132#498—24no—
133#486—24no—
134#486—24no—
135C2×#106—22no—
136C2×#103—24no—
137C2×#102—24no—
138C2×#102—22no—
139C2×C4×C8C22×C12712yes—
140C2×C4×C8C22×C12711yescharacteristic
141#384C22×C12711yescharacteristic
142C22×Q8—22no—
143C22×D4—21no—
144C22×D4—22no—
145C22×C8D4×C12714yes—
146C22×C8D4×C12712yes—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8D4×C12714yes—
151#127—22no—
152#127—21no—
153#127D4×C12712yes—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C42C23×C12711yescharacteristic
159#106—28no—
160#103—48no—
161#102—216no—
162#102—44no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C24—41no—
167C2×Q8—24no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—44no—
171C2×D4—24no—
172C2×D4—22no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—42no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4C2×D4×C12711yescharacteristic
182C22×C4C2×D4×C12712yes—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—24no—
186C2×C8—24no—
187C2×C8—24no—
188C2×C8SD16×C12714yes—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C42C2×D4×C12712yes—
196C42C2×D4×C12711yescharacteristic
197C42#41×C12711yescharacteristic
198C23—42no—
199C23—82no—
200C23—81no—
201C23—81no—
202C23—82no—
203C23#127×C12711yescharacteristic
204Q8—48no—
205Q8—84no—
206D4—48no—
207D4—44no—
208D4—82no—
209D4—84no—
210C2×C4—44no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—81no—
215C2×C4#120×C12711yescharacteristic
216C2×C4#120×C12712yes—
217C2×C4#121×C12711yescharacteristic
218C2×C4#121×C12712yes—
219C2×C4C2×SD16×C12712yes—
220C2×C4#135×C12714yes—
221C2×C4AGL(1,ℤ/8)×C12714yes—
222C2×C4#137×C12712yes—
223C8—24no—
224C8—44no—
225C22—84no—
226C22—82no—
227C22—161no—
228C22—161no—
229C22—162no—
230C22#392×C12711yescharacteristic
231C22#492×C12711yescharacteristic
232C22#495×C12712yes—
233C22#496×C12712yes—
234C22#497×C12711yescharacteristic
235C4—84no—
236C4—161no—
237C4#448×C12712yes—
238C4#449×C12714yes—
239C4#461×C12712yes—
240C4#463×C12714yes—
241C2—162no—
242C2C127×#170811yescharacteristic
243C2C127×#171511yescharacteristic
244C2C127×#172012yes—
245C2C127×#201212yes—
246C2C127×#201311yescharacteristic
subgroupG/H
center, socleC23×C127#127
commutator, Frattini, cosocleC2×C42C23×C127
prime radicalC2×#127×C127C22
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
1. C2×#498×C127C2semidirect
4. C127×#1956C2semidirect
5. C127×#1273C2semidirect
117. #10312C127direct (N × H)
124. C2×#498C2×C127semidirect
127. #1956C2×C127semidirect
128. #1273C2×C127semidirect
129. C127#10312direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#498×C1272
C2×#486×C1272
C127×#19572
C127×#19562
C127×#12732
#10312127