Finite Group Databasebeta

← C127×#11745 #44575418

Subgroups — C127×#11745 #44575418

Order 32512.

346 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3170C211yesmaximal, characteristic
2C127×#2163C211yesmaximal, characteristic
3C4×#106×C127C211yesmaximal, characteristic
4C127×#1348C214yesmaximal
5#533×C127—21no—
6#500×C127C411yescharacteristic, complemented
7#494×C127C411yescharacteristic, complemented
8#487×C127—21no—
9#467×C127—21no—
10#467×C127C2212yes—
11C4×Q16×C127—21no—
12C4×Q16×C127C412yescomplemented
13C2×#106×C127—21no—
14C2×C4×C8×C127C2211yescharacteristic
15C4×C4⋊C4×C127—21no—
16C4⋊C16×C127—21no—
17C22⋊C16×C127C2212yes—
18#346×C127—21no—
19#340×C127C2212yes—
20C2×Q16×C127—21no—
21C2×Q16×C127C2×C412yes—
22C22×C8×C127D411yescharacteristic
23C22×C8×C127C2311yescharacteristic
24C4⋊Q8×C127—21no—
25C4⋊Q8×C127—21no—
26#125×C127—41no—
27C22⋊Q8×C127—22no—
28C22⋊Q8×C127—41no—
29C4×Q8×C127—22no—
30C4×Q8×C127—41no—
31#117×C127—41no—
32C2×C4⋊C4×C127—22no—
33C2×C42×C127—41no—
34C2×C42×C127D411yescharacteristic
35C2×C16×C127—22no—
36#107×C127—21no—
37#107×C127C2×C412yes—
38#106×C127—24no—
39#103×C127—22no—
40#103×C127—22no—
41C4×C8×C127C2×C411yescharacteristic
42C4×C8×C127C2×C411yescharacteristic
43C4×C8×C127D411yescharacteristic
44C4×C8×C127D411yescharacteristic
45#95×C127—41no—
46C2×Q8×C127—22no—
47C2×Q8×C127—41no—
48C22×C4×C127—42no—
49C22×C4×C127D811yescharacteristic
50C22×C4×C127D811yescharacteristic
51C22×C4×C127C2×D411yescharacteristic
52Q16×C127—42no—
53Q16×C127—42no—
54C2×C8×C127—22no—
55C2×C8×C127—41no—
56C2×C8×C127C22⋊C411yescharacteristic
57C2×C8×C127C22⋊C411yescharacteristic
58C2×C8×C127C22×C411yescharacteristic
59C2×C8×C127C2×D411yescharacteristic
60C4⋊C4×C127—24no—
61C4⋊C4×C127—22no—
62C4⋊C4×C127—42no—
63C4⋊C4×C127—41no—
64C4⋊C4×C127—41no—
65C4⋊C4×C127—42no—
66C4⋊C4×C127—41no—
67C4⋊C4×C127—42no—
68C22⋊C4×C127—42no—
69C42×C127—42no—
70C42×C127—82no—
71C42×C127C22⋊C411yescharacteristic
72C42×C127C22⋊C411yescharacteristic
73C42×C127SD1611yescharacteristic
74C42×C127SD1611yescharacteristic
75C16×C127—42no—
76C23×C127C2×D811yescharacteristic
77Q8×C127—42no—
78Q8×C127—42no—
79Q8×C127—82no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—44no—
85C2×C4×C127—42no—
86C2×C4×C127—42no—
87C2×C4×C127—82no—
88C2×C4×C127—82no—
89C2×C4×C127#10211yescharacteristic
90C2×C4×C127#10211yescharacteristic
91C2×C4×C127#10211yescharacteristic
92C2×C4×C127#10211yescharacteristic
93C2×C4×C127C2×C22⋊C411yescharacteristic
94C2×C4×C127C2×SD1611yescharacteristic
95C8×C127—21no—
96C8×C127—21no—
97C8×C127—21no—
98C8×C127#10412yes—
99C22×C127—21no—
100C22×C127—21no—
101C22×C127C2×#10211yescharacteristic
102C22×C127#51012yes—
103C4×C127—21no—
104C4×C127—21no—
105C4×C127—21no—
106C4×C127—21no—
107C4×C127—84no—
108C4×C127—82no—
109C4×C127—82no—
110C4×C127#36012yes—
111C4×C127#41011yescharacteristic
112C4×C127C2×#10411yescharacteristic
113#11745C12711yesmaximal, characteristic, direct product factor, Sylow
114C2×C127—22no—
115C2×C127#150211yescharacteristic
116C2×C127#214111yescharacteristic
117C2×C127C2×#36011yescharacteristic
118#3170C2×C12711yescharacteristic
119#2163C2×C12711yescharacteristic
120C4×#106C2×C12711yescharacteristic
121#1348C2×C12714yes—
122C127#1174511yescharacteristic, direct product factor, Sylow
123#533—21no—
124#500C4×C12711yescharacteristic, complemented
125#494C4×C12711yescharacteristic, complemented
126#487—21no—
127#467—21no—
128#467C22×C12712yes—
129C4×Q16—21no—
130C4×Q16C4×C12712yescomplemented
131C2×#106—21no—
132C2×C4×C8C22×C12711yescharacteristic
133C4×C4⋊C4—21no—
134C4⋊C16—21no—
135C22⋊C16C22×C12712yes—
136#346—21no—
137#340C22×C12712yes—
138C2×Q16—21no—
139C2×Q16C2×C4×C12712yes—
140C22×C8D4×C12711yescharacteristic
141C22×C8C23×C12711yescharacteristic
142C4⋊Q8—21no—
143C4⋊Q8—21no—
144#125—41no—
145C22⋊Q8—22no—
146C22⋊Q8—41no—
147C4×Q8—22no—
148C4×Q8—41no—
149#117—41no—
150C2×C4⋊C4—22no—
151C2×C42—41no—
152C2×C42D4×C12711yescharacteristic
153C2×C16—22no—
154#107—21no—
155#107C2×C4×C12712yes—
156#106—24no—
157#103—22no—
158#103—22no—
159C4×C8C2×C4×C12711yescharacteristic
160C4×C8C2×C4×C12711yescharacteristic
161C4×C8D4×C12711yescharacteristic
162C4×C8D4×C12711yescharacteristic
163#95—41no—
164C2×Q8—22no—
165C2×Q8—41no—
166C22×C4—42no—
167C22×C4D8×C12711yescharacteristic
168C22×C4D8×C12711yescharacteristic
169C22×C4C2×D4×C12711yescharacteristic
170Q16—42no—
171Q16—42no—
172C2×C8—22no—
173C2×C8—41no—
174C2×C8C22⋊C4×C12711yescharacteristic
175C2×C8C22⋊C4×C12711yescharacteristic
176C2×C8C22×C4×C12711yescharacteristic
177C2×C8C2×D4×C12711yescharacteristic
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—42no—
186C22⋊C4—42no—
187C42—42no—
188C42—82no—
189C42C22⋊C4×C12711yescharacteristic
190C42C22⋊C4×C12711yescharacteristic
191C42SD16×C12711yescharacteristic
192C42SD16×C12711yescharacteristic
193C16—42no—
194C23C2×D8×C12711yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—82no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4#102×C12711yescharacteristic
208C2×C4#102×C12711yescharacteristic
209C2×C4#102×C12711yescharacteristic
210C2×C4#102×C12711yescharacteristic
211C2×C4C2×C22⋊C4×C12711yescharacteristic
212C2×C4C2×SD16×C12711yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8#104×C12712yes—
217C22—21no—
218C22—21no—
219C22C2×#102×C12711yescharacteristic
220C22#510×C12712yes—
221C4—21no—
222C4—21no—
223C4—21no—
224C4—21no—
225C4—84no—
226C4—82no—
227C4—82no—
228C4#360×C12712yes—
229C4#410×C12711yescharacteristic
230C4C2×#104×C12711yescharacteristic
231C2—22no—
232C2C127×#150211yescharacteristic
233C2C127×#214111yescharacteristic
234C2C2×#360×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#102
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC23×C127C2×D8
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
6. #500×C127C4semidirect
7. #494×C127C4semidirect
12. C4×Q16×C127C4semidirect
113. #11745C127direct (N × H)
122. C127#11745direct (N × H)
124. #500C4×C127semidirect
125. #494C4×C127semidirect
130. C4×Q16C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31702
C127×#21632
C4×#106×C1272
C127×#13482
#11745127