Finite Group Databasebeta

← C127×#7564 #44571237

Subgroups — C127×#7564 #44571237

Order 32512.

550 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 174 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×#2296C211yesmaximal, characteristic
2C4×C22⋊C8×C127C212yesmaximal
3C127×#1271C214yesmaximal, complemented
4C2×#117×C127C411yescharacteristic
5C2×#117×C127C411yescharacteristic
6C22×C42×C127C2211yescharacteristic
7C2×C22⋊C8×C127—22no—
8C2×C4×C8×C127—22no—
9#389×C127C412yes—
10#389×C127C2212yes—
11#385×C127—22no—
12C4×C4⋊C4×C127—22no—
13#336×C127C2214yes—
14C23×C4×C127C2×C411yescharacteristic
15C23×C4×C127D412yes—
16C22×C8×C127—24no—
17#117×C127—22no—
18#117×C127—42no—
19#117×C127C814yescomplemented
20C2×C4⋊C4×C127—22no—
21C2×C4⋊C4×C127—22no—
22C2×C4⋊C4×C127C2×C412yes—
23C2×C4⋊C4×C127C2×C412yes—
24C2×C22⋊C4×C127—21no—
25C2×C22⋊C4×C127—21no—
26C2×C42×C127—21no—
27C2×C42×C127—21no—
28C2×C42×C127—42no—
29C2×C42×C127C2×C411yescharacteristic
30C2×C42×C127D412yes—
31C2×C42×C127C2311yescharacteristic
32C22⋊C8×C127—28no—
33C4×C8×C127—44no—
34#95×C127—22no—
35#95×C127—22no—
36C24×C127C22⋊C411yescharacteristic
37C22×C4×C127—22no—
38C22×C4×C127—22no—
39C22×C4×C127—22no—
40C22×C4×C127—24no—
41C22×C4×C127—24no—
42C22×C4×C127—22no—
43C22×C4×C127—42no—
44C22×C4×C127C22⋊C412yes—
45C22×C4×C127C22⋊C411yescharacteristic
46C22×C4×C127C2×C812yes—
47C22×C4×C127M1612yes—
48C22×C4×C127C22×C411yescharacteristic
49C22×C4×C127C2×D412yes—
50C2×C8×C127—28no—
51C2×C8×C127—44no—
52C2×C8×C127—44no—
53C4⋊C4×C127—22no—
54C4⋊C4×C127—42no—
55C4⋊C4×C127—44no—
56C4⋊C4×C127—44no—
57C4⋊C4×C127C2×C814yes—
58C22⋊C4×C127—24no—
59C22⋊C4×C127—42no—
60C42×C127—24no—
61C42×C127—24no—
62C42×C127—41no—
63C42×C127—41no—
64C42×C127—42no—
65C23×C127—21no—
66C23×C127—21no—
67C23×C127—22no—
68C23×C127—22no—
69C23×C127C22⋊C812yes—
70C23×C127C2×C22⋊C411yescharacteristic
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—24no—
78C2×C4×C127—24no—
79C2×C4×C127—24no—
80C2×C4×C127—48no—
81C2×C4×C127—42no—
82C2×C4×C127—42no—
83C2×C4×C127—42no—
84C2×C4×C127—42no—
85C2×C4×C127—42no—
86C2×C4×C127C22⋊C812yes—
87C2×C4×C127#9912yes—
88C2×C4×C127#10112yes—
89C2×C4×C127#10414yes—
90C2×C4×C127C22×C811yescharacteristic
91C2×C4×C127C2×M1611yescharacteristic
92C2×C4×C127#13514yes—
93C8×C127—48no—
94C22×C127—22no—
95C22×C127—24no—
96C22×C127—41no—
97C22×C127—42no—
98C22×C127—41no—
99C22×C127C2×C22⋊C811yescharacteristic
100C22×C127C2×#9911yescharacteristic
101C22×C127C2×#10111yescharacteristic
102C22×C127#41612yes—
103C22×C127C2×#10412yes—
104C4×C127—21no—
105C4×C127—21no—
106C4×C127—24no—
107C4×C127—24no—
108C4×C127—44no—
109C4×C127—82no—
110C4×C127#32414yes—
111#7564C12711yesmaximal, characteristic, direct product factor, Sylow
112C2×C127—42no—
113C2×C127C2×#32411yescharacteristic
114C2×C127#147612yes—
115C2×C127#150112yes—
116C2×C127#152611yescharacteristic
117C2×C127#154411yescharacteristic
118#2296C2×C12711yescharacteristic
119C4×C22⋊C8C2×C12712yes—
120#1271C2×C12714yescomplemented
121C127#756411yescharacteristic, direct product factor, Sylow
122C2×#117C4×C12711yescharacteristic
123C2×#117C4×C12711yescharacteristic
124C22×C42C22×C12711yescharacteristic
125C2×C22⋊C8—22no—
126C2×C4×C8—22no—
127#389C4×C12712yes—
128#389C22×C12712yes—
129#385—22no—
130C4×C4⋊C4—22no—
131#336C22×C12714yes—
132C23×C4C2×C4×C12711yescharacteristic
133C23×C4D4×C12712yes—
134C22×C8—24no—
135#117—22no—
136#117—42no—
137#117C8×C12714yescomplemented
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4C2×C4×C12712yes—
141C2×C4⋊C4C2×C4×C12712yes—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—42no—
147C2×C42C2×C4×C12711yescharacteristic
148C2×C42D4×C12712yes—
149C2×C42C23×C12711yescharacteristic
150C22⋊C8—28no—
151C4×C8—44no—
152#95—22no—
153#95—22no—
154C24C22⋊C4×C12711yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4C22⋊C4×C12712yes—
163C22×C4C22⋊C4×C12711yescharacteristic
164C22×C4C2×C8×C12712yes—
165C22×C4M16×C12712yes—
166C22×C4C22×C4×C12711yescharacteristic
167C22×C4C2×D4×C12712yes—
168C2×C8—28no—
169C2×C8—44no—
170C2×C8—44no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4C2×C8×C12714yes—
176C22⋊C4—24no—
177C22⋊C4—42no—
178C42—24no—
179C42—24no—
180C42—41no—
181C42—41no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23C22⋊C8×C12712yes—
188C23C2×C22⋊C4×C12711yescharacteristic
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4C22⋊C8×C12712yes—
205C2×C4#99×C12712yes—
206C2×C4#101×C12712yes—
207C2×C4#104×C12714yes—
208C2×C4C22×C8×C12711yescharacteristic
209C2×C4C2×M16×C12711yescharacteristic
210C2×C4#135×C12714yes—
211C8—48no—
212C22—22no—
213C22—24no—
214C22—41no—
215C22—42no—
216C22—41no—
217C22C2×C22⋊C8×C12711yescharacteristic
218C22C2×#99×C12711yescharacteristic
219C22C2×#101×C12711yescharacteristic
220C22#416×C12712yes—
221C22C2×#104×C12712yes—
222C4—21no—
223C4—21no—
224C4—24no—
225C4—24no—
226C4—44no—
227C4—82no—
228C4#324×C12714yes—
229C2—42no—
230C2C2×#324×C12711yescharacteristic
231C2C127×#147612yes—
232C2C127×#150112yes—
233C2C127×#152611yescharacteristic
234C2C127×#154411yescharacteristic
subgroupG/H
centerC22×C4×C127C22⋊C4
commutatorC2×C4C22×C8×C127
Frattini, cosocleC2×C42C23×C127
socleC23×C127C2×C22⋊C4
prime radicalC24×C127C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1271C2semidirect
19. #117×C127C8semidirect
111. #7564C127direct (N × H)
120. #1271C2×C127semidirect
121. C127#7564direct (N × H)
137. #117C8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#22962
C4×C22⋊C8×C1272
C127×#12712
#7564127