Finite Group Databasebeta

← C127×#8308 #44571981

Subgroups — C127×#8308 #44571981

Order 32512.

1174 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 190 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
1C22×C22⋊Q8×C127C211yesmaximal, characteristic
2C127×#1779C212yesmaximal
3C2×#324×C127C214yesmaximal, complemented
4C23×Q8×C127C411yescharacteristic
5C24×C4×C127C2211yescharacteristic
6C2×C22⋊Q8×C127—24no—
7C2×C22⋊Q8×C127—28no—
8C22×C4⋊C4×C127C411yescharacteristic
9C22×C4⋊C4×C127C2212yes—
10C22×C22⋊C4×C127C412yes—
11C2×C22⋊C8×C127—24no—
12C2×C22⋊C8×C127C2214yes—
13#324×C127—28no—
14C25×C127C2×C411yescharacteristic
15C22×Q8×C127—24no—
16C22×Q8×C127—22no—
17C22×Q8×C127—44no—
18C23×C4×C127—24no—
19C23×C4×C127—44no—
20C23×C4×C127C2×C412yes—
21C23×C4×C127C2×C411yescharacteristic
22C23×C4×C127C2×C412yes—
23C23×C4×C127D414yes—
24C23×C4×C127C2311yescharacteristic
25C22×C8×C127—24no—
26C22⋊Q8×C127—232no—
27C2×C4⋊C4×C127—24no—
28C2×C4⋊C4×C127—22no—
29C2×C4⋊C4×C127—24no—
30C2×C4⋊C4×C127—24no—
31C2×C4⋊C4×C127D418yes—
32C2×C22⋊C4×C127—24no—
33C2×C22⋊C4×C127—24no—
34C2×C22⋊C4×C127—24no—
35C22⋊C8×C127—28no—
36C22⋊C8×C127—48no—
37C24×C127—24no—
38C24×C127—22no—
39C24×C127—24no—
40C24×C127—42no—
41C24×C127C22⋊C412yes—
42C24×C127C22×C411yescharacteristic
43C2×Q8×C127—28no—
44C2×Q8×C127—416no—
45C2×Q8×C127—44no—
46C2×Q8×C127—44no—
47C22×C4×C127—22no—
48C22×C4×C127—24no—
49C22×C4×C127—22no—
50C22×C4×C127—22no—
51C22×C4×C127—22no—
52C22×C4×C127—22no—
53C22×C4×C127—28no—
54C22×C4×C127—28no—
55C22×C4×C127—28no—
56C22×C4×C127—44no—
57C22×C4×C127—42no—
58C22×C4×C127—416no—
59C22×C4×C127—44no—
60C22×C4×C127C22⋊C414yes—
61C22×C4×C127C22⋊C412yes—
62C22×C4×C127C22⋊C414yes—
63C22×C4×C127C2×D412yes—
64C22×C4×C127C2×D414yes—
65C2×C8×C127—48no—
66C2×C8×C127—44no—
67C4⋊C4×C127—28no—
68C4⋊C4×C127—216no—
69C22⋊C4×C127—216no—
70C23×C127—22no—
71C23×C127—22no—
72C23×C127—28no—
73C23×C127—22no—
74C23×C127—24no—
75C23×C127—24no—
76C23×C127—22no—
77C23×C127—416no—
78C23×C127—44no—
79C23×C127—44no—
80C23×C127#10114yes—
81C23×C127C2×C22⋊C412yes—
82C23×C127C2×C22⋊C411yescharacteristic
83C23×C127#13114yes—
84Q8×C127—416no—
85C2×C4×C127—28no—
86C2×C4×C127—22no—
87C2×C4×C127—24no—
88C2×C4×C127—22no—
89C2×C4×C127—24no—
90C2×C4×C127—24no—
91C2×C4×C127—28no—
92C2×C4×C127—44no—
93C2×C4×C127—416no—
94C2×C4×C127—44no—
95C2×C4×C127—44no—
96C2×C4×C127—44no—
97C2×C4×C127—44no—
98C2×C4×C127—44no—
99C2×C4×C127#9914yes—
100C2×C4×C127#12014yes—
101C8×C127—84no—
102C22×C127—28no—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—28no—
106C22×C127—24no—
107C22×C127—416no—
108C22×C127—44no—
109C22×C127—42no—
110C22×C127—42no—
111C22×C127#37911yescharacteristic
112C22×C127#40814yes—
113C22×C127#40812yes—
114C22×C127C2×#9912yes—
115C22×C127C2×#10112yes—
116C4×C127—24no—
117C4×C127—24no—
118C4×C127—44no—
119C4×C127—44no—
120#8308C12711yesmaximal, characteristic, direct product factor, Sylow
121C2×C127—24no—
122C2×C127—44no—
123C2×C127#147214yes—
124C2×C127#178011yescharacteristic
125C2×C127#178111yescharacteristic
126C2×C127#178411yescharacteristic
127C22×C22⋊Q8C2×C12711yescharacteristic
128#1779C2×C12712yes—
129C2×#324C2×C12714yescomplemented
130C127#830811yescharacteristic, direct product factor, Sylow
131C23×Q8C4×C12711yescharacteristic
132C24×C4C22×C12711yescharacteristic
133C2×C22⋊Q8—24no—
134C2×C22⋊Q8—28no—
135C22×C4⋊C4C4×C12711yescharacteristic
136C22×C4⋊C4C22×C12712yes—
137C22×C22⋊C4C4×C12712yes—
138C2×C22⋊C8—24no—
139C2×C22⋊C8C22×C12714yes—
140#324—28no—
141C25C2×C4×C12711yescharacteristic
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—44no—
145C23×C4—24no—
146C23×C4—44no—
147C23×C4C2×C4×C12712yes—
148C23×C4C2×C4×C12711yescharacteristic
149C23×C4C2×C4×C12712yes—
150C23×C4D4×C12714yes—
151C23×C4C23×C12711yescharacteristic
152C22×C8—24no—
153C22⋊Q8—232no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4D4×C12718yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C22⋊C8—28no—
163C22⋊C8—48no—
164C24—24no—
165C24—22no—
166C24—24no—
167C24—42no—
168C24C22⋊C4×C12712yes—
169C24C22×C4×C12711yescharacteristic
170C2×Q8—28no—
171C2×Q8—416no—
172C2×Q8—44no—
173C2×Q8—44no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—28no—
181C22×C4—28no—
182C22×C4—28no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4C22⋊C4×C12714yes—
188C22×C4C22⋊C4×C12712yes—
189C22×C4C22⋊C4×C12714yes—
190C22×C4C2×D4×C12712yes—
191C22×C4C2×D4×C12714yes—
192C2×C8—48no—
193C2×C8—44no—
194C4⋊C4—28no—
195C4⋊C4—216no—
196C22⋊C4—216no—
197C23—22no—
198C23—22no—
199C23—28no—
200C23—22no—
201C23—24no—
202C23—24no—
203C23—22no—
204C23—416no—
205C23—44no—
206C23—44no—
207C23#101×C12714yes—
208C23C2×C22⋊C4×C12712yes—
209C23C2×C22⋊C4×C12711yescharacteristic
210C23#131×C12714yes—
211Q8—416no—
212C2×C4—28no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—28no—
219C2×C4—44no—
220C2×C4—416no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4#99×C12714yes—
227C2×C4#120×C12714yes—
228C8—84no—
229C22—28no—
230C22—22no—
231C22—22no—
232C22—28no—
233C22—24no—
234C22—416no—
235C22—44no—
236C22—42no—
237C22—42no—
238C22#379×C12711yescharacteristic
239C22#408×C12714yes—
240C22#408×C12712yes—
241C22C2×#99×C12712yes—
242C22C2×#101×C12712yes—
243C4—24no—
244C4—24no—
245C4—44no—
246C4—44no—
247C2—24no—
248C2—44no—
249C2C127×#147214yes—
250C2C127×#178011yescharacteristic
251C2C127×#178111yescharacteristic
252C2C127×#178411yescharacteristic
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC24C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC25×C127C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#324×C127C2semidirect
120. #8308C127direct (N × H)
129. C2×#324C2×C127semidirect
130. C127#8308direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊Q8×C1272
C127×#17792
C2×#324×C1272
#8308127