Finite Group Databasebeta

← C127×#8342 #44572015

Subgroups — C127×#8342 #44572015

Order 32512.

1030 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 238 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×C4⋊Q8×C127C211yesmaximal, characteristic
2C127×#1797C212yesmaximal
3C2×#333×C127C214yesmaximal, complemented
4C23×Q8×C127C412yes—
5C2×C4⋊Q8×C127—28no—
6C2×C4⋊Q8×C127—22no—
7C2×C4⋊Q8×C127C412yes—
8C2×C4⋊Q8×C127C2212yes—
9C22×C4⋊C4×C127—22no—
10C22×C42×C127C2211yescharacteristic
11C2×C4⋊C8×C127C2214yes—
12C2×C22⋊C8×C127—22no—
13#336×C127—22no—
14#333×C127—28no—
15C22×Q8×C127—28no—
16C22×Q8×C127—22no—
17C22×Q8×C127—44no—
18C22×Q8×C127—44no—
19C22×Q8×C127C2×C414yes—
20C23×C4×C127—22no—
21C23×C4×C127C2×C411yescharacteristic
22C23×C4×C127D412yes—
23C22×C8×C127—24no—
24C4⋊Q8×C127—216no—
25C4⋊Q8×C127—28no—
26C4⋊Q8×C127—24no—
27C4⋊Q8×C127D418yes—
28C2×C4⋊C4×C127—216no—
29C2×C4⋊C4×C127—24no—
30C2×C4⋊C4×C127—24no—
31C2×C42×C127—24no—
32C2×C42×C127C2×C411yescharacteristic
33C2×C42×C127D412yes—
34C2×C42×C127C2311yescharacteristic
35C4⋊C8×C127—28no—
36C22⋊C8×C127—44no—
37C24×C127C22⋊C411yescharacteristic
38C2×Q8×C127—28no—
39C2×Q8×C127—24no—
40C2×Q8×C127—432no—
41C2×Q8×C127—44no—
42C2×Q8×C127—44no—
43C2×Q8×C127—44no—
44C2×Q8×C127—44no—
45C2×Q8×C127C22⋊C418yes—
46C22×C4×C127—22no—
47C22×C4×C127—22no—
48C22×C4×C127—22no—
49C22×C4×C127—24no—
50C22×C4×C127—24no—
51C22×C4×C127—28no—
52C22×C4×C127—24no—
53C22×C4×C127—44no—
54C22×C4×C127—44no—
55C22×C4×C127C22⋊C411yescharacteristic
56C22×C4×C127C22⋊C412yes—
57C22×C4×C127SD1614yes—
58C22×C4×C127Q1614yes—
59C22×C4×C127C22×C411yescharacteristic
60C22×C4×C127C2×D412yes—
61C2×C8×C127—44no—
62C2×C8×C127—44no—
63C2×C8×C127—44no—
64C4⋊C4×C127—216no—
65C4⋊C4×C127—216no—
66C42×C127—22no—
67C42×C127—24no—
68C42×C127C2×D414yes—
69C23×C127—21no—
70C23×C127—21no—
71C23×C127—22no—
72C23×C127#10012yes—
73C23×C127#10314yes—
74C23×C127C2×C22⋊C411yescharacteristic
75Q8×C127—48no—
76Q8×C127—416no—
77Q8×C127—48no—
78C2×C4×C127—28no—
79C2×C4×C127—22no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—21no—
84C2×C4×C127—22no—
85C2×C4×C127—28no—
86C2×C4×C127—28no—
87C2×C4×C127—28no—
88C2×C4×C127—416no—
89C2×C4×C127—44no—
90C2×C4×C127—44no—
91C2×C4×C127#10012yes—
92C2×C4×C127#10314yes—
93C2×C4×C127C2×C22⋊C412yes—
94C2×C4×C127#12014yes—
95C2×C4×C127C2×SD1612yes—
96C2×C4×C127C2×Q1612yes—
97C2×C4×C127#13714yes—
98C8×C127—84no—
99C22×C127—24no—
100C22×C127—22no—
101C22×C127—22no—
102C22×C127—24no—
103C22×C127#33314yescomplemented
104C22×C127#37911yescharacteristic
105C22×C127C2×#10011yescharacteristic
106C22×C127C2×#10011yescharacteristic
107C22×C127C2×#10312yes—
108C22×C127#41912yes—
109C4×C127—22no—
110C4×C127—22no—
111C4×C127—24no—
112C4×C127—48no—
113C4×C127#44814yes—
114C4×C127#45114yes—
115#8342C12711yesmaximal, characteristic, direct product factor, Sylow
116C2×C127—24no—
117C2×C127C2×#33311yescharacteristic
118C2×C127#154812yes—
119C2×C127#155811yescharacteristic
120C2×C127#178311yescharacteristic
121C2×C127#178712yes—
122C22×C4⋊Q8C2×C12711yescharacteristic
123#1797C2×C12712yes—
124C2×#333C2×C12714yescomplemented
125C127#834211yescharacteristic, direct product factor, Sylow
126C23×Q8C4×C12712yes—
127C2×C4⋊Q8—28no—
128C2×C4⋊Q8—22no—
129C2×C4⋊Q8C4×C12712yes—
130C2×C4⋊Q8C22×C12712yes—
131C22×C4⋊C4—22no—
132C22×C42C22×C12711yescharacteristic
133C2×C4⋊C8C22×C12714yes—
134C2×C22⋊C8—22no—
135#336—22no—
136#333—28no—
137C22×Q8—28no—
138C22×Q8—22no—
139C22×Q8—44no—
140C22×Q8—44no—
141C22×Q8C2×C4×C12714yes—
142C23×C4—22no—
143C23×C4C2×C4×C12711yescharacteristic
144C23×C4D4×C12712yes—
145C22×C8—24no—
146C4⋊Q8—216no—
147C4⋊Q8—28no—
148C4⋊Q8—24no—
149C4⋊Q8D4×C12718yes—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C42—24no—
154C2×C42C2×C4×C12711yescharacteristic
155C2×C42D4×C12712yes—
156C2×C42C23×C12711yescharacteristic
157C4⋊C8—28no—
158C22⋊C8—44no—
159C24C22⋊C4×C12711yescharacteristic
160C2×Q8—28no—
161C2×Q8—24no—
162C2×Q8—432no—
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8C22⋊C4×C12718yes—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—24no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4C22⋊C4×C12711yescharacteristic
178C22×C4C22⋊C4×C12712yes—
179C22×C4SD16×C12714yes—
180C22×C4Q16×C12714yes—
181C22×C4C22×C4×C12711yescharacteristic
182C22×C4C2×D4×C12712yes—
183C2×C8—44no—
184C2×C8—44no—
185C2×C8—44no—
186C4⋊C4—216no—
187C4⋊C4—216no—
188C42—22no—
189C42—24no—
190C42C2×D4×C12714yes—
191C23—21no—
192C23—21no—
193C23—22no—
194C23#100×C12712yes—
195C23#103×C12714yes—
196C23C2×C22⋊C4×C12711yescharacteristic
197Q8—48no—
198Q8—416no—
199Q8—48no—
200C2×C4—28no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—28no—
208C2×C4—28no—
209C2×C4—28no—
210C2×C4—416no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4#100×C12712yes—
214C2×C4#103×C12714yes—
215C2×C4C2×C22⋊C4×C12712yes—
216C2×C4#120×C12714yes—
217C2×C4C2×SD16×C12712yes—
218C2×C4C2×Q16×C12712yes—
219C2×C4#137×C12714yes—
220C8—84no—
221C22—24no—
222C22—22no—
223C22—22no—
224C22—24no—
225C22#333×C12714yescomplemented
226C22#379×C12711yescharacteristic
227C22C2×#100×C12711yescharacteristic
228C22C2×#100×C12711yescharacteristic
229C22C2×#103×C12712yes—
230C22#419×C12712yes—
231C4—22no—
232C4—22no—
233C4—24no—
234C4—48no—
235C4#448×C12714yes—
236C4#451×C12714yes—
237C2—24no—
238C2C2×#333×C12711yescharacteristic
239C2C127×#154812yes—
240C2C127×#155811yescharacteristic
241C2C127×#178311yescharacteristic
242C2C127×#178712yes—
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×#333×C127C2semidirect
103. C22×C127#333semidirect
115. #8342C127direct (N × H)
124. C2×#333C2×C127semidirect
125. C127#8342direct (N × H)
225. C22#333×C127semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Q8×C1272
C127×#17972
C2×#333×C1272
#8342127