Finite Group Databasebeta

← C127×#8345 #44572018

Subgroups — C127×#8345 #44572018

Order 32512.

802 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 198 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×#533×C127C211yesmaximal, characteristic
2C127×#1797C212yesmaximal
3C2×#332×C127C214yesmaximal, complemented
4#533×C127—28no—
5C2×C4⋊Q8×C127C2212yes—
6C2×#125×C127C412yes—
7C2×C22⋊Q8×C127—22no—
8C2×C4×Q8×C127—22no—
9C2×#117×C127C412yes—
10C22×C42×C127C2211yescharacteristic
11C2×C4⋊C8×C127C2214yes—
12C2×C22⋊C8×C127—22no—
13#336×C127—22no—
14#332×C127—28no—
15C22×Q8×C127—22no—
16C23×C4×C127C2×C411yescharacteristic
17C23×C4×C127D412yes—
18C22×C8×C127—24no—
19C4⋊Q8×C127—24no—
20C4⋊Q8×C127D418yes—
21#125×C127—28no—
22C22⋊Q8×C127—216no—
23C4×Q8×C127—216no—
24#117×C127—28no—
25C2×C4⋊C4×C127—24no—
26C2×C4⋊C4×C127—22no—
27C2×C4⋊C4×C127C2×C414yes—
28C2×C22⋊C4×C127—22no—
29C2×C42×C127—22no—
30C2×C42×C127—24no—
31C2×C42×C127C2×C411yescharacteristic
32C2×C42×C127D412yes—
33C2×C42×C127C2311yescharacteristic
34C4⋊C8×C127—28no—
35C22⋊C8×C127—44no—
36C24×C127C22⋊C411yescharacteristic
37C2×Q8×C127—28no—
38C2×Q8×C127—44no—
39C2×Q8×C127—42no—
40C2×Q8×C127—42no—
41C22×C4×C127—22no—
42C22×C4×C127—22no—
43C22×C4×C127—22no—
44C22×C4×C127—24no—
45C22×C4×C127—24no—
46C22×C4×C127—24no—
47C22×C4×C127—22no—
48C22×C4×C127—22no—
49C22×C4×C127C22⋊C412yes—
50C22×C4×C127C22⋊C411yescharacteristic
51C22×C4×C127C22×C411yescharacteristic
52C22×C4×C127C2×D412yes—
53C2×C8×C127—44no—
54C2×C8×C127—44no—
55C2×C8×C127—44no—
56C4⋊C4×C127—216no—
57C4⋊C4×C127—24no—
58C4⋊C4×C127—28no—
59C4⋊C4×C127C22⋊C418yes—
60C22⋊C4×C127—28no—
61C42×C127—28no—
62C42×C127—22no—
63C42×C127—24no—
64C42×C127C2×D414yes—
65C23×C127—21no—
66C23×C127—21no—
67C23×C127—22no—
68C23×C127—22no—
69C23×C127#10112yes—
70C23×C127C2×C22⋊C411yescharacteristic
71Q8×C127—48no—
72Q8×C127—48no—
73C2×C4×C127—28no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—22no—
79C2×C4×C127—28no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—44no—
83C2×C4×C127—44no—
84C2×C4×C127—44no—
85C2×C4×C127—44no—
86C2×C4×C127#10112yes—
87C2×C4×C127C2×C22⋊C412yes—
88C2×C4×C127#12014yes—
89C2×C4×C127#13514yes—
90C2×C4×C127AGL(1,ℤ/8)12yes—
91C2×C4×C127#13712yes—
92C8×C127—84no—
93C22×C127—24no—
94C22×C127—22no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—42no—
98C22×C127#37911yescharacteristic
99C22×C127C2×#10111yescharacteristic
100C22×C127C2×#10111yescharacteristic
101C22×C127#41612yes—
102C22×C127#41712yes—
103C4×C127—22no—
104C4×C127—22no—
105C4×C127—42no—
106C4×C127—48no—
107C4×C127#44914yes—
108C4×C127#45214yes—
109#8345C12711yesmaximal, characteristic, direct product factor, Sylow
110C2×C127—42no—
111C2×C127#154411yescharacteristic
112C2×C127#155212yes—
113C2×C127#156111yescharacteristic
114C2×C127#178411yescharacteristic
115C2×C127#178812yes—
116C2×#533C2×C12711yescharacteristic
117#1797C2×C12712yes—
118C2×#332C2×C12714yescomplemented
119C127#834511yescharacteristic, direct product factor, Sylow
120#533—28no—
121C2×C4⋊Q8C22×C12712yes—
122C2×#125C4×C12712yes—
123C2×C22⋊Q8—22no—
124C2×C4×Q8—22no—
125C2×#117C4×C12712yes—
126C22×C42C22×C12711yescharacteristic
127C2×C4⋊C8C22×C12714yes—
128C2×C22⋊C8—22no—
129#336—22no—
130#332—28no—
131C22×Q8—22no—
132C23×C4C2×C4×C12711yescharacteristic
133C23×C4D4×C12712yes—
134C22×C8—24no—
135C4⋊Q8—24no—
136C4⋊Q8D4×C12718yes—
137#125—28no—
138C22⋊Q8—216no—
139C4×Q8—216no—
140#117—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4C2×C4×C12714yes—
144C2×C22⋊C4—22no—
145C2×C42—22no—
146C2×C42—24no—
147C2×C42C2×C4×C12711yescharacteristic
148C2×C42D4×C12712yes—
149C2×C42C23×C12711yescharacteristic
150C4⋊C8—28no—
151C22⋊C8—44no—
152C24C22⋊C4×C12711yescharacteristic
153C2×Q8—28no—
154C2×Q8—44no—
155C2×Q8—42no—
156C2×Q8—42no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4C22⋊C4×C12712yes—
166C22×C4C22⋊C4×C12711yescharacteristic
167C22×C4C22×C4×C12711yescharacteristic
168C22×C4C2×D4×C12712yes—
169C2×C8—44no—
170C2×C8—44no—
171C2×C8—44no—
172C4⋊C4—216no—
173C4⋊C4—24no—
174C4⋊C4—28no—
175C4⋊C4C22⋊C4×C12718yes—
176C22⋊C4—28no—
177C42—28no—
178C42—22no—
179C42—24no—
180C42C2×D4×C12714yes—
181C23—21no—
182C23—21no—
183C23—22no—
184C23—22no—
185C23#101×C12712yes—
186C23C2×C22⋊C4×C12711yescharacteristic
187Q8—48no—
188Q8—48no—
189C2×C4—28no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—28no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—44no—
202C2×C4#101×C12712yes—
203C2×C4C2×C22⋊C4×C12712yes—
204C2×C4#120×C12714yes—
205C2×C4#135×C12714yes—
206C2×C4AGL(1,ℤ/8)×C12712yes—
207C2×C4#137×C12712yes—
208C8—84no—
209C22—24no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22—42no—
214C22#379×C12711yescharacteristic
215C22C2×#101×C12711yescharacteristic
216C22C2×#101×C12711yescharacteristic
217C22#416×C12712yes—
218C22#417×C12712yes—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—48no—
223C4#449×C12714yes—
224C4#452×C12714yes—
225C2—42no—
226C2C127×#154411yescharacteristic
227C2C127×#155212yes—
228C2C127×#156111yescharacteristic
229C2C127×#178411yescharacteristic
230C2C127×#178812yes—
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×#332×C127C2semidirect
109. #8345C127direct (N × H)
118. C2×#332C2×C127semidirect
119. C127#8345direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#533×C1272
C127×#17972
C2×#332×C1272
#8345127