Finite Group Databasebeta

← C127×#10317 #44573990

Subgroups — C127×#10317 #44573990

Order 32512.

554 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 170 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×#498×C127C211yesmaximal, characteristic
2C2×#487×C127C212yesmaximal
3C127×#1990C212yesmaximal
4C127×#1956C211yesmaximal, characteristic
5C127×#1273C211yesmaximal, characteristic
6C2×C4⋊Q8×C127C2211yescharacteristic
7C2×C4⋊Q8×C127C2212yes—
8#498×C127—24no—
9#487×C127—24no—
10#487×C127—24no—
11C2×#106×C127—22no—
12C2×#103×C127—24no—
13C2×#103×C127—22no—
14C2×C4×C8×C127C2212yes—
15C2×C4×C8×C127C2211yescharacteristic
16#384×C127C2211yescharacteristic
17#340×C127—24no—
18C22×Q8×C127—21no—
19C22×Q8×C127—22no—
20C22×C8×C127D414yes—
21C22×C8×C127D412yes—
22C4⋊Q8×C127—22no—
23C4⋊Q8×C127—22no—
24C4⋊Q8×C127—22no—
25C4⋊Q8×C127—22no—
26C4⋊Q8×C127—21no—
27C4⋊Q8×C127D414yes—
28C4⋊Q8×C127D412yes—
29C2×C4⋊C4×C127—22no—
30C2×C4⋊C4×C127—22no—
31C2×C4⋊C4×C127—22no—
32C2×C4⋊C4×C127—22no—
33C2×C4⋊C4×C127—21no—
34C2×C42×C127C2311yescharacteristic
35#106×C127—28no—
36#103×C127—216no—
37#103×C127—44no—
38C4×C8×C127—22no—
39C4×C8×C127—22no—
40C4×C8×C127—22no—
41C2×Q8×C127—24no—
42C2×Q8×C127—22no—
43C2×Q8×C127—41no—
44C2×Q8×C127—42no—
45C2×Q8×C127—42no—
46C2×Q8×C127—44no—
47C2×Q8×C127—42no—
48C2×Q8×C127—44no—
49C22×C4×C127—41no—
50C22×C4×C127—42no—
51C22×C4×C127—41no—
52C22×C4×C127C2×D411yescharacteristic
53C22×C4×C127C2×D412yes—
54C2×C8×C127—22no—
55C2×C8×C127—22no—
56C2×C8×C127—24no—
57C2×C8×C127—24no—
58C2×C8×C127—24no—
59C2×C8×C127SD1614yes—
60C4⋊C4×C127—24no—
61C4⋊C4×C127—24no—
62C4⋊C4×C127—24no—
63C4⋊C4×C127—42no—
64C4⋊C4×C127—42no—
65C4⋊C4×C127—42no—
66C4⋊C4×C127—42no—
67C4⋊C4×C127—44no—
68C42×C127C2×D412yes—
69C42×C127C2×D411yescharacteristic
70C42×C127#4111yescharacteristic
71C23×C127#12711yescharacteristic
72Q8×C127—48no—
73Q8×C127—44no—
74Q8×C127—82no—
75Q8×C127—84no—
76C2×C4×C127—42no—
77C2×C4×C127—44no—
78C2×C4×C127—81no—
79C2×C4×C127—82no—
80C2×C4×C127—81no—
81C2×C4×C127—82no—
82C2×C4×C127—81no—
83C2×C4×C127—82no—
84C2×C4×C127#12011yescharacteristic
85C2×C4×C127#12012yes—
86C2×C4×C127#12111yescharacteristic
87C2×C4×C127#12112yes—
88C2×C4×C127C2×SD1612yes—
89C2×C4×C127#13514yes—
90C2×C4×C127#13712yes—
91C2×C4×C127#13714yes—
92C8×C127—24no—
93C8×C127—44no—
94C22×C127#39211yescharacteristic
95C22×C127#49211yescharacteristic
96C22×C127#49512yes—
97C22×C127#49712yes—
98C22×C127#49711yescharacteristic
99C4×C127—82no—
100C4×C127—84no—
101C4×C127—161no—
102C4×C127#44812yes—
103C4×C127#45214yes—
104C4×C127#46112yes—
105C4×C127#46414yes—
106#10317C12711yesmaximal, characteristic, direct product factor, Sylow
107C2×C127#170811yescharacteristic
108C2×C127#171611yescharacteristic
109C2×C127#171812yes—
110C2×C127#201311yescharacteristic
111C2×C127#201512yes—
112C2×#498C2×C12711yescharacteristic
113C2×#487C2×C12712yes—
114#1990C2×C12712yes—
115#1956C2×C12711yescharacteristic
116#1273C2×C12711yescharacteristic
117C127#1031711yescharacteristic, direct product factor, Sylow
118C2×C4⋊Q8C22×C12711yescharacteristic
119C2×C4⋊Q8C22×C12712yes—
120#498—24no—
121#487—24no—
122#487—24no—
123C2×#106—22no—
124C2×#103—24no—
125C2×#103—22no—
126C2×C4×C8C22×C12712yes—
127C2×C4×C8C22×C12711yescharacteristic
128#384C22×C12711yescharacteristic
129#340—24no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×C8D4×C12714yes—
133C22×C8D4×C12712yes—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—21no—
139C4⋊Q8D4×C12712yes—
140C4⋊Q8D4×C12714yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C42C23×C12711yescharacteristic
147#106—28no—
148#103—216no—
149#103—44no—
150C4×C8—22no—
151C4×C8—22no—
152C4×C8—22no—
153C2×Q8—24no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—44no—
159C2×Q8—42no—
160C2×Q8—44no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4C2×D4×C12711yescharacteristic
165C22×C4C2×D4×C12712yes—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—24no—
169C2×C8—24no—
170C2×C8—24no—
171C2×C8SD16×C12714yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C42C2×D4×C12712yes—
181C42C2×D4×C12711yescharacteristic
182C42#41×C12711yescharacteristic
183C23#127×C12711yescharacteristic
184Q8—48no—
185Q8—44no—
186Q8—82no—
187Q8—84no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4—81no—
193C2×C4—82no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4#120×C12711yescharacteristic
197C2×C4#120×C12712yes—
198C2×C4#121×C12711yescharacteristic
199C2×C4#121×C12712yes—
200C2×C4C2×SD16×C12712yes—
201C2×C4#135×C12714yes—
202C2×C4#137×C12712yes—
203C2×C4#137×C12714yes—
204C8—24no—
205C8—44no—
206C22#392×C12711yescharacteristic
207C22#492×C12711yescharacteristic
208C22#495×C12712yes—
209C22#497×C12712yes—
210C22#497×C12711yescharacteristic
211C4—82no—
212C4—84no—
213C4—161no—
214C4#448×C12712yes—
215C4#452×C12714yes—
216C4#461×C12712yes—
217C4#464×C12714yes—
218C2C127×#170811yescharacteristic
219C2C127×#171611yescharacteristic
220C2C127×#171812yes—
221C2C127×#201311yescharacteristic
222C2C127×#201512yes—
subgroupG/H
center, socle, prime radicalC23×C127#127
commutator, Frattini, cosocleC2×C42C23×C127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
106. #10317C127direct (N × H)
117. C127#10317direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#498×C1272
C2×#487×C1272
C127×#19902
C127×#19562
C127×#12732
#10317127