Finite Group Databasebeta

← C127×#10309 #44573982

Subgroups — C127×#10309 #44573982

Order 32512.

618 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 186 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×#500×C127C212yesmaximal
2C2×#494×C127C211yesmaximal, characteristic
3C127×#1990C212yesmaximal
4C127×#1956C211yesmaximal, characteristic
5C127×#1273C211yesmaximal, characteristic
6C22×Q16×C127—22no—
7C2×C4⋊Q8×C127C2212yes—
8C2×C4⋊Q8×C127C2211yescharacteristic
9#500×C127—24no—
10#500×C127—24no—
11#494×C127—24no—
12C2×#107×C127—24no—
13C2×#103×C127—22no—
14C2×C4×C8×C127C2211yescharacteristic
15C2×C4×C8×C127C2212yes—
16#384×C127C2211yescharacteristic
17#340×C127—24no—
18C22×Q8×C127—22no—
19C22×Q8×C127—21no—
20C2×Q16×C127—28no—
21C2×Q16×C127—44no—
22C2×Q16×C127—44no—
23C22×C8×C127D412yes—
24C22×C8×C127D414yes—
25C4⋊Q8×C127—22no—
26C4⋊Q8×C127—22no—
27C4⋊Q8×C127—22no—
28C4⋊Q8×C127—22no—
29C4⋊Q8×C127—21no—
30C4⋊Q8×C127D412yes—
31C4⋊Q8×C127D414yes—
32C2×C4⋊C4×C127—22no—
33C2×C4⋊C4×C127—22no—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—22no—
36C2×C4⋊C4×C127—21no—
37C2×C42×C127C2311yescharacteristic
38#107×C127—216no—
39#103×C127—44no—
40C4×C8×C127—22no—
41C4×C8×C127—22no—
42C4×C8×C127—22no—
43C2×Q8×C127—24no—
44C2×Q8×C127—22no—
45C2×Q8×C127—41no—
46C2×Q8×C127—42no—
47C2×Q8×C127—42no—
48C2×Q8×C127—44no—
49C2×Q8×C127—44no—
50C2×Q8×C127—42no—
51C22×C4×C127—42no—
52C22×C4×C127—41no—
53C22×C4×C127—41no—
54C22×C4×C127C2×D411yescharacteristic
55C22×C4×C127C2×D412yes—
56Q16×C127—416no—
57C2×C8×C127—24no—
58C2×C8×C127—22no—
59C2×C8×C127—22no—
60C2×C8×C127—24no—
61C2×C8×C127D814yes—
62C2×C8×C127Q1618yescomplemented
63C4⋊C4×C127—24no—
64C4⋊C4×C127—24no—
65C4⋊C4×C127—24no—
66C4⋊C4×C127—42no—
67C4⋊C4×C127—42no—
68C4⋊C4×C127—42no—
69C4⋊C4×C127—42no—
70C4⋊C4×C127—44no—
71C42×C127C2×D412yes—
72C42×C127C2×D411yescharacteristic
73C42×C127#4111yescharacteristic
74C23×C127#12711yescharacteristic
75Q8×C127—48no—
76Q8×C127—44no—
77Q8×C127—82no—
78Q8×C127—84no—
79C2×C4×C127—42no—
80C2×C4×C127—44no—
81C2×C4×C127—81no—
82C2×C4×C127—82no—
83C2×C4×C127—82no—
84C2×C4×C127—81no—
85C2×C4×C127—81no—
86C2×C4×C127—82no—
87C2×C4×C127#12011yescharacteristic
88C2×C4×C127#12012yes—
89C2×C4×C127#12111yescharacteristic
90C2×C4×C127#12112yes—
91C2×C4×C127C2×D812yes—
92C2×C4×C127C2×Q1614yes—
93C2×C4×C127AGL(1,ℤ/8)12yes—
94C2×C4×C127#13714yes—
95C8×C127—24no—
96C8×C127—28no—
97C22×C127#39211yescharacteristic
98C22×C127#49311yescharacteristic
99C22×C127#49412yes—
100C22×C127#49611yescharacteristic
101C22×C127#49712yes—
102C4×C127—84no—
103C4×C127—82no—
104C4×C127—161no—
105C4×C127#44712yes—
106C4×C127#45114yes—
107C4×C127#45912yes—
108C4×C127#46214yes—
109#10309C12711yesmaximal, characteristic, direct product factor, Sylow
110C2×C127#171311yescharacteristic
111C2×C127#172211yescharacteristic
112C2×C127#172312yes—
113C2×C127#201111yescharacteristic
114C2×C127#201612yes—
115C2×#500C2×C12712yes—
116C2×#494C2×C12711yescharacteristic
117#1990C2×C12712yes—
118#1956C2×C12711yescharacteristic
119#1273C2×C12711yescharacteristic
120C127#1030911yescharacteristic, direct product factor, Sylow
121C22×Q16—22no—
122C2×C4⋊Q8C22×C12712yes—
123C2×C4⋊Q8C22×C12711yescharacteristic
124#500—24no—
125#500—24no—
126#494—24no—
127C2×#107—24no—
128C2×#103—22no—
129C2×C4×C8C22×C12711yescharacteristic
130C2×C4×C8C22×C12712yes—
131#384C22×C12711yescharacteristic
132#340—24no—
133C22×Q8—22no—
134C22×Q8—21no—
135C2×Q16—28no—
136C2×Q16—44no—
137C2×Q16—44no—
138C22×C8D4×C12712yes—
139C22×C8D4×C12714yes—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—21no—
145C4⋊Q8D4×C12714yes—
146C4⋊Q8D4×C12712yes—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C42C23×C12711yescharacteristic
153#107—216no—
154#103—44no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C2×Q8—24no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C2×D4×C12711yescharacteristic
170C22×C4C2×D4×C12712yes—
171Q16—416no—
172C2×C8—24no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—24no—
176C2×C8D8×C12714yes—
177C2×C8Q16×C12718yescomplemented
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C42C2×D4×C12712yes—
187C42C2×D4×C12711yescharacteristic
188C42#41×C12711yescharacteristic
189C23#127×C12711yescharacteristic
190Q8—48no—
191Q8—44no—
192Q8—82no—
193Q8—84no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4#120×C12711yescharacteristic
203C2×C4#120×C12712yes—
204C2×C4#121×C12711yescharacteristic
205C2×C4#121×C12712yes—
206C2×C4C2×D8×C12712yes—
207C2×C4C2×Q16×C12714yes—
208C2×C4AGL(1,ℤ/8)×C12712yes—
209C2×C4#137×C12714yes—
210C8—24no—
211C8—28no—
212C22#392×C12711yescharacteristic
213C22#493×C12711yescharacteristic
214C22#494×C12712yes—
215C22#496×C12711yescharacteristic
216C22#497×C12712yes—
217C4—84no—
218C4—82no—
219C4—161no—
220C4#447×C12712yes—
221C4#451×C12714yes—
222C4#459×C12712yes—
223C4#462×C12714yes—
224C2C127×#171311yescharacteristic
225C2C127×#172211yescharacteristic
226C2C127×#172312yes—
227C2C127×#201111yescharacteristic
228C2C127×#201612yes—
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
62. C2×C8×C127Q16semidirect
109. #10309C127direct (N × H)
120. C127#10309direct (N × H)
177. C2×C8Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#500×C1272
C2×#494×C1272
C127×#19902
C127×#19562
C127×#12732
#10309127