Finite Group Databasebeta

← C127×#12168 #44575841

Subgroups — C127×#12168 #44575841

Order 32512.

330 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C127×#3261C211yesmaximal, characteristic
2C127×#2188C212yesmaximal
3C127×#2047C211yesmaximal, characteristic
4C127×#1350C212yesmaximal
5C127×#1323C211yesmaximal, characteristic
6C22×Q16×C127C2211yescharacteristic
7#541×C127—21no—
8C2×Q32×C127—22no—
9#479×C127—22no—
10#470×C127—21no—
11#467×C127C2212yes—
12#462×C127—22no—
13#424×C127C2211yescharacteristic
14#419×C127—21no—
15C2×#101×C127—21no—
16#358×C127—22no—
17C22⋊C16×C127C2212yes—
18#341×C127C2211yescharacteristic
19C22×Q8×C127—21no—
20C2×Q16×C127—21no—
21C2×Q16×C127—22no—
22C2×Q16×C127—42no—
23C2×Q16×C127D412yes—
24C2×M16×C127—21no—
25C2×M16×C127D411yescharacteristic
26C22×C8×C127C2311yescharacteristic
27C4⋊Q8×C127—21no—
28#126×C127—42no—
29#125×C127—21no—
30#125×C127—42no—
31C22⋊Q8×C127—22no—
32C4×Q8×C127—22no—
33#117×C127D411yescharacteristic
34Q32×C127—44no—
35C2×C16×C127—22no—
36#107×C127D412yes—
37#106×C127—21no—
38C4⋊C8×C127—22no—
39#103×C127—22no—
40#103×C127—22no—
41#103×C127—42no—
42#101×C127—42no—
43C2×Q8×C127—22no—
44C2×Q8×C127—22no—
45C2×Q8×C127—41no—
46C2×Q8×C127—41no—
47C2×Q8×C127—41no—
48C2×Q8×C127—41no—
49C2×Q8×C127—41no—
50C22×C4×C127—41no—
51C22×C4×C127C2×D411yescharacteristic
52Q16×C127—24no—
53Q16×C127—42no—
54Q16×C127—42no—
55Q16×C127—81no—
56M16×C127—41no—
57M16×C127—41no—
58M16×C127—41no—
59C2×C8×C127—21no—
60C2×C8×C127—41no—
61C2×C8×C127—41no—
62C2×C8×C127C2×D412yes—
63C4⋊C4×C127—22no—
64C4⋊C4×C127—42no—
65C4⋊C4×C127—42no—
66C4⋊C4×C127—42no—
67C4⋊C4×C127—42no—
68C4⋊C4×C127D812yes—
69C22⋊C4×C127—21no—
70C22⋊C4×C127—42no—
71C42×C127—21no—
72C42×C127—42no—
73C16×C127—42no—
74C23×C127AGL(1,ℤ/8)11yescharacteristic
75Q8×C127—42no—
76Q8×C127—42no—
77Q8×C127—42no—
78Q8×C127—42no—
79Q8×C127—81no—
80Q8×C127—81no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—22no—
84C2×C4×C127—42no—
85C2×C4×C127—42no—
86C2×C4×C127—42no—
87C2×C4×C127—81no—
88C2×C4×C127—81no—
89C2×C4×C127#12011yescharacteristic
90C2×C4×C127C2×D811yescharacteristic
91C8×C127—22no—
92C8×C127—41no—
93C8×C127—81no—
94C22×C127—21no—
95C22×C127—22no—
96C22×C127#44711yescharacteristic
97C4×C127—21no—
98C4×C127—42no—
99C4×C127—82no—
100C4×C127—82no—
101C4×C127—82no—
102C4×C127#45411yescharacteristic
103C4×C127B(4,𝔽2)11yescharacteristic
104#12168C12711yesmaximal, characteristic, direct product factor, Sylow
105C2×C127—41no—
106C2×C127#159811yescharacteristic
107C2×C127#219512yes—
108#3261C2×C12711yescharacteristic
109#2188C2×C12712yes—
110#2047C2×C12711yescharacteristic
111#1350C2×C12712yes—
112#1323C2×C12711yescharacteristic
113C127#1216811yescharacteristic, direct product factor, Sylow
114C22×Q16C22×C12711yescharacteristic
115#541—21no—
116C2×Q32—22no—
117#479—22no—
118#470—21no—
119#467C22×C12712yes—
120#462—22no—
121#424C22×C12711yescharacteristic
122#419—21no—
123C2×#101—21no—
124#358—22no—
125C22⋊C16C22×C12712yes—
126#341C22×C12711yescharacteristic
127C22×Q8—21no—
128C2×Q16—21no—
129C2×Q16—22no—
130C2×Q16—42no—
131C2×Q16D4×C12712yes—
132C2×M16—21no—
133C2×M16D4×C12711yescharacteristic
134C22×C8C23×C12711yescharacteristic
135C4⋊Q8—21no—
136#126—42no—
137#125—21no—
138#125—42no—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117D4×C12711yescharacteristic
142Q32—44no—
143C2×C16—22no—
144#107D4×C12712yes—
145#106—21no—
146C4⋊C8—22no—
147#103—22no—
148#103—22no—
149#103—42no—
150#101—42no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C22×C4—41no—
159C22×C4C2×D4×C12711yescharacteristic
160Q16—24no—
161Q16—42no—
162Q16—42no—
163Q16—81no—
164M16—41no—
165M16—41no—
166M16—41no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8C2×D4×C12712yes—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4D8×C12712yes—
177C22⋊C4—21no—
178C22⋊C4—42no—
179C42—21no—
180C42—42no—
181C16—42no—
182C23AGL(1,ℤ/8)×C12711yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—42no—
186Q8—42no—
187Q8—81no—
188Q8—81no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4#120×C12711yescharacteristic
198C2×C4C2×D8×C12711yescharacteristic
199C8—22no—
200C8—41no—
201C8—81no—
202C22—21no—
203C22—22no—
204C22#447×C12711yescharacteristic
205C4—21no—
206C4—42no—
207C4—82no—
208C4—82no—
209C4—82no—
210C4#454×C12711yescharacteristic
211C4B(4,𝔽2)×C12711yescharacteristic
212C2—41no—
213C2C127×#159811yescharacteristic
214C2C127×#219512yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC23×C127AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #424×C127 ◁ G

Complements

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

NH ≅ G/Ntype
104. #12168C127direct (N × H)
113. C127#12168direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32612
C127×#21882
C127×#20472
C127×#13502
C127×#13232
#12168127