Finite Group Databasebeta

← C127×#12501 #44576174

Subgroups — C127×#12501 #44576174

Order 32512.

214 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 78 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×#2226C211yesmaximal, characteristic
2C127×#2224C211yesmaximal, characteristic
3C127×#1710C211yesmaximal, characteristic, complemented
4C127×#1589C211yesmaximal, characteristic
5C127×#1367C211yesmaximal, characteristic, complemented
6C127×#1365C211yesmaximal, characteristic, complemented
7C127×#1364C211yesmaximal, characteristic, complemented
8#500×C127C2211yescharacteristic
9#500×C127C2211yescharacteristic
10#487×C127—21no—
11#487×C127—21no—
12C4×D8×C127C2211yescharacteristic
13#436×C127—21no—
14#367×C127—21no—
15#366×C127—21no—
16C4⋊C16×C127C2211yescharacteristic
17C4⋊C16×C127C2211yescharacteristic
18#357×C127—21no—
19#357×C127—21no—
20#335×C127C2211yescharacteristic
21D4⋊C8×C127—21no—
22C82×C127C2211yescharacteristic
23C2×D8×C127D411yescharacteristic
24C2×M16×C127—41no—
25C4⋊Q8×C127—21no—
26C4⋊Q8×C127—21no—
27C4×D4×C127—21no—
28C2×C16×C127—21no—
29C2×C16×C127—21no—
30#107×C127—21no—
31#107×C127—21no—
32#107×C127—21no—
33#107×C127—21no—
34#107×C127D411yescharacteristic
35C4⋊C8×C127—21no—
36#103×C127—42no—
37#103×C127—41no—
38#103×C127—41no—
39#102×C127—21no—
40C22⋊C8×C127—41no—
41C4×C8×C127D411yescharacteristic
42C4×C8×C127D411yescharacteristic
43C4×C8×C127C2311yescharacteristic
44C2×Q8×C127—41no—
45C2×Q8×C127—41no—
46C2×D4×C127—21no—
47C22×C4×C127—41no—
48D8×C127—21no—
49D8×C127—21no—
50M16×C127—82no—
51C2×C8×C127—21no—
52C2×C8×C127—41no—
53C2×C8×C127D811yescharacteristic
54C2×C8×C127D811yescharacteristic
55C2×C8×C127C2×D411yescharacteristic
56C2×C8×C127#4111yescharacteristic
57C4⋊C4×C127—21no—
58C4⋊C4×C127—41no—
59C4⋊C4×C127—41no—
60C4⋊C4×C127—41no—
61C4⋊C4×C127—41no—
62C22⋊C4×C127—41no—
63C42×C127C2×D411yescharacteristic
64C16×C127—41no—
65C16×C127—41no—
66C23×C127—41no—
67Q8×C127—82no—
68Q8×C127—81no—
69Q8×C127—81no—
70D4×C127—41no—
71D4×C127—41no—
72C2×C4×C127—41no—
73C2×C4×C127—81no—
74C2×C4×C127—81no—
75C2×C4×C127—81no—
76C2×C4×C127—81no—
77C2×C4×C127#12111yescharacteristic
78C2×C4×C127C2×D811yescharacteristic
79C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
80C8×C127—21no—
81C8×C127—21no—
82C8×C127—21no—
83C8×C127—41no—
84C8×C127—81no—
85C8×C127#13711yescharacteristic
86C8×C127#13711yescharacteristic
87C22×C127—81no—
88C22×C127—81no—
89C22×C127—81no—
90C22×C127#45911yescharacteristic
91C4×C127—21no—
92C4×C127—81no—
93C4×C127—161no—
94C4×C127—161no—
95C4×C127#47011yescharacteristic
96C4×C127#47311yescharacteristic
97C4×C127#50811yescharacteristic
98C4×C127#50911yescharacteristic
99#12501C12711yesmaximal, characteristic, direct product factor, Sylow
100C2×C127—161no—
101C2×C127#168911yescharacteristic
102C2×C127#221011yescharacteristic
103C2×C127#221211yescharacteristic
104#2226C2×C12711yescharacteristic
105#2224C2×C12711yescharacteristic
106#1710C2×C12711yescharacteristic, complemented
107#1589C2×C12711yescharacteristic
108#1367C2×C12711yescharacteristic, complemented
109#1365C2×C12711yescharacteristic, complemented
110#1364C2×C12711yescharacteristic, complemented
111C127#1250111yescharacteristic, direct product factor, Sylow
112#500C22×C12711yescharacteristic
113#500C22×C12711yescharacteristic
114#487—21no—
115#487—21no—
116C4×D8C22×C12711yescharacteristic
117#436—21no—
118#367—21no—
119#366—21no—
120C4⋊C16C22×C12711yescharacteristic
121C4⋊C16C22×C12711yescharacteristic
122#357—21no—
123#357—21no—
124#335C22×C12711yescharacteristic
125D4⋊C8—21no—
126C82C22×C12711yescharacteristic
127C2×D8D4×C12711yescharacteristic
128C2×M16—41no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4×D4—21no—
132C2×C16—21no—
133C2×C16—21no—
134#107—21no—
135#107—21no—
136#107—21no—
137#107—21no—
138#107D4×C12711yescharacteristic
139C4⋊C8—21no—
140#103—42no—
141#103—41no—
142#103—41no—
143#102—21no—
144C22⋊C8—41no—
145C4×C8D4×C12711yescharacteristic
146C4×C8D4×C12711yescharacteristic
147C4×C8C23×C12711yescharacteristic
148C2×Q8—41no—
149C2×Q8—41no—
150C2×D4—21no—
151C22×C4—41no—
152D8—21no—
153D8—21no—
154M16—82no—
155C2×C8—21no—
156C2×C8—41no—
157C2×C8D8×C12711yescharacteristic
158C2×C8D8×C12711yescharacteristic
159C2×C8C2×D4×C12711yescharacteristic
160C2×C8#41×C12711yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C22⋊C4—41no—
167C42C2×D4×C12711yescharacteristic
168C16—41no—
169C16—41no—
170C23—41no—
171Q8—82no—
172Q8—81no—
173Q8—81no—
174D4—41no—
175D4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4#121×C12711yescharacteristic
182C2×C4C2×D8×C12711yescharacteristic
183C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
184C8—21no—
185C8—21no—
186C8—21no—
187C8—41no—
188C8—81no—
189C8#137×C12711yescharacteristic
190C8#137×C12711yescharacteristic
191C22—81no—
192C22—81no—
193C22—81no—
194C22#459×C12711yescharacteristic
195C4—21no—
196C4—81no—
197C4—161no—
198C4—161no—
199C4#470×C12711yescharacteristic
200C4#473×C12711yescharacteristic
201C4#508×C12711yescharacteristic
202C4#509×C12711yescharacteristic
203C2—161no—
204C2C127×#168911yescharacteristic
205C2C127×#221011yescharacteristic
206C2C127×#221211yescharacteristic
subgroupG/H
center, socleC22×C127#459
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC2×D8×C127D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C82×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1710C2semidirect
5. C127×#1367C2semidirect
6. C127×#1365C2semidirect
7. C127×#1364C2semidirect
99. #12501C127direct (N × H)
106. #1710C2×C127semidirect
108. #1367C2×C127semidirect
109. #1365C2×C127semidirect
110. #1364C2×C127semidirect
111. C127#12501direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#22262
C127×#22242
C127×#17102
C127×#15892
C127×#13672
C127×#13652
C127×#13642
#12501127