Finite Group Databasebeta

← C127×#12456 #44576129

Subgroups — C127×#12456 #44576129

Order 32512.

212 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2254C211yesmaximal, characteristic, complemented
2C127×#2169C211yesmaximal, characteristic
3C127×#1675C211yesmaximal, characteristic
4C127×#1673C211yesmaximal, characteristic
5C127×#1371C211yesmaximal, characteristic, complemented
6C127×#1365C211yesmaximal, characteristic, complemented
7C127×#1364C211yesmaximal, characteristic, complemented
8C2×M32×C127—21no—
9#500×C127C2211yescharacteristic
10#500×C127C2211yescharacteristic
11#476×C127—21no—
12#461×C127—21no—
13C8×D4×C127C2211yescharacteristic
14#367×C127—21no—
15#367×C127—21no—
16C4⋊C16×C127C2211yescharacteristic
17C22⋊C16×C127—21no—
18C4×C16×C127C2211yescharacteristic
19#335×C127C2211yescharacteristic
20#335×C127C2211yescharacteristic
21#333×C127—21no—
22#332×C127—21no—
23C2×SD16×C127—41no—
24C22×C8×C127—21no—
25C4⋊Q8×C127—21no—
26C4⋊Q8×C127—21no—
27C4×D4×C127D411yescharacteristic
28M32×C127—42no—
29C2×C16×C127—21no—
30C2×C16×C127D411yescharacteristic
31C2×C16×C127D411yescharacteristic
32#107×C127—21no—
33#107×C127—21no—
34#107×C127—21no—
35#107×C127—21no—
36#106×C127—41no—
37C4⋊C8×C127—21no—
38C4⋊C8×C127—21no—
39C4⋊C8×C127D411yescharacteristic
40#103×C127—41no—
41#102×C127—41no—
42C22⋊C8×C127—21no—
43C4×C8×C127C2311yescharacteristic
44C2×Q8×C127—41no—
45C2×Q8×C127—41no—
46C2×D4×C127D811yescharacteristic
47C22×C4×C127—21no—
48SD16×C127—82no—
49C2×C8×C127—21no—
50C2×C8×C127—41no—
51C2×C8×C127—41no—
52C2×C8×C127—41no—
53C2×C8×C127—41no—
54C2×C8×C127C2×D411yescharacteristic
55C2×C8×C127#4111yescharacteristic
56C4⋊C4×C127—41no—
57C4⋊C4×C127—41no—
58C4⋊C4×C127—41no—
59C4⋊C4×C127—41no—
60C4⋊C4×C127D811yescharacteristic
61C22⋊C4×C127—21no—
62C42×C127C2×D411yescharacteristic
63C16×C127—21no—
64C16×C127—41no—
65C16×C127SD1612yescomplemented
66C23×C127—21no—
67Q8×C127—81no—
68Q8×C127—81no—
69Q8×C127—161no—
70D4×C127—21no—
71D4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—81no—
76C2×C4×C127—81no—
77C2×C4×C127#12111yescharacteristic
78C2×C4×C127C2×D811yescharacteristic
79C2×C4×C127#13511yescharacteristic
80C8×C127—21no—
81C8×C127—41no—
82C8×C127—81no—
83C8×C127—81no—
84C8×C127C2×SD1611yescharacteristic
85C8×C127#13711yescharacteristic
86C22×C127—41no—
87C22×C127—41no—
88C22×C127—41no—
89C22×C127#46611yescharacteristic
90C4×C127—21no—
91C4×C127—41no—
92C4×C127—161no—
93C4×C127—161no—
94C4×C127#46111yescharacteristic
95C4×C127#47311yescharacteristic
96C4×C127#50911yescharacteristic
97C4×C127#51011yescharacteristic
98#12456C12711yesmaximal, characteristic, direct product factor, Sylow
99C2×C127—81no—
100C2×C127#167311yescharacteristic
101C2×C127#222011yescharacteristic
102C2×C127#222311yescharacteristic
103#2254C2×C12711yescharacteristic, complemented
104#2169C2×C12711yescharacteristic
105#1675C2×C12711yescharacteristic
106#1673C2×C12711yescharacteristic
107#1371C2×C12711yescharacteristic, complemented
108#1365C2×C12711yescharacteristic, complemented
109#1364C2×C12711yescharacteristic, complemented
110C127#1245611yescharacteristic, direct product factor, Sylow
111C2×M32—21no—
112#500C22×C12711yescharacteristic
113#500C22×C12711yescharacteristic
114#476—21no—
115#461—21no—
116C8×D4C22×C12711yescharacteristic
117#367—21no—
118#367—21no—
119C4⋊C16C22×C12711yescharacteristic
120C22⋊C16—21no—
121C4×C16C22×C12711yescharacteristic
122#335C22×C12711yescharacteristic
123#335C22×C12711yescharacteristic
124#333—21no—
125#332—21no—
126C2×SD16—41no—
127C22×C8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4×D4D4×C12711yescharacteristic
131M32—42no—
132C2×C16—21no—
133C2×C16D4×C12711yescharacteristic
134C2×C16D4×C12711yescharacteristic
135#107—21no—
136#107—21no—
137#107—21no—
138#107—21no—
139#106—41no—
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8D4×C12711yescharacteristic
143#103—41no—
144#102—41no—
145C22⋊C8—21no—
146C4×C8C23×C12711yescharacteristic
147C2×Q8—41no—
148C2×Q8—41no—
149C2×D4D8×C12711yescharacteristic
150C22×C4—21no—
151SD16—82no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8C2×D4×C12711yescharacteristic
158C2×C8#41×C12711yescharacteristic
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4D8×C12711yescharacteristic
164C22⋊C4—21no—
165C42C2×D4×C12711yescharacteristic
166C16—21no—
167C16—41no—
168C16SD16×C12712yescomplemented
169C23—21no—
170Q8—81no—
171Q8—81no—
172Q8—161no—
173D4—21no—
174D4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4#121×C12711yescharacteristic
181C2×C4C2×D8×C12711yescharacteristic
182C2×C4#135×C12711yescharacteristic
183C8—21no—
184C8—41no—
185C8—81no—
186C8—81no—
187C8C2×SD16×C12711yescharacteristic
188C8#137×C12711yescharacteristic
189C22—41no—
190C22—41no—
191C22—41no—
192C22#466×C12711yescharacteristic
193C4—21no—
194C4—41no—
195C4—161no—
196C4—161no—
197C4#461×C12711yescharacteristic
198C4#473×C12711yescharacteristic
199C4#509×C12711yescharacteristic
200C4#510×C12711yescharacteristic
201C2—81no—
202C2C127×#167311yescharacteristic
203C2C127×#222011yescharacteristic
204C2C127×#222311yescharacteristic
subgroupG/H
center, socleC22×C127#466
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC2×D4×C127D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2254C2semidirect
5. C127×#1371C2semidirect
6. C127×#1365C2semidirect
7. C127×#1364C2semidirect
65. C16×C127SD16semidirect
98. #12456C127direct (N × H)
103. #2254C2×C127semidirect
107. #1371C2×C127semidirect
108. #1365C2×C127semidirect
109. #1364C2×C127semidirect
110. C127#12456direct (N × H)
168. C16SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#22542
C127×#21692
C127×#16752
C127×#16732
C127×#13712
C127×#13652
C127×#13642
#12456127