Finite Group Databasebeta

← C127×#12505 #44576178

Subgroups — C127×#12505 #44576178

Order 32512.

284 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2248C211yesmaximal, characteristic
3C127×#2241C211yesmaximal, characteristic, complemented
4C127×#2240C211yesmaximal, characteristic
5C127×#1707C211yesmaximal, characteristic, complemented
6C127×#1706C211yesmaximal, characteristic
7C8×C16×C127C211yesmaximal, characteristic, complemented
8C2×SD32×C127—22no—
9#500×C127C2211yescharacteristic
10#500×C127C2211yescharacteristic
11#498×C127—21no—
12#494×C127C2211yescharacteristic
13#493×C127C2211yescharacteristic
14#492×C127—21no—
15#487×C127—21no—
16#486×C127—21no—
17#367×C127—22no—
18#358×C127—22no—
19#357×C127—22no—
20C4×C16×C127C2211yescharacteristic
21C4×C16×C127C2211yescharacteristic
22C82×C127C2211yescharacteristic
23C2×Q16×C127—21no—
24C2×Q16×C127—21no—
25C2×SD16×C127—42no—
26C2×D8×C127—21no—
27C2×D8×C127—21no—
28C4⋊Q8×C127—21no—
29C4⋊Q8×C127—21no—
30C4⋊Q8×C127—21no—
31#127×C127—21no—
32SD32×C127—44no—
33C2×C16×C127D412yes—
34C2×C16×C127D412yes—
35#107×C127—21no—
36#107×C127—21no—
37#107×C127—21no—
38#107×C127—21no—
39#106×C127—42no—
40#103×C127—42no—
41#102×C127—42no—
42C4×C8×C127D411yescharacteristic
43C4×C8×C127D411yescharacteristic
44C4×C8×C127C2311yescharacteristic
45C2×Q8×C127—41no—
46C2×Q8×C127—41no—
47C2×Q8×C127—41no—
48C2×D4×C127—41no—
49C2×D4×C127—41no—
50C2×D4×C127—41no—
51Q16×C127—42no—
52Q16×C127—42no—
53SD16×C127—84no—
54D8×C127—42no—
55D8×C127—42no—
56C2×C8×C127D812yes—
57C2×C8×C127D812yes—
58C2×C8×C127C2×D411yescharacteristic
59C2×C8×C127C2×D411yescharacteristic
60C4⋊C4×C127—41no—
61C4⋊C4×C127—41no—
62C4⋊C4×C127—41no—
63C4⋊C4×C127—41no—
64C4⋊C4×C127—41no—
65C4⋊C4×C127—41no—
66C42×C127C2×D411yescharacteristic
67C16×C127—22no—
68C16×C127SD1614yescomplemented
69C23×C127—81no—
70Q8×C127—81no—
71Q8×C127—81no—
72Q8×C127—82no—
73Q8×C127—82no—
74D4×C127—81no—
75D4×C127—81no—
76D4×C127—82no—
77D4×C127—82no—
78C2×C4×C127—81no—
79C2×C4×C127—81no—
80C2×C4×C127—81no—
81C2×C4×C127#12711yescharacteristic
82C2×C4×C127C2×D811yescharacteristic
83C2×C4×C127C2×D811yescharacteristic
84C8×C127—22no—
85C8×C127SD3214yescomplemented
86C8×C127C2×SD1612yes—
87C8×C127#13512yes—
88C22×C127—161no—
89C22×C127—161no—
90C22×C127—161no—
91C22×C127#49311yescharacteristic
92C4×C127—161no—
93C4×C127—161no—
94C4×C127—161no—
95C4×C127#49211yescharacteristic
96C4×C127#49511yescharacteristic
97C4×C127C2×SD3212yes—
98C4×C127#50812yes—
99#12505C12711yesmaximal, characteristic, direct product factor, Sylow
100C2×C127—321no—
101C2×C127#170611yescharacteristic
102C2×C127#224811yescharacteristic
103C2×C127#224911yescharacteristic
104#2254C2×C12711yescharacteristic, complemented
105#2248C2×C12711yescharacteristic
106#2241C2×C12711yescharacteristic, complemented
107#2240C2×C12711yescharacteristic
108#1707C2×C12711yescharacteristic, complemented
109#1706C2×C12711yescharacteristic
110C8×C16C2×C12711yescharacteristic, complemented
111C127#1250511yescharacteristic, direct product factor, Sylow
112C2×SD32—22no—
113#500C22×C12711yescharacteristic
114#500C22×C12711yescharacteristic
115#498—21no—
116#494C22×C12711yescharacteristic
117#493C22×C12711yescharacteristic
118#492—21no—
119#487—21no—
120#486—21no—
121#367—22no—
122#358—22no—
123#357—22no—
124C4×C16C22×C12711yescharacteristic
125C4×C16C22×C12711yescharacteristic
126C82C22×C12711yescharacteristic
127C2×Q16—21no—
128C2×Q16—21no—
129C2×SD16—42no—
130C2×D8—21no—
131C2×D8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135#127—21no—
136SD32—44no—
137C2×C16D4×C12712yes—
138C2×C16D4×C12712yes—
139#107—21no—
140#107—21no—
141#107—21no—
142#107—21no—
143#106—42no—
144#103—42no—
145#102—42no—
146C4×C8D4×C12711yescharacteristic
147C4×C8D4×C12711yescharacteristic
148C4×C8C23×C12711yescharacteristic
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155Q16—42no—
156Q16—42no—
157SD16—84no—
158D8—42no—
159D8—42no—
160C2×C8D8×C12712yes—
161C2×C8D8×C12712yes—
162C2×C8C2×D4×C12711yescharacteristic
163C2×C8C2×D4×C12711yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C42C2×D4×C12711yescharacteristic
171C16—22no—
172C16SD16×C12714yescomplemented
173C23—81no—
174Q8—82no—
175Q8—82no—
176Q8—81no—
177Q8—81no—
178D4—82no—
179D4—82no—
180D4—81no—
181D4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4#127×C12711yescharacteristic
186C2×C4C2×D8×C12711yescharacteristic
187C2×C4C2×D8×C12711yescharacteristic
188C8—22no—
189C8SD32×C12714yescomplemented
190C8C2×SD16×C12712yes—
191C8#135×C12712yes—
192C22—161no—
193C22—161no—
194C22—161no—
195C22#493×C12711yescharacteristic
196C4—161no—
197C4—161no—
198C4—161no—
199C4#492×C12711yescharacteristic
200C4#495×C12711yescharacteristic
201C4C2×SD32×C12712yes—
202C4#508×C12712yes—
203C2—321no—
204C2C127×#170611yescharacteristic
205C2C127×#224811yescharacteristic
206C2C127×#224911yescharacteristic
subgroupG/H
center, socleC22×C127#493
commutator, Frattini, cosocleC4×C8C23×C127
prime radical#493×C127C22
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
1. C127×#2254C2semidirect
3. C127×#2241C2semidirect
5. C127×#1707C2semidirect
7. C8×C16×C127C2semidirect
68. C16×C127SD16semidirect
85. C8×C127SD32semidirect
99. #12505C127direct (N × H)
104. #2254C2×C127semidirect
106. #2241C2×C127semidirect
108. #1707C2×C127semidirect
110. C8×C16C2×C127semidirect
111. C127#12505direct (N × H)
172. C16SD16×C127semidirect
189. C8SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#22542
C127×#22482
C127×#22412
C127×#22402
C127×#17072
C127×#17062
C8×C16×C1272
#12505127