Finite Group Databasebeta

← C127×#13199 #44576872

Subgroups — C127×#13199 #44576872

Order 32512.

406 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 138 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×#2955C211yesmaximal, characteristic
2C127×#2151C211yesmaximal, characteristic, complemented
3C2×#350×C127C212yesmaximal, complemented
4C127×#2115C212yesmaximal, complemented
5C127×#2110C211yesmaximal, characteristic, complemented
6#575×C127C411yescharacteristic
7C2×#131×C127C2212yes—
8C2×M32×C127C2211yescharacteristic
9C2×M32×C127C2212yes—
10C22×C16×C127C2211yescharacteristic
11#444×C127—22no—
12#443×C127—22no—
13#430×C127C411yescharacteristic
14#421×C127C412yes—
15#405×C127C2211yescharacteristic
16C4⋊C16×C127—22no—
17#350×C127—24no—
18#349×C127—24no—
19#347×C127—22no—
20C2×#41×C127C2×C412yes—
21#137×C127—22no—
22AGL(1,ℤ/8)×C127—22no—
23#135×C127—22no—
24#131×C127—22no—
25#131×C127—22no—
26#131×C127—22no—
27#131×C127—22no—
28C2×M16×C127—22no—
29C2×M16×C127C2×C412yes—
30C2×M16×C127C2×C411yescharacteristic
31C22×C8×C127—22no—
32C22×C8×C127C2311yescharacteristic
33#117×C127C2×C411yescharacteristic
34M32×C127—22no—
35M32×C127—22no—
36M32×C127D414yes—
37C2×C16×C127—22no—
38C2×C16×C127—22no—
39C2×C16×C127—21no—
40C2×C16×C127—21no—
41#108×C127—22no—
42#104×C127—24no—
43#97×C127D412yes—
44C4×C8×C127D412yes—
45#41×C127—22no—
46#41×C127—22no—
47#41×C127—22no—
48#41×C127—22no—
49C2×Q8×C127M1612yes—
50C2×D4×C127—22no—
51C2×D4×C127M1612yes—
52C22×C4×C127—22no—
53C22×C4×C127C22×C411yescharacteristic
54Q16×C127—22no—
55SD16×C127—24no—
56D8×C127—22no—
57M16×C127—22no—
58M16×C127—22no—
59M16×C127—22no—
60M16×C127—42no—
61M16×C127—42no—
62C2×C8×C127—22no—
63C2×C8×C127—42no—
64C2×C8×C127—42no—
65C2×C8×C127C22⋊C412yes—
66C2×C8×C127C2×D411yescharacteristic
67C2×C8×C127C2×D412yes—
68C2×C8×C127C2×D411yescharacteristic
69C2×C8×C127#4112yes—
70C4⋊C4×C127—21no—
71C22⋊C4×C127—21no—
72C42×C127C22⋊C412yes—
73C16×C127—24no—
74C16×C127—41no—
75C16×C127—41no—
76C23×C127—22no—
77C23×C127#13111yescharacteristic
78Q8×C127—22no—
79Q8×C127—22no—
80D4×C127—22no—
81D4×C127—22no—
82D4×C127—42no—
83D4×C127—42no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—42no—
87C2×C4×C127—42no—
88C2×C4×C127C2×C22⋊C411yescharacteristic
89C2×C4×C127C4×D412yes—
90C2×C4×C127C2×M1612yes—
91C2×C4×C127#13111yescharacteristic
92C8×C127—22no—
93C8×C127—42no—
94C8×C127#12112yes—
95C8×C127#12411yescharacteristic
96C8×C127#12711yescharacteristic
97C22×C127—41no—
98C22×C127—42no—
99C22×C127—82no—
100C22×C127#40811yescharacteristic
101C22×C127#43512yes—
102C4×C127—42no—
103C4×C127—42no—
104C4×C127#39011yescharacteristic
105C4×C127#40711yescharacteristic
106C4×C127#43512yes—
107#13199C12711yesmaximal, characteristic, direct product factor, Sylow
108C2×C127—21no—
109C2×C127—22no—
110C2×C127—82no—
111C2×C127#195511yescharacteristic
112#2955C2×C12711yescharacteristic
113#2151C2×C12711yescharacteristic, complemented
114C2×#350C2×C12712yescomplemented
115#2115C2×C12712yescomplemented
116#2110C2×C12711yescharacteristic, complemented
117C127#1319911yescharacteristic, direct product factor, Sylow
118#575C4×C12711yescharacteristic
119C2×#131C22×C12712yes—
120C2×M32C22×C12711yescharacteristic
121C2×M32C22×C12712yes—
122C22×C16C22×C12711yescharacteristic
123#444—22no—
124#443—22no—
125#430C4×C12711yescharacteristic
126#421C4×C12712yes—
127#405C22×C12711yescharacteristic
128C4⋊C16—22no—
129#350—24no—
130#349—24no—
131#347—22no—
132C2×#41C2×C4×C12712yes—
133#137—22no—
134AGL(1,ℤ/8)—22no—
135#135—22no—
136#131—22no—
137#131—22no—
138#131—22no—
139#131—22no—
140C2×M16—22no—
141C2×M16C2×C4×C12712yes—
142C2×M16C2×C4×C12711yescharacteristic
143C22×C8—22no—
144C22×C8C23×C12711yescharacteristic
145#117C2×C4×C12711yescharacteristic
146M32—22no—
147M32—22no—
148M32D4×C12714yes—
149C2×C16—22no—
150C2×C16—22no—
151C2×C16—21no—
152C2×C16—21no—
153#108—22no—
154#104—24no—
155#97D4×C12712yes—
156C4×C8D4×C12712yes—
157#41—22no—
158#41—22no—
159#41—22no—
160#41—22no—
161C2×Q8M16×C12712yes—
162C2×D4—22no—
163C2×D4M16×C12712yes—
164C22×C4—22no—
165C22×C4C22×C4×C12711yescharacteristic
166Q16—22no—
167SD16—24no—
168D8—22no—
169M16—22no—
170M16—22no—
171M16—22no—
172M16—42no—
173M16—42no—
174C2×C8—22no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8C22⋊C4×C12712yes—
178C2×C8C2×D4×C12711yescharacteristic
179C2×C8C2×D4×C12712yes—
180C2×C8C2×D4×C12711yescharacteristic
181C2×C8#41×C12712yes—
182C4⋊C4—21no—
183C22⋊C4—21no—
184C42C22⋊C4×C12712yes—
185C16—24no—
186C16—41no—
187C16—41no—
188C23—22no—
189C23#131×C12711yescharacteristic
190Q8—22no—
191Q8—22no—
192D4—22no—
193D4—22no—
194D4—42no—
195D4—42no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4C2×C22⋊C4×C12711yescharacteristic
201C2×C4C4×D4×C12712yes—
202C2×C4C2×M16×C12712yes—
203C2×C4#131×C12711yescharacteristic
204C8—22no—
205C8—42no—
206C8#121×C12712yes—
207C8#124×C12711yescharacteristic
208C8#127×C12711yescharacteristic
209C22—41no—
210C22—42no—
211C22—82no—
212C22#408×C12711yescharacteristic
213C22#435×C12712yes—
214C4—42no—
215C4—42no—
216C4#390×C12711yescharacteristic
217C4#407×C12711yescharacteristic
218C4#435×C12712yes—
219C2—21no—
220C2—22no—
221C2—82no—
222C2C127×#195511yescharacteristic
subgroupG/H
centerC8×C127#124
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×C8C23×C127
socleC2×C127#1955
prime radical#575×C127C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C8×C127 ◁ C22×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2151C2semidirect
3. C2×#350×C127C2semidirect
4. C127×#2115C2semidirect
5. C127×#2110C2semidirect
107. #13199C127direct (N × H)
113. #2151C2×C127semidirect
114. C2×#350C2×C127semidirect
115. #2115C2×C127semidirect
116. #2110C2×C127semidirect
117. C127#13199direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#29552
C127×#21512
C2×#350×C1272
C127×#21152
C127×#21102
#13199127