Finite Group Databasebeta

← C127×#13663 #44577336

Subgroups — C127×#13663 #44577336

Order 32512.

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

normal subgroups only

This group has 298 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
1C2×#508×C127C211yesmaximal, characteristic, complemented
2C2×SD64×C127C211yesmaximal, characteristic, complemented
3C2×D32×C127C211yesmaximal, characteristic, complemented
4C127×#2215C211yesmaximal, characteristic, complemented
5C127×#1416C211yesmaximal, characteristic, complemented
6C127×#1415C211yesmaximal, characteristic, complemented
7C127×C22⋊C32C211yesmaximal, characteristic, complemented
8C2×#135×C127—21no—
9#508×C127—22no—
10#508×C127—22no—
11C2×Q32×C127C2211yescharacteristic
12C2×SD32×C127—21no—
13C2×D16×C127C2211yescharacteristic
14C2×D16×C127C2211yescharacteristic
15C22×C16×C127C2211yescharacteristic
16#466×C127—21no—
17SD64×C127—22no—
18D32×C127—22no—
19C2×C32×C127C2211yescharacteristic
20C2×C32×C127C2211yescharacteristic
21#366×C127C2211yescharacteristic
22#357×C127—21no—
23C2×#41×C127—41no—
24#135×C127—24no—
25#135×C127—42no—
26C2×Q16×C127—21no—
27C2×SD16×C127—41no—
28C2×D8×C127—21no—
29C2×D8×C127—21no—
30C22×C8×C127D411yescharacteristic
31#121×C127—41no—
32Q32×C127—21no—
33Q32×C127D412yescomplemented
34SD32×C127—24no—
35D16×C127—21no—
36D16×C127—21no—
37D16×C127—21no—
38D16×C127D412yescomplemented
39C2×C16×C127—21no—
40C2×C16×C127—21no—
41C2×C16×C127D411yescharacteristic
42C2×C16×C127C2311yescharacteristic
43#107×C127—21no—
44#102×C127—41no—
45C32×C127—21no—
46C32×C127—21no—
47#41×C127—44no—
48#41×C127—82no—
49C2×Q8×C127—41no—
50C2×D4×C127—41no—
51C2×D4×C127—41no—
52C2×D4×C127—81no—
53C2×D4×C127—81no—
54C22×C4×C127—81no—
55C22×C4×C127D811yescharacteristic
56Q16×C127—22no—
57Q16×C127—41no—
58SD16×C127—44no—
59D8×C127—22no—
60D8×C127—41no—
61D8×C127—41no—
62D8×C127—41no—
63C2×C8×C127—21no—
64C2×C8×C127—21no—
65C2×C8×C127D811yescharacteristic
66C2×C8×C127C2×D411yescharacteristic
67C4⋊C4×C127—41no—
68C22⋊C4×C127—81no—
69C16×C127—21no—
70C16×C127C2×D411yescharacteristic
71C16×C127C2×D411yescharacteristic
72C23×C127—81no—
73C23×C127—81no—
74C23×C127D1611yescharacteristic
75Q8×C127—42no—
76Q8×C127—81no—
77D4×C127—42no—
78D4×C127—84no—
79D4×C127—81no—
80D4×C127—81no—
81D4×C127—81no—
82D4×C127—162no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—84no—
86C2×C4×C127—81no—
87C2×C4×C127—81no—
88C2×C4×C127D1611yescharacteristic
89C2×C4×C127C2×D811yescharacteristic
90C8×C127—21no—
91C8×C127#12011yescharacteristic
92C8×C127AGL(1,ℤ/8)11yescharacteristic
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—82no—
97C22×C127—161no—
98C22×C127—161no—
99C22×C127—161no—
100C22×C127—161no—
101C22×C127—161no—
102C22×C127C2×D1611yescharacteristic
103C4×C127—21no—
104C4×C127—82no—
105C4×C127—161no—
106C4×C127#44711yescharacteristic
107C4×C127#50911yescharacteristic
108#13663C12711yesmaximal, characteristic, direct product factor, Sylow
109C2×C127—41no—
110C2×C127—162no—
111C2×C127—321no—
112C2×C127#218411yescharacteristic
113C2×C127#226211yescharacteristic
114C2×C127#226311yescharacteristic
115C2×#508C2×C12711yescharacteristic, complemented
116C2×SD64C2×C12711yescharacteristic, complemented
117C2×D32C2×C12711yescharacteristic, complemented
118#2215C2×C12711yescharacteristic, complemented
119#1416C2×C12711yescharacteristic, complemented
120#1415C2×C12711yescharacteristic, complemented
121C22⋊C32C2×C12711yescharacteristic, complemented
122C127#1366311yescharacteristic, direct product factor, Sylow
123C2×#135—21no—
124#508—22no—
125#508—22no—
126C2×Q32C22×C12711yescharacteristic
127C2×SD32—21no—
128C2×D16C22×C12711yescharacteristic
129C2×D16C22×C12711yescharacteristic
130C22×C16C22×C12711yescharacteristic
131#466—21no—
132SD64—22no—
133D32—22no—
134C2×C32C22×C12711yescharacteristic
135C2×C32C22×C12711yescharacteristic
136#366C22×C12711yescharacteristic
137#357—21no—
138C2×#41—41no—
139#135—24no—
140#135—42no—
141C2×Q16—21no—
142C2×SD16—41no—
143C2×D8—21no—
144C2×D8—21no—
145C22×C8D4×C12711yescharacteristic
146#121—41no—
147Q32—21no—
148Q32D4×C12712yescomplemented
149SD32—24no—
150D16—21no—
151D16—21no—
152D16—21no—
153D16D4×C12712yescomplemented
154C2×C16—21no—
155C2×C16—21no—
156C2×C16D4×C12711yescharacteristic
157C2×C16C23×C12711yescharacteristic
158#107—21no—
159#102—41no—
160C32—21no—
161C32—21no—
162#41—44no—
163#41—82no—
164C2×Q8—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—81no—
168C2×D4—81no—
169C22×C4—81no—
170C22×C4D8×C12711yescharacteristic
171Q16—22no—
172Q16—41no—
173SD16—44no—
174D8—22no—
175D8—41no—
176D8—41no—
177D8—41no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8D8×C12711yescharacteristic
181C2×C8C2×D4×C12711yescharacteristic
182C4⋊C4—41no—
183C22⋊C4—81no—
184C16—21no—
185C16C2×D4×C12711yescharacteristic
186C16C2×D4×C12711yescharacteristic
187C23—81no—
188C23—81no—
189C23D16×C12711yescharacteristic
190Q8—42no—
191Q8—81no—
192D4—42no—
193D4—84no—
194D4—81no—
195D4—81no—
196D4—81no—
197D4—162no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—84no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4D16×C12711yescharacteristic
204C2×C4C2×D8×C12711yescharacteristic
205C8—21no—
206C8#120×C12711yescharacteristic
207C8AGL(1,ℤ/8)×C12711yescharacteristic
208C22—21no—
209C22—21no—
210C22—21no—
211C22—82no—
212C22—161no—
213C22—161no—
214C22—161no—
215C22—161no—
216C22—161no—
217C22C2×D16×C12711yescharacteristic
218C4—21no—
219C4—82no—
220C4—161no—
221C4#447×C12711yescharacteristic
222C4#509×C12711yescharacteristic
223C2—41no—
224C2—162no—
225C2—321no—
226C2C127×#218411yescharacteristic
227C2C127×#226211yescharacteristic
228C2C127×#226311yescharacteristic
subgroupG/H
center, socleC22×C127C2×D16
commutator, Frattini, cosocleC2×C16C23×C127
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×C8×C127 ◁ C22×C16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#508×C127C2semidirect
2. C2×SD64×C127C2semidirect
3. C2×D32×C127C2semidirect
4. C127×#2215C2semidirect
5. C127×#1416C2semidirect
6. C127×#1415C2semidirect
7. C127×C22⋊C32C2semidirect
33. Q32×C127D4semidirect
38. D16×C127D4semidirect
108. #13663C127direct (N × H)
115. C2×#508C2×C127semidirect
116. C2×SD64C2×C127semidirect
117. C2×D32C2×C127semidirect
118. #2215C2×C127semidirect
119. #1416C2×C127semidirect
120. #1415C2×C127semidirect
121. C22⋊C32C2×C127semidirect
122. C127#13663direct (N × H)
148. Q32D4×C127semidirect
153. D16D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#508×C1272
C2×SD64×C1272
C2×D32×C1272
C127×#22152
C127×#14162
C127×#14152
C127×C22⋊C322
#13663127