Finite Group Databasebeta

← C127×#12441 #44576114

Subgroups — C127×#12441 #44576114

Order 32512.

232 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2248C211yesmaximal, characteristic
2Q8×C16×C127C211yesmaximal, characteristic, complemented
3C127×#1670C211yesmaximal, characteristic, complemented
4C127×#1668C211yesmaximal, characteristic
5C127×#1372C211yesmaximal, characteristic, complemented
6C127×#1362C211yesmaximal, characteristic, complemented
7C127×#1361C211yesmaximal, characteristic
8C2×SD32×C127—21no—
9C2×SD32×C127—21no—
10#494×C127C2211yescharacteristic
11#493×C127C2211yescharacteristic
12#475×C127—21no—
13#460×C127—21no—
14C8×Q8×C127C2211yescharacteristic
15C4⋊C16×C127—21no—
16C4⋊C16×C127C2211yescharacteristic
17C4×C16×C127—21no—
18C4×C16×C127C2211yescharacteristic
19#335×C127C2211yescharacteristic
20#335×C127C2211yescharacteristic
21#332×C127—21no—
22#331×C127—21no—
23C2×Q16×C127—21no—
24C2×Q16×C127—21no—
25C2×SD16×C127—41no—
26C2×D8×C127—21no—
27C2×D8×C127—21no—
28C4⋊Q8×C127—21no—
29#127×C127—21no—
30C4×Q8×C127D411yescharacteristic
31SD32×C127—42no—
32SD32×C127—42no—
33C2×C16×C127—21no—
34C2×C16×C127D411yescharacteristic
35C2×C16×C127D411yescharacteristic
36#106×C127—41no—
37C4⋊C8×C127—21no—
38C4⋊C8×C127—21no—
39C4⋊C8×C127—21no—
40C4⋊C8×C127D411yescharacteristic
41#103×C127—41no—
42#102×C127—41no—
43C4×C8×C127—21no—
44C4×C8×C127C2311yescharacteristic
45C2×Q8×C127—41no—
46C2×Q8×C127D811yescharacteristic
47C2×D4×C127—41no—
48C2×D4×C127—41no—
49C2×D4×C127—41no—
50Q16×C127—41no—
51Q16×C127—41no—
52Q16×C127—81no—
53SD16×C127—82no—
54D8×C127—41no—
55D8×C127—41no—
56D8×C127—81no—
57C2×C8×C127—21no—
58C2×C8×C127—41no—
59C2×C8×C127—41no—
60C2×C8×C127C2×D411yescharacteristic
61C2×C8×C127#4111yescharacteristic
62C4⋊C4×C127—21no—
63C4⋊C4×C127—41no—
64C4⋊C4×C127—41no—
65C4⋊C4×C127D811yescharacteristic
66C42×C127—21no—
67C42×C127C2×D411yescharacteristic
68C16×C127—21no—
69C16×C127—41no—
70C16×C127SD1612yescomplemented
71C23×C127—81no—
72Q8×C127—21no—
73Q8×C127—81no—
74Q8×C127—81no—
75Q8×C127SD3212yescomplemented
76D4×C127—81no—
77D4×C127—81no—
78D4×C127—81no—
79D4×C127—81no—
80D4×C127—161no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—81no—
84C2×C4×C127#12111yescharacteristic
85C2×C4×C127C2×D811yescharacteristic
86C2×C4×C127#13511yescharacteristic
87C8×C127—21no—
88C8×C127—41no—
89C8×C127—81no—
90C8×C127—81no—
91C8×C127C2×SD1611yescharacteristic
92C8×C127AGL(1,ℤ/8)11yescharacteristic
93C22×C127—161no—
94C22×C127—161no—
95C22×C127—161no—
96C22×C127#46611yescharacteristic
97C4×C127—22no—
98C4×C127—21no—
99C4×C127—41no—
100C4×C127—161no—
101C4×C127#46011yescharacteristic
102C4×C127#47211yescharacteristic
103C4×C127C2×SD3211yescharacteristic
104C4×C127#50811yescharacteristic
105#12441C12711yesmaximal, characteristic, direct product factor, Sylow
106C2×C127—321no—
107C2×C127#166811yescharacteristic
108C2×C127#221711yescharacteristic
109C2×C127#222311yescharacteristic
110#2248C2×C12711yescharacteristic
111Q8×C16C2×C12711yescharacteristic, complemented
112#1670C2×C12711yescharacteristic, complemented
113#1668C2×C12711yescharacteristic
114#1372C2×C12711yescharacteristic, complemented
115#1362C2×C12711yescharacteristic, complemented
116#1361C2×C12711yescharacteristic
117C127#1244111yescharacteristic, direct product factor, Sylow
118C2×SD32—21no—
119C2×SD32—21no—
120#494C22×C12711yescharacteristic
121#493C22×C12711yescharacteristic
122#475—21no—
123#460—21no—
124C8×Q8C22×C12711yescharacteristic
125C4⋊C16—21no—
126C4⋊C16C22×C12711yescharacteristic
127C4×C16—21no—
128C4×C16C22×C12711yescharacteristic
129#335C22×C12711yescharacteristic
130#335C22×C12711yescharacteristic
131#332—21no—
132#331—21no—
133C2×Q16—21no—
134C2×Q16—21no—
135C2×SD16—41no—
136C2×D8—21no—
137C2×D8—21no—
138C4⋊Q8—21no—
139#127—21no—
140C4×Q8D4×C12711yescharacteristic
141SD32—42no—
142SD32—42no—
143C2×C16—21no—
144C2×C16D4×C12711yescharacteristic
145C2×C16D4×C12711yescharacteristic
146#106—41no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8D4×C12711yescharacteristic
151#103—41no—
152#102—41no—
153C4×C8—21no—
154C4×C8C23×C12711yescharacteristic
155C2×Q8—41no—
156C2×Q8D8×C12711yescharacteristic
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160Q16—41no—
161Q16—41no—
162Q16—81no—
163SD16—82no—
164D8—41no—
165D8—41no—
166D8—81no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8C2×D4×C12711yescharacteristic
171C2×C8#41×C12711yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4D8×C12711yescharacteristic
176C42—21no—
177C42C2×D4×C12711yescharacteristic
178C16—21no—
179C16—41no—
180C16SD16×C12712yescomplemented
181C23—81no—
182Q8—21no—
183Q8—81no—
184Q8—81no—
185Q8SD32×C12712yescomplemented
186D4—81no—
187D4—81no—
188D4—81no—
189D4—81no—
190D4—161no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—81no—
194C2×C4#121×C12711yescharacteristic
195C2×C4C2×D8×C12711yescharacteristic
196C2×C4#135×C12711yescharacteristic
197C8—21no—
198C8—41no—
199C8—81no—
200C8—81no—
201C8C2×SD16×C12711yescharacteristic
202C8AGL(1,ℤ/8)×C12711yescharacteristic
203C22—161no—
204C22—161no—
205C22—161no—
206C22#466×C12711yescharacteristic
207C4—22no—
208C4—21no—
209C4—41no—
210C4—161no—
211C4#460×C12711yescharacteristic
212C4#472×C12711yescharacteristic
213C4C2×SD32×C12711yescharacteristic
214C4#508×C12711yescharacteristic
215C2—321no—
216C2C127×#166811yescharacteristic
217C2C127×#221711yescharacteristic
218C2C127×#222311yescharacteristic
subgroupG/H
center, socleC22×C127#466
commutator, Frattini, cosocleC4×C8C23×C127
prime radical#493×C127C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. Q8×C16×C127C2semidirect
3. C127×#1670C2semidirect
5. C127×#1372C2semidirect
6. C127×#1362C2semidirect
70. C16×C127SD16semidirect
75. Q8×C127SD32semidirect
105. #12441C127direct (N × H)
111. Q8×C16C2×C127semidirect
112. #1670C2×C127semidirect
114. #1372C2×C127semidirect
115. #1362C2×C127semidirect
117. C127#12441direct (N × H)
180. C16SD16×C127semidirect
185. Q8SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#22482
Q8×C16×C1272
C127×#16702
C127×#16682
C127×#13722
C127×#13622
C127×#13612
#12441127