Finite Group Databasebeta

← C127×#12078 #44575751

Subgroups — C127×#12078 #44575751

Order 32512.

234 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2177C211yesmaximal, characteristic
2C127×#1594C211yesmaximal, characteristic, complemented
3C127×#1580C211yesmaximal, characteristic, complemented
4C127×#1368C211yesmaximal, characteristic, complemented
5C127×#1333C211yesmaximal, characteristic, complemented
6C127×#1333C211yesmaximal, characteristic, complemented
7C127×#1313C211yesmaximal, characteristic, complemented
8C2×D16×C127C411yescharacteristic
9C4×D8×C127C2211yescharacteristic
10C4×D8×C127C2211yescharacteristic
11#436×C127—21no—
12C8×D4×C127—21no—
13#367×C127C411yescharacteristic
14#357×C127C411yescharacteristic
15#357×C127C411yescharacteristic
16#346×C127C2211yescharacteristic
17#346×C127C2211yescharacteristic
18#334×C127C2211yescharacteristic
19#334×C127C2211yescharacteristic
20D4⋊C8×C127—21no—
21D4⋊C8×C127—21no—
22#322×C127C2211yescharacteristic
23C2×D8×C127C2×C411yescharacteristic
24C2×D8×C127C2×C411yescharacteristic
25C2×M16×C127—41no—
26C22×C8×C127—41no—
27C4×D4×C127—21no—
28C4×D4×C127—21no—
29D16×C127—22no—
30C2×C16×C127—21no—
31C2×C16×C127C2×C411yescharacteristic
32C2×C16×C127C2×C411yescharacteristic
33#107×C127C2×C411yescharacteristic
34#107×C127C2×C411yescharacteristic
35C4⋊C8×C127—21no—
36C4⋊C8×C127—21no—
37#102×C127—21no—
38#102×C127—21no—
39C22⋊C8×C127—41no—
40C22⋊C8×C127—41no—
41C4×C8×C127D411yescharacteristic
42C4×C8×C127D411yescharacteristic
43C4×C8×C127C2311yescharacteristic
44C2×D4×C127—21no—
45C2×D4×C127—21no—
46C22×C4×C127—41no—
47C22×C4×C127—41no—
48D8×C127—21no—
49D8×C127—21no—
50D8×C127—21no—
51D8×C127M1612yescomplemented
52M16×C127—82no—
53C2×C8×C127—21no—
54C2×C8×C127—21no—
55C2×C8×C127—41no—
56C2×C8×C127—41no—
57C2×C8×C127—82no—
58C2×C8×C127C22×C411yescharacteristic
59C2×C8×C127#4111yescharacteristic
60C4⋊C4×C127—21no—
61C4⋊C4×C127—21no—
62C22⋊C4×C127—41no—
63C22⋊C4×C127—41no—
64C42×C127C2×D411yescharacteristic
65C16×C127—22no—
66C16×C127—21no—
67C16×C127—21no—
68C23×C127—41no—
69C23×C127—41no—
70D4×C127—22no—
71D4×C127—41no—
72D4×C127—41no—
73D4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—81no—
77C2×C4×C127—81no—
78C2×C4×C127—81no—
79C2×C4×C127—81no—
80C2×C4×C127C4×D411yescharacteristic
81C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
83C8×C127—21no—
84C8×C127—41no—
85C8×C127—41no—
86C8×C127—81no—
87C8×C127—81no—
88C8×C127C2×M1611yescharacteristic
89C8×C127#13111yescharacteristic
90C22×C127—42no—
91C22×C127—81no—
92C22×C127—81no—
93C22×C127—81no—
94C22×C127—81no—
95C22×C127—81no—
96C22×C127#44211yescharacteristic
97C4×C127—21no—
98C4×C127—21no—
99C4×C127—81no—
100C4×C127—81no—
101C4×C127#43511yescharacteristic
102C4×C127#44311yescharacteristic
103#12078C12711yesmaximal, characteristic, direct product factor, Sylow
104C2×C127—82no—
105C2×C127—161no—
106C2×C127#158611yescharacteristic
107C2×C127AGL(1,ℤ/16)11yescharacteristic
108C2×C127AGL(1,ℤ/16)11yescharacteristic
109#2177C2×C12711yescharacteristic
110#1594C2×C12711yescharacteristic, complemented
111#1580C2×C12711yescharacteristic, complemented
112#1368C2×C12711yescharacteristic, complemented
113#1333C2×C12711yescharacteristic, complemented
114#1333C2×C12711yescharacteristic, complemented
115#1313C2×C12711yescharacteristic, complemented
116C127#1207811yescharacteristic, direct product factor, Sylow
117C2×D16C4×C12711yescharacteristic
118C4×D8C22×C12711yescharacteristic
119C4×D8C22×C12711yescharacteristic
120#436—21no—
121C8×D4—21no—
122#367C4×C12711yescharacteristic
123#357C4×C12711yescharacteristic
124#357C4×C12711yescharacteristic
125#346C22×C12711yescharacteristic
126#346C22×C12711yescharacteristic
127#334C22×C12711yescharacteristic
128#334C22×C12711yescharacteristic
129D4⋊C8—21no—
130D4⋊C8—21no—
131#322C22×C12711yescharacteristic
132C2×D8C2×C4×C12711yescharacteristic
133C2×D8C2×C4×C12711yescharacteristic
134C2×M16—41no—
135C22×C8—41no—
136C4×D4—21no—
137C4×D4—21no—
138D16—22no—
139C2×C16—21no—
140C2×C16C2×C4×C12711yescharacteristic
141C2×C16C2×C4×C12711yescharacteristic
142#107C2×C4×C12711yescharacteristic
143#107C2×C4×C12711yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—21no—
146#102—21no—
147#102—21no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150C4×C8D4×C12711yescharacteristic
151C4×C8D4×C12711yescharacteristic
152C4×C8C23×C12711yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C22×C4—41no—
156C22×C4—41no—
157D8—21no—
158D8—21no—
159D8—21no—
160D8M16×C12712yescomplemented
161M16—82no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—82no—
167C2×C8C22×C4×C12711yescharacteristic
168C2×C8#41×C12711yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—21no—
171C22⋊C4—41no—
172C22⋊C4—41no—
173C42C2×D4×C12711yescharacteristic
174C16—22no—
175C16—21no—
176C16—21no—
177C23—41no—
178C23—41no—
179D4—22no—
180D4—41no—
181D4—41no—
182D4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4C4×D4×C12711yescharacteristic
190C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
191C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
192C8—21no—
193C8—41no—
194C8—41no—
195C8—81no—
196C8—81no—
197C8C2×M16×C12711yescharacteristic
198C8#131×C12711yescharacteristic
199C22—42no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22#442×C12711yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—81no—
210C4#435×C12711yescharacteristic
211C4#443×C12711yescharacteristic
212C2—82no—
213C2—161no—
214C2C127×#158611yescharacteristic
215C2C127×AGL(1,ℤ/16)11yescharacteristic
216C2C127×AGL(1,ℤ/16)11yescharacteristic
subgroupG/H
center, socleC22×C127#442
commutatorC2×C8C22×C4×C127
Frattini, cosocleC4×C8C23×C127
prime radicalC2×D16×C127C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ #322×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1594C2semidirect
3. C127×#1580C2semidirect
4. C127×#1368C2semidirect
5. C127×#1333C2semidirect
6. C127×#1333C2semidirect
7. C127×#1313C2semidirect
51. D8×C127M16semidirect
103. #12078C127direct (N × H)
110. #1594C2×C127semidirect
111. #1580C2×C127semidirect
112. #1368C2×C127semidirect
113. #1333C2×C127semidirect
114. #1333C2×C127semidirect
115. #1313C2×C127semidirect
116. C127#12078direct (N × H)
160. D8M16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#21772
C127×#15942
C127×#15802
C127×#13682
C127×#13332
C127×#13132
#12078127