Finite Group Databasebeta

← C127×#12307 #44575980

Subgroups — C127×#12307 #44575980

Order 32512.

318 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3253C211yesmaximal, characteristic
2C127×#2238C212yesmaximal, complemented
3C127×#2102C211yesmaximal, characteristic, complemented
4C127×#1339C212yesmaximal, complemented
5C127×#1327C211yesmaximal, characteristic, complemented
6#532×C127—21no—
7#476×C127—22no—
8#469×C127—21no—
9#469×C127C2212yes—
10#459×C127—22no—
11#430×C127C2211yescharacteristic
12#423×C127C2211yescharacteristic
13#365×C127—22no—
14#361×C127—22no—
15#349×C127C2212yes—
16#344×C127—21no—
17#339×C127—21no—
18H(3,ℤ/4)×C127C2211yescharacteristic
19C2×#41×C127—41no—
20C2×D8×C127—21no—
21C2×D8×C127D412yes—
22C2×M16×C127—21no—
23C2×M16×C127D411yescharacteristic
24C2×M16×C127C2311yescharacteristic
25C4⋊Q8×C127—21no—
26#127×C127—21no—
27#124×C127—41no—
28#121×C127—22no—
29#121×C127—41no—
30C4×D4×C127—22no—
31C4×D4×C127—41no—
32#117×C127—21no—
33C2×C42×C127D411yescharacteristic
34M32×C127—22no—
35#108×C127—22no—
36#106×C127—21no—
37#106×C127D412yes—
38C4⋊C8×C127—22no—
39#102×C127—22no—
40#102×C127—22no—
41#41×C127—84no—
42C2×Q8×C127—41no—
43C2×D4×C127—22no—
44C2×D4×C127—41no—
45C2×D4×C127—41no—
46C2×D4×C127—42no—
47C22×C4×C127—21no—
48C22×C4×C127—42no—
49C22×C4×C127C2×D411yescharacteristic
50D8×C127—42no—
51D8×C127—81no—
52M16×C127—41no—
53M16×C127—41no—
54M16×C127—41no—
55M16×C127—41no—
56C2×C8×C127—21no—
57C2×C8×C127—41no—
58C2×C8×C127C2×D412yes—
59C4⋊C4×C127—22no—
60C4⋊C4×C127—41no—
61C4⋊C4×C127—41no—
62C22⋊C4×C127—42no—
63C22⋊C4×C127—42no—
64C22⋊C4×C127—41no—
65C42×C127—21no—
66C42×C127—41no—
67C42×C127SD1611yescharacteristic
68C42×C127SD1611yescharacteristic
69C16×C127—42no—
70C23×C127—42no—
71C23×C127AGL(1,ℤ/8)11yescharacteristic
72Q8×C127—82no—
73D4×C127—42no—
74D4×C127—42no—
75D4×C127—82no—
76D4×C127—82no—
77D4×C127—84no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—42no—
83C2×C4×C127—42no—
84C2×C4×C127—84no—
85C2×C4×C127—81no—
86C2×C4×C127#12011yescharacteristic
87C2×C4×C127C2×SD1611yescharacteristic
88C8×C127—22no—
89C8×C127—41no—
90C8×C127—81no—
91C22×C127—21no—
92C22×C127—41no—
93C22×C127—82no—
94C22×C127—82no—
95C22×C127—82no—
96C22×C127#45011yescharacteristic
97C4×C127—22no—
98C4×C127—21no—
99C4×C127—41no—
100C4×C127—82no—
101C4×C127—161no—
102C4×C127#45411yescharacteristic
103C4×C127#45811yescharacteristic
104#12307C12711yesmaximal, characteristic, direct product factor, Sylow
105C2×C127—21no—
106C2×C127—41no—
107C2×C127—162no—
108C2×C127#161611yescharacteristic
109#3253C2×C12711yescharacteristic
110#2238C2×C12712yescomplemented
111#2102C2×C12711yescharacteristic, complemented
112#1339C2×C12712yescomplemented
113#1327C2×C12711yescharacteristic, complemented
114C127#1230711yescharacteristic, direct product factor, Sylow
115#532—21no—
116#476—22no—
117#469—21no—
118#469C22×C12712yes—
119#459—22no—
120#430C22×C12711yescharacteristic
121#423C22×C12711yescharacteristic
122#365—22no—
123#361—22no—
124#349C22×C12712yes—
125#344—21no—
126#339—21no—
127H(3,ℤ/4)C22×C12711yescharacteristic
128C2×#41—41no—
129C2×D8—21no—
130C2×D8D4×C12712yes—
131C2×M16—21no—
132C2×M16D4×C12711yescharacteristic
133C2×M16C23×C12711yescharacteristic
134C4⋊Q8—21no—
135#127—21no—
136#124—41no—
137#121—22no—
138#121—41no—
139C4×D4—22no—
140C4×D4—41no—
141#117—21no—
142C2×C42D4×C12711yescharacteristic
143M32—22no—
144#108—22no—
145#106—21no—
146#106D4×C12712yes—
147C4⋊C8—22no—
148#102—22no—
149#102—22no—
150#41—84no—
151C2×Q8—41no—
152C2×D4—22no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—21no—
157C22×C4—42no—
158C22×C4C2×D4×C12711yescharacteristic
159D8—42no—
160D8—81no—
161M16—41no—
162M16—41no—
163M16—41no—
164M16—41no—
165C2×C8—21no—
166C2×C8—41no—
167C2×C8C2×D4×C12712yes—
168C4⋊C4—22no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C22⋊C4—42no—
172C22⋊C4—42no—
173C22⋊C4—41no—
174C42—21no—
175C42—41no—
176C42SD16×C12711yescharacteristic
177C42SD16×C12711yescharacteristic
178C16—42no—
179C23—42no—
180C23AGL(1,ℤ/8)×C12711yescharacteristic
181Q8—82no—
182D4—42no—
183D4—42no—
184D4—82no—
185D4—82no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4—81no—
195C2×C4#120×C12711yescharacteristic
196C2×C4C2×SD16×C12711yescharacteristic
197C8—22no—
198C8—41no—
199C8—81no—
200C22—21no—
201C22—41no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#450×C12711yescharacteristic
206C4—22no—
207C4—21no—
208C4—41no—
209C4—82no—
210C4—161no—
211C4#454×C12711yescharacteristic
212C4#458×C12711yescharacteristic
213C2—21no—
214C2—41no—
215C2—162no—
216C2C127×#161611yescharacteristic
subgroupG/H
center, socleC2×C127#1616
commutator, Frattini, cosocleC2×M16C23×C127
prime radicalC127×#3253C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2238C2semidirect
3. C127×#2102C2semidirect
4. C127×#1339C2semidirect
5. C127×#1327C2semidirect
104. #12307C127direct (N × H)
110. #2238C2×C127semidirect
111. #2102C2×C127semidirect
112. #1339C2×C127semidirect
113. #1327C2×C127semidirect
114. C127#12307direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32532
C127×#22382
C127×#21022
C127×#13392
C127×#13272
#12307127