Finite Group Databasebeta

← C127×#12304 #44575977

Subgroups — C127×#12304 #44575977

Order 32512.

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

normal subgroups only

This group has 286 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×#3259C211yesmaximal, characteristic
2C127×#2238C212yesmaximal, complemented
3C127×#2103C211yesmaximal, characteristic, complemented
4C127×#1341C212yesmaximal, complemented
5C127×#1328C211yesmaximal, characteristic, complemented
6#540×C127—21no—
7#478×C127—22no—
8#469×C127C2212yes—
9#466×C127—21no—
10#459×C127—22no—
11#430×C127C2211yescharacteristic
12#424×C127C2211yescharacteristic
13#365×C127—22no—
14#361×C127—22no—
15#349×C127C2212yes—
16#344×C127—21no—
17#341×C127—21no—
18#338×C127C2211yescharacteristic
19C2×D8×C127—21no—
20C2×D8×C127D412yes—
21C2×M16×C127—21no—
22C2×M16×C127—21no—
23C2×M16×C127C2311yescharacteristic
24C22×C8×C127D411yescharacteristic
25#127×C127—21no—
26#125×C127—21no—
27#123×C127—42no—
28#121×C127—22no—
29#121×C127—42no—
30C4×D4×C127—22no—
31#117×C127D411yescharacteristic
32M32×C127—22no—
33#108×C127—22no—
34#107×C127—21no—
35#106×C127D412yes—
36C4⋊C8×C127—22no—
37#102×C127—22no—
38#102×C127—22no—
39C2×D4×C127—22no—
40C2×D4×C127—42no—
41C2×D4×C127—42no—
42C22×C4×C127—42no—
43C22×C4×C127C2×D411yescharacteristic
44D8×C127—42no—
45D8×C127—81no—
46M16×C127—41no—
47M16×C127—41no—
48M16×C127—41no—
49M16×C127—41no—
50M16×C127—41no—
51C2×C8×C127—21no—
52C2×C8×C127—41no—
53C2×C8×C127—41no—
54C2×C8×C127—41no—
55C2×C8×C127C2×D412yes—
56C4⋊C4×C127—21no—
57C4⋊C4×C127—22no—
58C4⋊C4×C127—42no—
59C22⋊C4×C127—42no—
60C22⋊C4×C127—42no—
61C22⋊C4×C127SD1612yes—
62C42×C127—21no—
63C16×C127—42no—
64C23×C127—42no—
65C23×C127C2×SD1611yescharacteristic
66D4×C127—42no—
67D4×C127—42no—
68D4×C127—162no—
69D4×C127—162no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—22no—
73C2×C4×C127—42no—
74C2×C4×C127—162no—
75C2×C4×C127#12011yescharacteristic
76C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
77C8×C127—22no—
78C8×C127—41no—
79C8×C127—81no—
80C8×C127—81no—
81C22×C127—21no—
82C22×C127—41no—
83C22×C127—82no—
84C22×C127—82no—
85C22×C127—82no—
86C22×C127#45011yescharacteristic
87C4×C127—21no—
88C4×C127—42no—
89C4×C127—82no—
90C4×C127#45411yescharacteristic
91C4×C127#45811yescharacteristic
92#12304C12711yesmaximal, characteristic, direct product factor, Sylow
93C2×C127—21no—
94C2×C127—41no—
95C2×C127—162no—
96C2×C127#161611yescharacteristic
97#3259C2×C12711yescharacteristic
98#2238C2×C12712yescomplemented
99#2103C2×C12711yescharacteristic, complemented
100#1341C2×C12712yescomplemented
101#1328C2×C12711yescharacteristic, complemented
102C127#1230411yescharacteristic, direct product factor, Sylow
103#540—21no—
104#478—22no—
105#469C22×C12712yes—
106#466—21no—
107#459—22no—
108#430C22×C12711yescharacteristic
109#424C22×C12711yescharacteristic
110#365—22no—
111#361—22no—
112#349C22×C12712yes—
113#344—21no—
114#341—21no—
115#338C22×C12711yescharacteristic
116C2×D8—21no—
117C2×D8D4×C12712yes—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16C23×C12711yescharacteristic
121C22×C8D4×C12711yescharacteristic
122#127—21no—
123#125—21no—
124#123—42no—
125#121—22no—
126#121—42no—
127C4×D4—22no—
128#117D4×C12711yescharacteristic
129M32—22no—
130#108—22no—
131#107—21no—
132#106D4×C12712yes—
133C4⋊C8—22no—
134#102—22no—
135#102—22no—
136C2×D4—22no—
137C2×D4—42no—
138C2×D4—42no—
139C22×C4—42no—
140C22×C4C2×D4×C12711yescharacteristic
141D8—42no—
142D8—81no—
143M16—41no—
144M16—41no—
145M16—41no—
146M16—41no—
147M16—41no—
148C2×C8—21no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—41no—
152C2×C8C2×D4×C12712yes—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4SD16×C12712yes—
159C42—21no—
160C16—42no—
161C23—42no—
162C23C2×SD16×C12711yescharacteristic
163D4—42no—
164D4—42no—
165D4—162no—
166D4—162no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—42no—
171C2×C4—162no—
172C2×C4#120×C12711yescharacteristic
173C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
174C8—22no—
175C8—41no—
176C8—81no—
177C8—81no—
178C22—21no—
179C22—41no—
180C22—82no—
181C22—82no—
182C22—82no—
183C22#450×C12711yescharacteristic
184C4—21no—
185C4—42no—
186C4—82no—
187C4#454×C12711yescharacteristic
188C4#458×C12711yescharacteristic
189C2—21no—
190C2—41no—
191C2—162no—
192C2C127×#161611yescharacteristic
subgroupG/H
center, socleC2×C127#1616
commutator, Frattini, cosocleC2×M16C23×C127
prime radicalC127×#3259C2
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×#2103C2semidirect
4. C127×#1341C2semidirect
5. C127×#1328C2semidirect
92. #12304C127direct (N × H)
98. #2238C2×C127semidirect
99. #2103C2×C127semidirect
100. #1341C2×C127semidirect
101. #1328C2×C127semidirect
102. C127#12304direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32592
C127×#22382
C127×#21032
C127×#13412
C127×#13282
#12304127