Finite Group Databasebeta

← C127×#13533 #44577206

Subgroups — C127×#13533 #44577206

Order 32512.

346 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 114 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×#365×C127C212yesmaximal, complemented
2C2×#360×C127C212yesmaximal
3C127×#1974C211yesmaximal, characteristic
4C127×#1844C211yesmaximal, characteristic, complemented
5C127×#1376C211yesmaximal, characteristic, complemented
6C2×#135×C127C2211yescharacteristic
7C2×M32×C127C2212yes—
8C2×#108×C127C2211yescharacteristic
9C2×#106×C127—21no—
10C2×#106×C127C2212yes—
11C2×#104×C127—21no—
12#408×C127—21no—
13C2×C4×C8×C127C2211yescharacteristic
14#384×C127—21no—
15#365×C127—24no—
16#360×C127—24no—
17C2×#41×C127—21no—
18#135×C127—22no—
19#135×C127—22no—
20C2×Q16×C127Q811yescharacteristic
21C2×SD16×C127—21no—
22C2×D8×C127Q811yescharacteristic
23C2×M16×C127—21no—
24C22×C8×C127D411yescharacteristic
25C22×C8×C127C2311yescharacteristic
26C2×C4⋊C4×C127—22no—
27C2×C4⋊C4×C127—42no—
28C2×C42×C127D411yescharacteristic
29M32×C127—22no—
30M32×C127D414yes—
31C2×C16×C127—22no—
32#108×C127—22no—
33#106×C127—22no—
34#106×C127—22no—
35#106×C127—42no—
36#104×C127—42no—
37C22⋊C8×C127—42no—
38C4×C8×C127—22no—
39#41×C127—41no—
40#41×C127—41no—
41#41×C127—41no—
42#41×C127—41no—
43C2×Q8×C127—21no—
44C2×D4×C127—21no—
45C2×D4×C127—41no—
46C22×C4×C127—21no—
47C22×C4×C127—41no—
48C22×C4×C127—42no—
49C22×C4×C127C2×D411yescharacteristic
50Q16×C127—22no—
51SD16×C127—42no—
52D8×C127—22no—
53M16×C127—41no—
54M16×C127—41no—
55C2×C8×C127—21no—
56C2×C8×C127—21no—
57C2×C8×C127—41no—
58C2×C8×C127SD1612yes—
59C2×C8×C127C2×D412yes—
60C2×C8×C127C2×Q811yescharacteristic
61C2×C8×C127#4112yes—
62C2×C8×C127#4111yescharacteristic
63C4⋊C4×C127—42no—
64C4⋊C4×C127—42no—
65C4⋊C4×C127—84no—
66C42×C127—21no—
67C42×C127SD1612yes—
68C16×C127—24no—
69C23×C127—41no—
70C23×C127AGL(1,ℤ/8)11yescharacteristic
71Q8×C127—41no—
72Q8×C127—41no—
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—81no—
76D4×C127—81no—
77C2×C4×C127—24no—
78C2×C4×C127—22no—
79C2×C4×C127—41no—
80C2×C4×C127—81no—
81C2×C4×C127—81no—
82C2×C4×C127—84no—
83C2×C4×C127—82no—
84C2×C4×C127C22⋊Q811yescharacteristic
85C2×C4×C127#12311yescharacteristic
86C2×C4×C127#12711yescharacteristic
87C2×C4×C127C2×SD1611yescharacteristic
88C2×C4×C127C2×SD1611yescharacteristic
89C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
90C8×C127—22no—
91C8×C127—81no—
92C8×C127C22⋊Q812yes—
93C8×C127#12312yes—
94C22×C127—22no—
95C22×C127—81no—
96C22×C127—82no—
97C22×C127#47611yescharacteristic
98C22×C127#48111yescharacteristic
99C22×C127#49611yescharacteristic
100C4×C127—24no—
101C4×C127—81no—
102C4×C127—162no—
103C4×C127#39911yescharacteristic
104C4×C127#47611yescharacteristic
105C4×C127#48111yescharacteristic
106C4×C127#49211yescharacteristic
107#13533C12711yesmaximal, characteristic, direct product factor, Sylow
108C2×C127—21no—
109C2×C127—21no—
110C2×C127—161no—
111C2×C127#209411yescharacteristic
112C2×C127#223012yes—
113C2×#365C2×C12712yescomplemented
114C2×#360C2×C12712yes—
115#1974C2×C12711yescharacteristic
116#1844C2×C12711yescharacteristic, complemented
117#1376C2×C12711yescharacteristic, complemented
118C127#1353311yescharacteristic, direct product factor, Sylow
119C2×#135C22×C12711yescharacteristic
120C2×M32C22×C12712yes—
121C2×#108C22×C12711yescharacteristic
122C2×#106—21no—
123C2×#106C22×C12712yes—
124C2×#104—21no—
125#408—21no—
126C2×C4×C8C22×C12711yescharacteristic
127#384—21no—
128#365—24no—
129#360—24no—
130C2×#41—21no—
131#135—22no—
132#135—22no—
133C2×Q16Q8×C12711yescharacteristic
134C2×SD16—21no—
135C2×D8Q8×C12711yescharacteristic
136C2×M16—21no—
137C22×C8D4×C12711yescharacteristic
138C22×C8C23×C12711yescharacteristic
139C2×C4⋊C4—22no—
140C2×C4⋊C4—42no—
141C2×C42D4×C12711yescharacteristic
142M32—22no—
143M32D4×C12714yes—
144C2×C16—22no—
145#108—22no—
146#106—22no—
147#106—22no—
148#106—42no—
149#104—42no—
150C22⋊C8—42no—
151C4×C8—22no—
152#41—41no—
153#41—41no—
154#41—41no—
155#41—41no—
156C2×Q8—21no—
157C2×D4—21no—
158C2×D4—41no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4C2×D4×C12711yescharacteristic
163Q16—22no—
164SD16—42no—
165D8—22no—
166M16—41no—
167M16—41no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—41no—
171C2×C8SD16×C12712yes—
172C2×C8C2×D4×C12712yes—
173C2×C8C2×Q8×C12711yescharacteristic
174C2×C8#41×C12712yes—
175C2×C8#41×C12711yescharacteristic
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—84no—
179C42—21no—
180C42SD16×C12712yes—
181C16—24no—
182C23—41no—
183C23AGL(1,ℤ/8)×C12711yescharacteristic
184Q8—41no—
185Q8—41no—
186D4—41no—
187D4—41no—
188D4—81no—
189D4—81no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—84no—
196C2×C4—82no—
197C2×C4C22⋊Q8×C12711yescharacteristic
198C2×C4#123×C12711yescharacteristic
199C2×C4#127×C12711yescharacteristic
200C2×C4C2×SD16×C12711yescharacteristic
201C2×C4C2×SD16×C12711yescharacteristic
202C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
203C8—22no—
204C8—81no—
205C8C22⋊Q8×C12712yes—
206C8#123×C12712yes—
207C22—22no—
208C22—81no—
209C22—82no—
210C22#476×C12711yescharacteristic
211C22#481×C12711yescharacteristic
212C22#496×C12711yescharacteristic
213C4—24no—
214C4—81no—
215C4—162no—
216C4#399×C12711yescharacteristic
217C4#476×C12711yescharacteristic
218C4#481×C12711yescharacteristic
219C4#492×C12711yescharacteristic
220C2—21no—
221C2—21no—
222C2—161no—
223C2C127×#209411yescharacteristic
224C2C127×#223012yes—
subgroupG/H
center, socleC22×C127#476
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC2×#135×C127C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#365×C127C2semidirect
4. C127×#1844C2semidirect
5. C127×#1376C2semidirect
107. #13533C127direct (N × H)
113. C2×#365C2×C127semidirect
116. #1844C2×C127semidirect
117. #1376C2×C127semidirect
118. C127#13533direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#365×C1272
C2×#360×C1272
C127×#19742
C127×#18442
C127×#13762
#13533127