Finite Group Databasebeta

← C2×C127×#1359 #44575594

Subgroups — C2×C127×#1359 #44575594

Order 32512.

442 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 142 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×#492×C127C211yesmaximal, characteristic
2C2×#364×C127C212yesmaximal, complemented
3C127×#1359C214yesmaximal, direct product factor
4C22×SD16×C127—22no—
5C2×C4⋊Q8×C127C411yescharacteristic
6C2×#127×C127C411yescharacteristic
7C2×M32×C127—22no—
8#492×C127—22no—
9#492×C127C412yes—
10#492×C127C2212yes—
11C2×C4×C8×C127C2211yescharacteristic
12#364×C127—22no—
13#364×C127C2214yescomplemented
14C22×Q8×C127—21no—
15C22×D4×C127—21no—
16C22×D4×C127—41no—
17C2×SD16×C127—24no—
18C2×SD16×C127—24no—
19C2×SD16×C127—42no—
20C2×SD16×C127—42no—
21C2×SD16×C127—42no—
22C2×SD16×C127—42no—
23C22×C8×C127D412yes—
24C4⋊Q8×C127—21no—
25C4⋊Q8×C127—41no—
26C4⋊Q8×C127C2×C412yes—
27#127×C127—21no—
28#127×C127—41no—
29#127×C127C2×C412yes—
30C2×C4⋊C4×C127—41no—
31C2×C42×C127C2×C411yescharacteristic
32M32×C127—24no—
33M32×C127—42no—
34C2×C16×C127—42no—
35C4×C8×C127C2×C411yescharacteristic
36C4×C8×C127D412yes—
37C4×C8×C127C2311yescharacteristic
38C24×C127—41no—
39C2×Q8×C127—22no—
40C2×Q8×C127—41no—
41C2×Q8×C127—42no—
42C2×Q8×C127—42no—
43C2×D4×C127—22no—
44C2×D4×C127—42no—
45C2×D4×C127—42no—
46C2×D4×C127—41no—
47C2×D4×C127—42no—
48C2×D4×C127—42no—
49C2×D4×C127—84no—
50C22×C4×C127—21no—
51C22×C4×C127—41no—
52C22×C4×C127C22⋊C411yescharacteristic
53SD16×C127—44no—
54SD16×C127—44no—
55SD16×C127—44no—
56SD16×C127—44no—
57C2×C8×C127C22⋊C412yes—
58C2×C8×C127D812yes—
59C2×C8×C127D812yes—
60C2×C8×C127SD1612yes—
61C2×C8×C127SD1612yes—
62C2×C8×C127C2×D412yes—
63C4⋊C4×C127—42no—
64C4⋊C4×C127—42no—
65C42×C127—21no—
66C42×C127C22⋊C411yescharacteristic
67C42×C127C22×C411yescharacteristic
68C16×C127—44no—
69C23×C127—42no—
70C23×C127—84no—
71C23×C127—81no—
72C23×C127—81no—
73C23×C127#10011yescharacteristic
74Q8×C127—44no—
75Q8×C127—82no—
76D4×C127—44no—
77D4×C127—84no—
78D4×C127—84no—
79D4×C127—82no—
80C2×C4×C127—24no—
81C2×C4×C127—21no—
82C2×C4×C127—21no—
83C2×C4×C127—42no—
84C2×C4×C127—81no—
85C2×C4×C127—81no—
86C2×C4×C127#10011yescharacteristic
87C2×C4×C127#10212yes—
88C2×C4×C127#10212yes—
89C2×C4×C127C2×C22⋊C411yescharacteristic
90C8×C127#10212yes—
91C8×C127#10212yes—
92C8×C127C2×D812yes—
93C8×C127C2×SD1612yes—
94C22×C127—22no—
95C22×C127—84no—
96C22×C127—82no—
97C22×C127—162no—
98C22×C127—162no—
99C22×C127#33111yescharacteristic
100C22×C127#33111yescharacteristic
101C22×C127C2×#10011yescharacteristic
102C4×C127—22no—
103C4×C127—22no—
104C4×C127—82no—
105C4×C127#33112yes—
106C4×C127C2×#10212yes—
107C2×#1359C12711yesmaximal, characteristic, direct product factor, Sylow
108C2×C127—21no—
109C2×C127—21no—
110C2×C127—162no—
111C2×C127#135912yesdirect product factor
112C2×C127C2×#33111yescharacteristic
113C2×#492C2×C12711yescharacteristic
114C2×#364C2×C12712yescomplemented
115#1359C2×C12714yesdirect product factor
116C127C2×#135911yescharacteristic, direct product factor, Sylow
117C22×SD16—22no—
118C2×C4⋊Q8C4×C12711yescharacteristic
119C2×#127C4×C12711yescharacteristic
120C2×M32—22no—
121#492—22no—
122#492C4×C12712yes—
123#492C22×C12712yes—
124C2×C4×C8C22×C12711yescharacteristic
125#364—22no—
126#364C22×C12714yescomplemented
127C22×Q8—21no—
128C22×D4—21no—
129C22×D4—41no—
130C2×SD16—24no—
131C2×SD16—24no—
132C2×SD16—42no—
133C2×SD16—42no—
134C2×SD16—42no—
135C2×SD16—42no—
136C22×C8D4×C12712yes—
137C4⋊Q8—21no—
138C4⋊Q8—41no—
139C4⋊Q8C2×C4×C12712yes—
140#127—21no—
141#127—41no—
142#127C2×C4×C12712yes—
143C2×C4⋊C4—41no—
144C2×C42C2×C4×C12711yescharacteristic
145M32—24no—
146M32—42no—
147C2×C16—42no—
148C4×C8C2×C4×C12711yescharacteristic
149C4×C8D4×C12712yes—
150C4×C8C23×C12711yescharacteristic
151C24—41no—
152C2×Q8—22no—
153C2×Q8—41no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×D4—22no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—84no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4C22⋊C4×C12711yescharacteristic
166SD16—44no—
167SD16—44no—
168SD16—44no—
169SD16—44no—
170C2×C8C22⋊C4×C12712yes—
171C2×C8D8×C12712yes—
172C2×C8D8×C12712yes—
173C2×C8SD16×C12712yes—
174C2×C8SD16×C12712yes—
175C2×C8C2×D4×C12712yes—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C42—21no—
179C42C22⋊C4×C12711yescharacteristic
180C42C22×C4×C12711yescharacteristic
181C16—44no—
182C23—42no—
183C23—84no—
184C23—81no—
185C23—81no—
186C23#100×C12711yescharacteristic
187Q8—44no—
188Q8—82no—
189D4—44no—
190D4—84no—
191D4—84no—
192D4—82no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4#100×C12711yescharacteristic
200C2×C4#102×C12712yes—
201C2×C4#102×C12712yes—
202C2×C4C2×C22⋊C4×C12711yescharacteristic
203C8#102×C12712yes—
204C8#102×C12712yes—
205C8C2×D8×C12712yes—
206C8C2×SD16×C12712yes—
207C22—22no—
208C22—84no—
209C22—82no—
210C22—162no—
211C22—162no—
212C22#331×C12711yescharacteristic
213C22#331×C12711yescharacteristic
214C22C2×#100×C12711yescharacteristic
215C4—22no—
216C4—22no—
217C4—82no—
218C4#331×C12712yes—
219C4C2×#102×C12712yes—
220C2—21no—
221C2—21no—
222C2—162no—
223C2C127×#135912yesdirect product factor
224C2C2×#331×C12711yescharacteristic
subgroupG/H
center, socleC22×C127#331
commutatorC42C22×C4×C127
Frattini, cosocleC4×C8C23×C127
prime radicalC2×#127×C127C4
LCSG ▷ C42 ▷ C2×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
2. C2×#364×C127C2semidirect
3. C127×#1359C2direct (N × H)
13. #364×C127C22semidirect
107. C2×#1359C127direct (N × H)
111. C2×C127#1359direct (N × H)
114. C2×#364C2×C127semidirect
115. #1359C2×C127direct (N × H)
116. C127C2×#1359direct (N × H)
126. #364C22×C127semidirect
223. C2C127×#1359direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#492×C1272
C2×#364×C1272
C127×#13592
C2×#1359127