Finite Group Databasebeta

← C127×#13523 #44577196

Subgroups — C127×#13523 #44577196

Order 32512.

308 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2161C211yesmaximal, characteristic, complemented
2C127×#2155C211yesmaximal, characteristic, complemented
3C2×#361×C127C211yesmaximal, characteristic
4C127×#2140C211yesmaximal, characteristic
5C127×#1978C211yesmaximal, characteristic
6C127×#1833C211yesmaximal, characteristic, complemented
7C127×#1378C211yesmaximal, characteristic, complemented
8C22×D8×C127C2211yescharacteristic
9C2×M32×C127C2211yescharacteristic
10C2×M32×C127C2211yescharacteristic
11C2×#108×C127C2211yescharacteristic
12C2×#108×C127C2211yescharacteristic
13#427×C127C2211yescharacteristic
14#418×C127—21no—
15C2×#100×C127—21no—
16#405×C127C2211yescharacteristic
17#368×C127—22no—
18#365×C127—22no—
19#361×C127—22no—
20#357×C127—22no—
21#344×C127—21no—
22H(3,ℤ/4)×C127—21no—
23C22×D4×C127—21no—
24C2×D8×C127—21no—
25C2×D8×C127—21no—
26C2×D8×C127—21no—
27C2×D8×C127—21no—
28C2×D8×C127—21no—
29C2×D8×C127Q812yes—
30C2×M16×C127—21no—
31C2×M16×C127—21no—
32C2×M16×C127D411yescharacteristic
33C22×C8×C127C2311yescharacteristic
34#117×C127—21no—
35#117×C127D411yescharacteristic
36M32×C127—21no—
37M32×C127—21no—
38M32×C127—21no—
39M32×C127D412yes—
40C2×C16×C127—21no—
41C2×C16×C127D412yes—
42#108×C127—21no—
43#108×C127—21no—
44#108×C127—21no—
45#108×C127—21no—
46#107×C127—21no—
47#106×C127—21no—
48#102×C127—42no—
49#100×C127—42no—
50#97×C127—21no—
51C4×C8×C127—21no—
52C24×C127—41no—
53C2×D4×C127—22no—
54C2×D4×C127—41no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C22×C4×C127C2×D411yescharacteristic
60D8×C127—22no—
61D8×C127—22no—
62D8×C127—41no—
63D8×C127—41no—
64M16×C127—21no—
65M16×C127—21no—
66M16×C127—41no—
67M16×C127—41no—
68M16×C127—41no—
69M16×C127—41no—
70C2×C8×C127—21no—
71C2×C8×C127—41no—
72C2×C8×C127—41no—
73C2×C8×C127C2×D411yescharacteristic
74C2×C8×C127C2×D411yescharacteristic
75C2×C8×C127C2×Q811yescharacteristic
76C2×C8×C127#4111yescharacteristic
77C2×C8×C127#4111yescharacteristic
78C2×C8×C127#4111yescharacteristic
79C4⋊C4×C127—21no—
80C4⋊C4×C127—41no—
81C22⋊C4×C127—21no—
82C22⋊C4×C127—41no—
83C42×C127—21no—
84C42×C127—41no—
85C16×C127—22no—
86C16×C127—22no—
87C23×C127—42no—
88C23×C127—81no—
89C23×C127—81no—
90C23×C127—162no—
91C23×C127#13511yescharacteristic
92D4×C127—42no—
93D4×C127—42no—
94D4×C127—81no—
95D4×C127—81no—
96C2×C4×C127—41no—
97C2×C4×C127—81no—
98C2×C4×C127C22⋊Q811yescharacteristic
99C2×C4×C127#12311yescharacteristic
100C2×C4×C127#12711yescharacteristic
101C2×C4×C127#13511yescharacteristic
102C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
103C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
104C8×C127—41no—
105C8×C127—81no—
106C8×C127—81no—
107C8×C127C22⋊Q811yescharacteristic
108C8×C127C22⋊Q811yescharacteristic
109C8×C127#12311yescharacteristic
110C8×C127#12311yescharacteristic
111C22×C127—41no—
112C22×C127—82no—
113C22×C127—162no—
114C22×C127—162no—
115C22×C127—162no—
116C22×C127#47811yescharacteristic
117C22×C127#48211yescharacteristic
118C22×C127#49511yescharacteristic
119C4×C127—81no—
120C4×C127—161no—
121C4×C127#39911yescharacteristic
122C4×C127#47811yescharacteristic
123C4×C127#48211yescharacteristic
124C4×C127#49611yescharacteristic
125#13523C12711yesmaximal, characteristic, direct product factor, Sylow
126C2×C127—21no—
127C2×C127—21no—
128C2×C127—21no—
129C2×C127—162no—
130C2×C127#209211yescharacteristic
131#2161C2×C12711yescharacteristic, complemented
132#2155C2×C12711yescharacteristic, complemented
133C2×#361C2×C12711yescharacteristic
134#2140C2×C12711yescharacteristic
135#1978C2×C12711yescharacteristic
136#1833C2×C12711yescharacteristic, complemented
137#1378C2×C12711yescharacteristic, complemented
138C127#1352311yescharacteristic, direct product factor, Sylow
139C22×D8C22×C12711yescharacteristic
140C2×M32C22×C12711yescharacteristic
141C2×M32C22×C12711yescharacteristic
142C2×#108C22×C12711yescharacteristic
143C2×#108C22×C12711yescharacteristic
144#427C22×C12711yescharacteristic
145#418—21no—
146C2×#100—21no—
147#405C22×C12711yescharacteristic
148#368—22no—
149#365—22no—
150#361—22no—
151#357—22no—
152#344—21no—
153H(3,ℤ/4)—21no—
154C22×D4—21no—
155C2×D8—21no—
156C2×D8—21no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8Q8×C12712yes—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16D4×C12711yescharacteristic
164C22×C8C23×C12711yescharacteristic
165#117—21no—
166#117D4×C12711yescharacteristic
167M32—21no—
168M32—21no—
169M32—21no—
170M32D4×C12712yes—
171C2×C16—21no—
172C2×C16D4×C12712yes—
173#108—21no—
174#108—21no—
175#108—21no—
176#108—21no—
177#107—21no—
178#106—21no—
179#102—42no—
180#100—42no—
181#97—21no—
182C4×C8—21no—
183C24—41no—
184C2×D4—22no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C22×C4C2×D4×C12711yescharacteristic
191D8—22no—
192D8—22no—
193D8—41no—
194D8—41no—
195M16—21no—
196M16—21no—
197M16—41no—
198M16—41no—
199M16—41no—
200M16—41no—
201C2×C8—21no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8C2×D4×C12711yescharacteristic
205C2×C8C2×D4×C12711yescharacteristic
206C2×C8C2×Q8×C12711yescharacteristic
207C2×C8#41×C12711yescharacteristic
208C2×C8#41×C12711yescharacteristic
209C2×C8#41×C12711yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—41no—
212C22⋊C4—21no—
213C22⋊C4—41no—
214C42—21no—
215C42—41no—
216C16—22no—
217C16—22no—
218C23—42no—
219C23—81no—
220C23—81no—
221C23—162no—
222C23#135×C12711yescharacteristic
223D4—42no—
224D4—42no—
225D4—81no—
226D4—81no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4C22⋊Q8×C12711yescharacteristic
230C2×C4#123×C12711yescharacteristic
231C2×C4#127×C12711yescharacteristic
232C2×C4#135×C12711yescharacteristic
233C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
234C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
235C8—41no—
236C8—81no—
237C8—81no—
238C8C22⋊Q8×C12711yescharacteristic
239C8C22⋊Q8×C12711yescharacteristic
240C8#123×C12711yescharacteristic
241C8#123×C12711yescharacteristic
242C22—41no—
243C22—82no—
244C22—162no—
245C22—162no—
246C22—162no—
247C22#478×C12711yescharacteristic
248C22#482×C12711yescharacteristic
249C22#495×C12711yescharacteristic
250C4—81no—
251C4—161no—
252C4#399×C12711yescharacteristic
253C4#478×C12711yescharacteristic
254C4#482×C12711yescharacteristic
255C4#496×C12711yescharacteristic
256C2—21no—
257C2—21no—
258C2—21no—
259C2—162no—
260C2C127×#209211yescharacteristic
subgroupG/H
center, socleC2×C127#2092
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC22×D8×C127C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2161C2semidirect
2. C127×#2155C2semidirect
6. C127×#1833C2semidirect
7. C127×#1378C2semidirect
125. #13523C127direct (N × H)
131. #2161C2×C127semidirect
132. #2155C2×C127semidirect
136. #1833C2×C127semidirect
137. #1378C2×C127semidirect
138. C127#13523direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#21612
C127×#21552
C2×#361×C1272
C127×#21402
C127×#19782
C127×#18332
C127×#13782
#13523127