Finite Group Databasebeta

← C127×#12112 #44575785

Subgroups — C127×#12112 #44575785

Order 32512.

390 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3202C211yesmaximal, characteristic
2C127×#2188C212yesmaximal, complemented
3C127×#2020C211yesmaximal, characteristic
4C127×#1350C212yesmaximal, complemented
5C127×#1315C211yesmaximal, characteristic
6C2×#137×C127—21no—
7C22×Q16×C127C2211yescharacteristic
8#536×C127—21no—
9C2×Q32×C127—22no—
10#485×C127—22no—
11#470×C127—21no—
12#467×C127C2212yes—
13#451×C127—22no—
14C2×#101×C127—21no—
15#408×C127C2211yescharacteristic
16#358×C127—22no—
17C22⋊C16×C127C2212yes—
18#341×C127C2211yescharacteristic
19C2×#41×C127D411yescharacteristic
20C22×Q8×C127—21no—
21C22×Q8×C127—21no—
22#137×C127—42no—
23#137×C127—42no—
24C2×Q16×C127—21no—
25C2×Q16×C127—22no—
26C2×Q16×C127—42no—
27C2×Q16×C127—41no—
28C2×Q16×C127D412yes—
29C2×SD16×C127—41no—
30C2×M16×C127—21no—
31C2×M16×C127D411yescharacteristic
32C22×C8×C127C2311yescharacteristic
33C4⋊Q8×C127—41no—
34#124×C127—41no—
35#123×C127—41no—
36#123×C127—41no—
37C22⋊Q8×C127—22no—
38C22⋊Q8×C127—22no—
39#117×C127—21no—
40Q32×C127—44no—
41C2×C16×C127—22no—
42#107×C127D412yes—
43#106×C127—21no—
44#103×C127—22no—
45#103×C127—22no—
46#101×C127—42no—
47C22⋊C8×C127—22no—
48#41×C127—41no—
49#41×C127—41no—
50C2×Q8×C127—22no—
51C2×Q8×C127—22no—
52C2×Q8×C127—41no—
53C2×Q8×C127—41no—
54C2×Q8×C127—41no—
55C2×Q8×C127—41no—
56C2×Q8×C127—41no—
57C2×Q8×C127—41no—
58C2×Q8×C127—81no—
59C2×Q8×C127—81no—
60C2×Q8×C127D811yescharacteristic
61C2×D4×C127—21no—
62C2×D4×C127D811yescharacteristic
63C22×C4×C127—21no—
64C22×C4×C127—41no—
65C22×C4×C127—41no—
66C22×C4×C127C2×D411yescharacteristic
67Q16×C127—24no—
68Q16×C127—42no—
69Q16×C127—42no—
70Q16×C127—44no—
71Q16×C127—81no—
72SD16×C127—44no—
73M16×C127—22no—
74M16×C127—41no—
75M16×C127—41no—
76C2×C8×C127—21no—
77C2×C8×C127—41no—
78C2×C8×C127—41no—
79C2×C8×C127C2×D412yes—
80C4⋊C4×C127—22no—
81C4⋊C4×C127—42no—
82C4⋊C4×C127—42no—
83C22⋊C4×C127—41no—
84C22⋊C4×C127—42no—
85C22⋊C4×C127—42no—
86C42×C127—41no—
87C16×C127—42no—
88C23×C127—21no—
89C23×C127AGL(1,ℤ/8)11yescharacteristic
90Q8×C127—41no—
91Q8×C127—42no—
92Q8×C127—42no—
93Q8×C127—42no—
94Q8×C127—42no—
95Q8×C127—81no—
96Q8×C127—82no—
97Q8×C127—81no—
98D4×C127—41no—
99D4×C127—81no—
100C2×C4×C127—21no—
101C2×C4×C127—21no—
102C2×C4×C127—21no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127—42no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127—81no—
109C2×C4×C127—82no—
110C2×C4×C127—81no—
111C2×C4×C127#12011yescharacteristic
112C2×C4×C127C2×D811yescharacteristic
113C8×C127—22no—
114C8×C127—42no—
115C8×C127—41no—
116C22×C127—21no—
117C22×C127—22no—
118C22×C127—41no—
119C22×C127—42no—
120C22×C127#44711yescharacteristic
121C4×C127—21no—
122C4×C127—41no—
123C4×C127—82no—
124C4×C127—82no—
125C4×C127—82no—
126C4×C127#45311yescharacteristic
127C4×C127B(4,𝔽2)11yescharacteristic
128#12112C12711yesmaximal, characteristic, direct product factor, Sylow
129C2×C127—41no—
130C2×C127—81no—
131C2×C127#159511yescharacteristic
132C2×C127#219512yes—
133#3202C2×C12711yescharacteristic
134#2188C2×C12712yescomplemented
135#2020C2×C12711yescharacteristic
136#1350C2×C12712yescomplemented
137#1315C2×C12711yescharacteristic
138C127#1211211yescharacteristic, direct product factor, Sylow
139C2×#137—21no—
140C22×Q16C22×C12711yescharacteristic
141#536—21no—
142C2×Q32—22no—
143#485—22no—
144#470—21no—
145#467C22×C12712yes—
146#451—22no—
147C2×#101—21no—
148#408C22×C12711yescharacteristic
149#358—22no—
150C22⋊C16C22×C12712yes—
151#341C22×C12711yescharacteristic
152C2×#41D4×C12711yescharacteristic
153C22×Q8—21no—
154C22×Q8—21no—
155#137—42no—
156#137—42no—
157C2×Q16—21no—
158C2×Q16—22no—
159C2×Q16—42no—
160C2×Q16—41no—
161C2×Q16D4×C12712yes—
162C2×SD16—41no—
163C2×M16—21no—
164C2×M16D4×C12711yescharacteristic
165C22×C8C23×C12711yescharacteristic
166C4⋊Q8—41no—
167#124—41no—
168#123—41no—
169#123—41no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172#117—21no—
173Q32—44no—
174C2×C16—22no—
175#107D4×C12712yes—
176#106—21no—
177#103—22no—
178#103—22no—
179#101—42no—
180C22⋊C8—22no—
181#41—41no—
182#41—41no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—81no—
192C2×Q8—81no—
193C2×Q8D8×C12711yescharacteristic
194C2×D4—21no—
195C2×D4D8×C12711yescharacteristic
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C2×D4×C12711yescharacteristic
200Q16—24no—
201Q16—42no—
202Q16—42no—
203Q16—44no—
204Q16—81no—
205SD16—44no—
206M16—22no—
207M16—41no—
208M16—41no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8C2×D4×C12712yes—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C42—41no—
220C16—42no—
221C23—21no—
222C23AGL(1,ℤ/8)×C12711yescharacteristic
223Q8—41no—
224Q8—42no—
225Q8—42no—
226Q8—42no—
227Q8—42no—
228Q8—81no—
229Q8—82no—
230Q8—81no—
231D4—41no—
232D4—81no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—81no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4#120×C12711yescharacteristic
245C2×C4C2×D8×C12711yescharacteristic
246C8—22no—
247C8—42no—
248C8—41no—
249C22—21no—
250C22—22no—
251C22—41no—
252C22—42no—
253C22#447×C12711yescharacteristic
254C4—21no—
255C4—41no—
256C4—82no—
257C4—82no—
258C4—82no—
259C4#453×C12711yescharacteristic
260C4B(4,𝔽2)×C12711yescharacteristic
261C2—41no—
262C2—81no—
263C2C127×#159511yescharacteristic
264C2C127×#219512yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC2×#41×C127D4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #408×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2188C2semidirect
4. C127×#1350C2semidirect
128. #12112C127direct (N × H)
134. #2188C2×C127semidirect
136. #1350C2×C127semidirect
138. C127#12112direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32022
C127×#21882
C127×#20202
C127×#13502
C127×#13152
#12112127