Finite Group Databasebeta

← C127×#13682 #44577355

Subgroups — C127×#13682 #44577355

Order 32512.

384 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3293C211yesmaximal, characteristic, complemented
2C127×#2204C212yesmaximal, complemented
3C127×#1634C211yesmaximal, characteristic, complemented
4C127×#1411C211yesmaximal, characteristic, complemented
5C127×#1405C212yesmaximal, complemented
6#578×C127—22no—
7#555×C127C2211yescharacteristic
8#494×C127C2211yescharacteristic
9#476×C127—21no—
10#473×C127—22no—
11#456×C127—21no—
12#456×C127C2212yes—
13#454×C127—22no—
14#443×C127—21no—
15#356×C127C2212yes—
16#355×C127—22no—
17#333×C127C2211yescharacteristic
18D4⋊C8×C127—21no—
192−1+4×C127—21no—
202−1+4×C127D412yes—
21#137×C127—22no—
22#137×C127—22no—
23#137×C127—42no—
24#137×C127—42no—
25AGL(1,ℤ/8)×C127—42no—
26#135×C127—21no—
27#135×C127—42no—
28C2×Q16×C127—22no—
29C2×Q16×C127—42no—
30#131×C127—22no—
31C4⋊Q8×C127D411yescharacteristic
32C4⋊Q8×C127C2311yescharacteristic
33#124×C127—22no—
34#121×C127—41no—
35C4×D4×C127—21no—
36#117×C127—41no—
37#108×C127—21no—
38#106×C127—41no—
39C4⋊C8×C127—21no—
40#104×C127—22no—
41#104×C127—42no—
42#102×C127—41no—
43#101×C127—21no—
44#101×C127—22no—
45#101×C127D412yes—
46C4×C8×C127D411yescharacteristic
47#41×C127—22no—
48#41×C127—42no—
49#41×C127—42no—
50#41×C127—42no—
51#41×C127—81no—
52C2×Q8×C127—21no—
53C2×Q8×C127—42no—
54C2×Q8×C127—42no—
55C2×Q8×C127C2×D412yes—
56C2×D4×C127—21no—
57C2×D4×C127—22no—
58C22×C4×C127—41no—
59Q16×C127—21no—
60Q16×C127—42no—
61Q16×C127—42no—
62Q16×C127—42no—
63Q16×C127—42no—
64SD16×C127—22no—
65SD16×C127—42no—
66SD16×C127—42no—
67SD16×C127—82no—
68D8×C127—21no—
69D8×C127—82no—
70M16×C127—22no—
71M16×C127—42no—
72M16×C127—42no—
73C2×C8×C127—21no—
74C2×C8×C127—42no—
75C2×C8×C127—41no—
76C4⋊C4×C127—21no—
77C4⋊C4×C127—41no—
78C4⋊C4×C127—41no—
79C22⋊C4×C127—42no—
80C22⋊C4×C127—41no—
81C42×C127—22no—
82C42×C127C2×D411yescharacteristic
83C23×C127—41no—
84Q8×C127—22no—
85Q8×C127—42no—
86Q8×C127—42no—
87Q8×C127—42no—
88Q8×C127—81no—
89D4×C127—22no—
90D4×C127—41no—
91D4×C127—41no—
92D4×C127—42no—
93D4×C127—42no—
94D4×C127—82no—
95D4×C127—81no—
96C2×C4×C127—21no—
97C2×C4×C127—22no—
98C2×C4×C127—42no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—81no—
102C2×C4×C127—81no—
103C2×C4×C127—82no—
104C2×C4×C127—82no—
105C2×C4×C127#12011yescharacteristic
106C8×C127—21no—
107C8×C127—21no—
108C8×C127—42no—
109C8×C127—81no—
110C8×C127—82no—
111C22×C127—42no—
112C22×C127—81no—
113C22×C127—81no—
114C22×C127—81no—
115C22×C127B(4,𝔽2)11yescharacteristic
116C4×C127—21no—
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—42no—
120C4×C127—42no—
121C4×C127—81no—
122C4×C127—81no—
123#13682C12711yesmaximal, characteristic, direct product factor, Sylow
124C2×C127—21no—
125C2×C127—82no—
126C2×C127—161no—
127C2×C127#219611yescharacteristic
128#3293C2×C12711yescharacteristic, complemented
129#2204C2×C12712yescomplemented
130#1634C2×C12711yescharacteristic, complemented
131#1411C2×C12711yescharacteristic, complemented
132#1405C2×C12712yescomplemented
133C127#1368211yescharacteristic, direct product factor, Sylow
134#578—22no—
135#555C22×C12711yescharacteristic
136#494C22×C12711yescharacteristic
137#476—21no—
138#473—22no—
139#456—21no—
140#456C22×C12712yes—
141#454—22no—
142#443—21no—
143#356C22×C12712yes—
144#355—22no—
145#333C22×C12711yescharacteristic
146D4⋊C8—21no—
1472−1+4—21no—
1482−1+4D4×C12712yes—
149#137—22no—
150#137—22no—
151#137—42no—
152#137—42no—
153AGL(1,ℤ/8)—42no—
154#135—21no—
155#135—42no—
156C2×Q16—22no—
157C2×Q16—42no—
158#131—22no—
159C4⋊Q8D4×C12711yescharacteristic
160C4⋊Q8C23×C12711yescharacteristic
161#124—22no—
162#121—41no—
163C4×D4—21no—
164#117—41no—
165#108—21no—
166#106—41no—
167C4⋊C8—21no—
168#104—22no—
169#104—42no—
170#102—41no—
171#101—21no—
172#101—22no—
173#101D4×C12712yes—
174C4×C8D4×C12711yescharacteristic
175#41—22no—
176#41—42no—
177#41—42no—
178#41—42no—
179#41—81no—
180C2×Q8—21no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8C2×D4×C12712yes—
184C2×D4—21no—
185C2×D4—22no—
186C22×C4—41no—
187Q16—21no—
188Q16—42no—
189Q16—42no—
190Q16—42no—
191Q16—42no—
192SD16—22no—
193SD16—42no—
194SD16—42no—
195SD16—82no—
196D8—21no—
197D8—82no—
198M16—22no—
199M16—42no—
200M16—42no—
201C2×C8—21no—
202C2×C8—42no—
203C2×C8—41no—
204C4⋊C4—21no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C22⋊C4—42no—
208C22⋊C4—41no—
209C42—22no—
210C42C2×D4×C12711yescharacteristic
211C23—41no—
212Q8—22no—
213Q8—42no—
214Q8—42no—
215Q8—42no—
216Q8—81no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—42no—
221D4—42no—
222D4—82no—
223D4—81no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4#120×C12711yescharacteristic
234C8—21no—
235C8—21no—
236C8—42no—
237C8—81no—
238C8—82no—
239C22—42no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22B(4,𝔽2)×C12711yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C2—21no—
252C2—82no—
253C2—161no—
254C2C127×#219611yescharacteristic
subgroupG/H
center, socleC2×C127#2196
commutator, Frattini, cosocleC4⋊Q8C23×C127
LCSG ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×C4×C127 ◁ C4⋊Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3293C2semidirect
2. C127×#2204C2semidirect
3. C127×#1634C2semidirect
4. C127×#1411C2semidirect
5. C127×#1405C2semidirect
123. #13682C127direct (N × H)
128. #3293C2×C127semidirect
129. #2204C2×C127semidirect
130. #1634C2×C127semidirect
131. #1411C2×C127semidirect
132. #1405C2×C127semidirect
133. C127#13682direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32932
C127×#22042
C127×#16342
C127×#14112
C127×#14052
#13682127