Finite Group Databasebeta

← C127×#12345 #44576018

Subgroups — C127×#12345 #44576018

Order 32512.

290 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3342C211yesmaximal, characteristic
2C127×#2241C211yesmaximal, characteristic, complemented
3C127×#2209C211yesmaximal, characteristic, complemented
4C127×#1669C211yesmaximal, characteristic
5C127×#1364C211yesmaximal, characteristic, complemented
6C127×#1332C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic
8#556×C127—21no—
9C2×Q32×C127—21no—
10#500×C127C2211yescharacteristic
11#494×C127C2211yescharacteristic
12#485×C127—22no—
13#475×C127—21no—
14#474×C127—21no—
15#467×C127—21no—
16C4×Q16×C127C2211yescharacteristic
17C8×D4×C127C2211yescharacteristic
18#427×C127—21no—
19C4⋊C16×C127C2211yescharacteristic
20#358×C127—21no—
21#358×C127—22no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#332×C127—21no—
25C2×Q16×C127—21no—
26C2×Q16×C127—21no—
27C2×Q16×C127D411yescharacteristic
28C22×C8×C127—21no—
29C4⋊Q8×C127—21no—
30C4⋊Q8×C127—21no—
31#126×C127—42no—
32#124×C127—41no—
33C22⋊Q8×C127—22no—
34C4×Q8×C127—21no—
35C4×Q8×C127—41no—
36C4×D4×C127D411yescharacteristic
37#117×C127—22no—
38Q32×C127—42no—
39C2×C16×C127—21no—
40C2×C16×C127—21no—
41#107×C127—21no—
42#107×C127—41no—
43#107×C127D412yes—
44#107×C127D411yescharacteristic
45#106×C127—22no—
46C4⋊C8×C127—21no—
47C4⋊C8×C127D411yescharacteristic
48#103×C127—22no—
49#103×C127—21no—
50#102×C127—41no—
51C22⋊C8×C127—21no—
52C4×C8×C127C2311yescharacteristic
53C2×Q8×C127—21no—
54C2×Q8×C127—41no—
55C2×Q8×C127—41no—
56C2×D4×C127D811yescharacteristic
57C22×C4×C127—21no—
58Q16×C127—21no—
59Q16×C127—21no—
60Q16×C127—42no—
61Q16×C127—41no—
62Q16×C127—41no—
63C2×C8×C127—21no—
64C2×C8×C127—41no—
65C2×C8×C127—41no—
66C2×C8×C127—41no—
67C2×C8×C127C2×D411yescharacteristic
68C2×C8×C127C2×D411yescharacteristic
69C4⋊C4×C127—21no—
70C4⋊C4×C127—22no—
71C4⋊C4×C127—42no—
72C4⋊C4×C127—41no—
73C4⋊C4×C127—41no—
74C4⋊C4×C127—41no—
75C4⋊C4×C127—41no—
76C4⋊C4×C127D811yescharacteristic
77C22⋊C4×C127—21no—
78C22⋊C4×C127—42no—
79C22⋊C4×C127—42no—
80C42×C127—42no—
81C42×C127—41no—
82C42×C127C2×D411yescharacteristic
83C16×C127—41no—
84C16×C127—41no—
85C23×C127—21no—
86Q8×C127—41no—
87Q8×C127—41no—
88Q8×C127—81no—
89Q8×C127—81no—
90Q8×C127—81no—
91Q8×C127—81no—
92D4×C127—21no—
93D4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127—42no—
99C2×C4×C127—41no—
100C2×C4×C127—81no—
101C2×C4×C127#12011yescharacteristic
102C2×C4×C127C2×D811yescharacteristic
103C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
104C8×C127—21no—
105C8×C127—41no—
106C8×C127—81no—
107C8×C127#13511yescharacteristic
108C8×C127AGL(1,ℤ/8)11yescharacteristic
109C22×C127—41no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127#44711yescharacteristic
113C4×C127—21no—
114C4×C127—41no—
115C4×C127—81no—
116C4×C127—82no—
117C4×C127—81no—
118C4×C127—161no—
119C4×C127#44911yescharacteristic
120C4×C127#45411yescharacteristic
121C4×C127#50811yescharacteristic
122C4×C127#51011yescharacteristic
123#12345C12711yesmaximal, characteristic, direct product factor, Sylow
124C2×C127—81no—
125C2×C127#162511yescharacteristic
126C2×C127#218911yescharacteristic
127C2×C127#219511yescharacteristic
128#3342C2×C12711yescharacteristic
129#2241C2×C12711yescharacteristic, complemented
130#2209C2×C12711yescharacteristic, complemented
131#1669C2×C12711yescharacteristic
132#1364C2×C12711yescharacteristic, complemented
133#1332C2×C12711yescharacteristic, complemented
134D4⋊C16C2×C12711yescharacteristic
135C127#1234511yescharacteristic, direct product factor, Sylow
136#556—21no—
137C2×Q32—21no—
138#500C22×C12711yescharacteristic
139#494C22×C12711yescharacteristic
140#485—22no—
141#475—21no—
142#474—21no—
143#467—21no—
144C4×Q16C22×C12711yescharacteristic
145C8×D4C22×C12711yescharacteristic
146#427—21no—
147C4⋊C16C22×C12711yescharacteristic
148#358—21no—
149#358—22no—
150C4×C16C22×C12711yescharacteristic
151#335C22×C12711yescharacteristic
152#332—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16D4×C12711yescharacteristic
156C22×C8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159#126—42no—
160#124—41no—
161C22⋊Q8—22no—
162C4×Q8—21no—
163C4×Q8—41no—
164C4×D4D4×C12711yescharacteristic
165#117—22no—
166Q32—42no—
167C2×C16—21no—
168C2×C16—21no—
169#107—21no—
170#107—41no—
171#107D4×C12711yescharacteristic
172#107D4×C12712yes—
173#106—22no—
174C4⋊C8—21no—
175C4⋊C8D4×C12711yescharacteristic
176#103—22no—
177#103—21no—
178#102—41no—
179C22⋊C8—21no—
180C4×C8C23×C12711yescharacteristic
181C2×Q8—21no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×D4D8×C12711yescharacteristic
185C22×C4—21no—
186Q16—21no—
187Q16—21no—
188Q16—42no—
189Q16—41no—
190Q16—41no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8C2×D4×C12711yescharacteristic
196C2×C8C2×D4×C12711yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4D8×C12711yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—42no—
209C42—41no—
210C42C2×D4×C12711yescharacteristic
211C16—41no—
212C16—41no—
213C23—21no—
214Q8—41no—
215Q8—41no—
216Q8—81no—
217Q8—81no—
218Q8—81no—
219Q8—81no—
220D4—21no—
221D4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—81no—
229C2×C4#120×C12711yescharacteristic
230C2×C4C2×D8×C12711yescharacteristic
231C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
232C8—21no—
233C8—41no—
234C8—81no—
235C8#135×C12711yescharacteristic
236C8AGL(1,ℤ/8)×C12711yescharacteristic
237C22—41no—
238C22—41no—
239C22—41no—
240C22#447×C12711yescharacteristic
241C4—21no—
242C4—41no—
243C4—81no—
244C4—82no—
245C4—81no—
246C4—161no—
247C4#449×C12711yescharacteristic
248C4#454×C12711yescharacteristic
249C4#508×C12711yescharacteristic
250C4#510×C12711yescharacteristic
251C2—81no—
252C2C127×#162511yescharacteristic
253C2C127×#218911yescharacteristic
254C2C127×#219511yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC2×D4×C127D8
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2241C2semidirect
3. C127×#2209C2semidirect
5. C127×#1364C2semidirect
6. C127×#1332C2semidirect
123. #12345C127direct (N × H)
129. #2241C2×C127semidirect
130. #2209C2×C127semidirect
132. #1364C2×C127semidirect
133. #1332C2×C127semidirect
135. C127#12345direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#33422
C127×#22412
C127×#22092
C127×#16692
C127×#13642
C127×#13322
C127×D4⋊C162
#12345127