Finite Group Databasebeta

← C127×#12162 #44575835

Subgroups — C127×#12162 #44575835

Order 32512.

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

normal subgroups only

This group has 354 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×#3249C211yesmaximal, characteristic
2C127×#2188C212yesmaximal
3C127×#2056C211yesmaximal, characteristic
4C127×#1348C212yesmaximal
5C127×#1322C211yesmaximal, characteristic
6C22×Q16×C127C2211yescharacteristic
7#533×C127—21no—
8C2×Q32×C127—22no—
9#477×C127—22no—
10#467×C127—21no—
11#467×C127C2212yes—
12#462×C127—22no—
13C2×C4⋊C8×C127C2211yescharacteristic
14C2×#103×C127—21no—
15#391×C127—21no—
16#358×C127—22no—
17C22⋊C16×C127C2212yes—
18#340×C127C2211yescharacteristic
19C22×Q8×C127—21no—
20C2×Q16×C127—21no—
21C2×Q16×C127—22no—
22C2×Q16×C127—42no—
23C2×Q16×C127D412yes—
24C22×C8×C127D411yescharacteristic
25C22×C8×C127C2311yescharacteristic
26C4⋊Q8×C127—21no—
27C4⋊Q8×C127—21no—
28#125×C127—41no—
29C22⋊Q8×C127—22no—
30C22⋊Q8×C127—41no—
31C4×Q8×C127—22no—
32C4×Q8×C127—41no—
33#117×C127—41no—
34C2×C4⋊C4×C127—21no—
35C2×C42×C127D411yescharacteristic
36Q32×C127—44no—
37C2×C16×C127—22no—
38#107×C127—21no—
39#107×C127D412yes—
40C4⋊C8×C127—22no—
41#103×C127—22no—
42#103×C127—22no—
43#103×C127—42no—
44#95×C127—41no—
45#95×C127—41no—
46C2×Q8×C127—22no—
47C2×Q8×C127—22no—
48C2×Q8×C127—41no—
49C2×Q8×C127—41no—
50C2×Q8×C127—41no—
51C2×Q8×C127—41no—
52C2×Q8×C127—41no—
53C2×Q8×C127—41no—
54C22×C4×C127—41no—
55C22×C4×C127—41no—
56C22×C4×C127D811yescharacteristic
57C22×C4×C127D811yescharacteristic
58C22×C4×C127C2×D411yescharacteristic
59Q16×C127—24no—
60Q16×C127—42no—
61Q16×C127—42no—
62Q16×C127—81no—
63C2×C8×C127—21no—
64C2×C8×C127—41no—
65C2×C8×C127—41no—
66C2×C8×C127C2×D412yes—
67C4⋊C4×C127—22no—
68C4⋊C4×C127—41no—
69C4⋊C4×C127—41no—
70C4⋊C4×C127—42no—
71C4⋊C4×C127—41no—
72C4⋊C4×C127—41no—
73C4⋊C4×C127—41no—
74C4⋊C4×C127—42no—
75C22⋊C4×C127—42no—
76C42×C127—21no—
77C42×C127—21no—
78C42×C127—42no—
79C16×C127—42no—
80C23×C127C2×D811yescharacteristic
81Q8×C127—42no—
82Q8×C127—42no—
83Q8×C127—42no—
84Q8×C127—42no—
85Q8×C127—81no—
86Q8×C127—82no—
87Q8×C127—81no—
88C2×C4×C127—21no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—41no—
94C2×C4×C127—42no—
95C2×C4×C127—42no—
96C2×C4×C127—42no—
97C2×C4×C127—81no—
98C2×C4×C127—81no—
99C2×C4×C127—81no—
100C2×C4×C127—81no—
101C2×C4×C127—81no—
102C2×C4×C127Q3212yescomplemented
103C2×C4×C127#12011yescharacteristic
104C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
105C8×C127—22no—
106C8×C127—41no—
107C22×C127—21no—
108C22×C127—21no—
109C22×C127#44711yescharacteristic
110C22×C127C2×Q3211yescharacteristic
111C22×C127#51011yescharacteristic
112C4×C127—21no—
113C4×C127—22no—
114C4×C127—41no—
115C4×C127—82no—
116C4×C127—82no—
117C4×C127—82no—
118C4×C127—161no—
119C4×C127#45411yescharacteristic
120C4×C127B(4,𝔽2)11yescharacteristic
121#12162C12711yesmaximal, characteristic, direct product factor, Sylow
122C2×C127—21no—
123C2×C127—21no—
124C2×C127#159811yescharacteristic
125C2×C127#218811yescharacteristic
126C2×C127#219111yescharacteristic
127#3249C2×C12711yescharacteristic
128#2188C2×C12712yes—
129#2056C2×C12711yescharacteristic
130#1348C2×C12712yes—
131#1322C2×C12711yescharacteristic
132C127#1216211yescharacteristic, direct product factor, Sylow
133C22×Q16C22×C12711yescharacteristic
134#533—21no—
135C2×Q32—22no—
136#477—22no—
137#467—21no—
138#467C22×C12712yes—
139#462—22no—
140C2×C4⋊C8C22×C12711yescharacteristic
141C2×#103—21no—
142#391—21no—
143#358—22no—
144C22⋊C16C22×C12712yes—
145#340C22×C12711yescharacteristic
146C22×Q8—21no—
147C2×Q16—21no—
148C2×Q16—22no—
149C2×Q16—42no—
150C2×Q16D4×C12712yes—
151C22×C8D4×C12711yescharacteristic
152C22×C8C23×C12711yescharacteristic
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155#125—41no—
156C22⋊Q8—22no—
157C22⋊Q8—41no—
158C4×Q8—22no—
159C4×Q8—41no—
160#117—41no—
161C2×C4⋊C4—21no—
162C2×C42D4×C12711yescharacteristic
163Q32—44no—
164C2×C16—22no—
165#107—21no—
166#107D4×C12712yes—
167C4⋊C8—22no—
168#103—22no—
169#103—22no—
170#103—42no—
171#95—41no—
172#95—41no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4D8×C12711yescharacteristic
184C22×C4D8×C12711yescharacteristic
185C22×C4C2×D4×C12711yescharacteristic
186Q16—24no—
187Q16—42no—
188Q16—42no—
189Q16—81no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8C2×D4×C12712yes—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C22⋊C4—42no—
203C42—21no—
204C42—21no—
205C42—42no—
206C16—42no—
207C23C2×D8×C12711yescharacteristic
208Q8—42no—
209Q8—42no—
210Q8—42no—
211Q8—42no—
212Q8—81no—
213Q8—82no—
214Q8—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4Q32×C12712yescomplemented
230C2×C4#120×C12711yescharacteristic
231C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
232C8—22no—
233C8—41no—
234C22—21no—
235C22—21no—
236C22#447×C12711yescharacteristic
237C22C2×Q32×C12711yescharacteristic
238C22#510×C12711yescharacteristic
239C4—21no—
240C4—22no—
241C4—41no—
242C4—82no—
243C4—82no—
244C4—82no—
245C4—161no—
246C4#454×C12711yescharacteristic
247C4B(4,𝔽2)×C12711yescharacteristic
248C2—21no—
249C2—21no—
250C2C127×#159811yescharacteristic
251C2C127×#218811yescharacteristic
252C2C127×#219111yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC23×C127C2×D8
LCSG ▷ C22×C8 ▷ 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
102. C2×C4×C127Q32semidirect
121. #12162C127direct (N × H)
132. C127#12162direct (N × H)
229. C2×C4Q32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32492
C127×#21882
C127×#20562
C127×#13482
C127×#13222
#12162127