Finite Group Databasebeta

← C127×#12161 #44575834

Subgroups — C127×#12161 #44575834

Order 32512.

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

normal subgroups only

This group has 398 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×#3245C211yesmaximal, characteristic
2C127×#2185C212yesmaximal, complemented
3C127×#2056C211yesmaximal, characteristic, complemented
4C127×#1347C212yesmaximal, complemented
5C127×#1322C211yesmaximal, characteristic, complemented
6C22×Q16×C127C2211yescharacteristic
7#532×C127—21no—
8C2×SD32×C127—22no—
9#474×C127—22no—
10#466×C127—21no—
11#466×C127C2212yes—
12#459×C127—22no—
13C2×C4⋊C8×C127C2211yescharacteristic
14C2×#103×C127—21no—
15#391×C127—21no—
16#358×C127—22no—
17C22⋊C16×C127C2212yes—
18#340×C127C2211yescharacteristic
19C2×#41×C127—41no—
20C22×Q8×C127—21no—
21C2×Q16×C127—22no—
22C2×Q16×C127—42no—
23C2×D8×C127—21no—
24C2×D8×C127D412yes—
25C22×C8×C127D411yescharacteristic
26C22×C8×C127C2311yescharacteristic
27C4⋊Q8×C127—21no—
28#127×C127—21no—
29#124×C127—41no—
30#121×C127—22no—
31#121×C127—41no—
32C4×D4×C127—22no—
33C4×D4×C127—41no—
34C2×C4⋊C4×C127—21no—
35C2×C42×C127D411yescharacteristic
36SD32×C127—44no—
37C2×C16×C127—22no—
38#107×C127—21no—
39#107×C127D412yes—
40C4⋊C8×C127—22no—
41#103×C127—42no—
42#102×C127—22no—
43#102×C127—22no—
44#95×C127—41no—
45#95×C127—41no—
46#41×C127—82no—
47#41×C127—82no—
48C2×Q8×C127—22no—
49C2×Q8×C127—41no—
50C2×Q8×C127—41no—
51C2×Q8×C127—41no—
52C2×Q8×C127—41no—
53C2×Q8×C127—41no—
54C2×Q8×C127—41no—
55C2×D4×C127—22no—
56C2×D4×C127—41no—
57C2×D4×C127—41no—
58C2×D4×C127—42no—
59C22×C4×C127—42no—
60C22×C4×C127—41no—
61C22×C4×C127—41no—
62C22×C4×C127D811yescharacteristic
63C22×C4×C127D811yescharacteristic
64C22×C4×C127C2×D411yescharacteristic
65Q16×C127—24no—
66Q16×C127—42no—
67D8×C127—42no—
68D8×C127—81no—
69C2×C8×C127—21no—
70C2×C8×C127—41no—
71C2×C8×C127—41no—
72C2×C8×C127C2×D412yes—
73C4⋊C4×C127—22no—
74C4⋊C4×C127—41no—
75C4⋊C4×C127—41no—
76C4⋊C4×C127—41no—
77C22⋊C4×C127—42no—
78C22⋊C4×C127—42no—
79C42×C127—21no—
80C42×C127—21no—
81C16×C127—42no—
82C23×C127—42no—
83C23×C127C2×D811yescharacteristic
84Q8×C127—42no—
85Q8×C127—42no—
86Q8×C127—81no—
87Q8×C127—82no—
88Q8×C127—81no—
89D4×C127—42no—
90D4×C127—42no—
91D4×C127—82no—
92D4×C127—82no—
93D4×C127—82no—
94D4×C127—82no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—41no—
101C2×C4×C127—42no—
102C2×C4×C127—42no—
103C2×C4×C127—84no—
104C2×C4×C127—81no—
105C2×C4×C127—81no—
106C2×C4×C127—81no—
107C2×C4×C127—81no—
108C2×C4×C127—81no—
109C2×C4×C127SD3212yescomplemented
110C2×C4×C127#12011yescharacteristic
111C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
112C8×C127—22no—
113C8×C127—41no—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127—82no—
117C22×C127—82no—
118C22×C127—82no—
119C22×C127#44711yescharacteristic
120C22×C127C2×SD3211yescharacteristic
121C22×C127#50911yescharacteristic
122C4×C127—21no—
123C4×C127—22no—
124C4×C127—41no—
125C4×C127—82no—
126C4×C127—82no—
127C4×C127—161no—
128C4×C127#45411yescharacteristic
129C4×C127B(4,𝔽2)11yescharacteristic
130#12161C12711yesmaximal, characteristic, direct product factor, Sylow
131C2×C127—21no—
132C2×C127—21no—
133C2×C127—162no—
134C2×C127#159811yescharacteristic
135C2×C127#218711yescharacteristic
136C2×C127#219111yescharacteristic
137#3245C2×C12711yescharacteristic
138#2185C2×C12712yescomplemented
139#2056C2×C12711yescharacteristic, complemented
140#1347C2×C12712yescomplemented
141#1322C2×C12711yescharacteristic, complemented
142C127#1216111yescharacteristic, direct product factor, Sylow
143C22×Q16C22×C12711yescharacteristic
144#532—21no—
145C2×SD32—22no—
146#474—22no—
147#466—21no—
148#466C22×C12712yes—
149#459—22no—
150C2×C4⋊C8C22×C12711yescharacteristic
151C2×#103—21no—
152#391—21no—
153#358—22no—
154C22⋊C16C22×C12712yes—
155#340C22×C12711yescharacteristic
156C2×#41—41no—
157C22×Q8—21no—
158C2×Q16—22no—
159C2×Q16—42no—
160C2×D8—21no—
161C2×D8D4×C12712yes—
162C22×C8D4×C12711yescharacteristic
163C22×C8C23×C12711yescharacteristic
164C4⋊Q8—21no—
165#127—21no—
166#124—41no—
167#121—22no—
168#121—41no—
169C4×D4—22no—
170C4×D4—41no—
171C2×C4⋊C4—21no—
172C2×C42D4×C12711yescharacteristic
173SD32—44no—
174C2×C16—22no—
175#107—21no—
176#107D4×C12712yes—
177C4⋊C8—22no—
178#103—42no—
179#102—22no—
180#102—22no—
181#95—41no—
182#95—41no—
183#41—82no—
184#41—82no—
185C2×Q8—22no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×D4—22no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—42no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4D8×C12711yescharacteristic
200C22×C4D8×C12711yescharacteristic
201C22×C4C2×D4×C12711yescharacteristic
202Q16—24no—
203Q16—42no—
204D8—42no—
205D8—81no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×D4×C12712yes—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—21no—
217C42—21no—
218C16—42no—
219C23—42no—
220C23C2×D8×C12711yescharacteristic
221Q8—42no—
222Q8—42no—
223Q8—81no—
224Q8—82no—
225Q8—81no—
226D4—42no—
227D4—42no—
228D4—82no—
229D4—82no—
230D4—82no—
231D4—82no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—84no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4SD32×C12712yescomplemented
247C2×C4#120×C12711yescharacteristic
248C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
249C8—22no—
250C8—41no—
251C22—21no—
252C22—21no—
253C22—82no—
254C22—82no—
255C22—82no—
256C22#447×C12711yescharacteristic
257C22C2×SD32×C12711yescharacteristic
258C22#509×C12711yescharacteristic
259C4—21no—
260C4—22no—
261C4—41no—
262C4—82no—
263C4—82no—
264C4—161no—
265C4#454×C12711yescharacteristic
266C4B(4,𝔽2)×C12711yescharacteristic
267C2—21no—
268C2—21no—
269C2—162no—
270C2C127×#159811yescharacteristic
271C2C127×#218711yescharacteristic
272C2C127×#219111yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#3245C2
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
2. C127×#2185C2semidirect
3. C127×#2056C2semidirect
4. C127×#1347C2semidirect
5. C127×#1322C2semidirect
109. C2×C4×C127SD32semidirect
130. #12161C127direct (N × H)
138. #2185C2×C127semidirect
139. #2056C2×C127semidirect
140. #1347C2×C127semidirect
141. #1322C2×C127semidirect
142. C127#12161direct (N × H)
246. C2×C4SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32452
C127×#21852
C127×#20562
C127×#13472
C127×#13222
#12161127