Finite Group Databasebeta

← C127×#12282 #44575955

Subgroups — C127×#12282 #44575955

Order 32512.

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

normal subgroups only

This group has 344 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×#3189C211yesmaximal, characteristic
2C127×#2236C212yesmaximal, complemented
3C127×#2076C211yesmaximal, characteristic
4C127×#1348C212yesmaximal, complemented
5C127×#1314C211yesmaximal, characteristic
6C2×C22⋊Q8×C127—21no—
7#484×C127—22no—
8#467×C127—21no—
9#467×C127C2212yes—
10#451×C127—22no—
11C2×#107×C127C2211yescharacteristic
12C2×C22⋊C8×C127C2211yescharacteristic
13#380×C127—21no—
14#366×C127—22no—
15#358×C127—22no—
16C22⋊C16×C127C2212yes—
17#340×C127—21no—
18#340×C127C2211yescharacteristic
19C22×Q8×C127—21no—
20C23×C4×C127D411yescharacteristic
21C2×Q16×C127—21no—
22C2×Q16×C127D412yes—
23C22×C8×C127D411yescharacteristic
24C22×C8×C127C2311yescharacteristic
25C22⋊Q8×C127—22no—
26C22⋊Q8×C127—22no—
27C22⋊Q8×C127—41no—
28C22⋊Q8×C127—41no—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127—41no—
33C2×C22⋊C4×C127—41no—
34C2×C22⋊C4×C127—41no—
35C2×C16×C127—22no—
36#107×C127—21no—
37#107×C127—22no—
38#107×C127D412yes—
39#103×C127—22no—
40#103×C127—22no—
41C22⋊C8×C127—22no—
42#95×C127—41no—
43C24×C127D811yescharacteristic
44C2×Q8×C127—22no—
45C2×Q8×C127—41no—
46C2×Q8×C127—81no—
47C2×Q8×C127—81no—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127—41no—
51C22×C4×C127—41no—
52C22×C4×C127—41no—
53C22×C4×C127—41no—
54C22×C4×C127—41no—
55C22×C4×C127—41no—
56C22×C4×C127D811yescharacteristic
57C22×C4×C127C2×D411yescharacteristic
58Q16×C127—42no—
59Q16×C127—81no—
60C2×C8×C127—21no—
61C2×C8×C127—41no—
62C2×C8×C127—41no—
63C2×C8×C127C2×D412yes—
64C4⋊C4×C127—22no—
65C4⋊C4×C127—41no—
66C4⋊C4×C127—42no—
67C4⋊C4×C127—41no—
68C4⋊C4×C127—41no—
69C4⋊C4×C127—42no—
70C4⋊C4×C127—41no—
71C4⋊C4×C127—41no—
72C22⋊C4×C127—42no—
73C22⋊C4×C127—42no—
74C22⋊C4×C127—82no—
75C16×C127—42no—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—41no—
79C23×C127—41no—
80C23×C127Q3212yes—
81C23×C127C2×D811yescharacteristic
82Q8×C127—42no—
83Q8×C127—42no—
84Q8×C127—82no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—41no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—42no—
92C2×C4×C127—42no—
93C2×C4×C127—81no—
94C2×C4×C127—82no—
95C2×C4×C127—82no—
96C2×C4×C127—81no—
97C2×C4×C127—81no—
98C2×C4×C127—81no—
99C2×C4×C127—81no—
100C2×C4×C127—81no—
101C2×C4×C127—81no—
102C2×C4×C127#12011yescharacteristic
103C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
104C8×C127—22no—
105C8×C127—41no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127—22no—
109C22×C127—42no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—81no—
113C22×C127#44711yescharacteristic
114C22×C127C2×Q3211yescharacteristic
115C22×C127#51011yescharacteristic
116C4×C127—21no—
117C4×C127—41no—
118C4×C127—82no—
119C4×C127—82no—
120C4×C127—161no—
121C4×C127—161no—
122C4×C127#45511yescharacteristic
123C4×C127#45811yescharacteristic
124#12282C12711yesmaximal, characteristic, direct product factor, Sylow
125C2×C127—21no—
126C2×C127—21no—
127C2×C127—42no—
128C2×C127#160711yescharacteristic
129C2×C127#218811yescharacteristic
130C2×C127#219011yescharacteristic
131#3189C2×C12711yescharacteristic
132#2236C2×C12712yescomplemented
133#2076C2×C12711yescharacteristic
134#1348C2×C12712yescomplemented
135#1314C2×C12711yescharacteristic
136C127#1228211yescharacteristic, direct product factor, Sylow
137C2×C22⋊Q8—21no—
138#484—22no—
139#467—21no—
140#467C22×C12712yes—
141#451—22no—
142C2×#107C22×C12711yescharacteristic
143C2×C22⋊C8C22×C12711yescharacteristic
144#380—21no—
145#366—22no—
146#358—22no—
147C22⋊C16C22×C12712yes—
148#340—21no—
149#340C22×C12711yescharacteristic
150C22×Q8—21no—
151C23×C4D4×C12711yescharacteristic
152C2×Q16—21no—
153C2×Q16D4×C12712yes—
154C22×C8D4×C12711yescharacteristic
155C22×C8C23×C12711yescharacteristic
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C16—22no—
167#107—21no—
168#107—22no—
169#107D4×C12712yes—
170#103—22no—
171#103—22no—
172C22⋊C8—22no—
173#95—41no—
174C24D8×C12711yescharacteristic
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—81no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4D8×C12711yescharacteristic
188C22×C4C2×D4×C12711yescharacteristic
189Q16—42no—
190Q16—81no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8C2×D4×C12712yes—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—82no—
206C16—42no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23Q32×C12712yes—
212C23C2×D8×C12711yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4#120×C12711yescharacteristic
234C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#447×C12711yescharacteristic
245C22C2×Q32×C12711yescharacteristic
246C22#510×C12711yescharacteristic
247C4—21no—
248C4—41no—
249C4—82no—
250C4—82no—
251C4—161no—
252C4—161no—
253C4#455×C12711yescharacteristic
254C4#458×C12711yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2C127×#160711yescharacteristic
259C2C127×#218811yescharacteristic
260C2C127×#219011yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC24×C127D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C22⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2236C2semidirect
4. C127×#1348C2semidirect
124. #12282C127direct (N × H)
132. #2236C2×C127semidirect
134. #1348C2×C127semidirect
136. C127#12282direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31892
C127×#22362
C127×#20762
C127×#13482
C127×#13142
#12282127