Finite Group Databasebeta

← C127×#12108 #44575781

Subgroups — C127×#12108 #44575781

Order 32512.

442 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3199C211yesmaximal, characteristic, complemented
2C127×#2184C212yesmaximal, complemented
3C127×#2018C211yesmaximal, characteristic, complemented
4C127×#1349C212yesmaximal, complemented
5C127×#1315C211yesmaximal, characteristic, complemented
6C2×AGL(1,ℤ/8)×C127—21no—
7C22×D8×C127C2211yescharacteristic, complemented
8#535×C127—21no—
9C2×D16×C127—22no—
10#482×C127—22no—
11#469×C127—21no—
12#466×C127C2212yes—
13#447×C127—22no—
14C2×#100×C127—21no—
15#408×C127C2211yescharacteristic
16#357×C127—22no—
17C22⋊C16×C127C2212yescomplemented
18#341×C127C2211yescharacteristic, complemented
19C2×#41×C127D411yescharacteristic
20C22×D4×C127—21no—
21C22×D4×C127—21no—
22AGL(1,ℤ/8)×C127—42no—
23AGL(1,ℤ/8)×C127—42no—
24C2×SD16×C127—41no—
25C2×D8×C127—21no—
26C2×D8×C127—22no—
27C2×D8×C127—42no—
28C2×D8×C127—41no—
29C2×D8×C127D412yes—
30C2×M16×C127—21no—
31C2×M16×C127D411yescharacteristic
32C22×C8×C127C2311yescharacteristic
33#127×C127—41no—
34#124×C127—41no—
35#121×C127—22no—
36#121×C127—22no—
37#120×C127—41no—
38#120×C127—41no—
39#117×C127—21no—
40D16×C127—44no—
41C2×C16×C127—22no—
42#107×C127D412yes—
43#106×C127—21no—
44#102×C127—22no—
45#102×C127—22no—
46#100×C127—42no—
47C22⋊C8×C127—22no—
48C24×C127—41no—
49C24×C127—41no—
50#41×C127—41no—
51#41×C127—41no—
52C2×Q8×C127D811yescharacteristic
53C2×D4×C127—22no—
54C2×D4×C127—21no—
55C2×D4×C127—22no—
56C2×D4×C127—41no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C2×D4×C127—41no—
60C2×D4×C127—42no—
61C2×D4×C127—42no—
62C2×D4×C127—41no—
63C2×D4×C127—41no—
64C2×D4×C127—81no—
65C2×D4×C127—81no—
66C2×D4×C127D811yescharacteristic
67C22×C4×C127—21no—
68C22×C4×C127C2×D411yescharacteristic
69SD16×C127—44no—
70D8×C127—24no—
71D8×C127—42no—
72D8×C127—42no—
73D8×C127—44no—
74D8×C127—81no—
75M16×C127—22no—
76M16×C127—41no—
77M16×C127—41no—
78C2×C8×C127—21no—
79C2×C8×C127—41no—
80C2×C8×C127—41no—
81C2×C8×C127C2×D412yes—
82C4⋊C4×C127—22no—
83C22⋊C4×C127—42no—
84C22⋊C4×C127—41no—
85C22⋊C4×C127—42no—
86C42×C127—41no—
87C16×C127—42no—
88C23×C127—21no—
89C23×C127—42no—
90C23×C127—42no—
91C23×C127—81no—
92C23×C127—82no—
93C23×C127—81no—
94C23×C127—162no—
95C23×C127—162no—
96C23×C127AGL(1,ℤ/8)11yescharacteristic
97Q8×C127—41no—
98D4×C127—41no—
99D4×C127—42no—
100D4×C127—42no—
101D4×C127—42no—
102D4×C127—42no—
103D4×C127—81no—
104D4×C127—81no—
105D4×C127—82no—
106D4×C127—81no—
107D4×C127—162no—
108D4×C127—162no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127—42no—
115C2×C4×C127#12011yescharacteristic
116C2×C4×C127C2×D811yescharacteristic
117C8×C127—22no—
118C8×C127—42no—
119C8×C127—41no—
120C22×C127—21no—
121C22×C127—22no—
122C22×C127—41no—
123C22×C127—42no—
124C22×C127—82no—
125C22×C127—82no—
126C22×C127—82no—
127C22×C127—82no—
128C22×C127—162no—
129C22×C127—162no—
130C22×C127—164no—
131C22×C127—162no—
132C22×C127#44711yescharacteristic
133C4×C127—21no—
134C4×C127—41no—
135C4×C127—82no—
136C4×C127#45311yescharacteristic
137C4×C127B(4,𝔽2)11yescharacteristic
138#12108C12711yesmaximal, characteristic, direct product factor, Sylow
139C2×C127—41no—
140C2×C127—81no—
141C2×C127—162no—
142C2×C127—162no—
143C2×C127#159511yescharacteristic
144C2×C127#219212yes—
145#3199C2×C12711yescharacteristic, complemented
146#2184C2×C12712yescomplemented
147#2018C2×C12711yescharacteristic, complemented
148#1349C2×C12712yescomplemented
149#1315C2×C12711yescharacteristic, complemented
150C127#1210811yescharacteristic, direct product factor, Sylow
151C2×AGL(1,ℤ/8)—21no—
152C22×D8C22×C12711yescharacteristic, complemented
153#535—21no—
154C2×D16—22no—
155#482—22no—
156#469—21no—
157#466C22×C12712yes—
158#447—22no—
159C2×#100—21no—
160#408C22×C12711yescharacteristic
161#357—22no—
162C22⋊C16C22×C12712yescomplemented
163#341C22×C12711yescharacteristic, complemented
164C2×#41D4×C12711yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167AGL(1,ℤ/8)—42no—
168AGL(1,ℤ/8)—42no—
169C2×SD16—41no—
170C2×D8—21no—
171C2×D8—22no—
172C2×D8—42no—
173C2×D8—41no—
174C2×D8D4×C12712yes—
175C2×M16—21no—
176C2×M16D4×C12711yescharacteristic
177C22×C8C23×C12711yescharacteristic
178#127—41no—
179#124—41no—
180#121—22no—
181#121—22no—
182#120—41no—
183#120—41no—
184#117—21no—
185D16—44no—
186C2×C16—22no—
187#107D4×C12712yes—
188#106—21no—
189#102—22no—
190#102—22no—
191#100—42no—
192C22⋊C8—22no—
193C24—41no—
194C24—41no—
195#41—41no—
196#41—41no—
197C2×Q8D8×C12711yescharacteristic
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—81no—
210C2×D4—81no—
211C2×D4D8×C12711yescharacteristic
212C22×C4—21no—
213C22×C4C2×D4×C12711yescharacteristic
214SD16—44no—
215D8—24no—
216D8—42no—
217D8—42no—
218D8—44no—
219D8—81no—
220M16—22no—
221M16—41no—
222M16—41no—
223C2×C8—21no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8C2×D4×C12712yes—
227C4⋊C4—22no—
228C22⋊C4—42no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C42—41no—
232C16—42no—
233C23—21no—
234C23—42no—
235C23—42no—
236C23—81no—
237C23—82no—
238C23—81no—
239C23—162no—
240C23—162no—
241C23AGL(1,ℤ/8)×C12711yescharacteristic
242Q8—41no—
243D4—41no—
244D4—42no—
245D4—42no—
246D4—42no—
247D4—42no—
248D4—81no—
249D4—81no—
250D4—82no—
251D4—81no—
252D4—162no—
253D4—162no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4#120×C12711yescharacteristic
261C2×C4C2×D8×C12711yescharacteristic
262C8—22no—
263C8—42no—
264C8—41no—
265C22—21no—
266C22—22no—
267C22—41no—
268C22—42no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—162no—
274C22—162no—
275C22—164no—
276C22—162no—
277C22#447×C12711yescharacteristic
278C4—21no—
279C4—41no—
280C4—82no—
281C4#453×C12711yescharacteristic
282C4B(4,𝔽2)×C12711yescharacteristic
283C2—41no—
284C2—81no—
285C2—162no—
286C2—162no—
287C2C127×#159511yescharacteristic
288C2C127×#219212yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #408×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3199C2semidirect
2. C127×#2184C2semidirect
3. C127×#2018C2semidirect
4. C127×#1349C2semidirect
5. C127×#1315C2semidirect
7. C22×D8×C127C22semidirect
17. C22⋊C16×C127C22semidirect
18. #341×C127C22semidirect
138. #12108C127direct (N × H)
145. #3199C2×C127semidirect
146. #2184C2×C127semidirect
147. #2018C2×C127semidirect
148. #1349C2×C127semidirect
149. #1315C2×C127semidirect
150. C127#12108direct (N × H)
152. C22×D8C22×C127semidirect
162. C22⋊C16C22×C127semidirect
163. #341C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31992
C127×#21842
C127×#20182
C127×#13492
C127×#13152
#12108127