Finite Group Databasebeta

← C127×#10078 #44573751

Subgroups — C127×#10078 #44573751

Order 32512.

780 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C2×#496×C127C212yesmaximal, complemented
2C127×#2819C211yesmaximal, characteristic, complemented
3C127×#1961C211yesmaximal, characteristic, complemented
4C2×#329×C127C212yesmaximal, complemented
5C127×#1295C211yesmaximal, characteristic, complemented
6C22×SD16×C127—22no—
7C22×D8×C127—22no—
8C2×#127×C127C2211yescharacteristic
9C2×#124×C127C2212yes—
10#496×C127—24no—
11#496×C127—24no—
12C2×#102×C127—22no—
13C2×#97×C127C2211yescharacteristic
14C2×#97×C127C2212yes—
15#394×C127—22no—
16#393×C127—22no—
17#392×C127—22no—
18#382×C127C2211yescharacteristic
19#329×C127—24no—
20C22×Q8×C127D412yes—
21C22×D4×C127—22no—
22C22×D4×C127—22no—
23C22×D4×C127—21no—
24C22×D4×C127D412yes—
25C2×SD16×C127—28no—
26C2×SD16×C127—48no—
27C2×D8×C127—28no—
28C2×D8×C127—48no—
29C22×C8×C127—22no—
30C22×C8×C127—21no—
31#127×C127—22no—
32#127×C127—21no—
33#127×C127D412yescomplemented
34#124×C127—22no—
35#124×C127—22no—
36#124×C127—22no—
37#124×C127—22no—
38C2×C4⋊C4×C127D412yes—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127—24no—
42C2×C42×C127C2311yescharacteristic
43#102×C127—44no—
44#97×C127—22no—
45#97×C127—22no—
46#97×C127D414yescomplemented
47#95×C127—22no—
48#95×C127—22no—
49C24×C127—22no—
50C24×C127—41no—
51C2×Q8×C127—22no—
52C2×Q8×C127—22no—
53C2×Q8×C127—44no—
54C2×D4×C127—22no—
55C2×D4×C127—24no—
56C2×D4×C127—22no—
57C2×D4×C127—22no—
58C2×D4×C127—44no—
59C2×D4×C127—42no—
60C2×D4×C127—44no—
61C2×D4×C127—44no—
62C2×D4×C127—42no—
63C2×D4×C127—42no—
64C2×D4×C127—41no—
65C2×D4×C127—84no—
66C2×D4×C127—82no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127C2×D412yes—
70C22×C4×C127C2×D411yescharacteristic
71SD16×C127—416no—
72D8×C127—416no—
73C2×C8×C127—28no—
74C2×C8×C127—24no—
75C2×C8×C127—22no—
76C2×C8×C127—41no—
77C2×C8×C127—41no—
78C4⋊C4×C127—42no—
79C22⋊C4×C127—42no—
80C22⋊C4×C127—42no—
81C22⋊C4×C127—42no—
82C22⋊C4×C127—42no—
83C22⋊C4×C127—84no—
84C42×C127C2×D411yescharacteristic
85C42×C127C2×D412yes—
86C42×C127#4111yescharacteristic
87C23×C127—42no—
88C23×C127—42no—
89C23×C127—44no—
90C23×C127—44no—
91C23×C127—42no—
92C23×C127—42no—
93C23×C127—84no—
94C23×C127—81no—
95C23×C127—81no—
96C23×C127#12011yescharacteristic
97Q8×C127—44no—
98Q8×C127—82no—
99D4×C127—44no—
100D4×C127—44no—
101D4×C127—44no—
102D4×C127—44no—
103D4×C127—82no—
104D4×C127—84no—
105D4×C127—82no—
106D4×C127—88no—
107C2×C4×C127—21no—
108C2×C4×C127—24no—
109C2×C4×C127—21no—
110C2×C4×C127—42no—
111C2×C4×C127—44no—
112C2×C4×C127—44no—
113C2×C4×C127—42no—
114C2×C4×C127#12012yes—
115C2×C4×C127#12111yescharacteristic
116C2×C4×C127#12112yes—
117C2×C4×C127#12711yescharacteristic
118C8×C127—28no—
119C8×C127—42no—
120C22×C127—84no—
121C22×C127—84no—
122C22×C127—84no—
123C22×C127—82no—
124C22×C127—82no—
125C22×C127—82no—
126C22×C127—82no—
127C22×C127—162no—
128C22×C127—162no—
129C22×C127#39211yescharacteristic
130C22×C127#45311yescharacteristic
131C22×C127#45314yes—
132C22×C127#45411yescharacteristic
133C4×C127—22no—
134C4×C127—22no—
135C4×C127—21no—
136C4×C127—21no—
137C4×C127—82no—
138C4×C127—82no—
139#10078C12711yesmaximal, characteristic, direct product factor, Sylow
140C2×C127—162no—
141C2×C127—162no—
142C2×C127#165512yes—
143C2×C127#165612yes—
144C2×C127#200212yes—
145C2×C127#200611yescharacteristic
146C2×#496C2×C12712yescomplemented
147#2819C2×C12711yescharacteristic, complemented
148#1961C2×C12711yescharacteristic, complemented
149C2×#329C2×C12712yescomplemented
150#1295C2×C12711yescharacteristic, complemented
151C127#1007811yescharacteristic, direct product factor, Sylow
152C22×SD16—22no—
153C22×D8—22no—
154C2×#127C22×C12711yescharacteristic
155C2×#124C22×C12712yes—
156#496—24no—
157#496—24no—
158C2×#102—22no—
159C2×#97C22×C12711yescharacteristic
160C2×#97C22×C12712yes—
161#394—22no—
162#393—22no—
163#392—22no—
164#382C22×C12711yescharacteristic
165#329—24no—
166C22×Q8D4×C12712yes—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4D4×C12712yes—
171C2×SD16—28no—
172C2×SD16—48no—
173C2×D8—28no—
174C2×D8—48no—
175C22×C8—22no—
176C22×C8—21no—
177#127—22no—
178#127—21no—
179#127D4×C12712yescomplemented
180#124—22no—
181#124—22no—
182#124—22no—
183#124—22no—
184C2×C4⋊C4D4×C12712yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C42C23×C12711yescharacteristic
189#102—44no—
190#97—22no—
191#97—22no—
192#97D4×C12714yescomplemented
193#95—22no—
194#95—22no—
195C24—22no—
196C24—41no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—44no—
200C2×D4—22no—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—44no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—41no—
211C2×D4—84no—
212C2×D4—82no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4C2×D4×C12712yes—
216C22×C4C2×D4×C12711yescharacteristic
217SD16—416no—
218D8—416no—
219C2×C8—28no—
220C2×C8—24no—
221C2×C8—22no—
222C2×C8—41no—
223C2×C8—41no—
224C4⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—84no—
230C42C2×D4×C12711yescharacteristic
231C42C2×D4×C12712yes—
232C42#41×C12711yescharacteristic
233C23—42no—
234C23—42no—
235C23—44no—
236C23—44no—
237C23—42no—
238C23—42no—
239C23—84no—
240C23—81no—
241C23—81no—
242C23#120×C12711yescharacteristic
243Q8—44no—
244Q8—82no—
245D4—44no—
246D4—44no—
247D4—44no—
248D4—44no—
249D4—82no—
250D4—84no—
251D4—82no—
252D4—88no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4#120×C12712yes—
261C2×C4#121×C12711yescharacteristic
262C2×C4#121×C12712yes—
263C2×C4#127×C12711yescharacteristic
264C8—28no—
265C8—42no—
266C22—84no—
267C22—84no—
268C22—84no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—82no—
273C22—162no—
274C22—162no—
275C22#392×C12711yescharacteristic
276C22#453×C12711yescharacteristic
277C22#453×C12714yes—
278C22#454×C12711yescharacteristic
279C4—22no—
280C4—22no—
281C4—21no—
282C4—21no—
283C4—82no—
284C4—82no—
285C2—162no—
286C2—162no—
287C2C127×#165512yes—
288C2C127×#165612yes—
289C2C127×#200212yes—
290C2C127×#200611yescharacteristic
subgroupG/H
center, socleC23×C127#120
commutator, Frattini, cosocleC2×C42C23×C127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#496×C127C2semidirect
2. C127×#2819C2semidirect
3. C127×#1961C2semidirect
4. C2×#329×C127C2semidirect
5. C127×#1295C2semidirect
33. #127×C127D4semidirect
46. #97×C127D4semidirect
139. #10078C127direct (N × H)
146. C2×#496C2×C127semidirect
147. #2819C2×C127semidirect
148. #1961C2×C127semidirect
149. C2×#329C2×C127semidirect
150. #1295C2×C127semidirect
151. C127#10078direct (N × H)
179. #127D4×C127semidirect
192. #97D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#496×C1272
C127×#28192
C127×#19612
C2×#329×C1272
C127×#12952
#10078127