Finite Group Databasebeta

← C127×#10327 #44574000

Subgroups — C127×#10327 #44574000

Order 32512.

806 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3192C211yesmaximal, characteristic
2C127×#2436C212yesmaximal, complemented
3C127×#1784C211yesmaximal, characteristic, complemented
4C127×#1507C212yesmaximal, complemented
5C127×#1297C211yesmaximal, characteristic, complemented
6C23×Q8×C127C2211yescharacteristic
7C2×#124×C127—22no—
8C2×#121×C127C2212yes—
9#481×C127—22no—
10#469×C127—22no—
11#447×C127—22no—
12C2×#101×C127—22no—
13#407×C127C2211yescharacteristic
14#407×C127C2211yescharacteristic
15#393×C127—24no—
16#391×C127—22no—
17#386×C127—24no—
18C4×C22⋊C4×C127C2212yes—
19#327×C127—24no—
20C22×Q8×C127—22no—
21C22×Q8×C127—21no—
22C22×Q8×C127—22no—
23C22×Q8×C127—21no—
24C22×Q8×C127—44no—
25C22×D4×C127—22no—
26C22×D4×C127D412yes—
27C23×C4×C127D412yes—
28C23×C4×C127C2311yescharacteristic
29C2×D8×C127—22no—
30C2×M16×C127—21no—
31C2×M16×C127D412yes—
32#124×C127—48no—
33#121×C127—22no—
34#121×C127—22no—
35#121×C127—22no—
36#121×C127—22no—
37C2×C4⋊C4×C127D412yes—
38C2×C22⋊C4×C127—28no—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C42×C127—22no—
42#106×C127—22no—
43#102×C127—24no—
44#101×C127—44no—
45C22⋊C8×C127—22no—
46C22⋊C8×C127—22no—
47#95×C127—28no—
48#95×C127D414yes—
49C24×C127—22no—
50C24×C127C2×D411yescharacteristic
51C2×Q8×C127—24no—
52C2×Q8×C127—28no—
53C2×Q8×C127—44no—
54C2×Q8×C127—42no—
55C2×Q8×C127—48no—
56C2×Q8×C127—44no—
57C2×Q8×C127—41no—
58C2×Q8×C127—42no—
59C2×Q8×C127—41no—
60C2×D4×C127—22no—
61C2×D4×C127—22no—
62C2×D4×C127—44no—
63C2×D4×C127—48no—
64C2×D4×C127—42no—
65C2×D4×C127—42no—
66C22×C4×C127—24no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127—24no—
70C22×C4×C127—44no—
71C22×C4×C127—42no—
72C22×C4×C127C2×D412yes—
73C22×C4×C127C2×D411yescharacteristic
74C22×C4×C127C2×D412yes—
75C22×C4×C127#4111yescharacteristic
76D8×C127—82no—
77M16×C127—42no—
78M16×C127—42no—
79C2×C8×C127—22no—
80C2×C8×C127—41no—
81C4⋊C4×C127—22no—
82C4⋊C4×C127—22no—
83C22⋊C4×C127—416no—
84C22⋊C4×C127—44no—
85C22⋊C4×C127—42no—
86C22⋊C4×C127—42no—
87C42×C127—44no—
88C23×C127—24no—
89C23×C127—41no—
90C23×C127—44no—
91C23×C127—44no—
92C23×C127—42no—
93C23×C127—44no—
94C23×C127#12012yes—
95C23×C127#12111yescharacteristic
96Q8×C127—44no—
97Q8×C127—48no—
98Q8×C127—44no—
99D4×C127—44no—
100D4×C127—82no—
101D4×C127—84no—
102D4×C127—84no—
103C2×C4×C127—24no—
104C2×C4×C127—22no—
105C2×C4×C127—28no—
106C2×C4×C127—24no—
107C2×C4×C127—42no—
108C2×C4×C127—41no—
109C2×C4×C127—44no—
110C2×C4×C127—42no—
111C2×C4×C127—42no—
112C2×C4×C127—44no—
113C2×C4×C127—44no—
114C2×C4×C127—44no—
115C2×C4×C127—42no—
116C2×C4×C127—44no—
117C2×C4×C127#12011yescharacteristic
118C2×C4×C127#12112yes—
119C2×C4×C127#12711yescharacteristic
120C8×C127—42no—
121C8×C127—81no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—21no—
126C22×C127—44no—
127C22×C127—88no—
128C22×C127—82no—
129C22×C127—82no—
130C22×C127—82no—
131C22×C127#39211yescharacteristic
132C22×C127#45414yes—
133C4×C127—21no—
134C4×C127—22no—
135C4×C127—44no—
136C4×C127—84no—
137C4×C127—82no—
138C4×C127B(4,𝔽2)12yes—
139#10327C12711yesmaximal, characteristic, direct product factor, Sylow
140C2×C127—24no—
141C2×C127—41no—
142C2×C127—162no—
143C2×C127#200312yes—
144C2×C127#202111yescharacteristic
145#3192C2×C12711yescharacteristic
146#2436C2×C12712yescomplemented
147#1784C2×C12711yescharacteristic, complemented
148#1507C2×C12712yescomplemented
149#1297C2×C12711yescharacteristic, complemented
150C127#1032711yescharacteristic, direct product factor, Sylow
151C23×Q8C22×C12711yescharacteristic
152C2×#124—22no—
153C2×#121C22×C12712yes—
154#481—22no—
155#469—22no—
156#447—22no—
157C2×#101—22no—
158#407C22×C12711yescharacteristic
159#407C22×C12711yescharacteristic
160#393—24no—
161#391—22no—
162#386—24no—
163C4×C22⋊C4C22×C12712yes—
164#327—24no—
165C22×Q8—22no—
166C22×Q8—21no—
167C22×Q8—22no—
168C22×Q8—21no—
169C22×Q8—44no—
170C22×D4—22no—
171C22×D4D4×C12712yes—
172C23×C4D4×C12712yes—
173C23×C4C23×C12711yescharacteristic
174C2×D8—22no—
175C2×M16—21no—
176C2×M16D4×C12712yes—
177#124—48no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—22no—
182C2×C4⋊C4D4×C12712yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C42—22no—
187#106—22no—
188#102—24no—
189#101—44no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192#95—28no—
193#95D4×C12714yes—
194C24—22no—
195C24C2×D4×C12711yescharacteristic
196C2×Q8—24no—
197C2×Q8—28no—
198C2×Q8—44no—
199C2×Q8—42no—
200C2×Q8—48no—
201C2×Q8—44no—
202C2×Q8—41no—
203C2×Q8—42no—
204C2×Q8—41no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—44no—
208C2×D4—48no—
209C2×D4—42no—
210C2×D4—42no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4C2×D4×C12712yes—
218C22×C4C2×D4×C12711yescharacteristic
219C22×C4C2×D4×C12712yes—
220C22×C4#41×C12711yescharacteristic
221D8—82no—
222M16—42no—
223M16—42no—
224C2×C8—22no—
225C2×C8—41no—
226C4⋊C4—22no—
227C4⋊C4—22no—
228C22⋊C4—416no—
229C22⋊C4—44no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C42—44no—
233C23—24no—
234C23—41no—
235C23—44no—
236C23—44no—
237C23—42no—
238C23—44no—
239C23#120×C12712yes—
240C23#121×C12711yescharacteristic
241Q8—44no—
242Q8—48no—
243Q8—44no—
244D4—44no—
245D4—82no—
246D4—84no—
247D4—84no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4#120×C12711yescharacteristic
263C2×C4#121×C12712yes—
264C2×C4#127×C12711yescharacteristic
265C8—42no—
266C8—81no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—21no—
271C22—44no—
272C22—88no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22#392×C12711yescharacteristic
277C22#454×C12714yes—
278C4—21no—
279C4—22no—
280C4—44no—
281C4—84no—
282C4—82no—
283C4B(4,𝔽2)×C12712yes—
284C2—24no—
285C2—41no—
286C2—162no—
287C2C127×#200312yes—
288C2C127×#202111yescharacteristic
subgroupG/H
center, socleC22×C127#392
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC127×#3192C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2436C2semidirect
3. C127×#1784C2semidirect
4. C127×#1507C2semidirect
5. C127×#1297C2semidirect
139. #10327C127direct (N × H)
146. #2436C2×C127semidirect
147. #1784C2×C127semidirect
148. #1507C2×C127semidirect
149. #1297C2×C127semidirect
150. C127#10327direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31922
C127×#24362
C127×#17842
C127×#15072
C127×#12972
#10327127