Finite Group Databasebeta

← C127×#8474 #44572147

Subgroups — C127×#8474 #44572147

Order 32512.

682 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 150 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×#3509C211yesmaximal, characteristic
2C127×#1997C212yesmaximal, complemented
3C127×#1561C211yesmaximal, characteristic, complemented
4C127×#1526C211yesmaximal, characteristic, complemented
5C127×#1489C212yesmaximal
6#557×C127—22no—
7#556×C127—22no—
8#552×C127—24no—
9#541×C127—22no—
10#536×C127C412yes—
11#533×C127—22no—
12#533×C127C2211yescharacteristic
13C2×C4⋊Q8×C127C2211yescharacteristic
14#519×C127C412yes—
15C4×#41×C127C2211yescharacteristic
16#423×C127C2212yes—
17C4×M16×C127C2212yes—
18#343×C127—24no—
19#332×C127—22no—
20#330×C127—22no—
21Q8⋊C8×C127—22no—
22D4⋊C8×C127—22no—
23C2×#41×C127C2×C411yescharacteristic
24C22×Q8×C127C2×C412yes—
25C2×M16×C127—22no—
26C2×M16×C127—22no—
27C4⋊Q8×C127—22no—
28C4⋊Q8×C127—21no—
29C4⋊Q8×C127—21no—
30C4⋊Q8×C127—22no—
31C4⋊Q8×C127D412yes—
32#126×C127—28no—
33#125×C127—24no—
34#125×C127—28no—
35#125×C127D412yes—
36#124×C127—22no—
37#123×C127—24no—
38C22⋊Q8×C127—24no—
39C22⋊Q8×C127—22no—
40C22⋊Q8×C127—24no—
41C4×Q8×C127—24no—
42C4×Q8×C127—24no—
43C4×Q8×C127—24no—
44C4×Q8×C127—22no—
45C4×Q8×C127D412yes—
46C4×D4×C127—22no—
47C4×D4×C127D412yes—
48#117×C127—24no—
49#117×C127—24no—
50#117×C127—21no—
51#117×C127C2×C411yescharacteristic
52#117×C127C2×C412yes—
53C2×C4⋊C4×C127D414yes—
54C2×C42×C127—21no—
55C2×C42×C127C2311yescharacteristic
56C4⋊C8×C127—22no—
57C4⋊C8×C127—22no—
58#97×C127—22no—
59C4×C8×C127—22no—
60#41×C127—42no—
61C2×Q8×C127—22no—
62C2×Q8×C127—22no—
63C2×Q8×C127—22no—
64C2×Q8×C127—44no—
65C2×Q8×C127C22⋊C411yescharacteristic
66C2×D4×C127—21no—
67C2×D4×C127C22⋊C411yescharacteristic
68C22×C4×C127—21no—
69C22×C4×C127—22no—
70C22×C4×C127C22⋊C414yes—
71C22×C4×C127C22×C411yescharacteristic
72C22×C4×C127C2×D412yes—
73M16×C127—24no—
74M16×C127—42no—
75M16×C127—44no—
76C2×C8×C127—42no—
77C2×C8×C127—42no—
78C4⋊C4×C127—28no—
79C4⋊C4×C127—28no—
80C4⋊C4×C127—24no—
81C4⋊C4×C127—24no—
82C4⋊C4×C127—22no—
83C4⋊C4×C127—22no—
84C4⋊C4×C127—24no—
85C4⋊C4×C127C22⋊C412yes—
86C4⋊C4×C127C22⋊C414yes—
87C22⋊C4×C127—28no—
88C22⋊C4×C127—21no—
89C22⋊C4×C127—22no—
90C22⋊C4×C127—22no—
91C42×C127—28no—
92C42×C127—21no—
93C42×C127—22no—
94C42×C127—22no—
95C42×C127C2×D412yes—
96C42×C127C2×D411yescharacteristic
97C42×C127C2×D411yescharacteristic
98C23×C127—21no—
99C23×C127C2×C22⋊C411yescharacteristic
100Q8×C127—22no—
101Q8×C127—44no—
102Q8×C127—44no—
103Q8×C127—42no—
104Q8×C127—42no—
105D4×C127—22no—
106D4×C127—42no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—24no—
112C2×C4×C127—24no—
113C2×C4×C127—21no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127—44no—
118C2×C4×C127C2×C22⋊C411yescharacteristic
119C2×C4×C127C2×C22⋊C411yescharacteristic
120C2×C4×C127#12012yes—
121C2×C4×C127#12012yes—
122C8×C127—44no—
123C8×C127—82no—
124C22×C127—22no—
125C22×C127—21no—
126C22×C127—42no—
127C22×C127—41no—
128C22×C127#37911yescharacteristic
129C4×C127—22no—
130C4×C127—21no—
131C4×C127—42no—
132C4×C127—44no—
133C4×C127—41no—
134C4×C127—44no—
135C4×C127#41312yes—
136C4×C127#45412yes—
137C4×C127#45612yes—
138#8474C12711yesmaximal, characteristic, direct product factor, Sylow
139C2×C127—41no—
140C2×C127—81no—
141C2×C127#178511yescharacteristic
142C2×C127#179312yes—
143#3509C2×C12711yescharacteristic
144#1997C2×C12712yescomplemented
145#1561C2×C12711yescharacteristic, complemented
146#1526C2×C12711yescharacteristic, complemented
147#1489C2×C12712yes—
148C127#847411yescharacteristic, direct product factor, Sylow
149#557—22no—
150#556—22no—
151#552—24no—
152#541—22no—
153#536C4×C12712yes—
154#533—22no—
155#533C22×C12711yescharacteristic
156C2×C4⋊Q8C22×C12711yescharacteristic
157#519C4×C12712yes—
158C4×#41C22×C12711yescharacteristic
159#423C22×C12712yes—
160C4×M16C22×C12712yes—
161#343—24no—
162#332—22no—
163#330—22no—
164Q8⋊C8—22no—
165D4⋊C8—22no—
166C2×#41C2×C4×C12711yescharacteristic
167C22×Q8C2×C4×C12712yes—
168C2×M16—22no—
169C2×M16—22no—
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—22no—
174C4⋊Q8D4×C12712yes—
175#126—28no—
176#125—24no—
177#125—28no—
178#125D4×C12712yes—
179#124—22no—
180#123—24no—
181C22⋊Q8—24no—
182C22⋊Q8—22no—
183C22⋊Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8D4×C12712yes—
189C4×D4—22no—
190C4×D4D4×C12712yes—
191#117—24no—
192#117—24no—
193#117—21no—
194#117C2×C4×C12711yescharacteristic
195#117C2×C4×C12712yes—
196C2×C4⋊C4D4×C12714yes—
197C2×C42—21no—
198C2×C42C23×C12711yescharacteristic
199C4⋊C8—22no—
200C4⋊C8—22no—
201#97—22no—
202C4×C8—22no—
203#41—42no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—44no—
208C2×Q8C22⋊C4×C12711yescharacteristic
209C2×D4—21no—
210C2×D4C22⋊C4×C12711yescharacteristic
211C22×C4—21no—
212C22×C4—22no—
213C22×C4C22⋊C4×C12714yes—
214C22×C4C22×C4×C12711yescharacteristic
215C22×C4C2×D4×C12712yes—
216M16—24no—
217M16—42no—
218M16—44no—
219C2×C8—42no—
220C2×C8—42no—
221C4⋊C4—28no—
222C4⋊C4—28no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4C22⋊C4×C12712yes—
229C4⋊C4C22⋊C4×C12714yes—
230C22⋊C4—28no—
231C22⋊C4—21no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—28no—
235C42—21no—
236C42—22no—
237C42—22no—
238C42C2×D4×C12712yes—
239C42C2×D4×C12711yescharacteristic
240C42C2×D4×C12711yescharacteristic
241C23—21no—
242C23C2×C22⋊C4×C12711yescharacteristic
243Q8—22no—
244Q8—44no—
245Q8—44no—
246Q8—42no—
247Q8—42no—
248D4—22no—
249D4—42no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4C2×C22⋊C4×C12711yescharacteristic
262C2×C4C2×C22⋊C4×C12711yescharacteristic
263C2×C4#120×C12712yes—
264C2×C4#120×C12712yes—
265C8—44no—
266C8—82no—
267C22—21no—
268C22—22no—
269C22—41no—
270C22—42no—
271C22#379×C12711yescharacteristic
272C4—21no—
273C4—22no—
274C4—41no—
275C4—44no—
276C4—42no—
277C4—44no—
278C4#413×C12712yes—
279C4#454×C12712yes—
280C4#456×C12712yes—
281C2—41no—
282C2—81no—
283C2C127×#178511yescharacteristic
284C2C127×#179312yes—
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC2×#41×C127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1997C2semidirect
3. C127×#1561C2semidirect
4. C127×#1526C2semidirect
138. #8474C127direct (N × H)
144. #1997C2×C127semidirect
145. #1561C2×C127semidirect
146. #1526C2×C127semidirect
148. C127#8474direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#35092
C127×#19972
C127×#15612
C127×#15262
C127×#14892
#8474127