Finite Group Databasebeta

← C127×#12709 #44576382

Subgroups — C127×#12709 #44576382

Order 32512.

986 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3485C211yesmaximal, characteristic
2C127×#2133C214yesmaximal, complemented
3C127×#1785C212yesmaximal, complemented
4C2×H(5,𝔽2)×C127C411yescharacteristic
5C22×#41×C127C2211yescharacteristic
6#547×C127—24no—
7#546×C127—24no—
8#536×C127C2212yes—
9#534×C127—22no—
10#525×C127C412yes—
11C2×#123×C127—22no—
12#518×C127C411yescharacteristic
13C2×#101×C127C2214yescomplemented
14C2×#100×C127—22no—
15#407×C127—22no—
16#356×C127—24no—
17#355×C127—24no—
18H(5,𝔽2)×C127—44no—
19C2×#41×C127—22no—
20C2×#41×C127—24no—
21C2×#41×C127—44no—
22C2×#41×C127C2×C412yes—
23C22×Q8×C127C2311yescharacteristic
24C22×D4×C127—22no—
25C22×D4×C127—22no—
26C22×D4×C127C2×C411yescharacteristic
27C22×D4×C127D412yes—
28C23×C4×C127C2×C411yescharacteristic
29C23×C4×C127D412yes—
30C2×M16×C127—24no—
31C4⋊Q8×C127D414yes—
32#124×C127D414yes—
33#123×C127—28no—
34#123×C127—24no—
35#123×C127—22no—
36#123×C127—24no—
37C22⋊Q8×C127—22no—
38C22⋊Q8×C127—24no—
39#121×C127—28no—
40#121×C127—22no—
41#120×C127—24no—
42#120×C127—22no—
43C4×D4×C127—24no—
44C4×D4×C127—24no—
45#117×C127C2×C412yes—
46C2×C4⋊C4×C127—24no—
47C2×C22⋊C4×C127—24no—
48C2×C22⋊C4×C127—22no—
49#101×C127—28no—
50#100×C127—44no—
51C22⋊C8×C127—44no—
52C24×C127—22no—
53C24×C127C22⋊C411yescharacteristic
54C24×C127C22⋊C412yes—
55#41×C127—44no—
56#41×C127—416no—
57#41×C127—48no—
58C2×Q8×C127—42no—
59C2×Q8×C127C22⋊C412yes—
60C2×Q8×C127C2×D414yes—
61C2×D4×C127—24no—
62C2×D4×C127—24no—
63C2×D4×C127—24no—
64C2×D4×C127—22no—
65C2×D4×C127—24no—
66C2×D4×C127—22no—
67C2×D4×C127—24no—
68C2×D4×C127—42no—
69C2×D4×C127—48no—
70C2×D4×C127—44no—
71C2×D4×C127—44no—
72C2×D4×C127C22⋊C412yes—
73C22×C4×C127—22no—
74C22×C4×C127—24no—
75C22×C4×C127—24no—
76C22×C4×C127—22no—
77C22×C4×C127—22no—
78C22×C4×C127—24no—
79C22×C4×C127—42no—
80C22×C4×C127—42no—
81C22×C4×C127C22⋊C411yescharacteristic
82C22×C4×C127C22×C411yescharacteristic
83C22×C4×C127C2×D412yes—
84M16×C127—44no—
85M16×C127—44no—
86C2×C8×C127—44no—
87C4⋊C4×C127—24no—
88C4⋊C4×C127—22no—
89C4⋊C4×C127—28no—
90C22⋊C4×C127—24no—
91C22⋊C4×C127—24no—
92C22⋊C4×C127—24no—
93C22⋊C4×C127—22no—
94C22⋊C4×C127—28no—
95C42×C127C22⋊C414yes—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—22no—
99C23×C127—21no—
100C23×C127—22no—
101C23×C127—44no—
102C23×C127—82no—
103C23×C127—81no—
104C23×C127—84no—
105C23×C127#9914yes—
106C23×C127C2×C22⋊C411yescharacteristic
107Q8×C127—44no—
108Q8×C127—42no—
109D4×C127—28no—
110D4×C127—48no—
111D4×C127—42no—
112D4×C127—48no—
113D4×C127—44no—
114D4×C127—84no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127—22no—
118C2×C4×C127—24no—
119C2×C4×C127—21no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—48no—
123C2×C4×C127—44no—
124C2×C4×C127—44no—
125C2×C4×C127—44no—
126C2×C4×C127—84no—
127C2×C4×C127C2×C22⋊C412yes—
128C2×C4×C127#12014yes—
129C8×C127—84no—
130C22×C127—24no—
131C22×C127—41no—
132C22×C127—44no—
133C22×C127—44no—
134C22×C127—42no—
135C22×C127—44no—
136C22×C127—41no—
137C22×C127—42no—
138C22×C127—82no—
139C22×C127—82no—
140C22×C127—164no—
141C22×C127#37911yescharacteristic
142C22×C127C2×#9912yes—
143C4×C127—24no—
144C4×C127—22no—
145C4×C127—44no—
146C4×C127—41no—
147C4×C127—42no—
148#12709C12711yesmaximal, characteristic, direct product factor, Sylow
149C2×C127—21no—
150C2×C127—22no—
151C2×C127—44no—
152C2×C127—81no—
153C2×C127—82no—
154C2×C127#178111yescharacteristic
155#3485C2×C12711yescharacteristic
156#2133C2×C12714yescomplemented
157#1785C2×C12712yescomplemented
158C127#1270911yescharacteristic, direct product factor, Sylow
159C2×H(5,𝔽2)C4×C12711yescharacteristic
160C22×#41C22×C12711yescharacteristic
161#547—24no—
162#546—24no—
163#536C22×C12712yes—
164#534—22no—
165#525C4×C12712yes—
166C2×#123—22no—
167#518C4×C12711yescharacteristic
168C2×#101C22×C12714yescomplemented
169C2×#100—22no—
170#407—22no—
171#356—24no—
172#355—24no—
173H(5,𝔽2)—44no—
174C2×#41—22no—
175C2×#41—24no—
176C2×#41—44no—
177C2×#41C2×C4×C12712yes—
178C22×Q8C23×C12711yescharacteristic
179C22×D4—22no—
180C22×D4—22no—
181C22×D4C2×C4×C12711yescharacteristic
182C22×D4D4×C12712yes—
183C23×C4C2×C4×C12711yescharacteristic
184C23×C4D4×C12712yes—
185C2×M16—24no—
186C4⋊Q8D4×C12714yes—
187#124D4×C12714yes—
188#123—28no—
189#123—24no—
190#123—22no—
191#123—24no—
192C22⋊Q8—22no—
193C22⋊Q8—24no—
194#121—28no—
195#121—22no—
196#120—24no—
197#120—22no—
198C4×D4—24no—
199C4×D4—24no—
200#117C2×C4×C12712yes—
201C2×C4⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—22no—
204#101—28no—
205#100—44no—
206C22⋊C8—44no—
207C24—22no—
208C24C22⋊C4×C12711yescharacteristic
209C24C22⋊C4×C12712yes—
210#41—44no—
211#41—416no—
212#41—48no—
213C2×Q8—42no—
214C2×Q8C22⋊C4×C12712yes—
215C2×Q8C2×D4×C12714yes—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—48no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4C22⋊C4×C12712yes—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊C4×C12711yescharacteristic
237C22×C4C22×C4×C12711yescharacteristic
238C22×C4C2×D4×C12712yes—
239M16—44no—
240M16—44no—
241C2×C8—44no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—22no—
249C22⋊C4—28no—
250C42C22⋊C4×C12714yes—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—44no—
257C23—82no—
258C23—81no—
259C23—84no—
260C23#99×C12714yes—
261C23C2×C22⋊C4×C12711yescharacteristic
262Q8—44no—
263Q8—42no—
264D4—28no—
265D4—48no—
266D4—42no—
267D4—48no—
268D4—44no—
269D4—84no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—84no—
282C2×C4C2×C22⋊C4×C12712yes—
283C2×C4#120×C12714yes—
284C8—84no—
285C22—24no—
286C22—41no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—44no—
291C22—41no—
292C22—42no—
293C22—82no—
294C22—82no—
295C22—164no—
296C22#379×C12711yescharacteristic
297C22C2×#99×C12712yes—
298C4—24no—
299C4—22no—
300C4—44no—
301C4—41no—
302C4—42no—
303C2—21no—
304C2—22no—
305C2—44no—
306C2—81no—
307C2—82no—
308C2C127×#178111yescharacteristic
subgroupG/H
center, socleC2×C127#1781
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×Q8C23×C127
prime radicalC127×#3485C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C23×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2133C2semidirect
3. C127×#1785C2semidirect
13. C2×#101×C127C22semidirect
148. #12709C127direct (N × H)
156. #2133C2×C127semidirect
157. #1785C2×C127semidirect
158. C127#12709direct (N × H)
168. C2×#101C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34852
C127×#21332
C127×#17852
#12709127