Finite Group Databasebeta

← C127×#12729 #44576402

Subgroups — C127×#12729 #44576402

Order 32512.

690 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3510C211yesmaximal, characteristic
2C127×#2133C212yesmaximal, complemented
3C2×#355×C127C212yesmaximal, complemented
4C127×#1830C212yesmaximal, complemented
5#559×C127—22no—
6#556×C127—22no—
7#551×C127—24no—
8#542×C127—22no—
9#536×C127C2211yescharacteristic
10#535×C127C411yescharacteristic
11#533×C127C411yescharacteristic
12#529×C127—22no—
13C2×#124×C127C2211yescharacteristic
14#519×C127C2211yescharacteristic
15#518×C127C411yescharacteristic
16C4×#41×C127C411yescharacteristic
17C2×#101×C127C2214yes—
18#356×C127—22no—
19#355×C127—22no—
20#355×C127—22no—
21#355×C127—22no—
22#344×C127—24no—
23C2×#41×C127C2×C411yescharacteristic
24C22×Q8×C127C2311yescharacteristic
25C22×D4×C127C2×C411yescharacteristic
26C2×M16×C127—24no—
27C4⋊Q8×C127—21no—
28C4⋊Q8×C127D412yes—
29#127×C127—21no—
30#126×C127—24no—
31#126×C127—24no—
32#125×C127—21no—
33#124×C127—24no—
34#124×C127—21no—
35#124×C127—22no—
36#124×C127D412yes—
37#124×C127D412yes—
38#124×C127D412yes—
39#123×C127—24no—
40#123×C127—24no—
41#123×C127—22no—
42C22⋊Q8×C127—24no—
43C22⋊Q8×C127—22no—
44C22⋊Q8×C127—22no—
45#121×C127—22no—
46#120×C127—22no—
47C4×Q8×C127—21no—
48C4×Q8×C127—22no—
49C4×Q8×C127—22no—
50C4×Q8×C127—22no—
51C4×D4×C127—24no—
52C4×D4×C127—21no—
53C4×D4×C127—22no—
54C4×D4×C127—22no—
55C4×D4×C127—22no—
56#117×C127—24no—
57#117×C127—21no—
58#117×C127C2×C411yescharacteristic
59#117×C127C2×C411yescharacteristic
60#117×C127C2×C411yescharacteristic
61#117×C127D414yes—
62C2×C22⋊C4×C127—22no—
63C2×C22⋊C4×C127—22no—
64C2×C42×C127—21no—
65C2×C42×C127C2×C411yescharacteristic
66#101×C127—24no—
67#101×C127—24no—
68C24×C127—21no—
69#41×C127—42no—
70C2×Q8×C127—22no—
71C2×Q8×C127—42no—
72C2×Q8×C127C22⋊C411yescharacteristic
73C2×Q8×C127C2×D412yes—
74C2×Q8×C127C2×D412yes—
75C2×D4×C127—21no—
76C2×D4×C127—24no—
77C2×D4×C127—21no—
78C2×D4×C127—42no—
79C2×D4×C127C22⋊C412yes—
80C2×D4×C127C22⋊C411yescharacteristic
81C22×C4×C127—22no—
82C22×C4×C127—24no—
83C22×C4×C127—21no—
84C22×C4×C127—21no—
85C22×C4×C127C22×C411yescharacteristic
86C22×C4×C127C2×D412yes—
87M16×C127—44no—
88M16×C127—44no—
89C2×C8×C127—44no—
90C4⋊C4×C127—22no—
91C4⋊C4×C127—22no—
92C4⋊C4×C127—22no—
93C4⋊C4×C127—22no—
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—24no—
98C4⋊C4×C127—24no—
99C22⋊C4×C127—24no—
100C22⋊C4×C127—22no—
101C22⋊C4×C127—24no—
102C22⋊C4×C127—21no—
103C22⋊C4×C127—24no—
104C22⋊C4×C127—24no—
105C22⋊C4×C127—24no—
106C22⋊C4×C127C22⋊C412yes—
107C22⋊C4×C127C22⋊C412yes—
108C42×C127—22no—
109C42×C127—21no—
110C42×C127—21no—
111C42×C127—21no—
112C42×C127—24no—
113C42×C127C22⋊C411yescharacteristic
114C42×C127C22⋊C411yescharacteristic
115C42×C127C22⋊C412yes—
116C23×C127—22no—
117C23×C127—21no—
118C23×C127—42no—
119C23×C127—82no—
120C23×C127C2×C22⋊C411yescharacteristic
121Q8×C127—42no—
122Q8×C127—42no—
123Q8×C127—41no—
124Q8×C127—82no—
125D4×C127—42no—
126D4×C127—41no—
127D4×C127—42no—
128D4×C127—42no—
129D4×C127—84no—
130C2×C4×C127—24no—
131C2×C4×C127—22no—
132C2×C4×C127—22no—
133C2×C4×C127—21no—
134C2×C4×C127—22no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—22no—
138C2×C4×C127—42no—
139C2×C4×C127—82no—
140C2×C4×C127—84no—
141C2×C4×C127—81no—
142C2×C4×C127C2×C22⋊C411yescharacteristic
143C2×C4×C127C2×C22⋊C411yescharacteristic
144C2×C4×C127#12014yes—
145C8×C127—84no—
146C22×C127—41no—
147C22×C127—42no—
148C22×C127—44no—
149C22×C127—41no—
150C22×C127—42no—
151C22×C127—162no—
152C22×C127#37911yescharacteristic
153C22×C127#41012yes—
154C4×C127—24no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—42no—
158C4×C127—41no—
159C4×C127—42no—
160C4×C127—41no—
161C4×C127—42no—
162C4×C127—41no—
163#12729C12711yesmaximal, characteristic, direct product factor, Sylow
164C2×C127—21no—
165C2×C127—22no—
166C2×C127—81no—
167C2×C127—82no—
168C2×C127#178211yescharacteristic
169#3510C2×C12711yescharacteristic
170#2133C2×C12712yescomplemented
171C2×#355C2×C12712yescomplemented
172#1830C2×C12712yescomplemented
173C127#1272911yescharacteristic, direct product factor, Sylow
174#559—22no—
175#556—22no—
176#551—24no—
177#542—22no—
178#536C22×C12711yescharacteristic
179#535C4×C12711yescharacteristic
180#533C4×C12711yescharacteristic
181#529—22no—
182C2×#124C22×C12711yescharacteristic
183#519C22×C12711yescharacteristic
184#518C4×C12711yescharacteristic
185C4×#41C4×C12711yescharacteristic
186C2×#101C22×C12714yes—
187#356—22no—
188#355—22no—
189#355—22no—
190#355—22no—
191#344—24no—
192C2×#41C2×C4×C12711yescharacteristic
193C22×Q8C23×C12711yescharacteristic
194C22×D4C2×C4×C12711yescharacteristic
195C2×M16—24no—
196C4⋊Q8—21no—
197C4⋊Q8D4×C12712yes—
198#127—21no—
199#126—24no—
200#126—24no—
201#125—21no—
202#124—24no—
203#124—21no—
204#124—22no—
205#124D4×C12712yes—
206#124D4×C12712yes—
207#124D4×C12712yes—
208#123—24no—
209#123—24no—
210#123—22no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214#121—22no—
215#120—22no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×D4—24no—
221C4×D4—21no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225#117—24no—
226#117—21no—
227#117C2×C4×C12711yescharacteristic
228#117C2×C4×C12711yescharacteristic
229#117C2×C4×C12711yescharacteristic
230#117D4×C12714yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C42—21no—
234C2×C42C2×C4×C12711yescharacteristic
235#101—24no—
236#101—24no—
237C24—21no—
238#41—42no—
239C2×Q8—22no—
240C2×Q8—42no—
241C2×Q8C22⋊C4×C12711yescharacteristic
242C2×Q8C2×D4×C12712yes—
243C2×Q8C2×D4×C12712yes—
244C2×D4—21no—
245C2×D4—24no—
246C2×D4—21no—
247C2×D4—42no—
248C2×D4C22⋊C4×C12712yes—
249C2×D4C22⋊C4×C12711yescharacteristic
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C22×C4×C12711yescharacteristic
255C22×C4C2×D4×C12712yes—
256M16—44no—
257M16—44no—
258C2×C8—44no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—22no—
270C22⋊C4—24no—
271C22⋊C4—21no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4C22⋊C4×C12712yes—
276C22⋊C4C22⋊C4×C12712yes—
277C42—22no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—24no—
282C42C22⋊C4×C12711yescharacteristic
283C42C22⋊C4×C12711yescharacteristic
284C42C22⋊C4×C12712yes—
285C23—22no—
286C23—21no—
287C23—42no—
288C23—82no—
289C23C2×C22⋊C4×C12711yescharacteristic
290Q8—42no—
291Q8—42no—
292Q8—41no—
293Q8—82no—
294D4—42no—
295D4—41no—
296D4—42no—
297D4—42no—
298D4—84no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—81no—
311C2×C4C2×C22⋊C4×C12711yescharacteristic
312C2×C4C2×C22⋊C4×C12711yescharacteristic
313C2×C4#120×C12714yes—
314C8—84no—
315C22—41no—
316C22—42no—
317C22—44no—
318C22—41no—
319C22—42no—
320C22—162no—
321C22#379×C12711yescharacteristic
322C22#410×C12712yes—
323C4—24no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C2—21no—
333C2—22no—
334C2—82no—
335C2—81no—
336C2C127×#178211yescharacteristic
subgroupG/H
center, socleC2×C127#1782
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×Q8C23×C127
prime radical#535×C127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C23×C127 ◁ #519×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2133C2semidirect
3. C2×#355×C127C2semidirect
4. C127×#1830C2semidirect
163. #12729C127direct (N × H)
170. #2133C2×C127semidirect
171. C2×#355C2×C127semidirect
172. #1830C2×C127semidirect
173. C127#12729direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#35102
C127×#21332
C2×#355×C1272
C127×#18302
#12729127