Finite Group Databasebeta

← C127×#12327 #44576000

Subgroups — C127×#12327 #44576000

Order 32512.

368 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3277C211yesmaximal, characteristic
2C127×#2251C211yesmaximal, characteristic, complemented
3C127×#2209C211yesmaximal, characteristic, complemented
4C127×#1689C211yesmaximal, characteristic, complemented
5C127×#1360C211yesmaximal, characteristic, complemented
6C127×#1334C211yesmaximal, characteristic, complemented
7C127×#1330C211yesmaximal, characteristic, complemented
8C2×#137×C127—21no—
9#548×C127—21no—
10C2×Q32×C127—21no—
11C2×Q32×C127—21no—
12C2×SD32×C127—21no—
13#495×C127C2211yescharacteristic
14#494×C127C2211yescharacteristic
15#474×C127—21no—
16#468×C127—21no—
17#464×C127—21no—
18#451×C127—21no—
19#449×C127—21no—
20C4×Q16×C127C2211yescharacteristic
21#436×C127C2211yescharacteristic
22C4⋊C16×C127C2211yescharacteristic
23#358×C127—21no—
24#346×C127C2211yescharacteristic
25#334×C127C2211yescharacteristic
26#332×C127—21no—
27C2×#41×C127—21no—
28C22×Q8×C127—21no—
29#137×C127—42no—
30#137×C127—42no—
31C2×Q16×C127—21no—
32C2×Q16×C127—21no—
33C2×Q16×C127—21no—
34C2×Q16×C127D411yescharacteristic
35C2×Q16×C127D411yescharacteristic
36C2×SD16×C127—21no—
37C2×SD16×C127—21no—
38C2×D8×C127D411yescharacteristic
39C2×M16×C127—21no—
40C4⋊Q8×C127—21no—
41#124×C127—21no—
42#124×C127—41no—
43C22⋊Q8×C127—21no—
44C22⋊Q8×C127—41no—
45#121×C127—21no—
46#121×C127—41no—
47C4×Q8×C127—21no—
48C4×D4×C127—41no—
49C4×D4×C127D411yescharacteristic
50Q32×C127—42no—
51Q32×C127—42no—
52SD32×C127—42no—
53C2×C16×C127—21no—
54C2×C16×C127—21no—
55#107×C127—41no—
56#107×C127D411yescharacteristic
57C4⋊C8×C127—21no—
58C4⋊C8×C127D411yescharacteristic
59#103×C127—21no—
60#103×C127—21no—
61#102×C127—21no—
62#102×C127—41no—
63C22⋊C8×C127—21no—
64C4×C8×C127C2311yescharacteristic
65#41×C127—42no—
66#41×C127—81no—
67C2×Q8×C127—21no—
68C2×Q8×C127—21no—
69C2×Q8×C127—41no—
70C2×Q8×C127—42no—
71C2×Q8×C127—41no—
72C2×Q8×C127—81no—
73C2×D4×C127—21no—
74C2×D4×C127—41no—
75C2×D4×C127—41no—
76C2×D4×C127D811yescharacteristic
77C22×C4×C127—21no—
78C22×C4×C127—41no—
79C22×C4×C127—41no—
80Q16×C127—21no—
81Q16×C127—41no—
82Q16×C127—41no—
83Q16×C127—41no—
84Q16×C127—42no—
85Q16×C127—41no—
86Q16×C127—41no—
87Q16×C127D812yes—
88SD16×C127—42no—
89SD16×C127—41no—
90SD16×C127—41no—
91D8×C127—41no—
92M16×C127—41no—
93M16×C127—41no—
94C2×C8×C127—21no—
95C2×C8×C127—41no—
96C2×C8×C127C2×D411yescharacteristic
97C2×C8×C127C2×D411yescharacteristic
98C4⋊C4×C127—21no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127D811yescharacteristic
104C22⋊C4×C127—21no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C42×C127—41no—
110C42×C127C2×D411yescharacteristic
111C16×C127—22no—
112C16×C127—41no—
113C23×C127—21no—
114C23×C127—41no—
115Q8×C127—22no—
116Q8×C127—41no—
117Q8×C127—41no—
118Q8×C127—41no—
119Q8×C127—82no—
120Q8×C127—81no—
121Q8×C127—81no—
122D4×C127—41no—
123D4×C127—41no—
124D4×C127—41no—
125D4×C127—82no—
126D4×C127—82no—
127C2×C4×C127—21no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—41no—
133C2×C4×C127—82no—
134C2×C4×C127—82no—
135C2×C4×C127—81no—
136C2×C4×C127#12011yescharacteristic
137C2×C4×C127C2×D811yescharacteristic
138C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
139C8×C127—21no—
140C8×C127—41no—
141C8×C127—81no—
142C8×C127C2×D811yescharacteristic
143C8×C127AGL(1,ℤ/8)11yescharacteristic
144C22×C127—41no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127—81no—
148C22×C127—81no—
149C22×C127—81no—
150C22×C127#44711yescharacteristic
151C4×C127—21no—
152C4×C127—21no—
153C4×C127—41no—
154C4×C127—42no—
155C4×C127—81no—
156C4×C127—81no—
157C4×C127—161no—
158C4×C127#44711yescharacteristic
159C4×C127#45311yescharacteristic
160#12327C12711yesmaximal, characteristic, direct product factor, Sylow
161C2×C127—81no—
162C2×C127—161no—
163C2×C127#161911yescharacteristic
164C2×C127#219111yescharacteristic
165C2×C127#219511yescharacteristic
166#3277C2×C12711yescharacteristic
167#2251C2×C12711yescharacteristic, complemented
168#2209C2×C12711yescharacteristic, complemented
169#1689C2×C12711yescharacteristic, complemented
170#1360C2×C12711yescharacteristic, complemented
171#1334C2×C12711yescharacteristic, complemented
172#1330C2×C12711yescharacteristic, complemented
173C127#1232711yescharacteristic, direct product factor, Sylow
174C2×#137—21no—
175#548—21no—
176C2×Q32—21no—
177C2×Q32—21no—
178C2×SD32—21no—
179#495C22×C12711yescharacteristic
180#494C22×C12711yescharacteristic
181#474—21no—
182#468—21no—
183#464—21no—
184#451—21no—
185#449—21no—
186C4×Q16C22×C12711yescharacteristic
187#436C22×C12711yescharacteristic
188C4⋊C16C22×C12711yescharacteristic
189#358—21no—
190#346C22×C12711yescharacteristic
191#334C22×C12711yescharacteristic
192#332—21no—
193C2×#41—21no—
194C22×Q8—21no—
195#137—42no—
196#137—42no—
197C2×Q16—21no—
198C2×Q16—21no—
199C2×Q16—21no—
200C2×Q16D4×C12711yescharacteristic
201C2×Q16D4×C12711yescharacteristic
202C2×SD16—21no—
203C2×SD16—21no—
204C2×D8D4×C12711yescharacteristic
205C2×M16—21no—
206C4⋊Q8—21no—
207#124—21no—
208#124—41no—
209C22⋊Q8—21no—
210C22⋊Q8—41no—
211#121—21no—
212#121—41no—
213C4×Q8—21no—
214C4×D4—41no—
215C4×D4D4×C12711yescharacteristic
216Q32—42no—
217Q32—42no—
218SD32—42no—
219C2×C16—21no—
220C2×C16—21no—
221#107—41no—
222#107D4×C12711yescharacteristic
223C4⋊C8—21no—
224C4⋊C8D4×C12711yescharacteristic
225#103—21no—
226#103—21no—
227#102—21no—
228#102—41no—
229C22⋊C8—21no—
230C4×C8C23×C12711yescharacteristic
231#41—42no—
232#41—81no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—41no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—81no—
239C2×D4—21no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4D8×C12711yescharacteristic
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246Q16—21no—
247Q16—41no—
248Q16—41no—
249Q16—41no—
250Q16—42no—
251Q16—41no—
252Q16—41no—
253Q16D8×C12712yes—
254SD16—42no—
255SD16—41no—
256SD16—41no—
257D8—41no—
258M16—41no—
259M16—41no—
260C2×C8—21no—
261C2×C8—41no—
262C2×C8C2×D4×C12711yescharacteristic
263C2×C8C2×D4×C12711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4D8×C12711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C42—41no—
276C42C2×D4×C12711yescharacteristic
277C16—22no—
278C16—41no—
279C23—21no—
280C23—41no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285Q8—82no—
286Q8—81no—
287Q8—81no—
288D4—41no—
289D4—41no—
290D4—41no—
291D4—82no—
292D4—82no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4#120×C12711yescharacteristic
303C2×C4C2×D8×C12711yescharacteristic
304C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
305C8—21no—
306C8—41no—
307C8—81no—
308C8C2×D8×C12711yescharacteristic
309C8AGL(1,ℤ/8)×C12711yescharacteristic
310C22—41no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22#447×C12711yescharacteristic
317C4—21no—
318C4—21no—
319C4—41no—
320C4—42no—
321C4—81no—
322C4—81no—
323C4—161no—
324C4#447×C12711yescharacteristic
325C4#453×C12711yescharacteristic
326C2—81no—
327C2—161no—
328C2C127×#161911yescharacteristic
329C2C127×#219111yescharacteristic
330C2C127×#219511yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3277C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ #436×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2251C2semidirect
3. C127×#2209C2semidirect
4. C127×#1689C2semidirect
5. C127×#1360C2semidirect
6. C127×#1334C2semidirect
7. C127×#1330C2semidirect
160. #12327C127direct (N × H)
167. #2251C2×C127semidirect
168. #2209C2×C127semidirect
169. #1689C2×C127semidirect
170. #1360C2×C127semidirect
171. #1334C2×C127semidirect
172. #1330C2×C127semidirect
173. C127#12327direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32772
C127×#22512
C127×#22092
C127×#16892
C127×#13602
C127×#13342
C127×#13302
#12327127