Finite Group Databasebeta

← C127×#10566 #44574239

Subgroups — C127×#10566 #44574239

Order 32512.

490 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3189C211yesmaximal, characteristic
2C127×#1996C212yesmaximal
3C127×#1892C212yesmaximal
4C127×#1850C211yesmaximal, characteristic
5C127×#1289C211yesmaximal, characteristic
6C23×C8×C127C2211yescharacteristic
7C2×C22⋊Q8×C127C2212yes—
8#484×C127—22no—
9#467×C127—21no—
10#467×C127—21no—
11#451×C127—22no—
12C2×C4⋊C8×C127—21no—
13C2×#103×C127—22no—
14C2×C22⋊C8×C127C2211yescharacteristic
15#380×C127C2212yes—
16C4×C22⋊C4×C127C2211yescharacteristic
17#342×C127—22no—
18#342×C127—22no—
19#340×C127—22no—
20#336×C127—21no—
21C22×Q8×C127D412yes—
22C23×C4×C127C2311yescharacteristic
23C2×Q16×C127—21no—
24C2×Q16×C127—21no—
25C22×C8×C127—21no—
26C22×C8×C127—22no—
27C22×C8×C127—42no—
28C22×C8×C127D411yescharacteristic
29C22×C8×C127D411yescharacteristic
30C22⋊Q8×C127—22no—
31C22⋊Q8×C127—22no—
32C22⋊Q8×C127—22no—
33C22⋊Q8×C127—22no—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—22no—
36C2×C4⋊C4×C127D412yes—
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127D411yescharacteristic
40C2×C22⋊C4×C127D411yescharacteristic
41C2×C42×C127—21no—
42#107×C127—21no—
43#107×C127—21no—
44C4⋊C8×C127—42no—
45#103×C127—22no—
46#103×C127—22no—
47#103×C127—44no—
48C22⋊C8×C127—22no—
49#95×C127—21no—
50#95×C127—22no—
51C24×C127C2×D411yescharacteristic
52C2×Q8×C127—22no—
53C2×Q8×C127—22no—
54C2×Q8×C127—42no—
55C2×Q8×C127—42no—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127—22no—
59C22×C4×C127—22no—
60C22×C4×C127—41no—
61C22×C4×C127—41no—
62C22×C4×C127—42no—
63C22×C4×C127C2×D412yes—
64C22×C4×C127C2×D411yescharacteristic
65C22×C4×C127#4111yescharacteristic
66C22×C4×C127#4112yes—
67Q16×C127—81no—
68Q16×C127—81no—
69C2×C8×C127—24no—
70C2×C8×C127—22no—
71C2×C8×C127—21no—
72C2×C8×C127—21no—
73C2×C8×C127—41no—
74C2×C8×C127—42no—
75C2×C8×C127—41no—
76C2×C8×C127—41no—
77C2×C8×C127—41no—
78C4⋊C4×C127—22no—
79C4⋊C4×C127—22no—
80C4⋊C4×C127—42no—
81C4⋊C4×C127—42no—
82C4⋊C4×C127—42no—
83C4⋊C4×C127—42no—
84C22⋊C4×C127—21no—
85C22⋊C4×C127—22no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—42no—
90C22⋊C4×C127Q1612yescomplemented
91C42×C127—41no—
92C42×C127—41no—
93C23×C127—41no—
94C23×C127—41no—
95C23×C127#12112yes—
96C23×C127#12111yescharacteristic
97C23×C127C2×Q1611yescharacteristic
98C23×C127#13511yescharacteristic
99C23×C127#13712yes—
100Q8×C127—44no—
101Q8×C127—42no—
102Q8×C127—42no—
103C2×C4×C127—21no—
104C2×C4×C127—22no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—22no—
108C2×C4×C127—41no—
109C2×C4×C127—42no—
110C2×C4×C127—41no—
111C2×C4×C127—41no—
112C2×C4×C127—41no—
113C2×C4×C127—41no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
116C2×C4×C127—41no—
117C2×C4×C127—82no—
118C2×C4×C127—82no—
119C2×C4×C127—82no—
120C2×C4×C127—82no—
121C2×C4×C127—82no—
122C2×C4×C127#12011yescharacteristic
123C2×C4×C127#12312yes—
124C2×C4×C127#12411yescharacteristic
125C8×C127—42no—
126C8×C127—41no—
127C8×C127—41no—
128C22×C127—21no—
129C22×C127—22no—
130C22×C127—21no—
131C22×C127—22no—
132C22×C127—41no—
133C22×C127—41no—
134C22×C127—42no—
135C22×C127#39411yescharacteristic
136C22×C127#46212yes—
137C22×C127#46412yes—
138C22×C127#46711yescharacteristic
139C22×C127#47011yescharacteristic
140C4×C127—22no—
141C4×C127—21no—
142C4×C127—42no—
143C4×C127—81no—
144C4×C127—82no—
145C4×C127—82no—
146C4×C127—162no—
147C4×C127B(4,𝔽2)11yescharacteristic
148C4×C127#45811yescharacteristic
149#10566C12711yesmaximal, characteristic, direct product factor, Sylow
150C2×C127—21no—
151C2×C127—21no—
152C2×C127—22no—
153C2×C127—22no—
154C2×C127#204411yescharacteristic
155C2×C127#205311yescharacteristic
156C2×C127#205611yescharacteristic
157#3189C2×C12711yescharacteristic
158#1996C2×C12712yes—
159#1892C2×C12712yes—
160#1850C2×C12711yescharacteristic
161#1289C2×C12711yescharacteristic
162C127#1056611yescharacteristic, direct product factor, Sylow
163C23×C8C22×C12711yescharacteristic
164C2×C22⋊Q8C22×C12712yes—
165#484—22no—
166#467—21no—
167#467—21no—
168#451—22no—
169C2×C4⋊C8—21no—
170C2×#103—22no—
171C2×C22⋊C8C22×C12711yescharacteristic
172#380C22×C12712yes—
173C4×C22⋊C4C22×C12711yescharacteristic
174#342—22no—
175#342—22no—
176#340—22no—
177#336—21no—
178C22×Q8D4×C12712yes—
179C23×C4C23×C12711yescharacteristic
180C2×Q16—21no—
181C2×Q16—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—42no—
185C22×C8D4×C12711yescharacteristic
186C22×C8D4×C12711yescharacteristic
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4D4×C12712yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4D4×C12711yescharacteristic
197C2×C22⋊C4D4×C12711yescharacteristic
198C2×C42—21no—
199#107—21no—
200#107—21no—
201C4⋊C8—42no—
202#103—22no—
203#103—22no—
204#103—44no—
205C22⋊C8—22no—
206#95—21no—
207#95—22no—
208C24C2×D4×C12711yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4C2×D4×C12712yes—
221C22×C4C2×D4×C12711yescharacteristic
222C22×C4#41×C12711yescharacteristic
223C22×C4#41×C12712yes—
224Q16—81no—
225Q16—81no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C22⋊C4—21no—
242C22⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4Q16×C12712yescomplemented
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23#121×C12712yes—
253C23#121×C12711yescharacteristic
254C23C2×Q16×C12711yescharacteristic
255C23#135×C12711yescharacteristic
256C23#137×C12712yes—
257Q8—44no—
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4#120×C12711yescharacteristic
280C2×C4#123×C12712yes—
281C2×C4#124×C12711yescharacteristic
282C8—42no—
283C8—41no—
284C8—41no—
285C22—21no—
286C22—22no—
287C22—21no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#394×C12711yescharacteristic
293C22#462×C12712yes—
294C22#464×C12712yes—
295C22#467×C12711yescharacteristic
296C22#470×C12711yescharacteristic
297C4—22no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—82no—
302C4—82no—
303C4—162no—
304C4B(4,𝔽2)×C12711yescharacteristic
305C4#458×C12711yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—22no—
310C2C127×#204411yescharacteristic
311C2C127×#205311yescharacteristic
312C2C127×#205611yescharacteristic
subgroupG/H
center, socleC22×C127#394
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC24×C127C2×D4
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
90. C22⋊C4×C127Q16semidirect
149. #10566C127direct (N × H)
162. C127#10566direct (N × H)
247. C22⋊C4Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31892
C127×#19962
C127×#18922
C127×#18502
C127×#12892
#10566127