Finite Group Databasebeta

← C127×#12101 #44575774

Subgroups — C127×#12101 #44575774

Order 32512.

442 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2734C211yesmaximal, characteristic, complemented
2C127×#2007C211yesmaximal, characteristic
3C127×#1610C212yesmaximal
4C127×#1346C212yesmaximal, complemented
5C127×#1319C211yesmaximal, characteristic, complemented
6C2×#137×C127—21no—
7#537×C127C2211yescharacteristic
8C2×C4⋊Q8×C127—21no—
9#480×C127—22no—
10C2×#104×C127—21no—
11#398×C127C2212yes—
12#395×C127—22no—
13#391×C127C2211yescharacteristic
14#389×C127—21no—
15#387×C127—22no—
16#343×C127C2211yescharacteristic
17#327×C127—22no—
18#324×C127C2212yes—
19C2×#41×C127—21no—
20C22×Q8×C127—21no—
21C22×Q8×C127D411yescharacteristic
22#137×C127—42no—
23#137×C127—42no—
24C2×Q16×C127—41no—
25C2×SD16×C127—41no—
26C2×M16×C127—21no—
27C4⋊Q8×C127—42no—
28C4⋊Q8×C127—42no—
29C22⋊Q8×C127—41no—
30#121×C127—22no—
31#121×C127—41no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—22no—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127D412yes—
37C2×C4⋊C4×C127C2311yescharacteristic
38C2×C42×C127—21no—
39C2×C42×C127D411yescharacteristic
40#107×C127—42no—
41#104×C127—42no—
42#102×C127—42no—
43C22⋊C8×C127—22no—
44#95×C127—22no—
45#95×C127—22no—
46#95×C127—22no—
47#95×C127D412yes—
48#41×C127—41no—
49#41×C127—41no—
50#41×C127—41no—
51#41×C127—41no—
52C2×Q8×C127—21no—
53C2×Q8×C127—21no—
54C2×Q8×C127—42no—
55C2×Q8×C127—41no—
56C2×Q8×C127—41no—
57C2×Q8×C127—81no—
58C2×Q8×C127—81no—
59C2×Q8×C127D812yes—
60C2×D4×C127—21no—
61C2×D4×C127—22no—
62C2×D4×C127—41no—
63C22×C4×C127—21no—
64C22×C4×C127—22no—
65C22×C4×C127—21no—
66C22×C4×C127—22no—
67C22×C4×C127—41no—
68C22×C4×C127C2×D411yescharacteristic
69C22×C4×C127C2×D412yes—
70Q16×C127—44no—
71SD16×C127—44no—
72M16×C127—22no—
73M16×C127—41no—
74C2×C8×C127—41no—
75C2×C8×C127—42no—
76C4⋊C4×C127—24no—
77C4⋊C4×C127—22no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—42no—
80C4⋊C4×C127—44no—
81C4⋊C4×C127—42no—
82C4⋊C4×C127—42no—
83C4⋊C4×C127—81no—
84C22⋊C4×C127—42no—
85C42×C127—41no—
86C42×C127—42no—
87C23×C127—41no—
88C23×C127#12011yescharacteristic
89Q8×C127—42no—
90Q8×C127—41no—
91Q8×C127—41no—
92Q8×C127—81no—
93Q8×C127—84no—
94D4×C127—41no—
95D4×C127—41no—
96D4×C127—82no—
97D4×C127—81no—
98D4×C127—81no—
99C2×C4×C127—21no—
100C2×C4×C127—21no—
101C2×C4×C127—22no—
102C2×C4×C127—24no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127—41no—
106C2×C4×C127—42no—
107C2×C4×C127—42no—
108C2×C4×C127—41no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—42no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127—41no—
115C2×C4×C127—42no—
116C2×C4×C127—81no—
117C2×C4×C127—81no—
118C2×C4×C127C2×D811yescharacteristic
119C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
120C8×C127—42no—
121C8×C127—82no—
122C22×C127—21no—
123C22×C127—21no—
124C22×C127—81no—
125C22×C127—81no—
126C22×C127—81no—
127C22×C127#44711yescharacteristic
128C22×C127#45311yescharacteristic
129C22×C127B(4,𝔽2)11yescharacteristic
130C4×C127—21no—
131C4×C127—21no—
132C4×C127—44no—
133C4×C127—42no—
134C4×C127—42no—
135C4×C127—42no—
136C4×C127—82no—
137C4×C127—81no—
138#12101C12711yesmaximal, characteristic, direct product factor, Sylow
139C2×C127—21no—
140C2×C127—21no—
141C2×C127—161no—
142C2×C127#159511yescharacteristic
143C2×C127#220111yescharacteristic
144C2×C127#220511yescharacteristic
145#2734C2×C12711yescharacteristic, complemented
146#2007C2×C12711yescharacteristic
147#1610C2×C12712yes—
148#1346C2×C12712yescomplemented
149#1319C2×C12711yescharacteristic, complemented
150C127#1210111yescharacteristic, direct product factor, Sylow
151C2×#137—21no—
152#537C22×C12711yescharacteristic
153C2×C4⋊Q8—21no—
154#480—22no—
155C2×#104—21no—
156#398C22×C12712yes—
157#395—22no—
158#391C22×C12711yescharacteristic
159#389—21no—
160#387—22no—
161#343C22×C12711yescharacteristic
162#327—22no—
163#324C22×C12712yes—
164C2×#41—21no—
165C22×Q8—21no—
166C22×Q8D4×C12711yescharacteristic
167#137—42no—
168#137—42no—
169C2×Q16—41no—
170C2×SD16—41no—
171C2×M16—21no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C22⋊Q8—41no—
175#121—22no—
176#121—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4D4×C12712yes—
182C2×C4⋊C4C23×C12711yescharacteristic
183C2×C42—21no—
184C2×C42D4×C12711yescharacteristic
185#107—42no—
186#104—42no—
187#102—42no—
188C22⋊C8—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95D4×C12712yes—
193#41—41no—
194#41—41no—
195#41—41no—
196#41—41no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—42no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×Q8D8×C12712yes—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4C2×D4×C12711yescharacteristic
214C22×C4C2×D4×C12712yes—
215Q16—44no—
216SD16—44no—
217M16—22no—
218M16—41no—
219C2×C8—41no—
220C2×C8—42no—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—81no—
229C22⋊C4—42no—
230C42—41no—
231C42—42no—
232C23—41no—
233C23#120×C12711yescharacteristic
234Q8—42no—
235Q8—41no—
236Q8—41no—
237Q8—81no—
238Q8—84no—
239D4—41no—
240D4—41no—
241D4—82no—
242D4—81no—
243D4—81no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4C2×D8×C12711yescharacteristic
264C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
265C8—42no—
266C8—82no—
267C22—21no—
268C22—21no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22#447×C12711yescharacteristic
273C22#453×C12711yescharacteristic
274C22B(4,𝔽2)×C12711yescharacteristic
275C4—21no—
276C4—21no—
277C4—44no—
278C4—42no—
279C4—42no—
280C4—42no—
281C4—82no—
282C4—81no—
283C2—21no—
284C2—21no—
285C2—161no—
286C2C127×#159511yescharacteristic
287C2C127×#220111yescharacteristic
288C2C127×#220511yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radical#537×C127C22
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2734C2semidirect
4. C127×#1346C2semidirect
5. C127×#1319C2semidirect
138. #12101C127direct (N × H)
145. #2734C2×C127semidirect
148. #1346C2×C127semidirect
149. #1319C2×C127semidirect
150. C127#12101direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#27342
C127×#20072
C127×#16102
C127×#13462
C127×#13192
#12101127