Finite Group Databasebeta

← C127×#12201 #44575874

Subgroups — C127×#12201 #44575874

Order 32512.

380 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2833C211yesmaximal, characteristic, complemented
2C127×#2007C211yesmaximal, characteristic
3C127×#1616C211yesmaximal, characteristic
4C127×#1610C211yesmaximal, characteristic
5C127×#1353C211yesmaximal, characteristic, complemented
6C127×#1346C211yesmaximal, characteristic, complemented
7C127×#1319C211yesmaximal, characteristic, complemented
8C2×#137×C127—21no—
9#537×C127C2211yescharacteristic
10#481×C127—21no—
11#480×C127—21no—
12C2×#104×C127—21no—
13#401×C127—21no—
14#400×C127—21no—
15#400×C127—21no—
16#400×C127C2211yescharacteristic
17#398×C127—21no—
18#398×C127C2211yescharacteristic
19#395×C127—21no—
20#391×C127C2211yescharacteristic
21#387×C127—21no—
22#343×C127C2211yescharacteristic
23#327×C127—21no—
24#327×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C22×Q8×C127D411yescharacteristic
29#137×C127—42no—
30#137×C127—42no—
31C2×Q16×C127—41no—
32C2×SD16×C127—41no—
33C2×M16×C127—21no—
34C22⋊Q8×C127—41no—
35#121×C127—21no—
36#121×C127—21no—
37#121×C127—41no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127D411yescharacteristic
43C2×C4⋊C4×C127D411yescharacteristic
44C2×C4⋊C4×C127C2311yescharacteristic
45C2×C42×C127D411yescharacteristic
46#107×C127—41no—
47#106×C127—41no—
48#104×C127—42no—
49#102×C127—41no—
50#102×C127—41no—
51C22⋊C8×C127—21no—
52C22⋊C8×C127—21no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—21no—
59#95×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127—21no—
63#95×C127D411yescharacteristic
64#95×C127D411yescharacteristic
65#41×C127—41no—
66#41×C127—41no—
67#41×C127—41no—
68#41×C127—41no—
69C2×Q8×C127—21no—
70C2×Q8×C127—21no—
71C2×Q8×C127—21no—
72C2×Q8×C127—41no—
73C2×Q8×C127—41no—
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127—21no—
77C2×D4×C127—41no—
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—41no—
85C22×C4×C127C2×D411yescharacteristic
86C22×C4×C127C2×D411yescharacteristic
87C22×C4×C127C2×D411yescharacteristic
88Q16×C127—44no—
89SD16×C127—44no—
90M16×C127—22no—
91M16×C127—41no—
92C2×C8×C127—41no—
93C2×C8×C127—41no—
94C2×C8×C127—41no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—41no—
98C4⋊C4×C127—41no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—81no—
103C4⋊C4×C127—81no—
104C4⋊C4×C127—81no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C42×C127—41no—
108C23×C127—41no—
109C23×C127#12011yescharacteristic
110Q8×C127—42no—
111Q8×C127—41no—
112Q8×C127—41no—
113Q8×C127—81no—
114D4×C127—41no—
115D4×C127—41no—
116D4×C127—81no—
117D4×C127—81no—
118D4×C127—81no—
119D4×C127—81no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—42no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—81no—
145C2×C4×C127—81no—
146C2×C4×C127#13511yescharacteristic
147C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
148C8×C127—42no—
149C8×C127—81no—
150C8×C127—81no—
151C22×C127—21no—
152C22×C127—21no—
153C22×C127—81no—
154C22×C127—81no—
155C22×C127—81no—
156C22×C127#44911yescharacteristic
157C22×C127#45311yescharacteristic
158C22×C127B(4,𝔽2)11yescharacteristic
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—81no—
164C4×C127—81no—
165C4×C127—81no—
166C4×C127—81no—
167C4×C127—81no—
168C4×C127—81no—
169C4×C127—81no—
170#12201C12711yesmaximal, characteristic, direct product factor, Sylow
171C2×C127—21no—
172C2×C127—21no—
173C2×C127—161no—
174C2×C127#160011yescharacteristic
175C2×C127#220111yescharacteristic
176C2×C127#220311yescharacteristic
177#2833C2×C12711yescharacteristic, complemented
178#2007C2×C12711yescharacteristic
179#1616C2×C12711yescharacteristic
180#1610C2×C12711yescharacteristic
181#1353C2×C12711yescharacteristic, complemented
182#1346C2×C12711yescharacteristic, complemented
183#1319C2×C12711yescharacteristic, complemented
184C127#1220111yescharacteristic, direct product factor, Sylow
185C2×#137—21no—
186#537C22×C12711yescharacteristic
187#481—21no—
188#480—21no—
189C2×#104—21no—
190#401—21no—
191#400—21no—
192#400—21no—
193#400C22×C12711yescharacteristic
194#398—21no—
195#398C22×C12711yescharacteristic
196#395—21no—
197#391C22×C12711yescharacteristic
198#387—21no—
199#343C22×C12711yescharacteristic
200#327—21no—
201#327—21no—
202#324C22×C12711yescharacteristic
203#324C22×C12711yescharacteristic
204C2×#41—21no—
205C22×Q8D4×C12711yescharacteristic
206#137—42no—
207#137—42no—
208C2×Q16—41no—
209C2×SD16—41no—
210C2×M16—21no—
211C22⋊Q8—41no—
212#121—21no—
213#121—21no—
214#121—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4D4×C12711yescharacteristic
220C2×C4⋊C4D4×C12711yescharacteristic
221C2×C4⋊C4C23×C12711yescharacteristic
222C2×C42D4×C12711yescharacteristic
223#107—41no—
224#106—41no—
225#104—42no—
226#102—41no—
227#102—41no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95D4×C12711yescharacteristic
241#95D4×C12711yescharacteristic
242#41—41no—
243#41—41no—
244#41—41no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4C2×D4×C12711yescharacteristic
263C22×C4C2×D4×C12711yescharacteristic
264C22×C4C2×D4×C12711yescharacteristic
265Q16—44no—
266SD16—44no—
267M16—22no—
268M16—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C42—41no—
285C23—41no—
286C23#120×C12711yescharacteristic
287Q8—42no—
288Q8—41no—
289Q8—41no—
290Q8—81no—
291D4—41no—
292D4—41no—
293D4—81no—
294D4—81no—
295D4—81no—
296D4—81no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4#135×C12711yescharacteristic
324C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
325C8—42no—
326C8—81no—
327C8—81no—
328C22—21no—
329C22—21no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#449×C12711yescharacteristic
334C22#453×C12711yescharacteristic
335C22B(4,𝔽2)×C12711yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C2—21no—
348C2—21no—
349C2—161no—
350C2C127×#160011yescharacteristic
351C2C127×#220111yescharacteristic
352C2C127×#220311yescharacteristic
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×#2833C2semidirect
5. C127×#1353C2semidirect
6. C127×#1346C2semidirect
7. C127×#1319C2semidirect
170. #12201C127direct (N × H)
177. #2833C2×C127semidirect
181. #1353C2×C127semidirect
182. #1346C2×C127semidirect
183. #1319C2×C127semidirect
184. C127#12201direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#28332
C127×#20072
C127×#16162
C127×#16102
C127×#13532
C127×#13462
C127×#13192
#12201127