Finite Group Databasebeta

← C127×#12197 #44575870

Subgroups — C127×#12197 #44575870

Order 32512.

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

normal subgroups only

This group has 426 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×#2586C211yesmaximal, characteristic, complemented
2C127×#2007C211yesmaximal, characteristic
3C127×#1610C211yesmaximal, characteristic
4C127×#1604C211yesmaximal, characteristic
5C127×#1354C211yesmaximal, characteristic, complemented
6C127×#1346C211yesmaximal, characteristic, complemented
7C127×#1319C211yesmaximal, characteristic, complemented
8C2×#137×C127—21no—
9#537×C127C2211yescharacteristic
10#480×C127—21no—
11#448×C127—21no—
12C2×#104×C127—21no—
13#398×C127C2211yescharacteristic
14#395×C127—21no—
15#395×C127C2211yescharacteristic
16#391×C127—21no—
17#391×C127C2211yescharacteristic
18#389×C127—21no—
19#387×C127—21no—
20#387×C127—21no—
21C4×C4⋊C4×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
29C22×Q8×C127D411yescharacteristic
30#137×C127—42no—
31#137×C127—42no—
32C2×Q16×C127—41no—
33C2×SD16×C127—41no—
34C2×SD16×C127—41no—
35C2×M16×C127—21no—
36C22⋊Q8×C127—41no—
37#121×C127—21no—
38#121×C127—21no—
39#121×C127—41no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127—21no—
45C2×C4⋊C4×C127D411yescharacteristic
46C2×C4⋊C4×C127C2311yescharacteristic
47C2×C42×C127—21no—
48C2×C42×C127—21no—
49C2×C42×C127D411yescharacteristic
50#107×C127—41no—
51#104×C127—42no—
52#103×C127—41no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C22⋊C8×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—42no—
73C2×Q8×C127—41no—
74C2×Q8×C127—41no—
75C2×Q8×C127—41no—
76C2×D4×C127—21no—
77C2×D4×C127—21no—
78C2×D4×C127—21no—
79C2×D4×C127—41no—
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—21no—
85C22×C4×C127—21no—
86C22×C4×C127—41no—
87C22×C4×C127C2×D411yescharacteristic
88C22×C4×C127C2×D411yescharacteristic
89C22×C4×C127C2×D411yescharacteristic
90Q16×C127—44no—
91SD16×C127—44no—
92SD16×C127—82no—
93M16×C127—22no—
94M16×C127—41no—
95C2×C8×C127—41no—
96C2×C8×C127—41no—
97C2×C8×C127—41no—
98C4⋊C4×C127—24no—
99C4⋊C4×C127—21no—
100C4⋊C4×C127—21no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127—42no—
104C4⋊C4×C127—41no—
105C4⋊C4×C127—41no—
106C4⋊C4×C127—81no—
107C4⋊C4×C127—81no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—41no—
110C42×C127—41no—
111C42×C127—42no—
112C42×C127—81no—
113C23×C127—41no—
114C23×C127#12011yescharacteristic
115Q8×C127—42no—
116Q8×C127—41no—
117Q8×C127—41no—
118Q8×C127—42no—
119Q8×C127—81no—
120Q8×C127—81no—
121D4×C127—41no—
122D4×C127—41no—
123D4×C127—81no—
124D4×C127—81no—
125D4×C127—81no—
126D4×C127—81no—
127C2×C4×C127—21no—
128C2×C4×C127—21no—
129C2×C4×C127—22no—
130C2×C4×C127—22no—
131C2×C4×C127—22no—
132C2×C4×C127—22no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
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—41no—
145C2×C4×C127—41no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—41no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—81no—
153C2×C4×C127—81no—
154C2×C4×C127#13511yescharacteristic
155C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
156C8×C127—42no—
157C8×C127—81no—
158C8×C127—81no—
159C22×C127—21no—
160C22×C127—21no—
161C22×C127—81no—
162C22×C127—81no—
163C22×C127—81no—
164C22×C127#44911yescharacteristic
165C22×C127#45311yescharacteristic
166C22×C127B(4,𝔽2)11yescharacteristic
167C4×C127—21no—
168C4×C127—21no—
169C4×C127—42no—
170C4×C127—42no—
171C4×C127—42no—
172C4×C127—41no—
173C4×C127—41no—
174C4×C127—42no—
175C4×C127—81no—
176C4×C127—81no—
177C4×C127—81no—
178#12197C12711yesmaximal, characteristic, direct product factor, Sylow
179C2×C127—21no—
180C2×C127—21no—
181C2×C127—161no—
182C2×C127#160011yescharacteristic
183C2×C127#219711yescharacteristic
184C2×C127#220511yescharacteristic
185#2586C2×C12711yescharacteristic, complemented
186#2007C2×C12711yescharacteristic
187#1610C2×C12711yescharacteristic
188#1604C2×C12711yescharacteristic
189#1354C2×C12711yescharacteristic, complemented
190#1346C2×C12711yescharacteristic, complemented
191#1319C2×C12711yescharacteristic, complemented
192C127#1219711yescharacteristic, direct product factor, Sylow
193C2×#137—21no—
194#537C22×C12711yescharacteristic
195#480—21no—
196#448—21no—
197C2×#104—21no—
198#398C22×C12711yescharacteristic
199#395—21no—
200#395C22×C12711yescharacteristic
201#391—21no—
202#391C22×C12711yescharacteristic
203#389—21no—
204#387—21no—
205#387—21no—
206C4×C4⋊C4—21no—
207#343C22×C12711yescharacteristic
208#327—21no—
209#327—21no—
210#324C22×C12711yescharacteristic
211#324C22×C12711yescharacteristic
212C2×#41—21no—
213C22×Q8D4×C12711yescharacteristic
214C22×Q8D4×C12711yescharacteristic
215#137—42no—
216#137—42no—
217C2×Q16—41no—
218C2×SD16—41no—
219C2×SD16—41no—
220C2×M16—21no—
221C22⋊Q8—41no—
222#121—21no—
223#121—21no—
224#121—41no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4D4×C12711yescharacteristic
231C2×C4⋊C4C23×C12711yescharacteristic
232C2×C42—21no—
233C2×C42—21no—
234C2×C42D4×C12711yescharacteristic
235#107—41no—
236#104—42no—
237#103—41no—
238#102—41no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95D4×C12711yescharacteristic
249#95D4×C12711yescharacteristic
250#41—41no—
251#41—41no—
252#41—41no—
253#41—41no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4C2×D4×C12711yescharacteristic
273C22×C4C2×D4×C12711yescharacteristic
274C22×C4C2×D4×C12711yescharacteristic
275Q16—44no—
276SD16—44no—
277SD16—82no—
278M16—22no—
279M16—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C4⋊C4—24no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C42—41no—
296C42—42no—
297C42—81no—
298C23—41no—
299C23#120×C12711yescharacteristic
300Q8—42no—
301Q8—41no—
302Q8—41no—
303Q8—42no—
304Q8—81no—
305Q8—81no—
306D4—41no—
307D4—41no—
308D4—81no—
309D4—81no—
310D4—81no—
311D4—81no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4#135×C12711yescharacteristic
340C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
341C8—42no—
342C8—81no—
343C8—81no—
344C22—21no—
345C22—21no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#449×C12711yescharacteristic
350C22#453×C12711yescharacteristic
351C22B(4,𝔽2)×C12711yescharacteristic
352C4—21no—
353C4—21no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—42no—
360C4—81no—
361C4—81no—
362C4—81no—
363C2—21no—
364C2—21no—
365C2—161no—
366C2C127×#160011yescharacteristic
367C2C127×#219711yescharacteristic
368C2C127×#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×#2586C2semidirect
5. C127×#1354C2semidirect
6. C127×#1346C2semidirect
7. C127×#1319C2semidirect
178. #12197C127direct (N × H)
185. #2586C2×C127semidirect
189. #1354C2×C127semidirect
190. #1346C2×C127semidirect
191. #1319C2×C127semidirect
192. C127#12197direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#25862
C127×#20072
C127×#16102
C127×#16042
C127×#13542
C127×#13462
C127×#13192
#12197127