Finite Group Databasebeta

← C127×#12253 #44575926

Subgroups — C127×#12253 #44575926

Order 32512.

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

normal subgroups only

This group has 398 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×#2763C211yesmaximal, characteristic
2C127×#2062C211yesmaximal, characteristic
3C127×#1602C211yesmaximal, characteristic
4C127×#1602C211yesmaximal, characteristic
5C127×#1351C211yesmaximal, characteristic
6C127×#1351C211yesmaximal, characteristic
7C127×#1326C211yesmaximal, characteristic
8#543×C127C2211yescharacteristic
9#451×C127—21no—
10#451×C127—21no—
11#400×C127—21no—
12#395×C127—21no—
13#393×C127C2211yescharacteristic
14#393×C127C2211yescharacteristic
15#391×C127—21no—
16#388×C127—21no—
17#385×C127—21no—
18#385×C127—21no—
19#385×C127C2211yescharacteristic
20#382×C127—21no—
21#342×C127—21no—
22#342×C127C2211yescharacteristic
23#328×C127—21no—
24#328×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C22×Q8×C127D411yescharacteristic
28C22×Q8×C127D411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C2×Q16×C127—41no—
31C2×Q16×C127—41no—
32C22⋊Q8×C127—21no—
33C22⋊Q8×C127—21no—
34C22⋊Q8×C127—41no—
35C22⋊Q8×C127—41no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127—41no—
41C2×C22⋊C4×C127—21no—
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127D411yescharacteristic
46C2×C22⋊C4×C127C2311yescharacteristic
47C2×C42×C127—21no—
48#103×C127—41no—
49#103×C127—41no—
50C22⋊C8×C127—21no—
51C22⋊C8×C127—21no—
52#95×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—41no—
60#95×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C24×C127C2×D411yescharacteristic
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—42no—
66C2×Q8×C127—41no—
67C2×Q8×C127—41no—
68C2×Q8×C127—81no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127—41no—
78C22×C4×C127—41no—
79C22×C4×C127—81no—
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82Q16×C127—82no—
83Q16×C127—82no—
84C2×C8×C127—41no—
85C2×C8×C127—41no—
86C4⋊C4×C127—21no—
87C4⋊C4×C127—21no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—81no—
91C4⋊C4×C127—81no—
92C4⋊C4×C127—81no—
93C4⋊C4×C127—81no—
94C4⋊C4×C127—82no—
95C22⋊C4×C127—21no—
96C22⋊C4×C127—22no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—42no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—81no—
103C22⋊C4×C127Q1612yescomplemented
104C42×C127—42no—
105C23×C127—21no—
106C23×C127—21no—
107C23×C127—41no—
108C23×C127—41no—
109C23×C127#12011yescharacteristic
110C23×C127C2×Q1611yescharacteristic
111C23×C127#13711yescharacteristic
112Q8×C127—42no—
113Q8×C127—42no—
114Q8×C127—81no—
115Q8×C127—81no—
116Q8×C127—81no—
117Q8×C127—81no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—42no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—42no—
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—81no—
147C2×C4×C127—81no—
148C2×C4×C127—81no—
149C8×C127—81no—
150C8×C127—81no—
151C22×C127—21no—
152C22×C127—21no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—41no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—81no—
160C22×C127#45111yescharacteristic
161C22×C127#45611yescharacteristic
162C22×C127B(4,𝔽2)11yescharacteristic
163C4×C127—42no—
164C4×C127—41no—
165C4×C127—42no—
166C4×C127—41no—
167C4×C127—81no—
168C4×C127—81no—
169C4×C127—82no—
170C4×C127—81no—
171C4×C127—81no—
172C4×C127—81no—
173C4×C127—81no—
174#12253C12711yesmaximal, characteristic, direct product factor, Sylow
175C2×C127—21no—
176C2×C127—21no—
177C2×C127—41no—
178C2×C127—41no—
179C2×C127#160511yescharacteristic
180C2×C127#219711yescharacteristic
181C2×C127#219911yescharacteristic
182#2763C2×C12711yescharacteristic
183#2062C2×C12711yescharacteristic
184#1602C2×C12711yescharacteristic
185#1602C2×C12711yescharacteristic
186#1351C2×C12711yescharacteristic
187#1351C2×C12711yescharacteristic
188#1326C2×C12711yescharacteristic
189C127#1225311yescharacteristic, direct product factor, Sylow
190#543C22×C12711yescharacteristic
191#451—21no—
192#451—21no—
193#400—21no—
194#395—21no—
195#393C22×C12711yescharacteristic
196#393C22×C12711yescharacteristic
197#391—21no—
198#388—21no—
199#385—21no—
200#385—21no—
201#385C22×C12711yescharacteristic
202#382—21no—
203#342—21no—
204#342C22×C12711yescharacteristic
205#328—21no—
206#328—21no—
207#323C22×C12711yescharacteristic
208#323C22×C12711yescharacteristic
209C22×Q8D4×C12711yescharacteristic
210C22×Q8D4×C12711yescharacteristic
211C23×C4D4×C12711yescharacteristic
212C2×Q16—41no—
213C2×Q16—41no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—41no—
217C22⋊Q8—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—41no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4D4×C12711yescharacteristic
228C2×C22⋊C4C23×C12711yescharacteristic
229C2×C42—21no—
230#103—41no—
231#103—41no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—41no—
242#95D4×C12711yescharacteristic
243#95D4×C12711yescharacteristic
244C24C2×D4×C12711yescharacteristic
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—81no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4C2×D4×C12711yescharacteristic
263C22×C4C2×D4×C12711yescharacteristic
264Q16—82no—
265Q16—82no—
266C2×C8—41no—
267C2×C8—41no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—81no—
276C4⋊C4—82no—
277C22⋊C4—21no—
278C22⋊C4—22no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—81no—
285C22⋊C4Q16×C12712yescomplemented
286C42—42no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23#120×C12711yescharacteristic
292C23C2×Q16×C12711yescharacteristic
293C23#137×C12711yescharacteristic
294Q8—42no—
295Q8—42no—
296Q8—81no—
297Q8—81no—
298Q8—81no—
299Q8—81no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C8—81no—
332C8—81no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22#451×C12711yescharacteristic
343C22#456×C12711yescharacteristic
344C22B(4,𝔽2)×C12711yescharacteristic
345C4—42no—
346C4—41no—
347C4—42no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—82no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C2—21no—
357C2—21no—
358C2—41no—
359C2—41no—
360C2C127×#160511yescharacteristic
361C2C127×#219711yescharacteristic
362C2C127×#219911yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC24×C127C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
103. C22⋊C4×C127Q16semidirect
174. #12253C127direct (N × H)
189. C127#12253direct (N × H)
285. C22⋊C4Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#27632
C127×#20622
C127×#16022
C127×#13512
C127×#13262
#12253127