Finite Group Databasebeta

← C127×#12219 #44575892

Subgroups — C127×#12219 #44575892

Order 32512.

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

normal subgroups only

This group has 370 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×#2841C211yesmaximal, characteristic
2C127×#2030C211yesmaximal, characteristic
3C127×#1614C211yesmaximal, characteristic
4C127×#1602C211yesmaximal, characteristic
5C127×#1352C211yesmaximal, characteristic
6C127×#1351C211yesmaximal, characteristic
7C127×#1317C211yesmaximal, characteristic
8#543×C127C2211yescharacteristic
9#484×C127—21no—
10#451×C127—21no—
11#400×C127—21no—
12#399×C127C2211yescharacteristic
13#398×C127—21no—
14#398×C127—21no—
15#396×C127—21no—
16#395×C127—21no—
17#393×C127C2211yescharacteristic
18#381×C127—21no—
19#381×C127C2211yescharacteristic
20#380×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
28C23×C4×C127D411yescharacteristic
29C2×Q16×C127—41no—
30C22⋊Q8×C127—21no—
31C22⋊Q8×C127—21no—
32C22⋊Q8×C127—41no—
33C22⋊Q8×C127—41no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127D411yescharacteristic
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
47#107×C127—41no—
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—41no—
59#95×C127—41no—
60#95×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C24×C127C2×D411yescharacteristic
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—41no—
66C2×Q8×C127—81no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
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—41no—
77C22×C4×C127—41no—
78C22×C4×C127—41no—
79C22×C4×C127Q1612yescomplemented
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82Q16×C127—82no—
83C2×C8×C127—41no—
84C2×C8×C127—41no—
85C4⋊C4×C127—21no—
86C4⋊C4×C127—21no—
87C4⋊C4×C127—41no—
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—81no—
95C4⋊C4×C127—81no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—81no—
103C22⋊C4×C127—81no—
104C23×C127—21no—
105C23×C127—21no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127#12011yescharacteristic
109C23×C127C2×Q1611yescharacteristic
110C23×C127#13711yescharacteristic
111Q8×C127—42no—
112Q8×C127—81no—
113Q8×C127—81no—
114Q8×C127—81no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—42no—
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—41no—
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—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—81no—
141C2×C4×C127—81no—
142C2×C4×C127—81no—
143C2×C4×C127—81no—
144C2×C4×C127—81no—
145C2×C4×C127—81no—
146C8×C127—81no—
147C8×C127—81no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127—41no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—41no—
156C22×C127—81no—
157C22×C127#45111yescharacteristic
158C22×C127#45511yescharacteristic
159C22×C127B(4,𝔽2)11yescharacteristic
160C4×C127—42no—
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—
170C4×C127—161no—
171#12219C12711yesmaximal, characteristic, direct product factor, Sylow
172C2×C127—21no—
173C2×C127—21no—
174C2×C127—41no—
175C2×C127—41no—
176C2×C127#160211yescharacteristic
177C2×C127#220011yescharacteristic
178C2×C127#220111yescharacteristic
179#2841C2×C12711yescharacteristic
180#2030C2×C12711yescharacteristic
181#1614C2×C12711yescharacteristic
182#1602C2×C12711yescharacteristic
183#1352C2×C12711yescharacteristic
184#1351C2×C12711yescharacteristic
185#1317C2×C12711yescharacteristic
186C127#1221911yescharacteristic, direct product factor, Sylow
187#543C22×C12711yescharacteristic
188#484—21no—
189#451—21no—
190#400—21no—
191#399C22×C12711yescharacteristic
192#398—21no—
193#398—21no—
194#396—21no—
195#395—21no—
196#393C22×C12711yescharacteristic
197#381—21no—
198#381C22×C12711yescharacteristic
199#380—21no—
200#342—21no—
201#342C22×C12711yescharacteristic
202#328—21no—
203#328—21no—
204#323C22×C12711yescharacteristic
205#323C22×C12711yescharacteristic
206C22×Q8D4×C12711yescharacteristic
207C23×C4D4×C12711yescharacteristic
208C2×Q16—41no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4D4×C12711yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4D4×C12711yescharacteristic
225C2×C22⋊C4C23×C12711yescharacteristic
226#107—41no—
227#103—41no—
228#103—41no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—41no—
238#95—41no—
239#95D4×C12711yescharacteristic
240#95D4×C12711yescharacteristic
241C24C2×D4×C12711yescharacteristic
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—81no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4Q16×C12712yescomplemented
259C22×C4C2×D4×C12711yescharacteristic
260C22×C4C2×D4×C12711yescharacteristic
261Q16—82no—
262C2×C8—41no—
263C2×C8—41no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C4⋊C4—81no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23#120×C12711yescharacteristic
288C23C2×Q16×C12711yescharacteristic
289C23#137×C12711yescharacteristic
290Q8—42no—
291Q8—81no—
292Q8—81no—
293Q8—81no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
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—41no—
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—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C8—81no—
326C8—81no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22#451×C12711yescharacteristic
337C22#455×C12711yescharacteristic
338C22B(4,𝔽2)×C12711yescharacteristic
339C4—42no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—161no—
350C2—21no—
351C2—21no—
352C2—41no—
353C2—41no—
354C2C127×#160211yescharacteristic
355C2C127×#220011yescharacteristic
356C2C127×#220111yescharacteristic
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
79. C22×C4×C127Q16semidirect
171. #12219C127direct (N × H)
186. C127#12219direct (N × H)
258. C22×C4Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#28412
C127×#20302
C127×#16142
C127×#16022
C127×#13522
C127×#13512
C127×#13172
#12219127