Finite Group Databasebeta

← C127×#12222 #44575895

Subgroups — C127×#12222 #44575895

Order 32512.

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

normal subgroups only

This group has 346 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×#2799C211yesmaximal, characteristic
2C127×#2028C211yesmaximal, characteristic
3C127×#1617C211yesmaximal, characteristic
4C127×#1605C211yesmaximal, characteristic
5C127×#1354C211yesmaximal, characteristic
6C127×#1353C211yesmaximal, characteristic
7C127×#1319C211yesmaximal, characteristic
8#544×C127C2211yescharacteristic
9#484×C127—21no—
10#451×C127—21no—
11#419×C127—21no—
12#400×C127—21no—
13#400×C127C2211yescharacteristic
14#398×C127—21no—
15#395×C127C2211yescharacteristic
16#391×C127—21no—
17#391×C127—21no—
18#391×C127C2211yescharacteristic
19#387×C127—21no—
20#387×C127—21no—
21#343×C127C2211yescharacteristic
22#339×C127—21no—
23#328×C127—21no—
24#328×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C22×Q8×C127D411yescharacteristic
28C22×Q8×C127D411yescharacteristic
29C2×Q16×C127—41no—
30C2×M16×C127—21no—
31C4⋊Q8×C127—41no—
32#125×C127—41no—
33C22⋊Q8×C127—21no—
34C22⋊Q8×C127—21no—
35#117×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×C4⋊C4×C127C2311yescharacteristic
42C2×C42×C127—21no—
43C2×C42×C127D411yescharacteristic
44#107×C127—41no—
45#103×C127—41no—
46#103×C127—41no—
47#103×C127—41no—
48#103×C127—41no—
49C22⋊C8×C127—21no—
50C22⋊C8×C127—21no—
51#95×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—21no—
60#95×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C2×Q8×C127—21no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—42no—
66C2×Q8×C127—41no—
67C2×Q8×C127—41no—
68C2×Q8×C127—41no—
69C2×Q8×C127Q1612yescomplemented
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×C127C2×D411yescharacteristic
77C22×C4×C127C2×D411yescharacteristic
78C22×C4×C127C2×D411yescharacteristic
79Q16×C127—82no—
80M16×C127—41no—
81M16×C127—41no—
82C2×C8×C127—41no—
83C2×C8×C127—41no—
84C2×C8×C127—41no—
85C4⋊C4×C127—21no—
86C4⋊C4×C127—21no—
87C4⋊C4×C127—21no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—42no—
92C4⋊C4×C127—42no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—81no—
96C4⋊C4×C127—81no—
97C22⋊C4×C127—41no—
98C42×C127—41no—
99C42×C127—41no—
100C42×C127—81no—
101C23×C127#12011yescharacteristic
102Q8×C127—42no—
103Q8×C127—42no—
104Q8×C127—81no—
105Q8×C127—81no—
106Q8×C127—81no—
107Q8×C127—81no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—42no—
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×C127C2×Q1611yescharacteristic
134C2×C4×C127#13711yescharacteristic
135C8×C127—81no—
136C8×C127—81no—
137C8×C127—81no—
138C22×C127—21no—
139C22×C127—21no—
140C22×C127#45111yescharacteristic
141C22×C127#45511yescharacteristic
142C22×C127B(4,𝔽2)11yescharacteristic
143C4×C127—21no—
144C4×C127—21no—
145C4×C127—42no—
146C4×C127—41no—
147C4×C127—41no—
148C4×C127—42no—
149C4×C127—81no—
150C4×C127—81no—
151C4×C127—81no—
152C4×C127—81no—
153C4×C127—81no—
154C4×C127—81no—
155#12222C12711yesmaximal, characteristic, direct product factor, Sylow
156C2×C127—21no—
157C2×C127—21no—
158C2×C127#160211yescharacteristic
159C2×C127#219711yescharacteristic
160C2×C127#220311yescharacteristic
161#2799C2×C12711yescharacteristic
162#2028C2×C12711yescharacteristic
163#1617C2×C12711yescharacteristic
164#1605C2×C12711yescharacteristic
165#1354C2×C12711yescharacteristic
166#1353C2×C12711yescharacteristic
167#1319C2×C12711yescharacteristic
168C127#1222211yescharacteristic, direct product factor, Sylow
169#544C22×C12711yescharacteristic
170#484—21no—
171#451—21no—
172#419—21no—
173#400—21no—
174#400C22×C12711yescharacteristic
175#398—21no—
176#395C22×C12711yescharacteristic
177#391—21no—
178#391—21no—
179#391C22×C12711yescharacteristic
180#387—21no—
181#387—21no—
182#343C22×C12711yescharacteristic
183#339—21no—
184#328—21no—
185#328—21no—
186#324C22×C12711yescharacteristic
187#324C22×C12711yescharacteristic
188C22×Q8D4×C12711yescharacteristic
189C22×Q8D4×C12711yescharacteristic
190C2×Q16—41no—
191C2×M16—21no—
192C4⋊Q8—41no—
193#125—41no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196#117—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4D4×C12711yescharacteristic
202C2×C4⋊C4C23×C12711yescharacteristic
203C2×C42—21no—
204C2×C42D4×C12711yescharacteristic
205#107—41no—
206#103—41no—
207#103—41no—
208#103—41no—
209#103—41no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95D4×C12711yescharacteristic
222#95D4×C12711yescharacteristic
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8Q16×C12712yescomplemented
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C2×D4×C12711yescharacteristic
238C22×C4C2×D4×C12711yescharacteristic
239C22×C4C2×D4×C12711yescharacteristic
240Q16—82no—
241M16—41no—
242M16—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C22⋊C4—41no—
259C42—41no—
260C42—41no—
261C42—81no—
262C23#120×C12711yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—81no—
266Q8—81no—
267Q8—81no—
268Q8—81no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4C2×Q16×C12711yescharacteristic
295C2×C4#137×C12711yescharacteristic
296C8—81no—
297C8—81no—
298C8—81no—
299C22—21no—
300C22—21no—
301C22#451×C12711yescharacteristic
302C22#455×C12711yescharacteristic
303C22B(4,𝔽2)×C12711yescharacteristic
304C4—21no—
305C4—21no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C2—21no—
317C2—21no—
318C2C127×#160211yescharacteristic
319C2C127×#219711yescharacteristic
320C2C127×#220311yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC23×C127#120
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
69. C2×Q8×C127Q16semidirect
155. #12222C127direct (N × H)
168. C127#12222direct (N × H)
230. C2×Q8Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#27992
C127×#20282
C127×#16172
C127×#16052
C127×#13542
C127×#13532
C127×#13192
#12222127