Finite Group Databasebeta

← C127×#11817 #44575490

Subgroups — C127×#11817 #44575490

Order 32512.

386 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 98 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×#2627C211yesmaximal, characteristic
2C127×#1821C211yesmaximal, characteristic
3C127×#1463C211yesmaximal, characteristic
4C127×#1354C212yesmaximal, complemented
5C127×#1353C212yesmaximal, complemented
6C2×C22⋊Q8×C127C411yescharacteristic
7C22×C4⋊C4×C127C2211yescharacteristic
8#407×C127—21no—
9#407×C127—21no—
10#400×C127C411yescharacteristic
11#400×C127C2211yescharacteristic
12#398×C127—21no—
13#398×C127—21no—
14#395×C127C411yescharacteristic
15#395×C127C2211yescharacteristic
16#385×C127—21no—
17#381×C127—21no—
18#380×C127C411yescharacteristic
19#343×C127—21no—
20#343×C127C2212yes—
21#324×C127—21no—
22#324×C127C2212yes—
23C22×Q8×C127C2×C411yescharacteristic
24C23×C4×C127C2×C411yescharacteristic
25C23×C4×C127D411yescharacteristic
26C23×C4×C127D411yescharacteristic
27C2×M16×C127—22no—
28C2×M16×C127—41no—
29C22⋊Q8×C127—41no—
30C22⋊Q8×C127—41no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127—41no—
36C2×C4⋊C4×C127—41no—
37C2×C4⋊C4×C127C2×C411yescharacteristic
38C2×C4⋊C4×C127C2×C411yescharacteristic
39C2×C4⋊C4×C127D411yescharacteristic
40C2×C4⋊C4×C127D411yescharacteristic
41C2×C4⋊C4×C127C2311yescharacteristic
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C22⋊C8×C127—22no—
45C22⋊C8×C127—41no—
46C22⋊C8×C127—41no—
47C22⋊C8×C127—41no—
48#95×C127—21no—
49#95×C127—21no—
50#95×C127—21no—
51#95×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127C2×C411yescharacteristic
55#95×C127C2×C411yescharacteristic
56C24×C127C22⋊C411yescharacteristic
57C2×Q8×C127—41no—
58C2×Q8×C127—81no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65C22×C4×C127—22no—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—41no—
71C22×C4×C127—81no—
72C22×C4×C127—81no—
73C22×C4×C127C22⋊C411yescharacteristic
74C22×C4×C127C22⋊C411yescharacteristic
75C22×C4×C127C22⋊C411yescharacteristic
76C22×C4×C127C22×C411yescharacteristic
77C22×C4×C127C2×D411yescharacteristic
78C22×C4×C127C2×D411yescharacteristic
79M16×C127—42no—
80M16×C127—42no—
81M16×C127—82no—
82C2×C8×C127—42no—
83C2×C8×C127—42no—
84C4⋊C4×C127—41no—
85C4⋊C4×C127—41no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—42no—
90C4⋊C4×C127—81no—
91C4⋊C4×C127—81no—
92C4⋊C4×C127—81no—
93C22⋊C4×C127—42no—
94C22⋊C4×C127—41no—
95C22⋊C4×C127—41no—
96C23×C127—21no—
97C23×C127—21no—
98C23×C127—41no—
99C23×C127—41no—
100C23×C127#9911yescharacteristic
101C23×C127#9911yescharacteristic
102C23×C127C2×C22⋊C411yescharacteristic
103Q8×C127—82no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—42no—
114C2×C4×C127—42no—
115C2×C4×C127—42no—
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—81no—
132C2×C4×C127—81no—
133C2×C4×C127—81no—
134C2×C4×C127#13712yes—
135C8×C127—82no—
136C8×C127—82no—
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—21no—
145C22×C127—42no—
146C22×C127—42no—
147C22×C127C2×#9911yescharacteristic
148C22×C127#41911yescharacteristic
149C22×C127#42111yescharacteristic
150C4×C127—22no—
151C4×C127—41no—
152C4×C127—41no—
153C4×C127—41no—
154C4×C127—41no—
155C4×C127—41no—
156C4×C127—81no—
157C4×C127—81no—
158C4×C127—81no—
159C4×C127—81no—
160#11817C12711yesmaximal, characteristic, direct product factor, Sylow
161C2×C127—21no—
162C2×C127—21no—
163C2×C127—41no—
164C2×C127—41no—
165C2×C127#151111yescharacteristic
166C2×C127#212111yescharacteristic
167C2×C127#213311yescharacteristic
168#2627C2×C12711yescharacteristic
169#1821C2×C12711yescharacteristic
170#1463C2×C12711yescharacteristic
171#1354C2×C12712yescomplemented
172#1353C2×C12712yescomplemented
173C127#1181711yescharacteristic, direct product factor, Sylow
174C2×C22⋊Q8C4×C12711yescharacteristic
175C22×C4⋊C4C22×C12711yescharacteristic
176#407—21no—
177#407—21no—
178#400C4×C12711yescharacteristic
179#400C22×C12711yescharacteristic
180#398—21no—
181#398—21no—
182#395C4×C12711yescharacteristic
183#395C22×C12711yescharacteristic
184#385—21no—
185#381—21no—
186#380C4×C12711yescharacteristic
187#343—21no—
188#343C22×C12712yes—
189#324—21no—
190#324C22×C12712yes—
191C22×Q8C2×C4×C12711yescharacteristic
192C23×C4C2×C4×C12711yescharacteristic
193C23×C4D4×C12711yescharacteristic
194C23×C4D4×C12711yescharacteristic
195C2×M16—22no—
196C2×M16—41no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×C4×C12711yescharacteristic
206C2×C4⋊C4C2×C4×C12711yescharacteristic
207C2×C4⋊C4D4×C12711yescharacteristic
208C2×C4⋊C4D4×C12711yescharacteristic
209C2×C4⋊C4C23×C12711yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C22⋊C8—22no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C2×C4×C12711yescharacteristic
223#95C2×C4×C12711yescharacteristic
224C24C22⋊C4×C12711yescharacteristic
225C2×Q8—41no—
226C2×Q8—81no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—81no—
240C22×C4—81no—
241C22×C4C22⋊C4×C12711yescharacteristic
242C22×C4C22⋊C4×C12711yescharacteristic
243C22×C4C22⋊C4×C12711yescharacteristic
244C22×C4C22×C4×C12711yescharacteristic
245C22×C4C2×D4×C12711yescharacteristic
246C22×C4C2×D4×C12711yescharacteristic
247M16—42no—
248M16—42no—
249M16—82no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—81no—
259C4⋊C4—81no—
260C4⋊C4—81no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#99×C12711yescharacteristic
269C23#99×C12711yescharacteristic
270C23C2×C22⋊C4×C12711yescharacteristic
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4#137×C12712yes—
303C8—82no—
304C8—82no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#99×C12711yescharacteristic
316C22#419×C12711yescharacteristic
317C22#421×C12711yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2C127×#151111yescharacteristic
333C2C127×#212111yescharacteristic
334C2C127×#213311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C22×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1354C2semidirect
5. C127×#1353C2semidirect
160. #11817C127direct (N × H)
171. #1354C2×C127semidirect
172. #1353C2×C127semidirect
173. C127#11817direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#26272
C127×#18212
C127×#14632
C127×#13542
C127×#13532
#11817127