Finite Group Databasebeta

← C127×#11874 #44575547

Subgroups — C127×#11874 #44575547

Order 32512.

472 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2486C211yesmaximal, characteristic
2C127×#1799C211yesmaximal, characteristic
3C127×#1460C211yesmaximal, characteristic
4C127×#1344C214yesmaximal, complemented
5C2×#123×C127—21no—
6C2×C4×D4×C127C2211yescharacteristic
7C22×C4⋊C4×C127C411yescharacteristic, complemented
8#431×C127—21no—
9C2×#99×C127—21no—
10#398×C127C412yescomplemented
11#397×C127C2212yes—
12#389×C127—21no—
13#381×C127—21no—
14#380×C127—21no—
15#380×C127—21no—
16#380×C127C411yescharacteristic, complemented
17#342×C127C2212yes—
18#323×C127—21no—
19#323×C127C2212yes—
20C22×D4×C127D411yescharacteristic
21C23×C4×C127—21no—
22C23×C4×C127C2×C411yescharacteristic
23C23×C4×C127D411yescharacteristic
24#123×C127—42no—
25#123×C127—42no—
26C4×D4×C127—41no—
27C4×D4×C127—41no—
28C2×C4⋊C4×C127—22no—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—41no—
32C2×C4⋊C4×C127—41no—
33C2×C4⋊C4×C127—41no—
34C2×C4⋊C4×C127C2×C412yes—
35C2×C4⋊C4×C127C2×C411yescharacteristic
36C2×C22⋊C4×C127—21no—
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127D411yescharacteristic
40C2×C22⋊C4×C127C2311yescharacteristic
41C2×C42×C127D411yescharacteristic
42C4⋊C8×C127—41no—
43#99×C127—42no—
44C22⋊C8×C127—22no—
45C4×C8×C127—41no—
46#95×C127—22no—
47#95×C127—22no—
48#95×C127—41no—
49#95×C127C2×C412yes—
50C24×C127C22⋊C411yescharacteristic
51C24×C127C2×D411yescharacteristic
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127D812yes—
55C2×D4×C127SD1612yes—
56C22×C4×C127—21no—
57C22×C4×C127—21no—
58C22×C4×C127—21no—
59C22×C4×C127—21no—
60C22×C4×C127—22no—
61C22×C4×C127—22no—
62C22×C4×C127—42no—
63C22×C4×C127—42no—
64C22×C4×C127—42no—
65C22×C4×C127—42no—
66C22×C4×C127—42no—
67C22×C4×C127—42no—
68C22×C4×C127C22⋊C411yescharacteristic
69C22×C4×C127C22⋊C411yescharacteristic
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22×C411yescharacteristic
72C22×C4×C127C2×D411yescharacteristic
73C2×C8×C127—42no—
74C4⋊C4×C127—41no—
75C4⋊C4×C127—42no—
76C4⋊C4×C127—42no—
77C4⋊C4×C127—41no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—44no—
80C4⋊C4×C127—82no—
81C22⋊C4×C127—41no—
82C22⋊C4×C127—41no—
83C22⋊C4×C127—42no—
84C22⋊C4×C127—42no—
85C22⋊C4×C127—42no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C42×C127—21no—
89C42×C127—21no—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—21no—
93C23×C127—21no—
94C23×C127—41no—
95C23×C127—41no—
96C23×C127—41no—
97C23×C127—41no—
98C23×C127#10212yes—
99C23×C127C2×C22⋊C411yescharacteristic
100C23×C127C2×D811yescharacteristic
101C23×C127C2×SD1611yescharacteristic
102D4×C127—42no—
103D4×C127—42no—
104C2×C4×C127—21no—
105C2×C4×C127—21no—
106C2×C4×C127—22no—
107C2×C4×C127—21no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127—42no—
111C2×C4×C127—42no—
112C2×C4×C127—42no—
113C2×C4×C127—41no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127—42no—
121C2×C4×C127—42no—
122C2×C4×C127—81no—
123C2×C4×C127—84no—
124C2×C4×C127—82no—
125C2×C4×C127—82no—
126C2×C4×C127—82no—
127C2×C4×C127#9911yescharacteristic
128C2×C4×C127#9911yescharacteristic
129C2×C4×C127#10212yes—
130C8×C127—82no—
131C22×C127—21no—
132C22×C127—21no—
133C22×C127—22no—
134C22×C127—21no—
135C22×C127—21no—
136C22×C127—42no—
137C22×C127—41no—
138C22×C127—41no—
139C22×C127—42no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—41no—
143C22×C127—41no—
144C22×C127—81no—
145C22×C127C2×#9911yescharacteristic
146C22×C127C2×#10211yescharacteristic
147C22×C127#42111yescharacteristic
148C4×C127—21no—
149C4×C127—22no—
150C4×C127—41no—
151C4×C127—41no—
152C4×C127—41no—
153C4×C127—82no—
154C4×C127—82no—
155C4×C127—162no—
156C4×C127#35212yes—
157#11874C12711yesmaximal, characteristic, direct product factor, Sylow
158C2×C127—21no—
159C2×C127—21no—
160C2×C127—41no—
161C2×C127—42no—
162C2×C127—41no—
163C2×C127#151911yescharacteristic
164C2×C127C2×#35211yescharacteristic
165C2×C127#212911yescharacteristic
166#2486C2×C12711yescharacteristic
167#1799C2×C12711yescharacteristic
168#1460C2×C12711yescharacteristic
169#1344C2×C12714yescomplemented
170C127#1187411yescharacteristic, direct product factor, Sylow
171C2×#123—21no—
172C2×C4×D4C22×C12711yescharacteristic
173C22×C4⋊C4C4×C12711yescharacteristic, complemented
174#431—21no—
175C2×#99—21no—
176#398C4×C12712yescomplemented
177#397C22×C12712yes—
178#389—21no—
179#381—21no—
180#380—21no—
181#380—21no—
182#380C4×C12711yescharacteristic, complemented
183#342C22×C12712yes—
184#323—21no—
185#323C22×C12712yes—
186C22×D4D4×C12711yescharacteristic
187C23×C4—21no—
188C23×C4C2×C4×C12711yescharacteristic
189C23×C4D4×C12711yescharacteristic
190#123—42no—
191#123—42no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×C4×C12712yes—
201C2×C4⋊C4C2×C4×C12711yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4D4×C12711yescharacteristic
206C2×C22⋊C4C23×C12711yescharacteristic
207C2×C42D4×C12711yescharacteristic
208C4⋊C8—41no—
209#99—42no—
210C22⋊C8—22no—
211C4×C8—41no—
212#95—22no—
213#95—22no—
214#95—41no—
215#95C2×C4×C12712yes—
216C24C22⋊C4×C12711yescharacteristic
217C24C2×D4×C12711yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4D8×C12712yes—
221C2×D4SD16×C12712yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4C22⋊C4×C12711yescharacteristic
235C22×C4C22⋊C4×C12711yescharacteristic
236C22×C4C22⋊C4×C12711yescharacteristic
237C22×C4C22×C4×C12711yescharacteristic
238C22×C4C2×D4×C12711yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—44no—
246C4⋊C4—82no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#102×C12712yes—
265C23C2×C22⋊C4×C12711yescharacteristic
266C23C2×D8×C12711yescharacteristic
267C23C2×SD16×C12711yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—84no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4#99×C12711yescharacteristic
294C2×C4#99×C12711yescharacteristic
295C2×C4#102×C12712yes—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#99×C12711yescharacteristic
312C22C2×#102×C12711yescharacteristic
313C22#421×C12711yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—162no—
322C4#352×C12712yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2C127×#151911yescharacteristic
329C2C2×#352×C12711yescharacteristic
330C2C127×#212911yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC22×D4×C127D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1344C2semidirect
7. C22×C4⋊C4×C127C4semidirect
10. #398×C127C4semidirect
16. #380×C127C4semidirect
157. #11874C127direct (N × H)
169. #1344C2×C127semidirect
170. C127#11874direct (N × H)
173. C22×C4⋊C4C4×C127semidirect
176. #398C4×C127semidirect
182. #380C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24862
C127×#17992
C127×#14602
C127×#13442
#11874127