Finite Group Databasebeta

← C127×#12091 #44575764

Subgroups — C127×#12091 #44575764

Order 32512.

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

normal subgroups only

This group has 618 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×#2483C211yesmaximal, characteristic, complemented
2C127×#2021C211yesmaximal, characteristic
3C127×#1607C212yesmaximal, complemented
4C127×#1344C212yesmaximal, complemented
5C127×#1316C211yesmaximal, characteristic, complemented
6#534×C127C2211yescharacteristic
7C2×#120×C127—21no—
8C22×C22⋊C4×C127—21no—
9#480×C127—22no—
10C2×#99×C127—21no—
11#397×C127—21no—
12#397×C127C2212yes—
13#385×C127—22no—
14#381×C127—22no—
15#379×C127C2211yescharacteristic
16#342×C127C2211yescharacteristic
17#327×C127—22no—
18#323×C127C2212yes—
19C25×C127D411yescharacteristic
20C22×D4×C127—21no—
21C22×D4×C127—41no—
22C23×C4×C127—21no—
23C23×C4×C127—21no—
24#123×C127—41no—
25#121×C127—22no—
26#120×C127—42no—
27#120×C127—41no—
28#120×C127—42no—
29C2×C4⋊C4×C127—21no—
30C2×C4⋊C4×C127D412yescomplemented
31C2×C22⋊C4×C127—21no—
32C2×C22⋊C4×C127—22no—
33C2×C22⋊C4×C127—22no—
34C2×C22⋊C4×C127—22no—
35C2×C22⋊C4×C127—41no—
36C2×C22⋊C4×C127—42no—
37C2×C22⋊C4×C127—42no—
38C2×C22⋊C4×C127D411yescharacteristic
39C2×C22⋊C4×C127C2311yescharacteristic
40#107×C127—42no—
41#102×C127—42no—
42#99×C127—42no—
43C22⋊C8×C127—22no—
44#95×C127—21no—
45#95×C127—22no—
46#95×C127—22no—
47#95×C127D412yescomplemented
48C24×C127—21no—
49C24×C127—21no—
50C24×C127—21no—
51C24×C127—41no—
52C24×C127—44no—
53C24×C127—41no—
54C24×C127D812yes—
55C24×C127C2×D411yescharacteristic
56C2×D4×C127—22no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C2×D4×C127—41no—
60C2×D4×C127—44no—
61C2×D4×C127—41no—
62C2×D4×C127—41no—
63C2×D4×C127—41no—
64C2×D4×C127—84no—
65C22×C4×C127—21no—
66C22×C4×C127—22no—
67C22×C4×C127—21no—
68C22×C4×C127—22no—
69C22×C4×C127—41no—
70C22×C4×C127—42no—
71C22×C4×C127—42no—
72C22×C4×C127—44no—
73C22×C4×C127—81no—
74C22×C4×C127—81no—
75C22×C4×C127C2×D412yes—
76C2×C8×C127—42no—
77C4⋊C4×C127—22no—
78C4⋊C4×C127—42no—
79C4⋊C4×C127—81no—
80C22⋊C4×C127—22no—
81C22⋊C4×C127—41no—
82C22⋊C4×C127—42no—
83C22⋊C4×C127—44no—
84C22⋊C4×C127—44no—
85C22⋊C4×C127—44no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—41no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—41no—
90C23×C127—24no—
91C23×C127—22no—
92C23×C127—21no—
93C23×C127—21no—
94C23×C127—42no—
95C23×C127—42no—
96C23×C127—44no—
97C23×C127—42no—
98C23×C127—44no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—41no—
102C23×C127—41no—
103C23×C127—44no—
104C23×C127—41no—
105C23×C127—81no—
106C23×C127—84no—
107C23×C127—82no—
108C23×C127—82no—
109C23×C127#12011yescharacteristic
110C23×C127C2×D811yescharacteristic
111C23×C127AGL(1,ℤ/8)11yescharacteristic
112D4×C127—82no—
113D4×C127—81no—
114D4×C127—81no—
115D4×C127—88no—
116D4×C127—81no—
117D4×C127—81no—
118C2×C4×C127—22no—
119C2×C4×C127—42no—
120C2×C4×C127—42no—
121C2×C4×C127—41no—
122C2×C4×C127—42no—
123C2×C4×C127—44no—
124C2×C4×C127—41no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127—41no—
128C2×C4×C127—42no—
129C2×C4×C127—41no—
130C2×C4×C127—42no—
131C2×C4×C127—84no—
132C2×C4×C127—84no—
133C2×C4×C127—82no—
134C8×C127—82no—
135C22×C127—22no—
136C22×C127—24no—
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—41no—
140C22×C127—41no—
141C22×C127—42no—
142C22×C127—44no—
143C22×C127—44no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127—44no—
147C22×C127—81no—
148C22×C127—82no—
149C22×C127—84no—
150C22×C127—82no—
151C22×C127—81no—
152C22×C127—82no—
153C22×C127—161no—
154C22×C127#44711yescharacteristic
155C22×C127#45311yescharacteristic
156C22×C127B(4,𝔽2)11yescharacteristic
157C4×C127—42no—
158C4×C127—81no—
159C4×C127—82no—
160C4×C127—82no—
161C4×C127—82no—
162C4×C127—81no—
163#12091C12711yesmaximal, characteristic, direct product factor, Sylow
164C2×C127—21no—
165C2×C127—21no—
166C2×C127—41no—
167C2×C127—44no—
168C2×C127—41no—
169C2×C127—161no—
170C2×C127#159511yescharacteristic
171C2×C127#219712yes—
172#2483C2×C12711yescharacteristic, complemented
173#2021C2×C12711yescharacteristic
174#1607C2×C12712yescomplemented
175#1344C2×C12712yescomplemented
176#1316C2×C12711yescharacteristic, complemented
177C127#1209111yescharacteristic, direct product factor, Sylow
178#534C22×C12711yescharacteristic
179C2×#120—21no—
180C22×C22⋊C4—21no—
181#480—22no—
182C2×#99—21no—
183#397—21no—
184#397C22×C12712yes—
185#385—22no—
186#381—22no—
187#379C22×C12711yescharacteristic
188#342C22×C12711yescharacteristic
189#327—22no—
190#323C22×C12712yes—
191C25D4×C12711yescharacteristic
192C22×D4—21no—
193C22×D4—41no—
194C23×C4—21no—
195C23×C4—21no—
196#123—41no—
197#121—22no—
198#120—42no—
199#120—41no—
200#120—42no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4D4×C12712yescomplemented
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4D4×C12711yescharacteristic
211C2×C22⋊C4C23×C12711yescharacteristic
212#107—42no—
213#102—42no—
214#99—42no—
215C22⋊C8—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95D4×C12712yescomplemented
220C24—21no—
221C24—21no—
222C24—21no—
223C24—41no—
224C24—44no—
225C24—41no—
226C24D8×C12712yes—
227C24C2×D4×C12711yescharacteristic
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—84no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×D4×C12712yes—
248C2×C8—42no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—81no—
252C22⋊C4—22no—
253C22⋊C4—41no—
254C22⋊C4—42no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—41no—
262C23—24no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—42no—
267C23—42no—
268C23—44no—
269C23—42no—
270C23—44no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—44no—
276C23—41no—
277C23—81no—
278C23—84no—
279C23—82no—
280C23—82no—
281C23#120×C12711yescharacteristic
282C23C2×D8×C12711yescharacteristic
283C23AGL(1,ℤ/8)×C12711yescharacteristic
284D4—82no—
285D4—81no—
286D4—81no—
287D4—88no—
288D4—81no—
289D4—81no—
290C2×C4—22no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—84no—
304C2×C4—84no—
305C2×C4—82no—
306C8—82no—
307C22—22no—
308C22—24no—
309C22—21no—
310C22—21no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—44no—
319C22—81no—
320C22—82no—
321C22—84no—
322C22—82no—
323C22—81no—
324C22—82no—
325C22—161no—
326C22#447×C12711yescharacteristic
327C22#453×C12711yescharacteristic
328C22B(4,𝔽2)×C12711yescharacteristic
329C4—42no—
330C4—82no—
331C4—81no—
332C4—82no—
333C4—82no—
334C4—81no—
335C2—21no—
336C2—21no—
337C2—41no—
338C2—44no—
339C2—41no—
340C2—161no—
341C2C127×#159511yescharacteristic
342C2C127×#219712yes—
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2021C2
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
1. C127×#2483C2semidirect
3. C127×#1607C2semidirect
4. C127×#1344C2semidirect
5. C127×#1316C2semidirect
30. C2×C4⋊C4×C127D4semidirect
47. #95×C127D4semidirect
163. #12091C127direct (N × H)
172. #2483C2×C127semidirect
174. #1607C2×C127semidirect
175. #1344C2×C127semidirect
176. #1316C2×C127semidirect
177. C127#12091direct (N × H)
202. C2×C4⋊C4D4×C127semidirect
219. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24832
C127×#20212
C127×#16072
C127×#13442
C127×#13162
#12091127