Finite Group Databasebeta

← C127×#9489 #44573162

Subgroups — C127×#9489 #44573162

Order 32512.

602 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2975C211yesmaximal, characteristic, complemented
2C127×#2535C211yesmaximal, characteristic
3C127×#1800C211yesmaximal, characteristic, complemented
4C127×#1486C212yesmaximal, complemented
5C127×#1286C212yesmaximal, complemented
6C2×#121×C127—21no—
7C2×C4×D4×C127C2211yescharacteristic
8C2×C4×D4×C127C2211yescharacteristic
9#436×C127—24no—
10#431×C127C2212yes—
11#423×C127C2212yes—
12C2×#100×C127—21no—
13#407×C127—22no—
14#397×C127C412yes—
15#392×C127C412yes—
16#386×C127—22no—
17#382×C127C2211yescharacteristic
18#380×C127—21no—
19#343×C127—21no—
20D4⋊C8×C127—24no—
21C22×D4×C127—22no—
22C22×D4×C127C2×C411yescharacteristic
23C22×D4×C127D411yescharacteristic
24C23×C4×C127—21no—
25C23×C4×C127D412yes—
26C2×M16×C127—24no—
27C2×M16×C127—22no—
28#121×C127—44no—
29C4×D4×C127—22no—
30C4×D4×C127—22no—
31C4×D4×C127—22no—
32C4×D4×C127—22no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127C2×C411yescharacteristic
35C2×C4⋊C4×C127D411yescharacteristic
36C2×C22⋊C4×C127—22no—
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127C2×C412yes—
40C2×C42×C127C2311yescharacteristic
41C4⋊C8×C127—22no—
42C4⋊C8×C127—22no—
43C4⋊C8×C127—22no—
44#100×C127—42no—
45C22⋊C8×C127—24no—
46C22⋊C8×C127—22no—
47C4×C8×C127D414yescomplemented
48#95×C127—22no—
49#95×C127—21no—
50#95×C127C2×C412yes—
51C24×C127—21no—
52C24×C127C22⋊C412yes—
53C2×D4×C127—22no—
54C2×D4×C127—22no—
55C2×D4×C127—21no—
56C2×D4×C127—22no—
57C2×D4×C127—21no—
58C2×D4×C127—42no—
59C2×D4×C127—44no—
60C2×D4×C127—44no—
61C2×D4×C127—82no—
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—21no—
66C22×C4×C127—42no—
67C22×C4×C127—42no—
68C22×C4×C127—41no—
69C22×C4×C127—41no—
70C22×C4×C127—82no—
71C22×C4×C127C22⋊C412yes—
72C22×C4×C127C22×C411yescharacteristic
73C22×C4×C127C2×D411yescharacteristic
74C22×C4×C127C2×D411yescharacteristic
75M16×C127—42no—
76M16×C127—42no—
77M16×C127—84no—
78C2×C8×C127—24no—
79C2×C8×C127—42no—
80C4⋊C4×C127—22no—
81C4⋊C4×C127—22no—
82C4⋊C4×C127—42no—
83C22⋊C4×C127—24no—
84C22⋊C4×C127—44no—
85C22⋊C4×C127—41no—
86C22⋊C4×C127—41no—
87C22⋊C4×C127—82no—
88C42×C127C2×D412yes—
89C42×C127#4112yes—
90C23×C127—22no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—42no—
95C23×C127—42no—
96C23×C127—41no—
97C23×C127—41no—
98C23×C127—41no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—41no—
102C23×C127C2×C22⋊C411yescharacteristic
103D4×C127—24no—
104D4×C127—42no—
105D4×C127—42no—
106D4×C127—42no—
107D4×C127—88no—
108C2×C4×C127—22no—
109C2×C4×C127—41no—
110C2×C4×C127—41no—
111C2×C4×C127—44no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127—42no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—42no—
119C2×C4×C127—82no—
120C2×C4×C127—82no—
121C2×C4×C127—82no—
122C2×C4×C127C4×D412yes—
123C2×C4×C127#12011yescharacteristic
124C2×C4×C127#12112yes—
125C2×C4×C127#12311yescharacteristic
126C2×C4×C127AGL(1,ℤ/8)12yes—
127C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C127#13711yescharacteristic
129C8×C127—44no—
130C8×C127—82no—
131C22×C127—21no—
132C22×C127—21no—
133C22×C127—42no—
134C22×C127—44no—
135C22×C127—42no—
136C22×C127—42no—
137C22×C127—42no—
138C22×C127—42no—
139C22×C127—42no—
140C22×C127—82no—
141C22×C127—81no—
142C22×C127—81no—
143C22×C127—81no—
144C22×C127—81no—
145C22×C127#38611yescharacteristic
146C22×C127#41711yescharacteristic
147C22×C127#41811yescharacteristic
148C4×C127—22no—
149C4×C127—21no—
150C4×C127—21no—
151C4×C127—42no—
152C4×C127—82no—
153C4×C127—81no—
154C4×C127#46812yes—
155C4×C127#46912yes—
156#9489C12711yesmaximal, characteristic, direct product factor, Sylow
157C2×C127—21no—
158C2×C127—21no—
159C2×C127—82no—
160C2×C127—82no—
161C2×C127#189111yescharacteristic
162C2×C127#189511yescharacteristic
163C2×C127#191111yescharacteristic
164#2975C2×C12711yescharacteristic, complemented
165#2535C2×C12711yescharacteristic
166#1800C2×C12711yescharacteristic, complemented
167#1486C2×C12712yescomplemented
168#1286C2×C12712yescomplemented
169C127#948911yescharacteristic, direct product factor, Sylow
170C2×#121—21no—
171C2×C4×D4C22×C12711yescharacteristic
172C2×C4×D4C22×C12711yescharacteristic
173#436—24no—
174#431C22×C12712yes—
175#423C22×C12712yes—
176C2×#100—21no—
177#407—22no—
178#397C4×C12712yes—
179#392C4×C12712yes—
180#386—22no—
181#382C22×C12711yescharacteristic
182#380—21no—
183#343—21no—
184D4⋊C8—24no—
185C22×D4—22no—
186C22×D4C2×C4×C12711yescharacteristic
187C22×D4D4×C12711yescharacteristic
188C23×C4—21no—
189C23×C4D4×C12712yes—
190C2×M16—24no—
191C2×M16—22no—
192#121—44no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×C4×C12711yescharacteristic
199C2×C4⋊C4D4×C12711yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C2×C4×C12712yes—
204C2×C42C23×C12711yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208#100—42no—
209C22⋊C8—24no—
210C22⋊C8—22no—
211C4×C8D4×C12714yescomplemented
212#95—22no—
213#95—21no—
214#95C2×C4×C12712yes—
215C24—21no—
216C24C22⋊C4×C12712yes—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—21no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—42no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—82no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—82no—
235C22×C4C22⋊C4×C12712yes—
236C22×C4C22×C4×C12711yescharacteristic
237C22×C4C2×D4×C12711yescharacteristic
238C22×C4C2×D4×C12711yescharacteristic
239M16—42no—
240M16—42no—
241M16—84no—
242C2×C8—24no—
243C2×C8—42no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—42no—
247C22⋊C4—24no—
248C22⋊C4—44no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—82no—
252C42C2×D4×C12712yes—
253C42#41×C12712yes—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊C4×C12711yescharacteristic
267D4—24no—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—88no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4C4×D4×C12712yes—
287C2×C4#120×C12711yescharacteristic
288C2×C4#121×C12712yes—
289C2×C4#123×C12711yescharacteristic
290C2×C4AGL(1,ℤ/8)×C12712yes—
291C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
292C2×C4#137×C12711yescharacteristic
293C8—44no—
294C8—82no—
295C22—21no—
296C22—21no—
297C22—42no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#386×C12711yescharacteristic
310C22#417×C12711yescharacteristic
311C22#418×C12711yescharacteristic
312C4—22no—
313C4—21no—
314C4—21no—
315C4—42no—
316C4—82no—
317C4—81no—
318C4#468×C12712yes—
319C4#469×C12712yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—82no—
324C2C127×#189111yescharacteristic
325C2C127×#189511yescharacteristic
326C2C127×#191111yescharacteristic
subgroupG/H
center, socleC22×C127#386
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#2535C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2975C2semidirect
3. C127×#1800C2semidirect
4. C127×#1486C2semidirect
5. C127×#1286C2semidirect
47. C4×C8×C127D4semidirect
156. #9489C127direct (N × H)
164. #2975C2×C127semidirect
166. #1800C2×C127semidirect
167. #1486C2×C127semidirect
168. #1286C2×C127semidirect
169. C127#9489direct (N × H)
211. C4×C8D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#29752
C127×#25352
C127×#18002
C127×#14862
C127×#12862
#9489127