Finite Group Databasebeta

← C127×#13217 #44576890

Subgroups — C127×#13217 #44576890

Order 32512.

536 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 126 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
2C2×#352×C127C214yesmaximal
3C127×#1799C211yesmaximal, characteristic
4C4×#100×C127C211yesmaximal, characteristic
5C2×#123×C127C2212yes—
6C2×C4×D4×C127C2211yescharacteristic
7C22×C4⋊C4×C127C411yescharacteristic, complemented
8C2×#100×C127C2212yes—
9C2×#99×C127C2212yes—
10C4×M16×C127—21no—
11#398×C127—21no—
12#397×C127—21no—
13#389×C127—21no—
14#381×C127C412yescomplemented
15#380×C127—21no—
16#380×C127—21no—
17#380×C127C411yescharacteristic, complemented
18#352×C127—28no—
19#343×C127—21no—
20#342×C127—21no—
21C22×D4×C127C2311yescharacteristic
22C23×C4×C127C2×C412yes—
23C23×C4×C127C2×C411yescharacteristic
24C23×C4×C127D411yescharacteristic
25C2×M16×C127—22no—
26#123×C127—24no—
27#123×C127—24no—
28C4×D4×C127—41no—
29C4×D4×C127—41no—
30C2×C4⋊C4×C127—21no—
31C2×C4⋊C4×C127—21no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—22no—
34C2×C4⋊C4×C127—41no—
35C2×C4⋊C4×C127—41no—
36C2×C4⋊C4×C127—41no—
37C2×C4⋊C4×C127D411yescharacteristic
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127C2×C412yes—
41C2×C22⋊C4×C127C2×C411yescharacteristic
42C2×C22⋊C4×C127D411yescharacteristic
43C2×C42×C127D411yescharacteristic
44#100×C127—22no—
45#100×C127D414yes—
46#99×C127—24no—
47#97×C127—41no—
48C4×C8×C127—41no—
49#95×C127—22no—
50#95×C127—21no—
51#95×C127—22no—
52#95×C127—41no—
53C24×C127C22×C411yescharacteristic
54C24×C127C2×D411yescharacteristic
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127C2×D412yes—
58C2×D4×C127#4112yes—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—24no—
66C22×C4×C127—42no—
67C22×C4×C127—42no—
68C22×C4×C127—42no—
69C22×C4×C127—42no—
70C22×C4×C127—42no—
71C22×C4×C127C22⋊C411yescharacteristic
72C22×C4×C127C22⋊C411yescharacteristic
73C22×C4×C127C22⋊C411yescharacteristic
74C22×C4×C127C22⋊C411yescharacteristic
75C22×C4×C127C2×D411yescharacteristic
76M16×C127—24no—
77M16×C127—42no—
78C2×C8×C127—42no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127—42no—
81C4⋊C4×C127—42no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—44no—
85C4⋊C4×C127—82no—
86C22⋊C4×C127—24no—
87C22⋊C4×C127—41no—
88C22⋊C4×C127—41no—
89C22⋊C4×C127—42no—
90C22⋊C4×C127—42no—
91C22⋊C4×C127—42no—
92C22⋊C4×C127—42no—
93C42×C127—21no—
94C42×C127—21no—
95C23×C127—21no—
96C23×C127—21no—
97C23×C127—21no—
98C23×C127—21no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—41no—
102C23×C127—41no—
103C23×C127C2×C22⋊C411yescharacteristic
104C23×C127C4×D412yes—
105C23×C127#12411yescharacteristic
106C23×C127#12711yescharacteristic
107D4×C127—42no—
108D4×C127—42no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—22no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—42no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127—81no—
128C2×C4×C127—84no—
129C2×C4×C127—82no—
130C2×C4×C127—82no—
131C2×C4×C127—82no—
132C2×C4×C127#9911yescharacteristic
133C2×C4×C127#9911yescharacteristic
134C2×C4×C127#12112yes—
135C8×C127—44no—
136C22×C127—21no—
137C22×C127—21no—
138C22×C127—22no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—42no—
142C22×C127—41no—
143C22×C127—41no—
144C22×C127—42no—
145C22×C127—41no—
146C22×C127—41no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—81no—
150C22×C127#39011yescharacteristic
151C22×C127C2×#9911yescharacteristic
152C22×C127#41011yescharacteristic
153C4×C127—21no—
154C4×C127—22no—
155C4×C127—41no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127—82no—
159C4×C127—82no—
160C4×C127—162no—
161C4×C127#35212yescomplemented
162#13217C12711yesmaximal, characteristic, direct product factor, Sylow
163C2×C127—21no—
164C2×C127—21no—
165C2×C127—41no—
166C2×C127—42no—
167C2×C127—41no—
168C2×C127#196411yescharacteristic
169C2×C127C2×#35211yescharacteristic
170C2×C127#212311yescharacteristic
171#2486C2×C12711yescharacteristic
172C2×#352C2×C12714yes—
173#1799C2×C12711yescharacteristic
174C4×#100C2×C12711yescharacteristic
175C127#1321711yescharacteristic, direct product factor, Sylow
176C2×#123C22×C12712yes—
177C2×C4×D4C22×C12711yescharacteristic
178C22×C4⋊C4C4×C12711yescharacteristic, complemented
179C2×#100C22×C12712yes—
180C2×#99C22×C12712yes—
181C4×M16—21no—
182#398—21no—
183#397—21no—
184#389—21no—
185#381C4×C12712yescomplemented
186#380—21no—
187#380—21no—
188#380C4×C12711yescharacteristic, complemented
189#352—28no—
190#343—21no—
191#342—21no—
192C22×D4C23×C12711yescharacteristic
193C23×C4C2×C4×C12712yes—
194C23×C4C2×C4×C12711yescharacteristic
195C23×C4D4×C12711yescharacteristic
196C2×M16—22no—
197#123—24no—
198#123—24no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4D4×C12711yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4C2×C4×C12712yes—
212C2×C22⋊C4C2×C4×C12711yescharacteristic
213C2×C22⋊C4D4×C12711yescharacteristic
214C2×C42D4×C12711yescharacteristic
215#100—22no—
216#100D4×C12714yes—
217#99—24no—
218#97—41no—
219C4×C8—41no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—41no—
224C24C22×C4×C12711yescharacteristic
225C24C2×D4×C12711yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4C2×D4×C12712yes—
229C2×D4#41×C12712yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4C22⋊C4×C12711yescharacteristic
243C22×C4C22⋊C4×C12711yescharacteristic
244C22×C4C22⋊C4×C12711yescharacteristic
245C22×C4C22⋊C4×C12711yescharacteristic
246C22×C4C2×D4×C12711yescharacteristic
247M16—24no—
248M16—42no—
249C2×C8—42no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—44no—
256C4⋊C4—82no—
257C22⋊C4—24no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×C22⋊C4×C12711yescharacteristic
275C23C4×D4×C12712yes—
276C23#124×C12711yescharacteristic
277C23#127×C12711yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4#99×C12711yescharacteristic
304C2×C4#99×C12711yescharacteristic
305C2×C4#121×C12712yes—
306C8—44no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#390×C12711yescharacteristic
322C22C2×#99×C12711yescharacteristic
323C22#410×C12711yescharacteristic
324C4—21no—
325C4—22no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—82no—
330C4—82no—
331C4—162no—
332C4#352×C12712yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2C127×#196411yescharacteristic
339C2C2×#352×C12711yescharacteristic
340C2C127×#212311yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC24C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC22×D4×C127C23
LCSG ▷ C24 ▷ C22 ▷ 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
7. C22×C4⋊C4×C127C4semidirect
14. #381×C127C4semidirect
17. #380×C127C4semidirect
161. C4×C127#352semidirect
162. #13217C127direct (N × H)
175. C127#13217direct (N × H)
178. C22×C4⋊C4C4×C127semidirect
185. #381C4×C127semidirect
188. #380C4×C127semidirect
332. C4#352×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24862
C2×#352×C1272
C127×#17992
C4×#100×C1272
#13217127