Finite Group Databasebeta

← C127×#12724 #44576397

Subgroups — C127×#12724 #44576397

Order 32512.

670 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3509C211yesmaximal, characteristic
2C127×#2135C211yesmaximal, characteristic
3C127×#2129C211yesmaximal, characteristic
4C127×#2123C212yesmaximal
5C127×#1827C212yesmaximal
6#557×C127—22no—
7#556×C127—22no—
8#552×C127—24no—
9#541×C127—22no—
10#536×C127C2211yescharacteristic
11#536×C127C2211yescharacteristic
12#533×C127—22no—
13#533×C127C411yescharacteristic, complemented
14C2×C4⋊Q8×C127C411yescharacteristic, complemented
15#519×C127C411yescharacteristic, complemented
16#519×C127C411yescharacteristic, complemented
17C4×#41×C127C2211yescharacteristic
18#413×C127C2212yes—
19#410×C127C2212yes—
20#356×C127—22no—
21#354×C127—22no—
22#352×C127—24no—
23#339×C127—22no—
24H(3,ℤ/4)×C127—22no—
25C2×#41×C127C2311yescharacteristic
26C22×Q8×C127C2×C411yescharacteristic
27C22×Q8×C127C2×C411yescharacteristic
28C2×M16×C127—22no—
29C4⋊Q8×C127—22no—
30C4⋊Q8×C127—21no—
31C4⋊Q8×C127—21no—
32C4⋊Q8×C127—21no—
33C4⋊Q8×C127—21no—
34C4⋊Q8×C127D412yes—
35#126×C127—24no—
36#126×C127—24no—
37#125×C127—24no—
38#125×C127—24no—
39#125×C127—24no—
40#125×C127—21no—
41#124×C127—21no—
42#124×C127D412yes—
43#123×C127—22no—
44#123×C127D414yes—
45C22⋊Q8×C127—24no—
46C22⋊Q8×C127—22no—
47C22⋊Q8×C127—22no—
48C22⋊Q8×C127—22no—
49C4×Q8×C127—24no—
50C4×Q8×C127—22no—
51C4×Q8×C127—21no—
52C4×Q8×C127—22no—
53C4×Q8×C127—22no—
54C4×Q8×C127—22no—
55C4×Q8×C127—22no—
56C4×D4×C127—22no—
57C4×D4×C127—21no—
58#117×C127—24no—
59#117×C127—22no—
60#117×C127—22no—
61#117×C127—22no—
62#117×C127C2×C411yescharacteristic
63#117×C127C2×C411yescharacteristic
64#117×C127C2×C411yescharacteristic
65#117×C127D412yes—
66C2×C4⋊C4×C127—22no—
67C2×C42×C127C2×C411yescharacteristic
68C2×C42×C127D412yes—
69#101×C127—22no—
70#100×C127—22no—
71#99×C127—22no—
72#99×C127—22no—
73#41×C127—42no—
74C2×Q8×C127—22no—
75C2×Q8×C127—21no—
76C2×Q8×C127—21no—
77C2×Q8×C127—21no—
78C2×Q8×C127—42no—
79C2×Q8×C127—42no—
80C2×Q8×C127C22⋊C412yes—
81C2×Q8×C127C2×D411yescharacteristic
82C2×D4×C127C2×D411yescharacteristic
83C2×D4×C127C2×D412yes—
84C22×C4×C127—21no—
85C22×C4×C127—21no—
86C22×C4×C127—22no—
87C22×C4×C127C22⋊C412yes—
88C22×C4×C127C22×C411yescharacteristic
89C22×C4×C127C2×D412yes—
90M16×C127—42no—
91M16×C127—42no—
92C2×C8×C127—42no—
93C4⋊C4×C127—24no—
94C4⋊C4×C127—24no—
95C4⋊C4×C127—22no—
96C4⋊C4×C127—24no—
97C4⋊C4×C127—24no—
98C4⋊C4×C127—22no—
99C4⋊C4×C127—22no—
100C4⋊C4×C127—21no—
101C4⋊C4×C127—21no—
102C4⋊C4×C127—21no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127—24no—
105C4⋊C4×C127—24no—
106C4⋊C4×C127—42no—
107C22⋊C4×C127—21no—
108C22⋊C4×C127—21no—
109C22⋊C4×C127—24no—
110C22⋊C4×C127—24no—
111C22⋊C4×C127—22no—
112C22⋊C4×C127—42no—
113C22⋊C4×C127C22⋊C412yes—
114C42×C127—24no—
115C42×C127—24no—
116C42×C127—21no—
117C42×C127—21no—
118C42×C127—22no—
119C42×C127—42no—
120C42×C127C22⋊C412yes—
121C42×C127C22⋊C411yescharacteristic
122C42×C127C22⋊C411yescharacteristic
123C42×C127C22⋊C412yes—
124C23×C127C2×C22⋊C411yescharacteristic
125C23×C127#12012yes—
126Q8×C127—41no—
127Q8×C127—42no—
128Q8×C127—44no—
129Q8×C127—42no—
130Q8×C127—42no—
131Q8×C127—42no—
132D4×C127—42no—
133D4×C127—41no—
134C2×C4×C127—21no—
135C2×C4×C127—22no—
136C2×C4×C127—22no—
137C2×C4×C127—22no—
138C2×C4×C127—22no—
139C2×C4×C127—22no—
140C2×C4×C127—22no—
141C2×C4×C127—22no—
142C2×C4×C127—22no—
143C2×C4×C127—21no—
144C2×C4×C127—42no—
145C2×C4×C127—44no—
146C2×C4×C127—42no—
147C2×C4×C127—42no—
148C2×C4×C127—82no—
149C2×C4×C127C2×C22⋊C411yescharacteristic
150C2×C4×C127C2×C22⋊C411yescharacteristic
151C2×C4×C127#12012yes—
152C8×C127—82no—
153C22×C127—21no—
154C22×C127—22no—
155C22×C127—41no—
156C22×C127—42no—
157C22×C127#37911yescharacteristic
158C4×C127—22no—
159C4×C127—22no—
160C4×C127—21no—
161C4×C127—42no—
162C4×C127—41no—
163C4×C127—42no—
164C4×C127—42no—
165C4×C127—42no—
166C4×C127—42no—
167C4×C127—162no—
168C4×C127#41012yes—
169#12724C12711yesmaximal, characteristic, direct product factor, Sylow
170C2×C127—21no—
171C2×C127—42no—
172C2×C127—41no—
173C2×C127#178211yescharacteristic
174#3509C2×C12711yescharacteristic
175#2135C2×C12711yescharacteristic
176#2129C2×C12711yescharacteristic
177#2123C2×C12712yes—
178#1827C2×C12712yes—
179C127#1272411yescharacteristic, direct product factor, Sylow
180#557—22no—
181#556—22no—
182#552—24no—
183#541—22no—
184#536C22×C12711yescharacteristic
185#536C22×C12711yescharacteristic
186#533—22no—
187#533C4×C12711yescharacteristic, complemented
188C2×C4⋊Q8C4×C12711yescharacteristic, complemented
189#519C4×C12711yescharacteristic, complemented
190#519C4×C12711yescharacteristic, complemented
191C4×#41C22×C12711yescharacteristic
192#413C22×C12712yes—
193#410C22×C12712yes—
194#356—22no—
195#354—22no—
196#352—24no—
197#339—22no—
198H(3,ℤ/4)—22no—
199C2×#41C23×C12711yescharacteristic
200C22×Q8C2×C4×C12711yescharacteristic
201C22×Q8C2×C4×C12711yescharacteristic
202C2×M16—22no—
203C4⋊Q8—22no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8D4×C12712yes—
209#126—24no—
210#126—24no—
211#125—24no—
212#125—24no—
213#125—24no—
214#125—21no—
215#124—21no—
216#124D4×C12712yes—
217#123—22no—
218#123D4×C12714yes—
219C22⋊Q8—24no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C4×Q8—24no—
224C4×Q8—22no—
225C4×Q8—21no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×D4—22no—
231C4×D4—21no—
232#117—24no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117C2×C4×C12711yescharacteristic
237#117C2×C4×C12711yescharacteristic
238#117C2×C4×C12711yescharacteristic
239#117D4×C12712yes—
240C2×C4⋊C4—22no—
241C2×C42C2×C4×C12711yescharacteristic
242C2×C42D4×C12712yes—
243#101—22no—
244#100—22no—
245#99—22no—
246#99—22no—
247#41—42no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8C22⋊C4×C12712yes—
255C2×Q8C2×D4×C12711yescharacteristic
256C2×D4C2×D4×C12711yescharacteristic
257C2×D4C2×D4×C12712yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4C22⋊C4×C12712yes—
262C22×C4C22×C4×C12711yescharacteristic
263C22×C4C2×D4×C12712yes—
264M16—42no—
265M16—42no—
266C2×C8—42no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—22no—
286C22⋊C4—42no—
287C22⋊C4C22⋊C4×C12712yes—
288C42—24no—
289C42—24no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—42no—
294C42C22⋊C4×C12711yescharacteristic
295C42C22⋊C4×C12711yescharacteristic
296C42C22⋊C4×C12712yes—
297C42C22⋊C4×C12712yes—
298C23C2×C22⋊C4×C12711yescharacteristic
299C23#120×C12712yes—
300Q8—41no—
301Q8—42no—
302Q8—42no—
303Q8—44no—
304Q8—42no—
305Q8—42no—
306D4—42no—
307D4—41no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4C2×C22⋊C4×C12711yescharacteristic
324C2×C4C2×C22⋊C4×C12711yescharacteristic
325C2×C4#120×C12712yes—
326C8—82no—
327C22—21no—
328C22—22no—
329C22—41no—
330C22—42no—
331C22#379×C12711yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—162no—
342C4#410×C12712yes—
343C2—21no—
344C2—42no—
345C2—41no—
346C2C127×#178211yescharacteristic
subgroupG/H
center, socleC2×C127#1782
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×#41C23×C127
prime radicalC2×#41×C127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
13. #533×C127C4semidirect
14. C2×C4⋊Q8×C127C4semidirect
15. #519×C127C4semidirect
16. #519×C127C4semidirect
169. #12724C127direct (N × H)
179. C127#12724direct (N × H)
187. #533C4×C127semidirect
188. C2×C4⋊Q8C4×C127semidirect
189. #519C4×C127semidirect
190. #519C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35092
C127×#21352
C127×#21292
C127×#21232
C127×#18272
#12724127