Finite Group Databasebeta

← C127×#13226 #44576899

Subgroups — C127×#13226 #44576899

Order 32512.

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

normal subgroups only

This group has 540 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×#2548C211yesmaximal, characteristic
2C2×#355×C127C214yesmaximal, complemented
3C127×#1801C211yesmaximal, characteristic, complemented
4C4×#101×C127C211yesmaximal, characteristic, complemented
5C2×#127×C127C411yescharacteristic
6C2×#124×C127C2212yes—
7C2×C4×Q8×C127C2211yescharacteristic
8#423×C127—21no—
9C2×#101×C127C2212yes—
10C2×#101×C127C2212yes—
11C4×M16×C127—21no—
12#394×C127—21no—
13#394×C127—21no—
14#390×C127—21no—
15#390×C127—21no—
16#390×C127C412yes—
17C4×C4⋊C4×C127C411yescharacteristic
18#355×C127—28no—
19#343×C127—21no—
20#343×C127—21no—
21C22×Q8×C127C2311yescharacteristic
22C22×D4×C127—21no—
23C22×D4×C127—21no—
24C22×D4×C127C2×C412yes—
25C2×M16×C127—22no—
26C2×M16×C127—22no—
27#127×C127—41no—
28#127×C127—41no—
29#124×C127—24no—
30#124×C127—24no—
31C4×Q8×C127—41no—
32C4×Q8×C127—41no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127C2×C411yescharacteristic
35C2×C4⋊C4×C127D411yescharacteristic
36C2×C4⋊C4×C127D411yescharacteristic
37C2×C22⋊C4×C127—22no—
38C2×C22⋊C4×C127—22no—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C42×C127C2×C412yes—
42C2×C42×C127C2×C411yescharacteristic
43C2×C42×C127D411yescharacteristic
44C2×C42×C127D411yescharacteristic
45C4⋊C8×C127—41no—
46C4⋊C8×C127—41no—
47#101×C127—22no—
48#101×C127—24no—
49#101×C127D414yescomplemented
50#97×C127—41no—
51C4×C8×C127—41no—
52#95×C127—21no—
53C24×C127—22no—
54C2×Q8×C127—41no—
55C2×Q8×C127—41no—
56C2×Q8×C127C2×D412yes—
57C2×Q8×C127#4112yes—
58C2×D4×C127—24no—
59C2×D4×C127—42no—
60C2×D4×C127—42no—
61C2×D4×C127—42no—
62C2×D4×C127—42no—
63C2×D4×C127—42no—
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C2×D4×C127—82no—
67C22×C4×C127—22no—
68C22×C4×C127C22⋊C411yescharacteristic
69C22×C4×C127C22⋊C411yescharacteristic
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22⋊C411yescharacteristic
72C22×C4×C127C22×C411yescharacteristic
73C22×C4×C127C2×D411yescharacteristic
74C22×C4×C127C2×D411yescharacteristic
75M16×C127—24no—
76M16×C127—42no—
77M16×C127—42no—
78M16×C127—42no—
79C2×C8×C127—42no—
80C2×C8×C127—42no—
81C4⋊C4×C127—41no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—42no—
85C22⋊C4×C127—44no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—42no—
88C22⋊C4×C127—82no—
89C22⋊C4×C127—82no—
90C42×C127—21no—
91C42×C127—21no—
92C42×C127—21no—
93C42×C127—21no—
94C42×C127—21no—
95C42×C127—21no—
96C42×C127—24no—
97C23×C127—42no—
98C23×C127—42no—
99C23×C127—42no—
100C23×C127—42no—
101C23×C127—42no—
102C23×C127—42no—
103C23×C127—42no—
104C23×C127C2×C22⋊C411yescharacteristic
105Q8×C127—42no—
106Q8×C127—42no—
107D4×C127—84no—
108D4×C127—82no—
109D4×C127—84no—
110D4×C127—82no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—22no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—42no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127#9911yescharacteristic
128C2×C4×C127#9911yescharacteristic
129C2×C4×C127C4×D412yes—
130C2×C4×C127#12112yes—
131C2×C4×C127#12411yescharacteristic
132C2×C4×C127#12711yescharacteristic
133C8×C127—44no—
134C8×C127—82no—
135C22×C127—21no—
136C22×C127—21no—
137C22×C127—82no—
138C22×C127—82no—
139C22×C127—82no—
140C22×C127—82no—
141C22×C127—82no—
142C22×C127—82no—
143C22×C127—82no—
144C22×C127#39011yescharacteristic
145C22×C127C2×#9911yescharacteristic
146C22×C127#41011yescharacteristic
147C4×C127—21no—
148C4×C127—22no—
149C4×C127—21no—
150C4×C127—22no—
151C4×C127—21no—
152C4×C127—41no—
153C4×C127—41no—
154C4×C127—41no—
155C4×C127—82no—
156C4×C127#35512yescomplemented
157#13226C12711yesmaximal, characteristic, direct product factor, Sylow
158C2×C127—21no—
159C2×C127—21no—
160C2×C127—162no—
161C2×C127#196411yescharacteristic
162C2×C127C2×#35511yescharacteristic
163C2×C127#213411yescharacteristic
164#2548C2×C12711yescharacteristic
165C2×#355C2×C12714yescomplemented
166#1801C2×C12711yescharacteristic, complemented
167C4×#101C2×C12711yescharacteristic, complemented
168C127#1322611yescharacteristic, direct product factor, Sylow
169C2×#127C4×C12711yescharacteristic
170C2×#124C22×C12712yes—
171C2×C4×Q8C22×C12711yescharacteristic
172#423—21no—
173C2×#101C22×C12712yes—
174C2×#101C22×C12712yes—
175C4×M16—21no—
176#394—21no—
177#394—21no—
178#390—21no—
179#390—21no—
180#390C4×C12712yes—
181C4×C4⋊C4C4×C12711yescharacteristic
182#355—28no—
183#343—21no—
184#343—21no—
185C22×Q8C23×C12711yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C2×C4×C12712yes—
189C2×M16—22no—
190C2×M16—22no—
191#127—41no—
192#127—41no—
193#124—24no—
194#124—24no—
195C4×Q8—41no—
196C4×Q8—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×C4×C12711yescharacteristic
199C2×C4⋊C4D4×C12711yescharacteristic
200C2×C4⋊C4D4×C12711yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42C2×C4×C12712yes—
206C2×C42C2×C4×C12711yescharacteristic
207C2×C42D4×C12711yescharacteristic
208C2×C42D4×C12711yescharacteristic
209C4⋊C8—41no—
210C4⋊C8—41no—
211#101—22no—
212#101—24no—
213#101D4×C12714yescomplemented
214#97—41no—
215C4×C8—41no—
216#95—21no—
217C24—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8C2×D4×C12712yes—
221C2×Q8#41×C12712yes—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—82no—
231C22×C4—22no—
232C22×C4C22⋊C4×C12711yescharacteristic
233C22×C4C22⋊C4×C12711yescharacteristic
234C22×C4C22⋊C4×C12711yescharacteristic
235C22×C4C22⋊C4×C12711yescharacteristic
236C22×C4C22×C4×C12711yescharacteristic
237C22×C4C2×D4×C12711yescharacteristic
238C22×C4C2×D4×C12711yescharacteristic
239M16—24no—
240M16—42no—
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—24no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23C2×C22⋊C4×C12711yescharacteristic
269Q8—42no—
270Q8—42no—
271D4—84no—
272D4—82no—
273D4—84no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4#99×C12711yescharacteristic
292C2×C4#99×C12711yescharacteristic
293C2×C4C4×D4×C12712yes—
294C2×C4#121×C12712yes—
295C2×C4#124×C12711yescharacteristic
296C2×C4#127×C12711yescharacteristic
297C8—44no—
298C8—82no—
299C22—21no—
300C22—21no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22#390×C12711yescharacteristic
309C22C2×#99×C12711yescharacteristic
310C22#410×C12711yescharacteristic
311C4—21no—
312C4—22no—
313C4—21no—
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4#355×C12712yescomplemented
321C2—21no—
322C2—21no—
323C2—162no—
324C2C127×#196411yescharacteristic
325C2C2×#355×C12711yescharacteristic
326C2C127×#213411yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×Q8C23×C127
prime radicalC2×#127×C127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#355×C127C2semidirect
3. C127×#1801C2semidirect
4. C4×#101×C127C2semidirect
49. #101×C127D4semidirect
156. C4×C127#355semidirect
157. #13226C127direct (N × H)
165. C2×#355C2×C127semidirect
166. #1801C2×C127semidirect
167. C4×#101C2×C127semidirect
168. C127#13226direct (N × H)
213. #101D4×C127semidirect
320. C4#355×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#25482
C2×#355×C1272
C127×#18012
C4×#101×C1272
#13226127