Finite Group Databasebeta

← C131×#12272 #45698156

Subgroups — C131×#12272 #45698156

Order 33536.

372 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#2593×C131C211yesmaximal, characteristic
2#2027×C131C211yesmaximal, characteristic
3#1617×C131C211yesmaximal, characteristic, complemented
4#1611×C131C211yesmaximal, characteristic, complemented
5#1353×C131C211yesmaximal, characteristic, complemented
6#1346×C131C211yesmaximal, characteristic, complemented
7#1318×C131C211yesmaximal, characteristic
8#544×C131C2211yescharacteristic
9#484×C131—21no—
10#483×C131—21no—
11#418×C131—21no—
12#400×C131C2211yescharacteristic
13#398×C131C2211yescharacteristic
14#394×C131—21no—
15#390×C131C2211yescharacteristic
16#388×C131—21no—
17#388×C131—21no—
18#388×C131—21no—
19#388×C131—21no—
20C4×C4⋊C4×C131—21no—
21#343×C131C2211yescharacteristic
22#339×C131—21no—
23#328×C131—21no—
24#328×C131—21no—
25#324×C131C2211yescharacteristic
26#324×C131C2211yescharacteristic
27C22×D4×C131D411yescharacteristic
28C2×M16×C131—21no—
29C4⋊Q8×C131—41no—
30#125×C131—41no—
31C22⋊Q8×C131—21no—
32C22⋊Q8×C131—21no—
33#117×C131—21no—
34C2×C4⋊C4×C131—21no—
35C2×C4⋊C4×C131—21no—
36C2×C4⋊C4×C131D411yescharacteristic
37C2×C4⋊C4×C131D411yescharacteristic
38C2×C4⋊C4×C131C2311yescharacteristic
39C2×C22⋊C4×C131—21no—
40C2×C22⋊C4×C131—21no—
41C2×C22⋊C4×C131—21no—
42C2×C22⋊C4×C131—21no—
43C2×C22⋊C4×C131—21no—
44C2×C22⋊C4×C131—21no—
45C2×C42×C131—21no—
46C2×C42×C131—21no—
47C2×C42×C131D411yescharacteristic
48#107×C131—41no—
49#106×C131—41no—
50#103×C131—41no—
51#103×C131—41no—
52#102×C131—41no—
53#102×C131—41no—
54C22⋊C8×C131—21no—
55C22⋊C8×C131—21no—
56#95×C131—21no—
57#95×C131—21no—
58#95×C131—21no—
59#95×C131—21no—
60#95×C131D411yescharacteristic
61#95×C131D411yescharacteristic
62C24×C131—21no—
63C2×Q8×C131—21no—
64C2×Q8×C131—21no—
65C2×D4×C131—21no—
66C2×D4×C131—21no—
67C2×D4×C131—41no—
68C2×D4×C131—41no—
69C22×C4×C131—21no—
70C22×C4×C131—21no—
71C22×C4×C131—21no—
72C22×C4×C131—21no—
73C22×C4×C131—21no—
74C22×C4×C131C2×D411yescharacteristic
75C22×C4×C131C2×D411yescharacteristic
76C22×C4×C131C2×D411yescharacteristic
77M16×C131—41no—
78M16×C131—41no—
79C2×C8×C131—41no—
80C2×C8×C131—41no—
81C2×C8×C131—41no—
82C4⋊C4×C131—21no—
83C4⋊C4×C131—21no—
84C4⋊C4×C131—21no—
85C4⋊C4×C131—21no—
86C4⋊C4×C131—41no—
87C4⋊C4×C131—41no—
88C4⋊C4×C131—41no—
89C4⋊C4×C131—42no—
90C4⋊C4×C131—41no—
91C4⋊C4×C131—41no—
92C4⋊C4×C131—41no—
93C4⋊C4×C131—41no—
94C22⋊C4×C131—42no—
95C22⋊C4×C131—42no—
96C22⋊C4×C131—42no—
97C22⋊C4×C131—42no—
98C22⋊C4×C131—41no—
99C22⋊C4×C131—81no—
100C22⋊C4×C131—81no—
101C42×C131—41no—
102C42×C131—42no—
103C42×C131—41no—
104C42×C131—81no—
105C23×C131—41no—
106C23×C131—41no—
107C23×C131—41no—
108C23×C131—41no—
109C23×C131—41no—
110C23×C131—41no—
111C23×C131—41no—
112C23×C131#12011yescharacteristic
113Q8×C131—81no—
114Q8×C131—81no—
115D4×C131—42no—
116D4×C131—81no—
117C2×C4×C131—21no—
118C2×C4×C131—21no—
119C2×C4×C131—22no—
120C2×C4×C131—22no—
121C2×C4×C131—21no—
122C2×C4×C131—21no—
123C2×C4×C131—41no—
124C2×C4×C131—41no—
125C2×C4×C131—41no—
126C2×C4×C131—41no—
127C2×C4×C131—41no—
128C2×C4×C131—41no—
129C2×C4×C131—41no—
130C2×C4×C131—41no—
131C2×C4×C131—41no—
132C2×C4×C131—41no—
133C2×C4×C131—41no—
134C2×C4×C131—41no—
135C2×C4×C131—41no—
136C2×C4×C131—41no—
137C2×C4×C131—41no—
138C2×C4×C131—41no—
139C2×C4×C131—41no—
140C2×C4×C131#13511yescharacteristic
141C2×C4×C131#13711yescharacteristic
142C8×C131—81no—
143C8×C131—81no—
144C8×C131—81no—
145C22×C131—21no—
146C22×C131—21no—
147C22×C131—81no—
148C22×C131—81no—
149C22×C131—81no—
150C22×C131—81no—
151C22×C131—81no—
152C22×C131—81no—
153C22×C131—81no—
154C22×C131#45211yescharacteristic
155C22×C131#45511yescharacteristic
156C22×C131B(4,𝔽2)11yescharacteristic
157C4×C131—21no—
158C4×C131—21no—
159C4×C131—42no—
160C4×C131—42no—
161C4×C131—41no—
162C4×C131—41no—
163C4×C131—81no—
164C4×C131—81no—
165C4×C131—81no—
166C4×C131—81no—
167C4×C131—81no—
168C2×C131—21no—
169C2×C131—21no—
170C2×C131—161no—
171C2×C131#160611yescharacteristic
172C2×C131#219811yescharacteristic
173C2×C131#220411yescharacteristic
174#12272C13111yesmaximal, characteristic, direct product factor, Sylow
175C131#1227211yescharacteristic, direct product factor, Sylow
176#2593C2×C13111yescharacteristic
177#2027C2×C13111yescharacteristic
178#1617C2×C13111yescharacteristic, complemented
179#1611C2×C13111yescharacteristic, complemented
180#1353C2×C13111yescharacteristic, complemented
181#1346C2×C13111yescharacteristic, complemented
182#1318C2×C13111yescharacteristic
183#544C22×C13111yescharacteristic
184#484—21no—
185#483—21no—
186#418—21no—
187#400C22×C13111yescharacteristic
188#398C22×C13111yescharacteristic
189#394—21no—
190#390C22×C13111yescharacteristic
191#388—21no—
192#388—21no—
193#388—21no—
194#388—21no—
195C4×C4⋊C4—21no—
196#343C22×C13111yescharacteristic
197#339—21no—
198#328—21no—
199#328—21no—
200#324C22×C13111yescharacteristic
201#324C22×C13111yescharacteristic
202C22×D4D4×C13111yescharacteristic
203C2×M16—21no—
204C4⋊Q8—41no—
205#125—41no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208#117—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4D4×C13111yescharacteristic
212C2×C4⋊C4D4×C13111yescharacteristic
213C2×C4⋊C4C23×C13111yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42D4×C13111yescharacteristic
223#107—41no—
224#106—41no—
225#103—41no—
226#103—41no—
227#102—41no—
228#102—41no—
229C22⋊C8—21no—
230C22⋊C8—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95D4×C13111yescharacteristic
236#95D4×C13111yescharacteristic
237C24—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—41no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4C2×D4×C13111yescharacteristic
250C22×C4C2×D4×C13111yescharacteristic
251C22×C4C2×D4×C13111yescharacteristic
252M16—41no—
253M16—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C42—41no—
277C42—42no—
278C42—41no—
279C42—81no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23#120×C13111yescharacteristic
288Q8—81no—
289Q8—81no—
290D4—42no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4#135×C13111yescharacteristic
316C2×C4#137×C13111yescharacteristic
317C8—81no—
318C8—81no—
319C8—81no—
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22#452×C13111yescharacteristic
330C22#455×C13111yescharacteristic
331C22B(4,𝔽2)×C13111yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C2—21no—
344C2—21no—
345C2—161no—
346C2#1606×C13111yescharacteristic
347C2#2198×C13111yescharacteristic
348C2#2204×C13111yescharacteristic
subgroupG/H
center, socleC22×C131B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C131
prime radicalC22×D4×C131D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C23×C131 ◁ C2×C4⋊C4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
3. #1617×C131C2semidirect
4. #1611×C131C2semidirect
5. #1353×C131C2semidirect
6. #1346×C131C2semidirect
174. #12272C131direct (N × H)
175. C131#12272direct (N × H)
178. #1617C2×C131semidirect
179. #1611C2×C131semidirect
180. #1353C2×C131semidirect
181. #1346C2×C131semidirect

Maximal subgroups

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

subgroupindex
#2593×C1312
#2027×C1312
#1617×C1312
#1611×C1312
#1353×C1312
#1346×C1312
#1318×C1312
#12272131