Finite Group Databasebeta

← C131×#12724 #45698608

Subgroups — C131×#12724 #45698608

Order 33536.

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
1#3509×C131C211yesmaximal, characteristic
2#2135×C131C211yesmaximal, characteristic
3#2129×C131C211yesmaximal, characteristic
4#2123×C131C212yesmaximal
5#1827×C131C212yesmaximal
6#557×C131—22no—
7#556×C131—22no—
8#552×C131—24no—
9#541×C131—22no—
10#536×C131C2211yescharacteristic
11#536×C131C2211yescharacteristic
12#533×C131—22no—
13#533×C131C411yescharacteristic, complemented
14C2×C4⋊Q8×C131C411yescharacteristic, complemented
15#519×C131C411yescharacteristic, complemented
16#519×C131C411yescharacteristic, complemented
17C4×#41×C131C2211yescharacteristic
18#413×C131C2212yes—
19#410×C131C2212yes—
20#356×C131—22no—
21#354×C131—22no—
22#352×C131—24no—
23#339×C131—22no—
24H(3,ℤ/4)×C131—22no—
25C2×#41×C131C2311yescharacteristic
26C22×Q8×C131C2×C411yescharacteristic
27C22×Q8×C131C2×C411yescharacteristic
28C2×M16×C131—22no—
29C4⋊Q8×C131—22no—
30C4⋊Q8×C131—21no—
31C4⋊Q8×C131—21no—
32C4⋊Q8×C131—21no—
33C4⋊Q8×C131—21no—
34C4⋊Q8×C131D412yes—
35#126×C131—24no—
36#126×C131—24no—
37#125×C131—24no—
38#125×C131—24no—
39#125×C131—24no—
40#125×C131—21no—
41#124×C131—21no—
42#124×C131D412yes—
43#123×C131—22no—
44#123×C131D414yes—
45C22⋊Q8×C131—24no—
46C22⋊Q8×C131—22no—
47C22⋊Q8×C131—22no—
48C22⋊Q8×C131—22no—
49C4×Q8×C131—24no—
50C4×Q8×C131—22no—
51C4×Q8×C131—21no—
52C4×Q8×C131—22no—
53C4×Q8×C131—22no—
54C4×Q8×C131—22no—
55C4×Q8×C131—22no—
56C4×D4×C131—22no—
57C4×D4×C131—21no—
58#117×C131—24no—
59#117×C131—22no—
60#117×C131—22no—
61#117×C131—22no—
62#117×C131C2×C411yescharacteristic
63#117×C131C2×C411yescharacteristic
64#117×C131C2×C411yescharacteristic
65#117×C131D412yes—
66C2×C4⋊C4×C131—22no—
67C2×C42×C131C2×C411yescharacteristic
68C2×C42×C131D412yes—
69#101×C131—22no—
70#100×C131—22no—
71#99×C131—22no—
72#99×C131—22no—
73#41×C131—42no—
74C2×Q8×C131—22no—
75C2×Q8×C131—21no—
76C2×Q8×C131—21no—
77C2×Q8×C131—21no—
78C2×Q8×C131—42no—
79C2×Q8×C131—42no—
80C2×Q8×C131C22⋊C412yes—
81C2×Q8×C131C2×D411yescharacteristic
82C2×D4×C131C2×D411yescharacteristic
83C2×D4×C131C2×D412yes—
84C22×C4×C131—21no—
85C22×C4×C131—21no—
86C22×C4×C131—22no—
87C22×C4×C131C22⋊C412yes—
88C22×C4×C131C22×C411yescharacteristic
89C22×C4×C131C2×D412yes—
90M16×C131—42no—
91M16×C131—42no—
92C2×C8×C131—42no—
93C4⋊C4×C131—24no—
94C4⋊C4×C131—24no—
95C4⋊C4×C131—22no—
96C4⋊C4×C131—24no—
97C4⋊C4×C131—24no—
98C4⋊C4×C131—22no—
99C4⋊C4×C131—22no—
100C4⋊C4×C131—21no—
101C4⋊C4×C131—21no—
102C4⋊C4×C131—21no—
103C4⋊C4×C131—22no—
104C4⋊C4×C131—24no—
105C4⋊C4×C131—24no—
106C4⋊C4×C131—42no—
107C22⋊C4×C131—21no—
108C22⋊C4×C131—21no—
109C22⋊C4×C131—24no—
110C22⋊C4×C131—24no—
111C22⋊C4×C131—22no—
112C22⋊C4×C131—42no—
113C22⋊C4×C131C22⋊C412yes—
114C42×C131—24no—
115C42×C131—24no—
116C42×C131—21no—
117C42×C131—21no—
118C42×C131—22no—
119C42×C131—42no—
120C42×C131C22⋊C412yes—
121C42×C131C22⋊C411yescharacteristic
122C42×C131C22⋊C411yescharacteristic
123C42×C131C22⋊C412yes—
124C23×C131C2×C22⋊C411yescharacteristic
125C23×C131#12012yes—
126Q8×C131—41no—
127Q8×C131—42no—
128Q8×C131—44no—
129Q8×C131—42no—
130Q8×C131—42no—
131Q8×C131—42no—
132D4×C131—42no—
133D4×C131—41no—
134C2×C4×C131—21no—
135C2×C4×C131—22no—
136C2×C4×C131—22no—
137C2×C4×C131—22no—
138C2×C4×C131—22no—
139C2×C4×C131—22no—
140C2×C4×C131—22no—
141C2×C4×C131—22no—
142C2×C4×C131—22no—
143C2×C4×C131—21no—
144C2×C4×C131—42no—
145C2×C4×C131—44no—
146C2×C4×C131—42no—
147C2×C4×C131—42no—
148C2×C4×C131—82no—
149C2×C4×C131C2×C22⋊C411yescharacteristic
150C2×C4×C131C2×C22⋊C411yescharacteristic
151C2×C4×C131#12012yes—
152C8×C131—82no—
153C22×C131—21no—
154C22×C131—22no—
155C22×C131—41no—
156C22×C131—42no—
157C22×C131#37911yescharacteristic
158C4×C131—22no—
159C4×C131—22no—
160C4×C131—21no—
161C4×C131—42no—
162C4×C131—41no—
163C4×C131—42no—
164C4×C131—42no—
165C4×C131—42no—
166C4×C131—42no—
167C4×C131—162no—
168C4×C131#41012yes—
169C2×C131—21no—
170C2×C131—42no—
171C2×C131—41no—
172C2×C131#178211yescharacteristic
173#12724C13111yesmaximal, characteristic, direct product factor, Sylow
174C131#1272411yescharacteristic, direct product factor, Sylow
175#3509C2×C13111yescharacteristic
176#2135C2×C13111yescharacteristic
177#2129C2×C13111yescharacteristic
178#2123C2×C13112yes—
179#1827C2×C13112yes—
180#557—22no—
181#556—22no—
182#552—24no—
183#541—22no—
184#536C22×C13111yescharacteristic
185#536C22×C13111yescharacteristic
186#533—22no—
187#533C4×C13111yescharacteristic, complemented
188C2×C4⋊Q8C4×C13111yescharacteristic, complemented
189#519C4×C13111yescharacteristic, complemented
190#519C4×C13111yescharacteristic, complemented
191C4×#41C22×C13111yescharacteristic
192#413C22×C13112yes—
193#410C22×C13112yes—
194#356—22no—
195#354—22no—
196#352—24no—
197#339—22no—
198H(3,ℤ/4)—22no—
199C2×#41C23×C13111yescharacteristic
200C22×Q8C2×C4×C13111yescharacteristic
201C22×Q8C2×C4×C13111yescharacteristic
202C2×M16—22no—
203C4⋊Q8—22no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8D4×C13112yes—
209#126—24no—
210#126—24no—
211#125—24no—
212#125—24no—
213#125—24no—
214#125—21no—
215#124—21no—
216#124D4×C13112yes—
217#123—22no—
218#123D4×C13114yes—
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×C13111yescharacteristic
237#117C2×C4×C13111yescharacteristic
238#117C2×C4×C13111yescharacteristic
239#117D4×C13112yes—
240C2×C4⋊C4—22no—
241C2×C42C2×C4×C13111yescharacteristic
242C2×C42D4×C13112yes—
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×C13112yes—
255C2×Q8C2×D4×C13111yescharacteristic
256C2×D4C2×D4×C13111yescharacteristic
257C2×D4C2×D4×C13112yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4C22⋊C4×C13112yes—
262C22×C4C22×C4×C13111yescharacteristic
263C22×C4C2×D4×C13112yes—
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×C13112yes—
288C42—24no—
289C42—24no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—42no—
294C42C22⋊C4×C13111yescharacteristic
295C42C22⋊C4×C13111yescharacteristic
296C42C22⋊C4×C13112yes—
297C42C22⋊C4×C13112yes—
298C23C2×C22⋊C4×C13111yescharacteristic
299C23#120×C13112yes—
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×C13111yescharacteristic
324C2×C4C2×C22⋊C4×C13111yescharacteristic
325C2×C4#120×C13112yes—
326C8—82no—
327C22—21no—
328C22—22no—
329C22—41no—
330C22—42no—
331C22#379×C13111yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—42no—
336C4—41no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—162no—
342C4#410×C13112yes—
343C2—21no—
344C2—42no—
345C2—41no—
346C2#1782×C13111yescharacteristic
subgroupG/H
center, socleC2×C131#1782
commutatorC22×C4C22×C4×C131
Frattini, cosocleC2×#41C23×C131
prime radicalC2×#41×C131C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C131 ◁ C2×C4×C131 ◁ C4×#41×C131 ◁ G

Complements

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

NH ≅ G/Ntype
13. #533×C131C4semidirect
14. C2×C4⋊Q8×C131C4semidirect
15. #519×C131C4semidirect
16. #519×C131C4semidirect
173. #12724C131direct (N × H)
174. C131#12724direct (N × H)
187. #533C4×C131semidirect
188. C2×C4⋊Q8C4×C131semidirect
189. #519C4×C131semidirect
190. #519C4×C131semidirect

Maximal subgroups

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

subgroupindex
#3509×C1312
#2135×C1312
#2129×C1312
#2123×C1312
#1827×C1312
#12724131