Finite Group Databasebeta

← C131×#12258 #45698142

Subgroups — C131×#12258 #45698142

Order 33536.

352 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 74 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#2612×C131C211yesmaximal, characteristic
2#2061×C131C211yesmaximal, characteristic
3#1617×C131C211yesmaximal, characteristic
4#1605×C131C211yesmaximal, characteristic
5#1354×C131C211yesmaximal, characteristic
6#1353×C131C211yesmaximal, characteristic
7#1324×C131C211yesmaximal, characteristic
8#544×C131C2211yescharacteristic
9#484×C131—21no—
10#451×C131—21no—
11#428×C131—21no—
12#400×C131C2211yescharacteristic
13#398×C131—21no—
14#395×C131C2211yescharacteristic
15#391×C131—21no—
16#389×C131—21no—
17#387×C131—21no—
18#387×C131—21no—
19#382×C131C2211yescharacteristic
20#376×C131—21no—
21#343×C131C2211yescharacteristic
22#339×C131—21no—
23#328×C131—21no—
24#328×C131—21no—
25#324×C131C2211yescharacteristic
26#324×C131C2211yescharacteristic
27C22×Q8×C131D411yescharacteristic
28C2×Q16×C131—41no—
29C2×M16×C131—21no—
30C4⋊Q8×C131—41no—
31#125×C131—41no—
32C22⋊Q8×C131—21no—
33C22⋊Q8×C131—21no—
34#117×C131—21no—
35C2×C4⋊C4×C131—21no—
36C2×C4⋊C4×C131—21no—
37C2×C4⋊C4×C131—21no—
38C2×C4⋊C4×C131—21no—
39C2×C4⋊C4×C131D411yescharacteristic
40C2×C4⋊C4×C131D411yescharacteristic
41C2×C4⋊C4×C131C2311yescharacteristic
42C2×C42×C131—21no—
43C2×C42×C131—21no—
44C2×C42×C131D411yescharacteristic
45#107×C131—41no—
46#107×C131—41no—
47#106×C131—41no—
48#103×C131—41no—
49#103×C131—41no—
50C22⋊C8×C131—21no—
51C22⋊C8×C131—21no—
52#95×C131—21no—
53#95×C131—21no—
54#95×C131—21no—
55#95×C131—21no—
56#95×C131—21no—
57#95×C131—21no—
58#95×C131—21no—
59#95×C131—21no—
60#95×C131D411yescharacteristic
61#95×C131D411yescharacteristic
62C2×Q8×C131—21no—
63C2×Q8×C131—21no—
64C2×Q8×C131—42no—
65C2×Q8×C131—41no—
66C22×C4×C131—21no—
67C22×C4×C131—21no—
68C22×C4×C131—21no—
69C22×C4×C131—21no—
70C22×C4×C131—21no—
71C22×C4×C131—21no—
72C22×C4×C131C2×D411yescharacteristic
73C22×C4×C131C2×D411yescharacteristic
74C22×C4×C131C2×D411yescharacteristic
75Q16×C131—82no—
76M16×C131—22no—
77M16×C131—41no—
78C2×C8×C131—41no—
79C2×C8×C131—41no—
80C2×C8×C131—41no—
81C4⋊C4×C131—21no—
82C4⋊C4×C131—21no—
83C4⋊C4×C131—21no—
84C4⋊C4×C131—21no—
85C4⋊C4×C131—41no—
86C4⋊C4×C131—41no—
87C4⋊C4×C131—41no—
88C4⋊C4×C131—42no—
89C4⋊C4×C131—42no—
90C4⋊C4×C131—42no—
91C4⋊C4×C131—41no—
92C4⋊C4×C131—41no—
93C4⋊C4×C131—81no—
94C22⋊C4×C131—41no—
95C42×C131—21no—
96C42×C131—42no—
97C42×C131—42no—
98C42×C131—41no—
99C42×C131Q1612yescomplemented
100C23×C131#12011yescharacteristic
101Q8×C131—42no—
102Q8×C131—81no—
103Q8×C131—81no—
104Q8×C131—81no—
105C2×C4×C131—21no—
106C2×C4×C131—21no—
107C2×C4×C131—22no—
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131—21no—
111C2×C4×C131—21no—
112C2×C4×C131—41no—
113C2×C4×C131—41no—
114C2×C4×C131—41no—
115C2×C4×C131—41no—
116C2×C4×C131—41no—
117C2×C4×C131—41no—
118C2×C4×C131—41no—
119C2×C4×C131—41no—
120C2×C4×C131—41no—
121C2×C4×C131—41no—
122C2×C4×C131—41no—
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×C131C2×Q1611yescharacteristic
132C2×C4×C131#13711yescharacteristic
133C8×C131—42no—
134C8×C131—81no—
135C8×C131—81no—
136C22×C131—21no—
137C22×C131—21no—
138C22×C131#45111yescharacteristic
139C22×C131#45611yescharacteristic
140C22×C131B(4,𝔽2)11yescharacteristic
141C4×C131—21no—
142C4×C131—21no—
143C4×C131—42no—
144C4×C131—42no—
145C4×C131—41no—
146C4×C131—41no—
147C4×C131—42no—
148C4×C131—81no—
149C4×C131—81no—
150C4×C131—81no—
151C4×C131—81no—
152C4×C131—81no—
153C2×C131—21no—
154C2×C131—21no—
155C2×C131#160511yescharacteristic
156C2×C131#219711yescharacteristic
157C2×C131#220411yescharacteristic
158#12258C13111yesmaximal, characteristic, direct product factor, Sylow
159C131#1225811yescharacteristic, direct product factor, Sylow
160#2612C2×C13111yescharacteristic
161#2061C2×C13111yescharacteristic
162#1617C2×C13111yescharacteristic
163#1605C2×C13111yescharacteristic
164#1354C2×C13111yescharacteristic
165#1353C2×C13111yescharacteristic
166#1324C2×C13111yescharacteristic
167#544C22×C13111yescharacteristic
168#484—21no—
169#451—21no—
170#428—21no—
171#400C22×C13111yescharacteristic
172#398—21no—
173#395C22×C13111yescharacteristic
174#391—21no—
175#389—21no—
176#387—21no—
177#387—21no—
178#382C22×C13111yescharacteristic
179#376—21no—
180#343C22×C13111yescharacteristic
181#339—21no—
182#328—21no—
183#328—21no—
184#324C22×C13111yescharacteristic
185#324C22×C13111yescharacteristic
186C22×Q8D4×C13111yescharacteristic
187C2×Q16—41no—
188C2×M16—21no—
189C4⋊Q8—41no—
190#125—41no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193#117—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4D4×C13111yescharacteristic
199C2×C4⋊C4D4×C13111yescharacteristic
200C2×C4⋊C4C23×C13111yescharacteristic
201C2×C42—21no—
202C2×C42—21no—
203C2×C42D4×C13111yescharacteristic
204#107—41no—
205#107—41no—
206#106—41no—
207#103—41no—
208#103—41no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95D4×C13111yescharacteristic
220#95D4×C13111yescharacteristic
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—42no—
224C2×Q8—41no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4C2×D4×C13111yescharacteristic
232C22×C4C2×D4×C13111yescharacteristic
233C22×C4C2×D4×C13111yescharacteristic
234Q16—82no—
235M16—22no—
236M16—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—81no—
253C22⋊C4—41no—
254C42—21no—
255C42—42no—
256C42—42no—
257C42—41no—
258C42Q16×C13112yescomplemented
259C23#120×C13111yescharacteristic
260Q8—42no—
261Q8—81no—
262Q8—81no—
263Q8—81no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4C2×Q16×C13111yescharacteristic
291C2×C4#137×C13111yescharacteristic
292C8—42no—
293C8—81no—
294C8—81no—
295C22—21no—
296C22—21no—
297C22#451×C13111yescharacteristic
298C22#456×C13111yescharacteristic
299C22B(4,𝔽2)×C13111yescharacteristic
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—42no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4—81no—
312C2—21no—
313C2—21no—
314C2#1605×C13111yescharacteristic
315C2#2197×C13111yescharacteristic
316C2#2204×C13111yescharacteristic
subgroupG/H
center, socleC22×C131B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C131
prime radicalC23×C131#120
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
99. C42×C131Q16semidirect
158. #12258C131direct (N × H)
159. C131#12258direct (N × H)
258. C42Q16×C131semidirect

Maximal subgroups

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

subgroupindex
#2612×C1312
#2061×C1312
#1617×C1312
#1605×C1312
#1354×C1312
#1353×C1312
#1324×C1312
#12258131