Finite Group Databasebeta

← C131×#12252 #45698136

Subgroups — C131×#12252 #45698136

Order 33536.

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

normal subgroups only

This group has 452 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#2677×C131C211yesmaximal, characteristic
2#2062×C131C211yesmaximal, characteristic
3#1614×C131C211yesmaximal, characteristic
4#1602×C131C211yesmaximal, characteristic
5#1352×C131C211yesmaximal, characteristic
6#1351×C131C211yesmaximal, characteristic
7#1326×C131C211yesmaximal, characteristic
8#543×C131C2211yescharacteristic
9C2×C4⋊Q8×C131—21no—
10C2×C22⋊Q8×C131—21no—
11#484×C131—21no—
12#451×C131—21no—
13#399×C131C2211yescharacteristic
14#398×C131—21no—
15#395×C131—21no—
16#393×C131C2211yescharacteristic
17#388×C131—21no—
18#387×C131—21no—
19#385×C131—21no—
20#385×C131C2211yescharacteristic
21#342×C131—21no—
22#342×C131C2211yescharacteristic
23#328×C131—21no—
24#328×C131—21no—
25#323×C131C2211yescharacteristic
26#323×C131C2211yescharacteristic
27C22×Q8×C131—21no—
28C22×Q8×C131D411yescharacteristic
29C23×C4×C131D411yescharacteristic
30C2×Q16×C131—41no—
31C4⋊Q8×C131—44no—
32C22⋊Q8×C131—21no—
33C22⋊Q8×C131—21no—
34C22⋊Q8×C131—41no—
35C22⋊Q8×C131—41no—
36C22⋊Q8×C131—44no—
37C2×C4⋊C4×C131—21no—
38C2×C4⋊C4×C131—21no—
39C2×C4⋊C4×C131—21no—
40C2×C4⋊C4×C131—21no—
41C2×C4⋊C4×C131—21no—
42C2×C4⋊C4×C131—21no—
43C2×C4⋊C4×C131—41no—
44C2×C4⋊C4×C131D411yescharacteristic
45C2×C22⋊C4×C131—21no—
46C2×C22⋊C4×C131—21no—
47C2×C22⋊C4×C131—21no—
48C2×C22⋊C4×C131—21no—
49C2×C22⋊C4×C131D411yescharacteristic
50C2×C22⋊C4×C131C2311yescharacteristic
51C2×C42×C131—21no—
52#107×C131—41no—
53#103×C131—41no—
54#103×C131—41no—
55C22⋊C8×C131—21no—
56C22⋊C8×C131—21no—
57#95×C131—21no—
58#95×C131—21no—
59#95×C131—21no—
60#95×C131—21no—
61#95×C131—41no—
62#95×C131D411yescharacteristic
63#95×C131D411yescharacteristic
64C24×C131C2×D411yescharacteristic
65C2×Q8×C131—21no—
66C2×Q8×C131—21no—
67C2×Q8×C131—42no—
68C2×Q8×C131—42no—
69C2×Q8×C131—42no—
70C2×Q8×C131—41no—
71C2×Q8×C131—81no—
72C22×C4×C131—21no—
73C22×C4×C131—21no—
74C22×C4×C131—21no—
75C22×C4×C131—21no—
76C22×C4×C131—21no—
77C22×C4×C131—21no—
78C22×C4×C131—21no—
79C22×C4×C131—21no—
80C22×C4×C131—42no—
81C22×C4×C131—41no—
82C22×C4×C131—41no—
83C22×C4×C131C2×D411yescharacteristic
84C22×C4×C131C2×D411yescharacteristic
85Q16×C131—82no—
86C2×C8×C131—41no—
87C2×C8×C131—41no—
88C4⋊C4×C131—24no—
89C4⋊C4×C131—21no—
90C4⋊C4×C131—21no—
91C4⋊C4×C131—41no—
92C4⋊C4×C131—42no—
93C4⋊C4×C131—42no—
94C4⋊C4×C131—42no—
95C4⋊C4×C131—42no—
96C4⋊C4×C131—42no—
97C4⋊C4×C131—41no—
98C4⋊C4×C131—41no—
99C4⋊C4×C131—82no—
100C22⋊C4×C131—21no—
101C22⋊C4×C131—22no—
102C22⋊C4×C131—41no—
103C22⋊C4×C131—42no—
104C22⋊C4×C131—42no—
105C22⋊C4×C131—41no—
106C22⋊C4×C131—41no—
107C22⋊C4×C131—41no—
108C22⋊C4×C131Q1612yescomplemented
109C42×C131—42no—
110C23×C131—21no—
111C23×C131—21no—
112C23×C131—41no—
113C23×C131—41no—
114C23×C131#12011yescharacteristic
115C23×C131C2×Q1611yescharacteristic
116C23×C131#13711yescharacteristic
117Q8×C131—42no—
118Q8×C131—81no—
119Q8×C131—84no—
120Q8×C131—81no—
121Q8×C131—81no—
122C2×C4×C131—22no—
123C2×C4×C131—22no—
124C2×C4×C131—22no—
125C2×C4×C131—22no—
126C2×C4×C131—21no—
127C2×C4×C131—21no—
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—42no—
139C2×C4×C131—42no—
140C2×C4×C131—41no—
141C2×C4×C131—41no—
142C2×C4×C131—41no—
143C2×C4×C131—41no—
144C2×C4×C131—41no—
145C2×C4×C131—41no—
146C2×C4×C131—41no—
147C2×C4×C131—41no—
148C2×C4×C131—81no—
149C2×C4×C131—81no—
150C2×C4×C131—81no—
151C2×C4×C131—81no—
152C2×C4×C131—81no—
153C8×C131—81no—
154C8×C131—81no—
155C22×C131—21no—
156C22×C131—21no—
157C22×C131—21no—
158C22×C131—21no—
159C22×C131—41no—
160C22×C131—41no—
161C22×C131—41no—
162C22×C131—41no—
163C22×C131—81no—
164C22×C131#45111yescharacteristic
165C22×C131#45611yescharacteristic
166C22×C131B(4,𝔽2)11yescharacteristic
167C4×C131—42no—
168C4×C131—42no—
169C4×C131—41no—
170C4×C131—42no—
171C4×C131—42no—
172C4×C131—41no—
173C4×C131—81no—
174C4×C131—82no—
175C4×C131—81no—
176C4×C131—81no—
177C4×C131—81no—
178C2×C131—21no—
179C2×C131—21no—
180C2×C131—41no—
181C2×C131—41no—
182C2×C131#160511yescharacteristic
183C2×C131#219611yescharacteristic
184C2×C131#220111yescharacteristic
185#12252C13111yesmaximal, characteristic, direct product factor, Sylow
186C131#1225211yescharacteristic, direct product factor, Sylow
187#2677C2×C13111yescharacteristic
188#2062C2×C13111yescharacteristic
189#1614C2×C13111yescharacteristic
190#1602C2×C13111yescharacteristic
191#1352C2×C13111yescharacteristic
192#1351C2×C13111yescharacteristic
193#1326C2×C13111yescharacteristic
194#543C22×C13111yescharacteristic
195C2×C4⋊Q8—21no—
196C2×C22⋊Q8—21no—
197#484—21no—
198#451—21no—
199#399C22×C13111yescharacteristic
200#398—21no—
201#395—21no—
202#393C22×C13111yescharacteristic
203#388—21no—
204#387—21no—
205#385—21no—
206#385C22×C13111yescharacteristic
207#342—21no—
208#342C22×C13111yescharacteristic
209#328—21no—
210#328—21no—
211#323C22×C13111yescharacteristic
212#323C22×C13111yescharacteristic
213C22×Q8—21no—
214C22×Q8D4×C13111yescharacteristic
215C23×C4D4×C13111yescharacteristic
216C2×Q16—41no—
217C4⋊Q8—44no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—41no—
221C22⋊Q8—41no—
222C22⋊Q8—44no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4D4×C13111yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4D4×C13111yescharacteristic
236C2×C22⋊C4C23×C13111yescharacteristic
237C2×C42—21no—
238#107—41no—
239#103—41no—
240#103—41no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—41no—
248#95D4×C13111yescharacteristic
249#95D4×C13111yescharacteristic
250C24C2×D4×C13111yescharacteristic
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×Q8—81no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4C2×D4×C13111yescharacteristic
270C22×C4C2×D4×C13111yescharacteristic
271Q16—82no—
272C2×C8—41no—
273C2×C8—41no—
274C4⋊C4—24no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—82no—
286C22⋊C4—21no—
287C22⋊C4—22no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4Q16×C13112yescomplemented
295C42—42no—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23#120×C13111yescharacteristic
301C23C2×Q16×C13111yescharacteristic
302C23#137×C13111yescharacteristic
303Q8—42no—
304Q8—81no—
305Q8—84no—
306Q8—81no—
307Q8—81no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C8—81no—
340C8—81no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22#451×C13111yescharacteristic
351C22#456×C13111yescharacteristic
352C22B(4,𝔽2)×C13111yescharacteristic
353C4—42no—
354C4—42no—
355C4—41no—
356C4—42no—
357C4—42no—
358C4—41no—
359C4—81no—
360C4—82no—
361C4—81no—
362C4—81no—
363C4—81no—
364C2—21no—
365C2—21no—
366C2—41no—
367C2—41no—
368C2#1605×C13111yescharacteristic
369C2#2196×C13111yescharacteristic
370C2#2201×C13111yescharacteristic
subgroupG/H
center, socleC22×C131B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C131
prime radicalC24×C131C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C23×C131 ◁ C2×C22⋊C4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
108. C22⋊C4×C131Q16semidirect
185. #12252C131direct (N × H)
186. C131#12252direct (N × H)
294. C22⋊C4Q16×C131semidirect

Maximal subgroups

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

subgroupindex
#2677×C1312
#2062×C1312
#1614×C1312
#1602×C1312
#1352×C1312
#1351×C1312
#1326×C1312
#12252131