Finite Group Databasebeta

← C131×#12230 #45698114

Subgroups — C131×#12230 #45698114

Order 33536.

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

normal subgroups only

This group has 422 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#2824×C131C211yesmaximal, characteristic
2#2062×C131C211yesmaximal, characteristic, complemented
3#1602×C131C211yesmaximal, characteristic, complemented
4#1596×C131C211yesmaximal, characteristic
5#1351×C131C211yesmaximal, characteristic, complemented
6#1343×C131C211yesmaximal, characteristic
7#1326×C131C211yesmaximal, characteristic, complemented
8#543×C131C2211yescharacteristic
9#451×C131—21no—
10#450×C131—21no—
11#400×C131—21no—
12#397×C131—21no—
13#394×C131—21no—
14#394×C131—21no—
15#393×C131—21no—
16#393×C131C2211yescharacteristic
17#392×C131C2211yescharacteristic
18#386×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×C131D411yescharacteristic
28C22×D4×C131—21no—
29C22×D4×C131D411yescharacteristic
30C23×C4×C131D411yescharacteristic
31C2×Q16×C131—41no—
32C2×SD16×C131—41no—
33C22⋊Q8×C131—21no—
34C22⋊Q8×C131—21no—
35C22⋊Q8×C131—41no—
36C22⋊Q8×C131—41no—
37C2×C4⋊C4×C131—21no—
38C2×C4⋊C4×C131—21no—
39C2×C4⋊C4×C131—41no—
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×C22⋊C4×C131—21no—
46C2×C22⋊C4×C131—21no—
47C2×C22⋊C4×C131—21no—
48C2×C22⋊C4×C131D411yescharacteristic
49C2×C22⋊C4×C131C2311yescharacteristic
50#103×C131—41no—
51#102×C131—41no—
52C22⋊C8×C131—21no—
53C22⋊C8×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—41no—
60#95×C131D411yescharacteristic
61#95×C131D411yescharacteristic
62C24×C131—21no—
63C24×C131C2×D411yescharacteristic
64C2×Q8×C131—21no—
65C2×Q8×C131—21no—
66C2×Q8×C131—41no—
67C2×Q8×C131—81no—
68C2×D4×C131—41no—
69C2×D4×C131—41no—
70C2×D4×C131—41no—
71C2×D4×C131—81no—
72C2×D4×C131—81no—
73C2×D4×C131—81no—
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—21no—
81C22×C4×C131—41no—
82C22×C4×C131—41no—
83C22×C4×C131—81no—
84C22×C4×C131C2×D411yescharacteristic
85C22×C4×C131C2×D411yescharacteristic
86Q16×C131—82no—
87SD16×C131—82no—
88C2×C8×C131—41no—
89C2×C8×C131—41no—
90C4⋊C4×C131—21no—
91C4⋊C4×C131—21no—
92C4⋊C4×C131—41no—
93C4⋊C4×C131—41no—
94C4⋊C4×C131—81no—
95C4⋊C4×C131—81no—
96C4⋊C4×C131—82no—
97C22⋊C4×C131—22no—
98C22⋊C4×C131—21no—
99C22⋊C4×C131—21no—
100C22⋊C4×C131—41no—
101C22⋊C4×C131—41no—
102C22⋊C4×C131—41no—
103C22⋊C4×C131—41no—
104C22⋊C4×C131—81no—
105C22⋊C4×C131—81no—
106C22⋊C4×C131—81no—
107C22⋊C4×C131—81no—
108C22⋊C4×C131—81no—
109C22⋊C4×C131—81no—
110C23×C131—21no—
111C23×C131—21no—
112C23×C131—41no—
113C23×C131—41no—
114C23×C131—41no—
115C23×C131—41no—
116C23×C131—41no—
117C23×C131—41no—
118C23×C131—41no—
119C23×C131—41no—
120C23×C131—41no—
121C23×C131#12011yescharacteristic
122C23×C131#13511yescharacteristic
123C23×C131#13711yescharacteristic
124Q8×C131—42no—
125Q8×C131—81no—
126Q8×C131—81no—
127Q8×C131—81no—
128D4×C131—42no—
129D4×C131—81no—
130D4×C131—82no—
131D4×C131—82no—
132C2×C4×C131—21no—
133C2×C4×C131—21no—
134C2×C4×C131—41no—
135C2×C4×C131—41no—
136C2×C4×C131—42no—
137C2×C4×C131—41no—
138C2×C4×C131—41no—
139C2×C4×C131—41no—
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—42no—
148C2×C4×C131—41no—
149C2×C4×C131—41no—
150C2×C4×C131—41no—
151C2×C4×C131—41no—
152C2×C4×C131—41no—
153C2×C4×C131—41no—
154C2×C4×C131—41no—
155C2×C4×C131—81no—
156C2×C4×C131—81no—
157C2×C4×C131—81no—
158C2×C4×C131—81no—
159C2×C4×C131—81no—
160C8×C131—81no—
161C8×C131—81no—
162C22×C131—21no—
163C22×C131—21no—
164C22×C131—21no—
165C22×C131—21no—
166C22×C131—41no—
167C22×C131—41no—
168C22×C131—41no—
169C22×C131—41no—
170C22×C131—81no—
171C22×C131—81no—
172C22×C131—81no—
173C22×C131—81no—
174C22×C131—81no—
175C22×C131—81no—
176C22×C131—81no—
177C22×C131—81no—
178C22×C131#45211yescharacteristic
179C22×C131#45611yescharacteristic
180C22×C131B(4,𝔽2)11yescharacteristic
181C4×C131—41no—
182C4×C131—41no—
183C4×C131—81no—
184C4×C131—81no—
185C4×C131—81no—
186C4×C131—82no—
187C4×C131—81no—
188C4×C131—81no—
189C4×C131—81no—
190C4×C131—81no—
191C2×C131—21no—
192C2×C131—21no—
193C2×C131—41no—
194C2×C131—41no—
195C2×C131—161no—
196C2×C131#160311yescharacteristic
197C2×C131#219911yescharacteristic
198C2×C131#220011yescharacteristic
199#12230C13111yesmaximal, characteristic, direct product factor, Sylow
200C131#1223011yescharacteristic, direct product factor, Sylow
201#2824C2×C13111yescharacteristic
202#2062C2×C13111yescharacteristic, complemented
203#1602C2×C13111yescharacteristic, complemented
204#1596C2×C13111yescharacteristic
205#1351C2×C13111yescharacteristic, complemented
206#1343C2×C13111yescharacteristic
207#1326C2×C13111yescharacteristic, complemented
208#543C22×C13111yescharacteristic
209#451—21no—
210#450—21no—
211#400—21no—
212#397—21no—
213#394—21no—
214#394—21no—
215#393—21no—
216#393C22×C13111yescharacteristic
217#392C22×C13111yescharacteristic
218#386—21no—
219#385—21no—
220#385C22×C13111yescharacteristic
221#342—21no—
222#342C22×C13111yescharacteristic
223#328—21no—
224#328—21no—
225#323C22×C13111yescharacteristic
226#323C22×C13111yescharacteristic
227C22×Q8D4×C13111yescharacteristic
228C22×D4—21no—
229C22×D4D4×C13111yescharacteristic
230C23×C4D4×C13111yescharacteristic
231C2×Q16—41no—
232C2×SD16—41no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—41no—
236C22⋊Q8—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—41no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4D4×C13111yescharacteristic
249C2×C22⋊C4C23×C13111yescharacteristic
250#103—41no—
251#102—41no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95D4×C13111yescharacteristic
261#95D4×C13111yescharacteristic
262C24—21no—
263C24C2×D4×C13111yescharacteristic
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—41no—
267C2×Q8—81no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—81no—
272C2×D4—81no—
273C2×D4—81no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—81no—
284C22×C4C2×D4×C13111yescharacteristic
285C22×C4C2×D4×C13111yescharacteristic
286Q16—82no—
287SD16—82no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C22⋊C4—22no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#120×C13111yescharacteristic
322C23#135×C13111yescharacteristic
323C23#137×C13111yescharacteristic
324Q8—42no—
325Q8—81no—
326Q8—81no—
327Q8—81no—
328D4—42no—
329D4—81no—
330D4—82no—
331D4—82no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C8—81no—
361C8—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#452×C13111yescharacteristic
379C22#456×C13111yescharacteristic
380C22B(4,𝔽2)×C13111yescharacteristic
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—161no—
396C2#1603×C13111yescharacteristic
397C2#2199×C13111yescharacteristic
398C2#2200×C13111yescharacteristic
subgroupG/H
center, socleC22×C131B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C131
prime radical#392×C131C22
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
2. #2062×C131C2semidirect
3. #1602×C131C2semidirect
5. #1351×C131C2semidirect
7. #1326×C131C2semidirect
199. #12230C131direct (N × H)
200. C131#12230direct (N × H)
202. #2062C2×C131semidirect
203. #1602C2×C131semidirect
205. #1351C2×C131semidirect
207. #1326C2×C131semidirect

Maximal subgroups

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

subgroupindex
#2824×C1312
#2062×C1312
#1602×C1312
#1596×C1312
#1351×C1312
#1343×C1312
#1326×C1312
#12230131