Finite Group Databasebeta

← C131×#10022 #45695906

Subgroups — C131×#10022 #45695906

Order 33536.

576 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 98 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#3140×C131C211yesmaximal, characteristic, complemented
2#3110×C131C211yesmaximal, characteristic, complemented
3#2631×C131C211yesmaximal, characteristic
4#1533×C131C211yesmaximal, characteristic, complemented
5#1515×C131C211yesmaximal, characteristic, complemented
6#1490×C131C211yesmaximal, characteristic, complemented
7#1288×C131C211yesmaximal, characteristic, complemented
8#533×C131C2211yescharacteristic
9C2×#124×C131C2211yescharacteristic
10C2×C4×D4×C131C2211yescharacteristic
11#489×C131—21no—
12#489×C131—21no—
13#488×C131—22no—
14#484×C131—21no—
15#474×C131—21no—
16#461×C131—21no—
17#450×C131—21no—
18#440×C131—22no—
19#432×C131C2211yescharacteristic
20#432×C131C2211yescharacteristic
21#419×C131—21no—
22#418×C131—21no—
23C4×M16×C131C2211yescharacteristic
24#397×C131—21no—
25#393×C131—21no—
26#392×C131—21no—
27#392×C131—21no—
28#386×C131—22no—
29#383×C131C2211yescharacteristic
30#329×C131—22no—
31#328×C131—22no—
32D4⋊C8×C131—22no—
33C22×Q8×C131D411yescharacteristic
34C22×D4×C131—22no—
35C22×D4×C131D411yescharacteristic
36C22×D4×C131D411yescharacteristic
37C23×C4×C131—21no—
38C2×SD16×C131—22no—
39C2×M16×C131—21no—
40C4⋊Q8×C131D411yescharacteristic, complemented
41C4⋊Q8×C131D411yescharacteristic, complemented
42#125×C131—21no—
43#124×C131—21no—
44#124×C131—21no—
45#124×C131—21no—
46#124×C131—21no—
47C22⋊Q8×C131—21no—
48C22⋊Q8×C131—21no—
49C4×Q8×C131—21no—
50C4×Q8×C131—21no—
51C4×D4×C131—21no—
52C4×D4×C131—21no—
53C4×D4×C131—21no—
54C4×D4×C131—21no—
55#117×C131—21no—
56C2×C4⋊C4×C131D411yescharacteristic
57C2×C22⋊C4×C131—21no—
58C2×C22⋊C4×C131—21no—
59C2×C22⋊C4×C131—21no—
60C2×C22⋊C4×C131—22no—
61C2×C22⋊C4×C131—22no—
62C2×C42×C131C2311yescharacteristic
63#107×C131—22no—
64C4⋊C8×C131—21no—
65C4⋊C8×C131—21no—
66#103×C131—22no—
67#103×C131—22no—
68#103×C131—22no—
69#102×C131—22no—
70#102×C131—22no—
71#102×C131—22no—
72C22⋊C8×C131—21no—
73C22⋊C8×C131—21no—
74#97×C131—21no—
75#97×C131—21no—
76#97×C131D412yescomplemented
77C4×C8×C131D412yescomplemented
78#95×C131—22no—
79#95×C131—21no—
80#95×C131—21no—
81#95×C131D412yes—
82C24×C131—21no—
83C24×C131—21no—
84C2×Q8×C131—21no—
85C2×Q8×C131—21no—
86C2×Q8×C131—21no—
87C2×Q8×C131—21no—
88C2×Q8×C131—42no—
89C2×D4×C131—21no—
90C2×D4×C131—21no—
91C2×D4×C131—21no—
92C2×D4×C131—21no—
93C2×D4×C131—42no—
94C2×D4×C131—42no—
95C2×D4×C131—44no—
96C2×D4×C131—82no—
97C2×D4×C131—82no—
98C22×C4×C131—22no—
99C22×C4×C131—21no—
100C22×C4×C131—21no—
101C22×C4×C131—41no—
102C22×C4×C131—41no—
103C22×C4×C131—82no—
104C22×C4×C131C2×D411yescharacteristic
105C22×C4×C131C2×D411yescharacteristic
106C22×C4×C131C2×D411yescharacteristic
107SD16×C131—44no—
108M16×C131—42no—
109C2×C8×C131—22no—
110C2×C8×C131—22no—
111C2×C8×C131—41no—
112C4⋊C4×C131—21no—
113C4⋊C4×C131—21no—
114C4⋊C4×C131—21no—
115C4⋊C4×C131—21no—
116C4⋊C4×C131—21no—
117C4⋊C4×C131—21no—
118C4⋊C4×C131—41no—
119C4⋊C4×C131—41no—
120C22⋊C4×C131—41no—
121C22⋊C4×C131—41no—
122C22⋊C4×C131—41no—
123C22⋊C4×C131—41no—
124C22⋊C4×C131—41no—
125C22⋊C4×C131—41no—
126C22⋊C4×C131—41no—
127C22⋊C4×C131—82no—
128C22⋊C4×C131—82no—
129C42×C131—41no—
130C42×C131C2×D411yescharacteristic
131C42×C131C2×D411yescharacteristic
132C42×C131C2×D411yescharacteristic
133C42×C131#4111yescharacteristic
134C23×C131—22no—
135C23×C131—42no—
136C23×C131—41no—
137C23×C131—41no—
138C23×C131—42no—
139C23×C131—41no—
140C23×C131—41no—
141C23×C131—42no—
142C23×C131—41no—
143C23×C131—42no—
144C23×C131#12011yescharacteristic
145Q8×C131—42no—
146Q8×C131—42no—
147Q8×C131—81no—
148Q8×C131—81no—
149D4×C131—22no—
150D4×C131—22no—
151D4×C131—41no—
152D4×C131—41no—
153D4×C131—42no—
154D4×C131—81no—
155D4×C131—88no—
156C2×C4×C131—22no—
157C2×C4×C131—41no—
158C2×C4×C131—41no—
159C2×C4×C131—42no—
160C2×C4×C131—42no—
161C2×C4×C131—42no—
162C2×C4×C131—41no—
163C2×C4×C131—41no—
164C2×C4×C131—41no—
165C2×C4×C131—41no—
166C2×C4×C131—42no—
167C2×C4×C131—42no—
168C2×C4×C131—82no—
169C2×C4×C131—82no—
170C2×C4×C131#12011yescharacteristic
171C2×C4×C131#12011yescharacteristic
172C2×C4×C131#12111yescharacteristic
173C2×C4×C131#12111yescharacteristic
174C2×C4×C131#12111yescharacteristic
175C2×C4×C131#12711yescharacteristic
176C8×C131—42no—
177C8×C131—42no—
178C8×C131—81no—
179C22×C131—21no—
180C22×C131—21no—
181C22×C131—44no—
182C22×C131—42no—
183C22×C131—82no—
184C22×C131—82no—
185C22×C131—82no—
186C22×C131—81no—
187C22×C131—81no—
188C22×C131—81no—
189C22×C131—81no—
190C22×C131—81no—
191C22×C131#39211yescharacteristic
192C22×C131#45512yes—
193C4×C131—21no—
194C4×C131—21no—
195C4×C131—21no—
196C4×C131—21no—
197C4×C131—21no—
198C4×C131—21no—
199C4×C131—82no—
200C4×C131—81no—
201C4×C131—81no—
202C4×C131—81no—
203C4×C131—81no—
204C2×C131—22no—
205C2×C131—82no—
206C2×C131—161no—
207C2×C131#200411yescharacteristic
208C2×C131#200811yescharacteristic
209C2×C131#201011yescharacteristic
210#10022C13111yesmaximal, characteristic, direct product factor, Sylow
211C131#1002211yescharacteristic, direct product factor, Sylow
212#3140C2×C13111yescharacteristic, complemented
213#3110C2×C13111yescharacteristic, complemented
214#2631C2×C13111yescharacteristic
215#1533C2×C13111yescharacteristic, complemented
216#1515C2×C13111yescharacteristic, complemented
217#1490C2×C13111yescharacteristic, complemented
218#1288C2×C13111yescharacteristic, complemented
219#533C22×C13111yescharacteristic
220C2×#124C22×C13111yescharacteristic
221C2×C4×D4C22×C13111yescharacteristic
222#489—21no—
223#489—21no—
224#488—22no—
225#484—21no—
226#474—21no—
227#461—21no—
228#450—21no—
229#440—22no—
230#432C22×C13111yescharacteristic
231#432C22×C13111yescharacteristic
232#419—21no—
233#418—21no—
234C4×M16C22×C13111yescharacteristic
235#397—21no—
236#393—21no—
237#392—21no—
238#392—21no—
239#386—22no—
240#383C22×C13111yescharacteristic
241#329—22no—
242#328—22no—
243D4⋊C8—22no—
244C22×Q8D4×C13111yescharacteristic
245C22×D4—22no—
246C22×D4D4×C13111yescharacteristic
247C22×D4D4×C13111yescharacteristic
248C23×C4—21no—
249C2×SD16—22no—
250C2×M16—21no—
251C4⋊Q8D4×C13111yescharacteristic, complemented
252C4⋊Q8D4×C13111yescharacteristic, complemented
253#125—21no—
254#124—21no—
255#124—21no—
256#124—21no—
257#124—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266#117—21no—
267C2×C4⋊C4D4×C13111yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C42C23×C13111yescharacteristic
274#107—22no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277#103—22no—
278#103—22no—
279#103—22no—
280#102—22no—
281#102—22no—
282#102—22no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285#97—21no—
286#97—21no—
287#97D4×C13112yescomplemented
288C4×C8D4×C13112yescomplemented
289#95—22no—
290#95—21no—
291#95—21no—
292#95D4×C13112yes—
293C24—21no—
294C24—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—82no—
308C2×D4—82no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—82no—
315C22×C4C2×D4×C13111yescharacteristic
316C22×C4C2×D4×C13111yescharacteristic
317C22×C4C2×D4×C13111yescharacteristic
318SD16—44no—
319M16—42no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—41no—
341C42C2×D4×C13111yescharacteristic
342C42C2×D4×C13111yescharacteristic
343C42C2×D4×C13111yescharacteristic
344C42#41×C13111yescharacteristic
345C23—22no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—41no—
354C23—42no—
355C23#120×C13111yescharacteristic
356Q8—42no—
357Q8—42no—
358Q8—81no—
359Q8—81no—
360D4—22no—
361D4—22no—
362D4—41no—
363D4—41no—
364D4—42no—
365D4—81no—
366D4—88no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4#120×C13111yescharacteristic
382C2×C4#120×C13111yescharacteristic
383C2×C4#121×C13111yescharacteristic
384C2×C4#121×C13111yescharacteristic
385C2×C4#121×C13111yescharacteristic
386C2×C4#127×C13111yescharacteristic
387C8—42no—
388C8—42no—
389C8—81no—
390C22—21no—
391C22—21no—
392C22—44no—
393C22—42no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22#392×C13111yescharacteristic
403C22#455×C13112yes—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—82no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—81no—
415C2—22no—
416C2—82no—
417C2—161no—
418C2#2004×C13111yescharacteristic
419C2#2008×C13111yescharacteristic
420C2#2010×C13111yescharacteristic
subgroupG/H
center, socleC22×C131#392
commutator, Frattini, cosocleC2×C42C23×C131
prime radical#2631×C131C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C131 ◁ C2×C42×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3140×C131C2semidirect
2. #3110×C131C2semidirect
4. #1533×C131C2semidirect
5. #1515×C131C2semidirect
6. #1490×C131C2semidirect
7. #1288×C131C2semidirect
40. C4⋊Q8×C131D4semidirect
41. C4⋊Q8×C131D4semidirect
76. #97×C131D4semidirect
77. C4×C8×C131D4semidirect
210. #10022C131direct (N × H)
211. C131#10022direct (N × H)
212. #3140C2×C131semidirect
213. #3110C2×C131semidirect
215. #1533C2×C131semidirect
216. #1515C2×C131semidirect
217. #1490C2×C131semidirect
218. #1288C2×C131semidirect
251. C4⋊Q8D4×C131semidirect
252. C4⋊Q8D4×C131semidirect
287. #97D4×C131semidirect
288. C4×C8D4×C131semidirect

Maximal subgroups

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

subgroupindex
#3140×C1312
#3110×C1312
#2631×C1312
#1533×C1312
#1515×C1312
#1490×C1312
#1288×C1312
#10022131