Finite Group Databasebeta

← C2×#2201×C131 #45719433

Subgroups — C2×#2201×C131 #45719433

Order 33536.

1136 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 222 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
1C2×#563×C131C211yesmaximal, characteristic, complemented
2C2×#458×C131C212yesmaximal
3C2×#455×C131C211yesmaximal, characteristic
4#2201×C131C218yesmaximal, direct product factor
5C2×#354×C131C211yesmaximal, characteristic, complemented
6C2×#352×C131C212yesmaximal, complemented
7C2×H(5,𝔽2)×C131C2211yescharacteristic
8C2×#137×C131—21no—
9#563×C131—22no—
10#563×C131—24no—
11#563×C131C2214yescomplemented
12C2×C4⋊Q8×C131—21no—
13C2×#126×C131—22no—
14C2×#125×C131—21no—
15C2×#124×C131C2211yescharacteristic
16C2×#123×C131—22no—
17C2×#123×C131C2212yes—
18C2×C22⋊Q8×C131—22no—
19#458×C131—24no—
20#458×C131C2218yes—
21#455×C131—22no—
22#455×C131C2214yes—
23C2×#104×C131—21no—
24C2×#100×C131C2211yescharacteristic
25C2×#99×C131—22no—
26C2×#99×C131C2212yes—
27#354×C131C2214yescomplemented
28#352×C131C2218yescomplemented
29H(5,𝔽2)×C131—22no—
30H(5,𝔽2)×C131—21no—
31H(5,𝔽2)×C131—42no—
32H(5,𝔽2)×C131C2312yes—
33C2×#41×C131—21no—
34C2×#41×C131—41no—
35C22×Q8×C131—21no—
36C22×Q8×C131D411yescharacteristic
37C22×D4×C131—22no—
38C22×D4×C131—41no—
39C22×D4×C131C2311yescharacteristic
40C23×C4×C131D412yes—
41#137×C131—24no—
42#137×C131—42no—
43C2×Q16×C131—41no—
44C2×SD16×C131—41no—
45C2×M16×C131—21no—
46C4⋊Q8×C131—24no—
47C4⋊Q8×C131—24no—
48#126×C131—28no—
49#126×C131—28no—
50#125×C131—24no—
51#125×C131—24no—
52#124×C131—22no—
53#124×C131D412yes—
54#124×C131C2312yes—
55#123×C131—22no—
56#123×C131—22no—
57#123×C131—28no—
58#123×C131—44no—
59#123×C131D414yes—
60#123×C131C2314yes—
61C22⋊Q8×C131—28no—
62C22⋊Q8×C131—28no—
63C2×C4⋊C4×C131—21no—
64C2×C4⋊C4×C131—22no—
65C2×C4⋊C4×C131—22no—
66C2×C4⋊C4×C131—21no—
67C2×C4⋊C4×C131—22no—
68C2×C4⋊C4×C131—42no—
69C2×C22⋊C4×C131—22no—
70C2×C22⋊C4×C131—22no—
71C2×C22⋊C4×C131—22no—
72C2×C22⋊C4×C131—42no—
73C2×C22⋊C4×C131D412yes—
74C2×C42×C131—21no—
75C2×C42×C131D411yescharacteristic
76#104×C131—24no—
77#100×C131—21no—
78#100×C131C2312yes—
79#99×C131—22no—
80#99×C131—28no—
81#99×C131C2314yes—
82C24×C131—22no—
83C24×C131C2×D412yes—
84#41×C131—22no—
85#41×C131—22no—
86#41×C131—42no—
87#41×C131—41no—
88#41×C131—44no—
89#41×C131—42no—
90#41×C131—41no—
91C2×Q8×C131—24no—
92C2×Q8×C131—21no—
93C2×Q8×C131—21no—
94C2×Q8×C131—41no—
95C2×Q8×C131—41no—
96C2×Q8×C131—81no—
97C2×Q8×C131—81no—
98C2×Q8×C131C2×D412yes—
99C2×D4×C131—21no—
100C2×D4×C131—24no—
101C2×D4×C131—24no—
102C2×D4×C131—41no—
103C2×D4×C131—41no—
104C2×D4×C131—42no—
105C2×D4×C131—42no—
106C2×D4×C131—42no—
107C2×D4×C131—42no—
108C2×D4×C131—44no—
109C2×D4×C131—42no—
110C2×D4×C131—41no—
111C2×D4×C131C2×D412yes—
112C2×D4×C131C2×D411yescharacteristic
113C2×D4×C131C2411yescharacteristic
114C22×C4×C131—22no—
115C22×C4×C131—21no—
116C22×C4×C131—22no—
117C22×C4×C131—21no—
118C22×C4×C131—22no—
119C22×C4×C131—22no—
120C22×C4×C131—41no—
121C22×C4×C131—42no—
122C22×C4×C131—42no—
123C22×C4×C131—42no—
124C22×C4×C131C2×D411yescharacteristic
125C22×C4×C131C2×D414yes—
126Q16×C131—44no—
127SD16×C131—44no—
128M16×C131—22no—
129M16×C131—41no—
130C2×C8×C131—41no—
131C4⋊C4×C131—24no—
132C4⋊C4×C131—28no—
133C4⋊C4×C131—28no—
134C4⋊C4×C131—24no—
135C4⋊C4×C131—24no—
136C4⋊C4×C131—24no—
137C4⋊C4×C131—48no—
138C22⋊C4×C131—24no—
139C22⋊C4×C131—24no—
140C22⋊C4×C131—22no—
141C22⋊C4×C131—24no—
142C22⋊C4×C131—24no—
143C22⋊C4×C131—24no—
144C22⋊C4×C131—42no—
145C22⋊C4×C131—48no—
146C22⋊C4×C131C2×D414yes—
147C42×C131—24no—
148C42×C131—21no—
149C42×C131C2×D412yes—
150C23×C131—22no—
151C23×C131—22no—
152C23×C131—24no—
153C23×C131—44no—
154C23×C131—42no—
155C23×C131—41no—
156C23×C131—84no—
157C23×C131#12012yes—
158C23×C131#12011yescharacteristic
159C23×C131C22×D412yes—
160Q8×C131—22no—
161Q8×C131—42no—
162Q8×C131—41no—
163Q8×C131—81no—
164Q8×C131—84no—
165D4×C131—22no—
166D4×C131—41no—
167D4×C131—42no—
168D4×C131—41no—
169D4×C131—41no—
170D4×C131—44no—
171D4×C131—44no—
172D4×C131—42no—
173D4×C131—42no—
174C2×C4×C131—21no—
175C2×C4×C131—21no—
176C2×C4×C131—24no—
177C2×C4×C131—22no—
178C2×C4×C131—24no—
179C2×C4×C131—22no—
180C2×C4×C131—24no—
181C2×C4×C131—42no—
182C2×C4×C131—42no—
183C2×C4×C131—44no—
184C2×C4×C131—42no—
185C2×C4×C131—42no—
186C2×C4×C131—42no—
187C2×C4×C131—41no—
188C2×C4×C131—41no—
189C2×C4×C131—42no—
190C2×C4×C131—41no—
191C2×C4×C131—44no—
192C2×C4×C131—44no—
193C2×C4×C131—82no—
194C2×C4×C131—84no—
195C2×C4×C131#12011yescharacteristic
196C2×C4×C131C22×D411yescharacteristic
197C8×C131—42no—
198C22×C131—22no—
199C22×C131—22no—
200C22×C131—22no—
201C22×C131—42no—
202C22×C131—42no—
203C22×C131—44no—
204C22×C131—42no—
205C22×C131—82no—
206C22×C131—84no—
207C22×C131—82no—
208C22×C131—162no—
209C22×C131B(4,𝔽2)11yescharacteristic
210C22×C131B(4,𝔽2)11yescharacteristic
211C22×C131C2×#12011yescharacteristic
212C4×C131—21no—
213C4×C131—21no—
214C4×C131—44no—
215C4×C131—42no—
216C4×C131—42no—
217C4×C131—44no—
218C4×C131—42no—
219C4×C131—84no—
220C2×C131—21no—
221C2×C131—21no—
222C2×C131—42no—
223C2×C131—42no—
224C2×C131—82no—
225C2×C131#220112yesdirect product factor
226C2×C131C2×B(4,𝔽2)11yescharacteristic
227C2×#2201C13111yesmaximal, characteristic, direct product factor, Sylow
228C131C2×#220111yescharacteristic, direct product factor, Sylow
229C2×#563C2×C13111yescharacteristic, complemented
230C2×#458C2×C13112yes—
231C2×#455C2×C13111yescharacteristic
232#2201C2×C13118yesdirect product factor
233C2×#354C2×C13111yescharacteristic, complemented
234C2×#352C2×C13112yescomplemented
235C2×H(5,𝔽2)C22×C13111yescharacteristic
236C2×#137—21no—
237#563—22no—
238#563—24no—
239#563C22×C13114yescomplemented
240C2×C4⋊Q8—21no—
241C2×#126—22no—
242C2×#125—21no—
243C2×#124C22×C13111yescharacteristic
244C2×#123—22no—
245C2×#123C22×C13112yes—
246C2×C22⋊Q8—22no—
247#458—24no—
248#458C22×C13118yes—
249#455—22no—
250#455C22×C13114yes—
251C2×#104—21no—
252C2×#100C22×C13111yescharacteristic
253C2×#99—22no—
254C2×#99C22×C13112yes—
255#354C22×C13114yescomplemented
256#352C22×C13118yescomplemented
257H(5,𝔽2)—22no—
258H(5,𝔽2)—21no—
259H(5,𝔽2)—42no—
260H(5,𝔽2)C23×C13112yes—
261C2×#41—21no—
262C2×#41—41no—
263C22×Q8—21no—
264C22×Q8D4×C13111yescharacteristic
265C22×D4—22no—
266C22×D4—41no—
267C22×D4C23×C13111yescharacteristic
268C23×C4D4×C13112yes—
269#137—24no—
270#137—42no—
271C2×Q16—41no—
272C2×SD16—41no—
273C2×M16—21no—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276#126—28no—
277#126—28no—
278#125—24no—
279#125—24no—
280#124—22no—
281#124D4×C13112yes—
282#124C23×C13112yes—
283#123—22no—
284#123—22no—
285#123—28no—
286#123—44no—
287#123D4×C13114yes—
288#123C23×C13114yes—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4D4×C13112yes—
302C2×C42—21no—
303C2×C42D4×C13111yescharacteristic
304#104—24no—
305#100—21no—
306#100C23×C13112yes—
307#99—22no—
308#99—28no—
309#99C23×C13114yes—
310C24—22no—
311C24C2×D4×C13112yes—
312#41—22no—
313#41—22no—
314#41—42no—
315#41—41no—
316#41—44no—
317#41—42no—
318#41—41no—
319C2×Q8—24no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—81no—
325C2×Q8—81no—
326C2×Q8C2×D4×C13112yes—
327C2×D4—21no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4C2×D4×C13112yes—
340C2×D4C2×D4×C13111yescharacteristic
341C2×D4C24×C13111yescharacteristic
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4C2×D4×C13111yescharacteristic
353C22×C4C2×D4×C13114yes—
354Q16—44no—
355SD16—44no—
356M16—22no—
357M16—41no—
358C2×C8—41no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—48no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—22no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—48no—
374C22⋊C4C2×D4×C13114yes—
375C42—24no—
376C42—21no—
377C42C2×D4×C13112yes—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—44no—
382C23—42no—
383C23—41no—
384C23—84no—
385C23#120×C13111yescharacteristic
386C23#120×C13112yes—
387C23C22×D4×C13112yes—
388Q8—22no—
389Q8—42no—
390Q8—41no—
391Q8—81no—
392Q8—84no—
393D4—22no—
394D4—41no—
395D4—42no—
396D4—41no—
397D4—41no—
398D4—44no—
399D4—44no—
400D4—42no—
401D4—42no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4#120×C13111yescharacteristic
424C2×C4C22×D4×C13111yescharacteristic
425C8—42no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—42no—
430C22—42no—
431C22—44no—
432C22—42no—
433C22—82no—
434C22—84no—
435C22—82no—
436C22—162no—
437C22B(4,𝔽2)×C13111yescharacteristic
438C22B(4,𝔽2)×C13111yescharacteristic
439C22C2×#120×C13111yescharacteristic
440C4—21no—
441C4—21no—
442C4—44no—
443C4—42no—
444C4—42no—
445C4—44no—
446C4—42no—
447C4—84no—
448C2—21no—
449C2—21no—
450C2—42no—
451C2—42no—
452C2—82no—
453C2#2201×C13112yesdirect product factor
454C2C2×B(4,𝔽2)×C13111yescharacteristic
subgroupG/H
center, socleC22×C131B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C131
prime radicalC2×H(5,𝔽2)×C131C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C23×C131 ◁ C22×D4×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#563×C131C2semidirect
4. #2201×C131C2direct (N × H)
5. C2×#354×C131C2semidirect
6. C2×#352×C131C2semidirect
11. #563×C131C22semidirect
27. #354×C131C22semidirect
28. #352×C131C22semidirect
225. C2×C131#2201direct (N × H)
227. C2×#2201C131direct (N × H)
228. C131C2×#2201direct (N × H)
229. C2×#563C2×C131semidirect
232. #2201C2×C131direct (N × H)
233. C2×#354C2×C131semidirect
234. C2×#352C2×C131semidirect
239. #563C22×C131semidirect
255. #354C22×C131semidirect
256. #352C22×C131semidirect
453. C2#2201×C131direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#563×C1312
C2×#458×C1312
C2×#455×C1312
#2201×C1312
C2×#354×C1312
C2×#352×C1312
C2×#2201131