Finite Group Databasebeta

← C2×#2203×C131 #45719447

Subgroups — C2×#2203×C131 #45719447

Order 33536.

1092 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1092 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×#456×C131C212yesmaximal, complemented
3C2×#454×C131C211yesmaximal, characteristic
4#2203×C131C218yesmaximal, direct product factor
5C2×#356×C131C212yesmaximal, complemented
6C2×#355×C131C211yesmaximal, characteristic, complemented
7C2×2−1+4×C131C2211yescharacteristic
8C2×#137×C131—22no—
9C2×AGL(1,ℤ/8)×C131—21no—
10#563×C131—24no—
11#563×C131—22no—
12#563×C131C2214yescomplemented
13C2×C4⋊Q8×C131C2212yes—
14C2×#126×C131—22no—
15C2×#125×C131—21no—
16C2×#124×C131C2211yescharacteristic
17C2×#123×C131—21no—
18C2×C22⋊Q8×C131—22no—
19#456×C131—24no—
20#456×C131C2218yescomplemented
21#454×C131—22no—
22#454×C131C2214yes—
23C2×#104×C131—21no—
24C2×#104×C131—22no—
25C2×#101×C131C2211yescharacteristic
26C2×#101×C131C2212yes—
27#356×C131C2218yescomplemented
28#355×C131C2214yescomplemented
292−1+4×C131—21no—
302−1+4×C131—22no—
312−1+4×C131—42no—
322−1+4×C131C2312yes—
33C2×#41×C131—21no—
34C2×#41×C131—22no—
35C2×#41×C131—41no—
36C22×Q8×C131—41no—
37C22×Q8×C131D412yes—
38C22×Q8×C131C2311yescharacteristic
39C22×D4×C131D411yescharacteristic
40C23×C4×C131—21no—
41#137×C131—28no—
42#137×C131—44no—
43AGL(1,ℤ/8)×C131—24no—
44AGL(1,ℤ/8)×C131—42no—
45C2×Q16×C131—42no—
46C2×SD16×C131—42no—
47C2×SD16×C131—41no—
48C2×D8×C131—41no—
49C2×M16×C131—21no—
50C2×M16×C131—22no—
51C4⋊Q8×C131—22no—
52C4⋊Q8×C131—22no—
53C4⋊Q8×C131D414yes—
54C4⋊Q8×C131C2314yes—
55#126×C131—28no—
56#126×C131—28no—
57#125×C131—24no—
58#125×C131—24no—
59#124×C131—22no—
60#124×C131D412yes—
61#124×C131C2312yes—
62#123×C131—24no—
63#123×C131—24no—
64C22⋊Q8×C131—28no—
65C22⋊Q8×C131—28no—
66C2×C4⋊C4×C131—21no—
67C2×C4⋊C4×C131—22no—
68C2×C4⋊C4×C131—21no—
69C2×C4⋊C4×C131—22no—
70C2×C4⋊C4×C131—22no—
71C2×C22⋊C4×C131—22no—
72C2×C22⋊C4×C131—21no—
73C2×C22⋊C4×C131—22no—
74C2×C42×C131D411yescharacteristic
75C2×C42×C131D412yes—
76#104×C131—24no—
77#104×C131—28no—
78#101×C131—21no—
79#101×C131—22no—
80#101×C131C2312yes—
81#101×C131C2314yes—
82C24×C131—21no—
83#41×C131—22no—
84#41×C131—24no—
85#41×C131—22no—
86#41×C131—24no—
87#41×C131—41no—
88#41×C131—42no—
89#41×C131—42no—
90#41×C131—42no—
91#41×C131—44no—
92#41×C131—42no—
93#41×C131—41no—
94C2×Q8×C131—22no—
95C2×Q8×C131—21no—
96C2×Q8×C131—22no—
97C2×Q8×C131—41no—
98C2×Q8×C131—41no—
99C2×Q8×C131—42no—
100C2×Q8×C131—42no—
101C2×Q8×C131—42no—
102C2×Q8×C131—42no—
103C2×Q8×C131—44no—
104C2×Q8×C131C2×D414yes—
105C2×Q8×C131C2×D412yes—
106C2×Q8×C131C2×D411yescharacteristic
107C2×Q8×C131C2411yescharacteristic
108C2×D4×C131—21no—
109C2×D4×C131—22no—
110C2×D4×C131—21no—
111C2×D4×C131—41no—
112C2×D4×C131—41no—
113C2×D4×C131—41no—
114C2×D4×C131C2×D412yes—
115C22×C4×C131—21no—
116C22×C4×C131—22no—
117C22×C4×C131—22no—
118C22×C4×C131—24no—
119C22×C4×C131—41no—
120C22×C4×C131—42no—
121C22×C4×C131—81no—
122C22×C4×C131—81no—
123C22×C4×C131C2×D412yes—
124C22×C4×C131C2×D411yescharacteristic
125Q16×C131—48no—
126SD16×C131—48no—
127SD16×C131—44no—
128D8×C131—44no—
129M16×C131—22no—
130M16×C131—24no—
131M16×C131—41no—
132M16×C131—42no—
133C2×C8×C131—41no—
134C2×C8×C131—42no—
135C4⋊C4×C131—24no—
136C4⋊C4×C131—28no—
137C4⋊C4×C131—24no—
138C4⋊C4×C131—24no—
139C4⋊C4×C131—24no—
140C4⋊C4×C131—24no—
141C4⋊C4×C131—24no—
142C22⋊C4×C131—28no—
143C22⋊C4×C131—24no—
144C22⋊C4×C131—24no—
145C22⋊C4×C131—24no—
146C42×C131—21no—
147C42×C131—22no—
148C42×C131C2×D412yes—
149C42×C131C2×D414yes—
150C23×C131—22no—
151C23×C131—44no—
152C23×C131—41no—
153C23×C131—41no—
154C23×C131—41no—
155C23×C131#12011yescharacteristic
156Q8×C131—22no—
157Q8×C131—24no—
158Q8×C131—41no—
159Q8×C131—42no—
160Q8×C131—41no—
161Q8×C131—44no—
162Q8×C131—41no—
163Q8×C131—42no—
164Q8×C131—82no—
165D4×C131—22no—
166D4×C131—24no—
167D4×C131—42no—
168D4×C131—41no—
169D4×C131—42no—
170D4×C131—42no—
171D4×C131—42no—
172D4×C131—81no—
173C2×C4×C131—21no—
174C2×C4×C131—22no—
175C2×C4×C131—21no—
176C2×C4×C131—22no—
177C2×C4×C131—22no—
178C2×C4×C131—24no—
179C2×C4×C131—24no—
180C2×C4×C131—42no—
181C2×C4×C131—44no—
182C2×C4×C131—42no—
183C2×C4×C131—44no—
184C2×C4×C131—42no—
185C2×C4×C131—44no—
186C2×C4×C131—42no—
187C2×C4×C131—42no—
188C2×C4×C131—41no—
189C2×C4×C131—84no—
190C2×C4×C131#12012yes—
191C2×C4×C131#12011yescharacteristic
192C2×C4×C131C22×D412yes—
193C2×C4×C131C22×D411yescharacteristic
194C8×C131—42no—
195C8×C131—44no—
196C22×C131—22no—
197C22×C131—44no—
198C22×C131—42no—
199C22×C131—42no—
200C22×C131—82no—
201C22×C131—82no—
202C22×C131—82no—
203C22×C131B(4,𝔽2)11yescharacteristic
204C22×C131B(4,𝔽2)11yescharacteristic
205C22×C131C2×#12011yescharacteristic
206C4×C131—21no—
207C4×C131—22no—
208C4×C131—21no—
209C4×C131—22no—
210C4×C131—42no—
211C4×C131—44no—
212C4×C131—44no—
213C4×C131—42no—
214C2×C131—21no—
215C2×C131—21no—
216C2×C131—82no—
217C2×C131—82no—
218C2×C131#220312yesdirect product factor
219C2×C131C2×B(4,𝔽2)11yescharacteristic
220C2×#2203C13111yesmaximal, characteristic, direct product factor, Sylow
221C131C2×#220311yescharacteristic, direct product factor, Sylow
222C2×#563C2×C13111yescharacteristic, complemented
223C2×#456C2×C13112yescomplemented
224C2×#454C2×C13111yescharacteristic
225#2203C2×C13118yesdirect product factor
226C2×#356C2×C13112yescomplemented
227C2×#355C2×C13111yescharacteristic, complemented
228C2×2−1+4C22×C13111yescharacteristic
229C2×#137—22no—
230C2×AGL(1,ℤ/8)—21no—
231#563—24no—
232#563—22no—
233#563C22×C13114yescomplemented
234C2×C4⋊Q8C22×C13112yes—
235C2×#126—22no—
236C2×#125—21no—
237C2×#124C22×C13111yescharacteristic
238C2×#123—21no—
239C2×C22⋊Q8—22no—
240#456—24no—
241#456C22×C13118yescomplemented
242#454—22no—
243#454C22×C13114yes—
244C2×#104—21no—
245C2×#104—22no—
246C2×#101C22×C13112yes—
247C2×#101C22×C13111yescharacteristic
248#356C22×C13118yescomplemented
249#355C22×C13114yescomplemented
2502−1+4—21no—
2512−1+4—22no—
2522−1+4—42no—
2532−1+4C23×C13112yes—
254C2×#41—21no—
255C2×#41—22no—
256C2×#41—41no—
257C22×Q8—41no—
258C22×Q8D4×C13112yes—
259C22×Q8C23×C13111yescharacteristic
260C22×D4D4×C13111yescharacteristic
261C23×C4—21no—
262#137—28no—
263#137—44no—
264AGL(1,ℤ/8)—24no—
265AGL(1,ℤ/8)—42no—
266C2×Q16—42no—
267C2×SD16—42no—
268C2×SD16—41no—
269C2×D8—41no—
270C2×M16—21no—
271C2×M16—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8D4×C13114yes—
275C4⋊Q8C23×C13114yes—
276#126—28no—
277#126—28no—
278#125—24no—
279#125—24no—
280#124—22no—
281#124D4×C13112yes—
282#124C23×C13112yes—
283#123—24no—
284#123—24no—
285C22⋊Q8—28no—
286C22⋊Q8—28no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C42D4×C13111yescharacteristic
296C2×C42D4×C13112yes—
297#104—24no—
298#104—28no—
299#101—21no—
300#101—22no—
301#101C23×C13114yes—
302#101C23×C13112yes—
303C24—21no—
304#41—22no—
305#41—24no—
306#41—22no—
307#41—24no—
308#41—41no—
309#41—42no—
310#41—42no—
311#41—42no—
312#41—44no—
313#41—42no—
314#41—41no—
315C2×Q8—22no—
316C2×Q8—21no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8C2×D4×C13114yes—
326C2×Q8C2×D4×C13112yes—
327C2×Q8C2×D4×C13111yescharacteristic
328C2×Q8C24×C13111yescharacteristic
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4C2×D4×C13112yes—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4C2×D4×C13112yes—
345C22×C4C2×D4×C13111yescharacteristic
346Q16—48no—
347SD16—48no—
348SD16—44no—
349D8—44no—
350M16—22no—
351M16—24no—
352M16—41no—
353M16—42no—
354C2×C8—41no—
355C2×C8—42no—
356C4⋊C4—24no—
357C4⋊C4—28no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C22⋊C4—28no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C42—21no—
368C42—22no—
369C42C2×D4×C13112yes—
370C42C2×D4×C13114yes—
371C23—22no—
372C23—44no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23#120×C13111yescharacteristic
377Q8—22no—
378Q8—24no—
379Q8—41no—
380Q8—42no—
381Q8—41no—
382Q8—44no—
383Q8—41no—
384Q8—42no—
385Q8—82no—
386D4—22no—
387D4—24no—
388D4—42no—
389D4—41no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—81no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—84no—
411C2×C4#120×C13112yes—
412C2×C4#120×C13111yescharacteristic
413C2×C4C22×D4×C13112yes—
414C2×C4C22×D4×C13111yescharacteristic
415C8—42no—
416C8—44no—
417C22—22no—
418C22—44no—
419C22—42no—
420C22—42no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22B(4,𝔽2)×C13111yescharacteristic
425C22B(4,𝔽2)×C13111yescharacteristic
426C22C2×#120×C13111yescharacteristic
427C4—21no—
428C4—22no—
429C4—21no—
430C4—22no—
431C4—42no—
432C4—44no—
433C4—44no—
434C4—42no—
435C2—21no—
436C2—21no—
437C2—82no—
438C2—82no—
439C2#2203×C13112yesdirect product factor
440C2C2×B(4,𝔽2)×C13111yescharacteristic
subgroupG/H
center, socleC22×C131B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C131
prime radicalC2×#454×C131C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C23×C131 ◁ C22×Q8×C131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#563×C131C2semidirect
2. C2×#456×C131C2semidirect
4. #2203×C131C2direct (N × H)
5. C2×#356×C131C2semidirect
6. C2×#355×C131C2semidirect
12. #563×C131C22semidirect
20. #456×C131C22semidirect
27. #356×C131C22semidirect
28. #355×C131C22semidirect
218. C2×C131#2203direct (N × H)
220. C2×#2203C131direct (N × H)
221. C131C2×#2203direct (N × H)
222. C2×#563C2×C131semidirect
223. C2×#456C2×C131semidirect
225. #2203C2×C131direct (N × H)
226. C2×#356C2×C131semidirect
227. C2×#355C2×C131semidirect
233. #563C22×C131semidirect
241. #456C22×C131semidirect
248. #356C22×C131semidirect
249. #355C22×C131semidirect
439. C2#2203×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×#456×C1312
C2×#454×C1312
#2203×C1312
C2×#356×C1312
C2×#355×C1312
C2×#2203131