Finite Group Databasebeta

← C113×#23047 #43291067

Subgroups — C113×#23047 #43291067

Order 28928.

840 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#473×C113C211yesmaximal, characteristic
2C113×#2950C211yesmaximal, characteristic, complemented
3C113×#2947C212yesmaximal
4C113×#2912C211yesmaximal, characteristic
5C113×#1971C211yesmaximal, characteristic, complemented
6C113×#1951C211yesmaximal, characteristic, complemented
7C113×#1930C212yesmaximal, complemented
8C113×#1882C212yesmaximal
9C113×#1857C212yesmaximal, complemented
10C113×#1773C211yesmaximal, characteristic, complemented
11C113×#1765C211yesmaximal, characteristic
12C2×#137×C113C2212yes—
13C22×Q16×C113C2211yescharacteristic
14C2×#131×C113C2211yescharacteristic
15#520×C113C2212yes—
16#519×C113C2212yes—
17#473×C113C418yescomplemented
18#441×C113—21no—
19#441×C113—21no—
20#441×C113—22no—
21#440×C113—22no—
22C4×Q16×C113—22no—
23C4×Q16×C113—22no—
24C4×SD16×C113—22no—
25C2×#108×C113C2211yescharacteristic
26#428×C113—22no—
27#428×C113C2212yes—
28#428×C113C2212yes—
29#427×C113—21no—
30#427×C113C2211yescharacteristic
31C2×#106×C113—21no—
32C2×#106×C113C2211yescharacteristic
33C2×#104×C113C2211yescharacteristic
34#419×C113C2212yes—
35#416×C113C2211yescharacteristic
36C2×#103×C113C2212yes—
37C2×#101×C113C2212yes—
38#405×C113C2211yescharacteristic
39#405×C113C2212yes—
40C2×#97×C113C2211yescharacteristic
41#391×C113C2212yes—
42#343×C113C2212yes—
43#341×C113C2211yescharacteristic
44#339×C113C2211yescharacteristic
45#339×C113C2212yes—
46C2×#41×C113C2311yescharacteristic
47C22×Q8×C113C2312yes—
48#137×C113—24no—
49#137×C113C2×C418yes—
50C2×Q16×C113—22no—
51C2×Q16×C113—21no—
52C2×Q16×C113—21no—
53C2×Q16×C113—22no—
54C2×Q16×C113C2×C414yescomplemented
55C2×SD16×C113—22no—
56#131×C113—22no—
57#131×C113C2×C414yes—
58C2×M16×C113—21no—
59C2×M16×C113C2311yescharacteristic
60C2×M16×C113C2311yescharacteristic
61C2×M16×C113C2312yes—
62C22×C8×C113—21no—
63C22×C8×C113C2311yescharacteristic
64C22×C8×C113C2311yescharacteristic
65C4×Q8×C113—22no—
66C4×Q8×C113—22no—
67C4×Q8×C113—22no—
68C4×Q8×C113—22no—
69C4×Q8×C113—22no—
70C4×D4×C113—22no—
71C4×D4×C113—42no—
72#117×C113—22no—
73#117×C113—42no—
74#117×C113C2311yescharacteristic
75#117×C113C2312yes—
76C2×C4⋊C4×C113—22no—
77C2×C4⋊C4×C113C2312yes—
78C2×C42×C113C2311yescharacteristic
79#108×C113C2×C414yescomplemented
80#107×C113—22no—
81#107×C113—22no—
82#107×C113D412yes—
83#106×C113—21no—
84#106×C113—21no—
85#106×C113—22no—
86#106×C113—22no—
87#106×C113D411yescharacteristic
88#106×C113D411yescharacteristic
89#104×C113—22no—
90#103×C113—22no—
91#103×C113—22no—
92#103×C113—22no—
93#103×C113—22no—
94#103×C113D412yes—
95#102×C113D412yes—
96#101×C113C2×C418yescomplemented
97#97×C113—22no—
98#97×C113—21no—
99#97×C113—22no—
100C4×C8×C113—22no—
101C4×C8×C113—21no—
102#95×C113—42no—
103#41×C113—21no—
104#41×C113—21no—
105#41×C113C22×C412yes—
106#41×C113#4112yes—
107C2×Q8×C113—22no—
108C2×Q8×C113—22no—
109C2×Q8×C113—22no—
110C2×Q8×C113—22no—
111C2×Q8×C113—22no—
112C2×Q8×C113C22×C414yes—
113C2×Q8×C113C2×D411yescharacteristic
114C2×D4×C113—21no—
115C2×D4×C113C2×D411yescharacteristic
116C22×C4×C113—21no—
117C22×C4×C113—22no—
118C22×C4×C113—21no—
119C22×C4×C113—22no—
120C22×C4×C113—42no—
121C22×C4×C113C2411yescharacteristic
122Q16×C113—28no—
123Q16×C113—24no—
124Q16×C113—24no—
125SD16×C113—28no—
126M16×C113—21no—
127M16×C113—22no—
128M16×C113—24no—
129M16×C113—21no—
130M16×C113—21no—
131M16×C113C22×C412yes—
132M16×C113C22×C414yes—
133C2×C8×C113—22no—
134C2×C8×C113—21no—
135C2×C8×C113—21no—
136C2×C8×C113—21no—
137C2×C8×C113—24no—
138C2×C8×C113—21no—
139C2×C8×C113C22×C412yes—
140C2×C8×C113C2×D412yes—
141C2×C8×C113C2×D411yescharacteristic
142C2×C8×C113C2×D411yescharacteristic
143C2×C8×C113#4112yes—
144C4⋊C4×C113—22no—
145C4⋊C4×C113—22no—
146C4⋊C4×C113—42no—
147C4⋊C4×C113—42no—
148C4⋊C4×C113—42no—
149C4⋊C4×C113C2×D412yes—
150C4⋊C4×C113C2×D412yes—
151C4⋊C4×C113C2×D412yes—
152C22⋊C4×C113—21no—
153C22⋊C4×C113—22no—
154C22⋊C4×C113—42no—
155C42×C113—22no—
156C42×C113—21no—
157C42×C113—21no—
158C42×C113—21no—
159C42×C113—42no—
160C42×C113—42no—
161C42×C113—42no—
162C23×C113—21no—
163C23×C113H(5,𝔽2)11yescharacteristic
164Q8×C113—21no—
165Q8×C113—24no—
166Q8×C113—24no—
167Q8×C113—42no—
168Q8×C113—42no—
169Q8×C113C4×D412yes—
170D4×C113—22no—
171D4×C113—22no—
172D4×C113—21no—
173D4×C113C4×D412yes—
174C2×C4×C113—21no—
175C2×C4×C113—22no—
176C2×C4×C113—22no—
177C2×C4×C113—24no—
178C2×C4×C113—22no—
179C2×C4×C113—22no—
180C2×C4×C113—22no—
181C2×C4×C113—42no—
182C2×C4×C113—42no—
183C2×C4×C113—42no—
184C2×C4×C113—42no—
185C2×C4×C113—41no—
186C2×C4×C113—82no—
187C2×C4×C113—162no—
188C2×C4×C113C23×C411yescharacteristic
189C2×C4×C113C22×D411yescharacteristic
190C2×C4×C113C22×D411yescharacteristic
191C2×C4×C113C2×#4111yescharacteristic
192C2×C4×C113C2×#4111yescharacteristic
193C2×C4×C1132−1+411yescharacteristic
194C8×C113—22no—
195C8×C113—24no—
196C8×C113—22no—
197C8×C113—41no—
198C8×C113C4×D412yes—
199C8×C113C4×D412yes—
200C22×C113—22no—
201C22×C113—22no—
202C22×C113—42no—
203C22×C113#52011yescharacteristic
204C22×C113D4211yescharacteristic
205C22×C113#55311yescharacteristic
206C4×C113—22no—
207C4×C113—44no—
208C4×C113—42no—
209C4×C113—42no—
210C4×C113—42no—
211C4×C113—81no—
212C4×C113—82no—
213C4×C113C2×C4×D411yescharacteristic
214C4×C113C2×C4×D411yescharacteristic
215C4×C113#54711yescharacteristic
216C4×C113#54811yescharacteristic
217#23047C11311yesmaximal, characteristic, direct product factor, Sylow
218C2×C113—21no—
219C2×C113—21no—
220C2×C113—42no—
221C2×C113#236111yescharacteristic
222C2×C113#328912yes—
223C2×#473C2×C11311yescharacteristic
224#2950C2×C11311yescharacteristic, complemented
225#2947C2×C11312yes—
226#2912C2×C11311yescharacteristic
227#1971C2×C11311yescharacteristic, complemented
228#1951C2×C11311yescharacteristic, complemented
229#1930C2×C11312yescomplemented
230#1882C2×C11312yes—
231#1857C2×C11312yescomplemented
232#1773C2×C11311yescharacteristic, complemented
233#1765C2×C11311yescharacteristic
234C113#2304711yescharacteristic, direct product factor, Sylow
235C2×#137C22×C11312yes—
236C22×Q16C22×C11311yescharacteristic
237C2×#131C22×C11311yescharacteristic
238#520C22×C11312yes—
239#519C22×C11312yes—
240#473C4×C11318yescomplemented
241#441—21no—
242#441—21no—
243#441—22no—
244#440—22no—
245C4×Q16—22no—
246C4×Q16—22no—
247C4×SD16—22no—
248C2×#108C22×C11311yescharacteristic
249#428—22no—
250#428C22×C11312yes—
251#428C22×C11312yes—
252#427—21no—
253#427C22×C11311yescharacteristic
254C2×#106—21no—
255C2×#106C22×C11311yescharacteristic
256C2×#104C22×C11311yescharacteristic
257#419C22×C11312yes—
258#416C22×C11311yescharacteristic
259C2×#103C22×C11312yes—
260C2×#101C22×C11312yes—
261#405C22×C11312yes—
262#405C22×C11311yescharacteristic
263C2×#97C22×C11311yescharacteristic
264#391C22×C11312yes—
265#343C22×C11312yes—
266#341C22×C11311yescharacteristic
267#339C22×C11311yescharacteristic
268#339C22×C11312yes—
269C2×#41C23×C11311yescharacteristic
270C22×Q8C23×C11312yes—
271#137—24no—
272#137C2×C4×C11318yes—
273C2×Q16—22no—
274C2×Q16—21no—
275C2×Q16—21no—
276C2×Q16—22no—
277C2×Q16C2×C4×C11314yescomplemented
278C2×SD16—22no—
279#131—22no—
280#131C2×C4×C11314yes—
281C2×M16—21no—
282C2×M16C23×C11311yescharacteristic
283C2×M16C23×C11311yescharacteristic
284C2×M16C23×C11312yes—
285C22×C8—21no—
286C22×C8C23×C11311yescharacteristic
287C22×C8C23×C11311yescharacteristic
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×D4—22no—
294C4×D4—42no—
295#117—22no—
296#117—42no—
297#117C23×C11311yescharacteristic
298#117C23×C11312yes—
299C2×C4⋊C4—22no—
300C2×C4⋊C4C23×C11312yes—
301C2×C42C23×C11311yescharacteristic
302#108C2×C4×C11314yescomplemented
303#107—22no—
304#107—22no—
305#107D4×C11312yes—
306#106—21no—
307#106—21no—
308#106—22no—
309#106—22no—
310#106D4×C11311yescharacteristic
311#106D4×C11311yescharacteristic
312#104—22no—
313#103—22no—
314#103—22no—
315#103—22no—
316#103—22no—
317#103D4×C11312yes—
318#102D4×C11312yes—
319#101C2×C4×C11318yescomplemented
320#97—22no—
321#97—21no—
322#97—22no—
323C4×C8—22no—
324C4×C8—21no—
325#95—42no—
326#41—21no—
327#41—21no—
328#41C22×C4×C11312yes—
329#41#41×C11312yes—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8C22×C4×C11314yes—
336C2×Q8C2×D4×C11311yescharacteristic
337C2×D4—21no—
338C2×D4C2×D4×C11311yescharacteristic
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4C24×C11311yescharacteristic
345Q16—28no—
346Q16—24no—
347Q16—24no—
348SD16—28no—
349M16—21no—
350M16—22no—
351M16—24no—
352M16—21no—
353M16—21no—
354M16C22×C4×C11312yes—
355M16C22×C4×C11314yes—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—24no—
361C2×C8—21no—
362C2×C8C22×C4×C11312yes—
363C2×C8C2×D4×C11312yes—
364C2×C8C2×D4×C11311yescharacteristic
365C2×C8C2×D4×C11311yescharacteristic
366C2×C8#41×C11312yes—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4C2×D4×C11312yes—
373C4⋊C4C2×D4×C11312yes—
374C4⋊C4C2×D4×C11312yes—
375C22⋊C4—21no—
376C22⋊C4—22no—
377C22⋊C4—42no—
378C42—22no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—42no—
383C42—42no—
384C42—42no—
385C23—21no—
386C23H(5,𝔽2)×C11311yescharacteristic
387Q8—21no—
388Q8—24no—
389Q8—24no—
390Q8—42no—
391Q8—42no—
392Q8C4×D4×C11312yes—
393D4—22no—
394D4—22no—
395D4—21no—
396D4C4×D4×C11312yes—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4—162no—
411C2×C4C23×C4×C11311yescharacteristic
412C2×C4C22×D4×C11311yescharacteristic
413C2×C4C22×D4×C11311yescharacteristic
414C2×C4C2×#41×C11311yescharacteristic
415C2×C4C2×#41×C11311yescharacteristic
416C2×C42−1+4×C11311yescharacteristic
417C8—22no—
418C8—24no—
419C8—22no—
420C8—41no—
421C8C4×D4×C11312yes—
422C8C4×D4×C11312yes—
423C22—22no—
424C22—22no—
425C22—42no—
426C22#520×C11311yescharacteristic
427C22D42×C11311yescharacteristic
428C22#553×C11311yescharacteristic
429C4—22no—
430C4—44no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—81no—
435C4—82no—
436C4C2×C4×D4×C11311yescharacteristic
437C4C2×C4×D4×C11311yescharacteristic
438C4#547×C11311yescharacteristic
439C4#548×C11311yescharacteristic
440C2—21no—
441C2—21no—
442C2—42no—
443C2C113×#236111yescharacteristic
444C2C113×#328912yes—
subgroupG/H
center, socleC22×C113D42
commutatorC2×C4C23×C4×C113
Frattini, cosocleC22×C4C24×C113
prime radicalC2×#41×C113C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C113 ◁ C22×C4×C113 ◁ G

Complements

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

NH ≅ G/Ntype
2. C113×#2950C2semidirect
5. C113×#1971C2semidirect
6. C113×#1951C2semidirect
7. C113×#1930C2semidirect
9. C113×#1857C2semidirect
10. C113×#1773C2semidirect
17. #473×C113C4semidirect
54. C2×Q16×C113C2×C4semidirect
79. #108×C113C2×C4semidirect
96. #101×C113C2×C4semidirect
217. #23047C113direct (N × H)
224. #2950C2×C113semidirect
227. #1971C2×C113semidirect
228. #1951C2×C113semidirect
229. #1930C2×C113semidirect
231. #1857C2×C113semidirect
232. #1773C2×C113semidirect
234. C113#23047direct (N × H)
240. #473C4×C113semidirect
277. C2×Q16C2×C4×C113semidirect
302. #108C2×C4×C113semidirect
319. #101C2×C4×C113semidirect

Maximal subgroups

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

subgroupindex
C2×#473×C1132
C113×#29502
C113×#29472
C113×#29122
C113×#19712
C113×#19512
C113×#19302
C113×#18822
C113×#18572
C113×#17732
C113×#17652
#23047113