Finite Group Databasebeta

← C113×#26081 #43294101

Subgroups — C113×#26081 #43294101

Order 28928.

690 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C113×#3163C211yesmaximal, characteristic, complemented
2C113×#3084C212yesmaximal
3C113×#2912C211yesmaximal, characteristic
4C113×#2090C212yesmaximal, complemented
5C113×#2060C212yesmaximal, complemented
6C113×#1951C211yesmaximal, characteristic, complemented
7C113×#1945C211yesmaximal, characteristic, complemented
8C113×#1859C212yesmaximal
9C113×#1844C211yesmaximal, characteristic, complemented
10C113×#1815C211yesmaximal, characteristic
11C113×#1764C211yesmaximal, characteristic
12C2×#131×C113C2211yescharacteristic
13#544×C113C2212yes—
14#520×C113C2212yes—
15#501×C113—22no—
16#499×C113—21no—
17#498×C113—21no—
18#479×C113—22no—
19#478×C113—22no—
20#476×C113—22no—
21#475×C113—22no—
22C2×#108×C113C2211yescharacteristic
23#428×C113—22no—
24#428×C113C2212yes—
25#428×C113C2212yes—
26#428×C113C2212yes—
27#427×C113—21no—
28#427×C113C2211yescharacteristic
29C2×#106×C113—21no—
30C2×#106×C113C2212yes—
31C2×#106×C113C2211yescharacteristic
32C2×#106×C113C2211yescharacteristic
33#424×C113C2212yes—
34C2×#104×C113C2211yescharacteristic
35#417×C113C2212yes—
36#416×C113C2211yescharacteristic
37#405×C113C2211yescharacteristic
38C2×C4×C8×C113C2211yescharacteristic
39#384×C113C2212yes—
40#343×C113C2212yes—
41#343×C113C2212yes—
42#341×C113C2211yescharacteristic
43#339×C113C2211yescharacteristic
44#339×C113C2212yes—
45C2×#41×C113C2311yescharacteristic
46#131×C113—22no—
47#131×C113Q814yescomplemented
48C2×M16×C113—21no—
49C2×M16×C113C2312yes—
50C2×M16×C113C2311yescharacteristic
51C2×M16×C113C2311yescharacteristic
52C22×C8×C113—21no—
53C22×C8×C113C2311yescharacteristic
54C22×C8×C113C2311yescharacteristic
55C4⋊Q8×C113—22no—
56C4⋊Q8×C113—22no—
57#125×C113—22no—
58#125×C113—22no—
59C22⋊Q8×C113—42no—
60C4×Q8×C113—22no—
61C4×D4×C113—22no—
62C4×D4×C113—42no—
63#117×C113—22no—
64#117×C113C2312yes—
65#117×C113C2311yescharacteristic
66C2×C4⋊C4×C113—22no—
67C2×C4⋊C4×C113—42no—
68C2×C4⋊C4×C113C2312yes—
69C2×C4⋊C4×C113C2312yes—
70C2×C42×C113C2311yescharacteristic
71#108×C113—22no—
72#107×C113—22no—
73#107×C113—22no—
74#107×C113—22no—
75#107×C113D412yes—
76#106×C113—21no—
77#106×C113—21no—
78#106×C113—22no—
79#106×C113—22no—
80#106×C113—21no—
81#106×C113—21no—
82#106×C113—22no—
83#106×C113—22no—
84#106×C113—22no—
85#106×C113D411yescharacteristic
86#106×C113D411yescharacteristic
87C4⋊C8×C113—22no—
88C4⋊C8×C113—22no—
89#104×C113Q814yescomplemented
90#103×C113—22no—
91#103×C113D412yes—
92#102×C113—22no—
93#102×C113D412yes—
94#97×C113—21no—
95C4×C8×C113—21no—
96C4×C8×C113—22no—
97#41×C113—21no—
98#41×C113—21no—
99#41×C113C2×Q812yes—
100#41×C113C2×Q812yes—
101C2×Q8×C113—42no—
102C2×Q8×C113C2×D411yescharacteristic
103C2×D4×C113—21no—
104C2×D4×C113C2×D411yescharacteristic
105C22×C4×C113—21no—
106C22×C4×C113—21no—
107C22×C4×C113—22no—
108C22×C4×C113—22no—
109C22×C4×C113—42no—
110C22×C4×C113C2411yescharacteristic
111M16×C113—21no—
112M16×C113—21no—
113M16×C113—24no—
114M16×C113—22no—
115M16×C113—22no—
116M16×C113C2×Q812yes—
117M16×C113C2×Q812yes—
118C2×C8×C113—21no—
119C2×C8×C113—21no—
120C2×C8×C113—21no—
121C2×C8×C113—21no—
122C2×C8×C113—24no—
123C2×C8×C113—22no—
124C2×C8×C113C2×D412yes—
125C2×C8×C113C2×D411yescharacteristic
126C2×C8×C113C2×D411yescharacteristic
127C2×C8×C113C2×Q812yes—
128C2×C8×C113#4112yes—
129C4⋊C4×C113—22no—
130C4⋊C4×C113—22no—
131C4⋊C4×C113—22no—
132C4⋊C4×C113—22no—
133C4⋊C4×C113—42no—
134C4⋊C4×C113—42no—
135C4⋊C4×C113—42no—
136C4⋊C4×C113—42no—
137C4⋊C4×C113—84no—
138C4⋊C4×C113C2×D412yes—
139C4⋊C4×C113C2×D412yes—
140C4⋊C4×C113C2×D412yes—
141C22⋊C4×C113—22no—
142C22⋊C4×C113—21no—
143C22⋊C4×C113—42no—
144C42×C113—21no—
145C42×C113—22no—
146C42×C113—21no—
147C42×C113—42no—
148C42×C113C2×Q812yes—
149C23×C113—21no—
150C23×C113H(5,𝔽2)11yescharacteristic
151Q8×C113—21no—
152Q8×C113—162no—
153Q8×C113C4⋊Q812yes—
154D4×C113—22no—
155D4×C113—22no—
156D4×C113—21no—
157D4×C113C4⋊Q812yes—
158C2×C4×C113—24no—
159C2×C4×C113—21no—
160C2×C4×C113—22no—
161C2×C4×C113—22no—
162C2×C4×C113—22no—
163C2×C4×C113—22no—
164C2×C4×C113—22no—
165C2×C4×C113—22no—
166C2×C4×C113—42no—
167C2×C4×C113—42no—
168C2×C4×C113—82no—
169C2×C4×C113—82no—
170C2×C4×C113—162no—
171C2×C4×C113C22×D411yescharacteristic
172C2×C4×C113C22×D411yescharacteristic
173C2×C4×C113C22×Q811yescharacteristic
174C2×C4×C113C22×Q811yescharacteristic
175C2×C4×C113C2×#4111yescharacteristic
176C2×C4×C1132−1+411yescharacteristic
177C8×C113—22no—
178C8×C113—22no—
179C8×C113—22no—
180C8×C113—42no—
181C8×C113C22⋊Q812yes—
182C8×C113C22⋊Q812yes—
183C22×C113—22no—
184C22×C113—22no—
185C22×C113—42no—
186C22×C113#54411yescharacteristic
187C22×C113D4211yescharacteristic
188C22×C113#55411yescharacteristic
189C4×C113—24no—
190C4×C113—22no—
191C4×C113—42no—
192C4×C113—42no—
193C4×C113—42no—
194C4×C113—82no—
195C4×C113—82no—
196C4×C113C2×C22⋊Q811yescharacteristic
197C4×C113C2×C4⋊Q811yescharacteristic
198C4×C113#54711yescharacteristic
199C4×C113D4×Q811yescharacteristic
200#26081C11311yesmaximal, characteristic, direct product factor, Sylow
201C2×C113—21no—
202C2×C113—21no—
203C2×C113—42no—
204C2×C113#244011yescharacteristic
205C2×C113#329112yes—
206#3163C2×C11311yescharacteristic, complemented
207#3084C2×C11312yes—
208#2912C2×C11311yescharacteristic
209#2090C2×C11312yescomplemented
210#2060C2×C11312yescomplemented
211#1951C2×C11311yescharacteristic, complemented
212#1945C2×C11311yescharacteristic, complemented
213#1859C2×C11312yes—
214#1844C2×C11311yescharacteristic, complemented
215#1815C2×C11311yescharacteristic
216#1764C2×C11311yescharacteristic
217C113#2608111yescharacteristic, direct product factor, Sylow
218C2×#131C22×C11311yescharacteristic
219#544C22×C11312yes—
220#520C22×C11312yes—
221#501—22no—
222#499—21no—
223#498—21no—
224#479—22no—
225#478—22no—
226#476—22no—
227#475—22no—
228C2×#108C22×C11311yescharacteristic
229#428—22no—
230#428C22×C11312yes—
231#428C22×C11312yes—
232#428C22×C11312yes—
233#427—21no—
234#427C22×C11311yescharacteristic
235C2×#106—21no—
236C2×#106C22×C11312yes—
237C2×#106C22×C11311yescharacteristic
238C2×#106C22×C11311yescharacteristic
239#424C22×C11312yes—
240C2×#104C22×C11311yescharacteristic
241#417C22×C11312yes—
242#416C22×C11311yescharacteristic
243#405C22×C11311yescharacteristic
244C2×C4×C8C22×C11311yescharacteristic
245#384C22×C11312yes—
246#343C22×C11312yes—
247#343C22×C11312yes—
248#341C22×C11311yescharacteristic
249#339C22×C11311yescharacteristic
250#339C22×C11312yes—
251C2×#41C23×C11311yescharacteristic
252#131—22no—
253#131Q8×C11314yescomplemented
254C2×M16—21no—
255C2×M16C23×C11312yes—
256C2×M16C23×C11311yescharacteristic
257C2×M16C23×C11311yescharacteristic
258C22×C8—21no—
259C22×C8C23×C11311yescharacteristic
260C22×C8C23×C11311yescharacteristic
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263#125—22no—
264#125—22no—
265C22⋊Q8—42no—
266C4×Q8—22no—
267C4×D4—22no—
268C4×D4—42no—
269#117—22no—
270#117C23×C11312yes—
271#117C23×C11311yescharacteristic
272C2×C4⋊C4—22no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C23×C11312yes—
275C2×C4⋊C4C23×C11312yes—
276C2×C42C23×C11311yescharacteristic
277#108—22no—
278#107—22no—
279#107—22no—
280#107—22no—
281#107D4×C11312yes—
282#106—21no—
283#106—21no—
284#106—22no—
285#106—22no—
286#106—21no—
287#106—21no—
288#106—22no—
289#106—22no—
290#106—22no—
291#106D4×C11311yescharacteristic
292#106D4×C11311yescharacteristic
293C4⋊C8—22no—
294C4⋊C8—22no—
295#104Q8×C11314yescomplemented
296#103—22no—
297#103D4×C11312yes—
298#102—22no—
299#102D4×C11312yes—
300#97—21no—
301C4×C8—21no—
302C4×C8—22no—
303#41—21no—
304#41—21no—
305#41C2×Q8×C11312yes—
306#41C2×Q8×C11312yes—
307C2×Q8—42no—
308C2×Q8C2×D4×C11311yescharacteristic
309C2×D4—21no—
310C2×D4C2×D4×C11311yescharacteristic
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—42no—
316C22×C4C24×C11311yescharacteristic
317M16—21no—
318M16—21no—
319M16—24no—
320M16—22no—
321M16—22no—
322M16C2×Q8×C11312yes—
323M16C2×Q8×C11312yes—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—24no—
329C2×C8—22no—
330C2×C8C2×D4×C11312yes—
331C2×C8C2×D4×C11311yescharacteristic
332C2×C8C2×D4×C11311yescharacteristic
333C2×C8C2×Q8×C11312yes—
334C2×C8#41×C11312yes—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—84no—
344C4⋊C4C2×D4×C11312yes—
345C4⋊C4C2×D4×C11312yes—
346C4⋊C4C2×D4×C11312yes—
347C22⋊C4—22no—
348C22⋊C4—21no—
349C22⋊C4—42no—
350C42—21no—
351C42—22no—
352C42—21no—
353C42—42no—
354C42C2×Q8×C11312yes—
355C23—21no—
356C23H(5,𝔽2)×C11311yescharacteristic
357Q8—21no—
358Q8—162no—
359Q8C4⋊Q8×C11312yes—
360D4—22no—
361D4—22no—
362D4—21no—
363D4C4⋊Q8×C11312yes—
364C2×C4—24no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—162no—
377C2×C4C22×D4×C11311yescharacteristic
378C2×C4C22×D4×C11311yescharacteristic
379C2×C4C22×Q8×C11311yescharacteristic
380C2×C4C22×Q8×C11311yescharacteristic
381C2×C4C2×#41×C11311yescharacteristic
382C2×C42−1+4×C11311yescharacteristic
383C8—22no—
384C8—22no—
385C8—22no—
386C8—42no—
387C8C22⋊Q8×C11312yes—
388C8C22⋊Q8×C11312yes—
389C22—22no—
390C22—22no—
391C22—42no—
392C22#544×C11311yescharacteristic
393C22D42×C11311yescharacteristic
394C22#554×C11311yescharacteristic
395C4—24no—
396C4—22no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—82no—
401C4—82no—
402C4C2×C22⋊Q8×C11311yescharacteristic
403C4C2×C4⋊Q8×C11311yescharacteristic
404C4#547×C11311yescharacteristic
405C4D4×Q8×C11311yescharacteristic
406C2—21no—
407C2—21no—
408C2—42no—
409C2C113×#244011yescharacteristic
410C2C113×#329112yes—
subgroupG/H
center, socleC22×C113D42
commutator, Frattini, cosocleC22×C4C24×C113
prime radicalC2×#41×C113C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C113 ◁ C22×C4×C113 ◁ G

Complements

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

NH ≅ G/Ntype
1. C113×#3163C2semidirect
4. C113×#2090C2semidirect
5. C113×#2060C2semidirect
6. C113×#1951C2semidirect
7. C113×#1945C2semidirect
9. C113×#1844C2semidirect
47. #131×C113Q8semidirect
89. #104×C113Q8semidirect
200. #26081C113direct (N × H)
206. #3163C2×C113semidirect
209. #2090C2×C113semidirect
210. #2060C2×C113semidirect
211. #1951C2×C113semidirect
212. #1945C2×C113semidirect
214. #1844C2×C113semidirect
217. C113#26081direct (N × H)
253. #131Q8×C113semidirect
295. #104Q8×C113semidirect

Maximal subgroups

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

subgroupindex
C113×#31632
C113×#30842
C113×#29122
C113×#20902
C113×#20602
C113×#19512
C113×#19452
C113×#18592
C113×#18442
C113×#18152
C113×#17642
#26081113