Finite Group Databasebeta

← C89×#13005 #36390853

Subgroups — C89×#13005 #36390853

Order 22784.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3067C211yesmaximal, characteristic
2C89×#2912C211yesmaximal, characteristic, complemented
3C89×#2145C211yesmaximal, characteristic, complemented
4C2×#360×C89C211yesmaximal, characteristic, complemented
5C89×#2117C211yesmaximal, characteristic, complemented
6C89×#1389C211yesmaximal, characteristic, complemented
7C89×#1383C211yesmaximal, characteristic, complemented
8#575×C89—21no—
9C2×#135×C89C2211yescharacteristic
10C2×#131×C89C2211yescharacteristic
11#520×C89—21no—
12C2×M32×C89C2211yescharacteristic
13C2×M32×C89C2211yescharacteristic
14#473×C89C412yescomplemented
15#472×C89C412yescomplemented
16#471×C89—22no—
17C2×#108×C89C2211yescharacteristic
18#428×C89—21no—
19#428×C89—22no—
20#427×C89—21no—
21#427×C89C2211yescharacteristic
22C2×#106×C89—21no—
23C2×#106×C89C2211yescharacteristic
24#413×C89—21no—
25#360×C89—22no—
26#360×C89—21no—
27#360×C89—21no—
28#350×C89—22no—
29C2×#41×C89—21no—
30C2×#41×C89D411yescharacteristic
31#137×C89—22no—
32#137×C89—41no—
33AGL(1,ℤ/8)×C89—22no—
34AGL(1,ℤ/8)×C89—41no—
35#135×C89—21no—
36#135×C89—21no—
37#135×C89—21no—
38#135×C89—21no—
39#135×C89—42no—
40C2×Q16×C89C2×C411yescharacteristic, complemented
41C2×SD16×C89—21no—
42C2×D8×C89C2×C411yescharacteristic, complemented
43#131×C89—21no—
44#131×C89—21no—
45#131×C89C2×C412yes—
46#131×C89D412yes—
47C2×M16×C89—21no—
48C2×M16×C89—21no—
49C2×M16×C89D411yescharacteristic
50C22×C8×C89—21no—
51C22×C8×C89C2311yescharacteristic
52C4×Q8×C89—41no—
53C4×D4×C89—42no—
54C4×D4×C89—41no—
55#117×C89—22no—
56#117×C89—21no—
57C2×C4⋊C4×C89—22no—
58C2×C4⋊C4×C89—21no—
59M32×C89—21no—
60M32×C89—21no—
61M32×C89—21no—
62M32×C89—21no—
63C2×C16×C89—21no—
64C2×C16×C89—21no—
65#108×C89—21no—
66#108×C89C2×C412yescomplemented
67#107×C89—22no—
68#107×C89—21no—
69#106×C89—22no—
70#106×C89—21no—
71#106×C89D411yescharacteristic
72#106×C89D412yes—
73#106×C89D411yescharacteristic
74#101×C89—22no—
75#100×C89—22no—
76#41×C89—21no—
77#41×C89—21no—
78#41×C89—41no—
79#41×C89—41no—
80#41×C89—41no—
81#41×C89—41no—
82#41×C89C22⋊C412yes—
83#41×C89SD1612yes—
84C2×Q8×C89—21no—
85C2×Q8×C89D811yescharacteristic
86C2×D4×C89—21no—
87C2×D4×C89—21no—
88C2×D4×C89—41no—
89C2×D4×C89D811yescharacteristic
90C22×C4×C89—21no—
91C22×C4×C89—42no—
92C22×C4×C89—41no—
93C22×C4×C89—41no—
94C22×C4×C89C2×D411yescharacteristic
95Q16×C89—21no—
96Q16×C89—21no—
97Q16×C89—42no—
98SD16×C89—42no—
99SD16×C89—42no—
100SD16×C89—41no—
101SD16×C89—41no—
102D8×C89—21no—
103D8×C89—21no—
104D8×C89—42no—
105M16×C89—21no—
106M16×C89—22no—
107M16×C89—22no—
108M16×C89—22no—
109M16×C89—41no—
110M16×C89C22⋊C412yes—
111C2×C8×C89—21no—
112C2×C8×C89—22no—
113C2×C8×C89—22no—
114C2×C8×C89—41no—
115C2×C8×C89C22×C411yescharacteristic
116C2×C8×C89C2×D411yescharacteristic
117C2×C8×C89C2×D411yescharacteristic
118C2×C8×C89C2×D411yescharacteristic
119C2×C8×C89#4111yescharacteristic
120C2×C8×C89#4111yescharacteristic
121C4⋊C4×C89—21no—
122C4⋊C4×C89—22no—
123C4⋊C4×C89—21no—
124C4⋊C4×C89—42no—
125C4⋊C4×C89—41no—
126C22⋊C4×C89—42no—
127C22⋊C4×C89—41no—
128C42×C89—42no—
129C42×C89—41no—
130C16×C89—41no—
131C16×C89—41no—
132C23×C89—21no—
133C23×C89—41no—
134C23×C89AGL(1,ℤ/8)11yescharacteristic
135Q8×C89—21no—
136Q8×C89—41no—
137Q8×C89—41no—
138Q8×C89#10212yes—
139D4×C89—22no—
140D4×C89—22no—
141D4×C89—21no—
142D4×C89—41no—
143D4×C89—41no—
144D4×C89—81no—
145D4×C89—81no—
146D4×C89#10212yes—
147C2×C4×C89—21no—
148C2×C4×C89—22no—
149C2×C4×C89—22no—
150C2×C4×C89—41no—
151C2×C4×C89—41no—
152C2×C4×C89—42no—
153C2×C4×C89—41no—
154C2×C4×C89—81no—
155C2×C4×C89—81no—
156C2×C4×C89—161no—
157C2×C4×C89—162no—
158C2×C4×C89C2×C22⋊C411yescharacteristic
159C2×C4×C89#12011yescharacteristic
160C2×C4×C89#12311yescharacteristic
161C2×C4×C89C2×D811yescharacteristic
162C2×C4×C89C2×SD1611yescharacteristic
163C2×C4×C89#13711yescharacteristic
164C8×C89—22no—
165C8×C89—42no—
166C8×C89C4×D411yescharacteristic
167C8×C89C4×D411yescharacteristic
168C8×C89#12111yescharacteristic
169C8×C89#12111yescharacteristic
170C22×C89—22no—
171C22×C89—41no—
172C22×C89—42no—
173C22×C89—81no—
174C22×C89—161no—
175C22×C89#41711yescharacteristic
176C22×C89#44711yescharacteristic
177C22×C89#48111yescharacteristic
178C4×C89—22no—
179C4×C89—81no—
180C4×C89—82no—
181C4×C89—81no—
182C4×C89—81no—
183C4×C89#38611yescharacteristic
184C4×C89C2×#10211yescharacteristic
185C4×C89#44811yescharacteristic
186C4×C89#48011yescharacteristic
187#13005C8911yesmaximal, characteristic, direct product factor, Sylow
188C2×C89—21no—
189C2×C89—21no—
190C2×C89—21no—
191C2×C89—42no—
192C2×C89—161no—
193C2×C89#187411yescharacteristic
194#3067C2×C8911yescharacteristic
195#2912C2×C8911yescharacteristic, complemented
196#2145C2×C8911yescharacteristic, complemented
197C2×#360C2×C8911yescharacteristic, complemented
198#2117C2×C8911yescharacteristic, complemented
199#1389C2×C8911yescharacteristic, complemented
200#1383C2×C8911yescharacteristic, complemented
201C89#1300511yescharacteristic, direct product factor, Sylow
202#575—21no—
203C2×#135C22×C8911yescharacteristic
204C2×#131C22×C8911yescharacteristic
205#520—21no—
206C2×M32C22×C8911yescharacteristic
207C2×M32C22×C8911yescharacteristic
208#473C4×C8912yescomplemented
209#472C4×C8912yescomplemented
210#471—22no—
211C2×#108C22×C8911yescharacteristic
212#428—21no—
213#428—22no—
214#427—21no—
215#427C22×C8911yescharacteristic
216C2×#106—21no—
217C2×#106C22×C8911yescharacteristic
218#413—21no—
219#360—22no—
220#360—21no—
221#360—21no—
222#350—22no—
223C2×#41—21no—
224C2×#41D4×C8911yescharacteristic
225#137—22no—
226#137—41no—
227AGL(1,ℤ/8)—22no—
228AGL(1,ℤ/8)—41no—
229#135—21no—
230#135—21no—
231#135—21no—
232#135—21no—
233#135—42no—
234C2×Q16C2×C4×C8911yescharacteristic, complemented
235C2×SD16—21no—
236C2×D8C2×C4×C8911yescharacteristic, complemented
237#131—21no—
238#131—21no—
239#131C2×C4×C8912yes—
240#131D4×C8912yes—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16D4×C8911yescharacteristic
244C22×C8—21no—
245C22×C8C23×C8911yescharacteristic
246C4×Q8—41no—
247C4×D4—42no—
248C4×D4—41no—
249#117—22no—
250#117—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253M32—21no—
254M32—21no—
255M32—21no—
256M32—21no—
257C2×C16—21no—
258C2×C16—21no—
259#108—21no—
260#108C2×C4×C8912yescomplemented
261#107—22no—
262#107—21no—
263#106—22no—
264#106—21no—
265#106D4×C8911yescharacteristic
266#106D4×C8912yes—
267#106D4×C8911yescharacteristic
268#101—22no—
269#100—22no—
270#41—21no—
271#41—21no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41C22⋊C4×C8912yes—
277#41SD16×C8912yes—
278C2×Q8—21no—
279C2×Q8D8×C8911yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C2×D4D8×C8911yescharacteristic
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4C2×D4×C8911yescharacteristic
289Q16—21no—
290Q16—21no—
291Q16—42no—
292SD16—42no—
293SD16—42no—
294SD16—41no—
295SD16—41no—
296D8—21no—
297D8—21no—
298D8—42no—
299M16—21no—
300M16—22no—
301M16—22no—
302M16—22no—
303M16—41no—
304M16C22⋊C4×C8912yes—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8C22×C4×C8911yescharacteristic
310C2×C8C2×D4×C8911yescharacteristic
311C2×C8C2×D4×C8911yescharacteristic
312C2×C8C2×D4×C8911yescharacteristic
313C2×C8#41×C8911yescharacteristic
314C2×C8#41×C8911yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—41no—
322C42—42no—
323C42—41no—
324C16—41no—
325C16—41no—
326C23—21no—
327C23—41no—
328C23AGL(1,ℤ/8)×C8911yescharacteristic
329Q8—21no—
330Q8—41no—
331Q8—41no—
332Q8#102×C8912yes—
333D4—22no—
334D4—22no—
335D4—21no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340D4#102×C8912yes—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—161no—
351C2×C4—162no—
352C2×C4C2×C22⋊C4×C8911yescharacteristic
353C2×C4#120×C8911yescharacteristic
354C2×C4#123×C8911yescharacteristic
355C2×C4C2×D8×C8911yescharacteristic
356C2×C4C2×SD16×C8911yescharacteristic
357C2×C4#137×C8911yescharacteristic
358C8—22no—
359C8—42no—
360C8C4×D4×C8911yescharacteristic
361C8C4×D4×C8911yescharacteristic
362C8#121×C8911yescharacteristic
363C8#121×C8911yescharacteristic
364C22—22no—
365C22—41no—
366C22—42no—
367C22—81no—
368C22—161no—
369C22#417×C8911yescharacteristic
370C22#447×C8911yescharacteristic
371C22#481×C8911yescharacteristic
372C4—22no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4#386×C8911yescharacteristic
378C4C2×#102×C8911yescharacteristic
379C4#448×C8911yescharacteristic
380C4#480×C8911yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—161no—
386C2C89×#187411yescharacteristic
subgroupG/H
center, socleC2×C89#1874
commutatorC2×C8C22×C4×C89
Frattini, cosocleC22×C8C23×C89
prime radicalC89×#3067C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C89 ◁ C22×C4×C89 ◁ C2×#131×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2912C2semidirect
3. C89×#2145C2semidirect
4. C2×#360×C89C2semidirect
5. C89×#2117C2semidirect
6. C89×#1389C2semidirect
7. C89×#1383C2semidirect
14. #473×C89C4semidirect
15. #472×C89C4semidirect
40. C2×Q16×C89C2×C4semidirect
42. C2×D8×C89C2×C4semidirect
66. #108×C89C2×C4semidirect
187. #13005C89direct (N × H)
195. #2912C2×C89semidirect
196. #2145C2×C89semidirect
197. C2×#360C2×C89semidirect
198. #2117C2×C89semidirect
199. #1389C2×C89semidirect
200. #1383C2×C89semidirect
201. C89#13005direct (N × H)
208. #473C4×C89semidirect
209. #472C4×C89semidirect
234. C2×Q16C2×C4×C89semidirect
236. C2×D8C2×C4×C89semidirect
260. #108C2×C4×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#30672
C89×#29122
C89×#21452
C2×#360×C892
C89×#21172
C89×#13892
C89×#13832
#1300589