Finite Group Databasebeta

← C89×#13021 #36390869

Subgroups — C89×#13021 #36390869

Order 22784.

462 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3414C211yesmaximal, characteristic
2C89×#2957C211yesmaximal, characteristic, complemented
3C2×#359×C89C211yesmaximal, characteristic, complemented
4C89×#2138C211yesmaximal, characteristic, complemented
5C89×#1976C211yesmaximal, characteristic, complemented
6C89×#1389C211yesmaximal, characteristic, complemented
7C89×#1379C211yesmaximal, characteristic, complemented
8C2×#135×C89C2211yescharacteristic
9C2×#135×C89C2211yescharacteristic
10#568×C89—21no—
11#510×C89—21no—
12#510×C89C412yescomplemented
13#509×C89—21no—
14#509×C89C412yescomplemented
15#508×C89—22no—
16C2×M32×C89C2211yescharacteristic
17C22×C16×C89C2211yescharacteristic
18#444×C89—22no—
19#443×C89—22no—
20C2×#108×C89C2211yescharacteristic
21#427×C89C2211yescharacteristic
22#421×C89—21no—
23#421×C89—21no—
24#410×C89—21no—
25#405×C89C2211yescharacteristic
26#359×C89—22no—
27#358×C89—21no—
28#357×C89—21no—
29C2×#41×C89—21no—
30C2×#41×C89—21no—
31#135×C89—21no—
32#135×C89—21no—
33#135×C89—21no—
34#135×C89—21no—
35#135×C89—21no—
36#135×C89—21no—
37#135×C89C2×C412yes—
38#135×C89D412yes—
39C2×Q16×C89C2×C411yescharacteristic
40C2×Q16×C89D411yescharacteristic
41C2×SD16×C89—21no—
42C2×SD16×C89—21no—
43C2×D8×C89C2×C411yescharacteristic
44C2×D8×C89D411yescharacteristic
45#131×C89—24no—
46#131×C89—42no—
47C2×M16×C89—21no—
48C2×M16×C89—42no—
49C2×M16×C89D411yescharacteristic
50C22×C8×C89C2311yescharacteristic
51#117×C89—21no—
52#117×C89D411yescharacteristic
53Q32×C89—22no—
54SD32×C89—22no—
55SD32×C89—22no—
56D16×C89—22no—
57M32×C89—21no—
58M32×C89C2×C412yes—
59C2×C16×C89—21no—
60C2×C16×C89—21no—
61C2×C16×C89—21no—
62C2×C16×C89—21no—
63#108×C89—21no—
64#108×C89D412yes—
65#107×C89—21no—
66#106×C89—21no—
67#104×C89—24no—
68#104×C89—42no—
69#99×C89—42no—
70#97×C89—21no—
71C4×C8×C89—21no—
72#41×C89—22no—
73#41×C89—22no—
74#41×C89—41no—
75#41×C89—41no—
76#41×C89—41no—
77#41×C89—41no—
78#41×C89—41no—
79#41×C89—41no—
80C2×Q8×C89—21no—
81C2×Q8×C89—21no—
82C2×D4×C89—21no—
83C2×D4×C89—21no—
84C2×D4×C89—41no—
85C2×D4×C89—41no—
86C22×C4×C89—41no—
87C22×C4×C89—41no—
88C22×C4×C89C2×D411yescharacteristic
89Q16×C89—21no—
90Q16×C89—21no—
91Q16×C89—21no—
92Q16×C89C22⋊C412yes—
93SD16×C89—22no—
94SD16×C89—22no—
95SD16×C89—41no—
96SD16×C89—41no—
97D8×C89—21no—
98D8×C89—21no—
99D8×C89—21no—
100D8×C89C22⋊C412yes—
101M16×C89—21no—
102M16×C89—22no—
103M16×C89—44no—
104M16×C89—41no—
105M16×C89D812yes—
106C2×C8×C89—21no—
107C2×C8×C89—44no—
108C2×C8×C89—41no—
109C2×C8×C89C22×C411yescharacteristic
110C2×C8×C89C2×D411yescharacteristic
111C2×C8×C89C2×D411yescharacteristic
112C2×C8×C89C2×D411yescharacteristic
113C2×C8×C89#4111yescharacteristic
114C2×C8×C89#4111yescharacteristic
115C4⋊C4×C89—21no—
116C4⋊C4×C89—41no—
117C22⋊C4×C89—21no—
118C22⋊C4×C89—41no—
119C42×C89—21no—
120C42×C89—41no—
121C16×C89—22no—
122C16×C89—41no—
123C23×C89—41no—
124C23×C89—41no—
125C23×C89AGL(1,ℤ/8)11yescharacteristic
126Q8×C89—22no—
127Q8×C89—41no—
128Q8×C89—41no—
129Q8×C89—41no—
130D4×C89—22no—
131D4×C89—42no—
132D4×C89—42no—
133D4×C89—41no—
134D4×C89—41no—
135D4×C89—41no—
136D4×C89—81no—
137D4×C89—81no—
138C2×C4×C89—41no—
139C2×C4×C89—42no—
140C2×C4×C89—42no—
141C2×C4×C89—41no—
142C2×C4×C89—41no—
143C2×C4×C89—81no—
144C2×C4×C89—81no—
145C2×C4×C89—81no—
146C2×C4×C89C4×D411yescharacteristic
147C2×C4×C89#12011yescharacteristic
148C2×C4×C89#12111yescharacteristic
149C2×C4×C89C2×D811yescharacteristic
150C2×C4×C89#13511yescharacteristic
151C2×C4×C89AGL(1,ℤ/8)11yescharacteristic
152C8×C89—22no—
153C8×C89—42no—
154C8×C89C2×C22⋊C411yescharacteristic
155C8×C89C4×D411yescharacteristic
156C8×C89#12111yescharacteristic
157C8×C89#12311yescharacteristic
158C22×C89—41no—
159C22×C89—42no—
160C22×C89—81no—
161C22×C89—161no—
162C22×C89—161no—
163C22×C89#44211yescharacteristic
164C22×C89#44911yescharacteristic
165C22×C89#45911yescharacteristic
166C4×C89—42no—
167C4×C89—81no—
168C4×C89—81no—
169C4×C89—161no—
170C4×C89#38611yescharacteristic
171C4×C89C4×D811yescharacteristic
172C4×C89#44711yescharacteristic
173C4×C89#46311yescharacteristic
174#13021C8911yesmaximal, characteristic, direct product factor, Sylow
175C2×C89—21no—
176C2×C89—21no—
177C2×C89—21no—
178C2×C89—82no—
179C2×C89—161no—
180C2×C89#187911yescharacteristic
181#3414C2×C8911yescharacteristic
182#2957C2×C8911yescharacteristic, complemented
183C2×#359C2×C8911yescharacteristic, complemented
184#2138C2×C8911yescharacteristic, complemented
185#1976C2×C8911yescharacteristic, complemented
186#1389C2×C8911yescharacteristic, complemented
187#1379C2×C8911yescharacteristic, complemented
188C89#1302111yescharacteristic, direct product factor, Sylow
189C2×#135C22×C8911yescharacteristic
190C2×#135C22×C8911yescharacteristic
191#568—21no—
192#510—21no—
193#510C4×C8912yescomplemented
194#509—21no—
195#509C4×C8912yescomplemented
196#508—22no—
197C2×M32C22×C8911yescharacteristic
198C22×C16C22×C8911yescharacteristic
199#444—22no—
200#443—22no—
201C2×#108C22×C8911yescharacteristic
202#427C22×C8911yescharacteristic
203#421—21no—
204#421—21no—
205#410—21no—
206#405C22×C8911yescharacteristic
207#359—22no—
208#358—21no—
209#357—21no—
210C2×#41—21no—
211C2×#41—21no—
212#135—21no—
213#135—21no—
214#135—21no—
215#135—21no—
216#135—21no—
217#135—21no—
218#135C2×C4×C8912yes—
219#135D4×C8912yes—
220C2×Q16C2×C4×C8911yescharacteristic
221C2×Q16D4×C8911yescharacteristic
222C2×SD16—21no—
223C2×SD16—21no—
224C2×D8C2×C4×C8911yescharacteristic
225C2×D8D4×C8911yescharacteristic
226#131—24no—
227#131—42no—
228C2×M16—21no—
229C2×M16—42no—
230C2×M16D4×C8911yescharacteristic
231C22×C8C23×C8911yescharacteristic
232#117—21no—
233#117D4×C8911yescharacteristic
234Q32—22no—
235SD32—22no—
236SD32—22no—
237D16—22no—
238M32—21no—
239M32C2×C4×C8912yes—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244#108—21no—
245#108D4×C8912yes—
246#107—21no—
247#106—21no—
248#104—24no—
249#104—42no—
250#99—42no—
251#97—21no—
252C4×C8—21no—
253#41—22no—
254#41—22no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—41no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4C2×D4×C8911yescharacteristic
270Q16—21no—
271Q16—21no—
272Q16—21no—
273Q16C22⋊C4×C8912yes—
274SD16—22no—
275SD16—22no—
276SD16—41no—
277SD16—41no—
278D8—21no—
279D8—21no—
280D8—21no—
281D8C22⋊C4×C8912yes—
282M16—21no—
283M16—22no—
284M16—44no—
285M16—41no—
286M16D8×C8912yes—
287C2×C8—21no—
288C2×C8—44no—
289C2×C8—41no—
290C2×C8C22×C4×C8911yescharacteristic
291C2×C8C2×D4×C8911yescharacteristic
292C2×C8C2×D4×C8911yescharacteristic
293C2×C8C2×D4×C8911yescharacteristic
294C2×C8#41×C8911yescharacteristic
295C2×C8#41×C8911yescharacteristic
296C4⋊C4—21no—
297C4⋊C4—41no—
298C22⋊C4—21no—
299C22⋊C4—41no—
300C42—21no—
301C42—41no—
302C16—22no—
303C16—41no—
304C23—41no—
305C23—41no—
306C23AGL(1,ℤ/8)×C8911yescharacteristic
307Q8—22no—
308Q8—41no—
309Q8—41no—
310Q8—41no—
311D4—22no—
312D4—42no—
313D4—42no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4C4×D4×C8911yescharacteristic
328C2×C4#120×C8911yescharacteristic
329C2×C4#121×C8911yescharacteristic
330C2×C4C2×D8×C8911yescharacteristic
331C2×C4#135×C8911yescharacteristic
332C2×C4AGL(1,ℤ/8)×C8911yescharacteristic
333C8—22no—
334C8—42no—
335C8C2×C22⋊C4×C8911yescharacteristic
336C8C4×D4×C8911yescharacteristic
337C8#121×C8911yescharacteristic
338C8#123×C8911yescharacteristic
339C22—41no—
340C22—42no—
341C22—81no—
342C22—161no—
343C22—161no—
344C22#442×C8911yescharacteristic
345C22#449×C8911yescharacteristic
346C22#459×C8911yescharacteristic
347C4—42no—
348C4—81no—
349C4—81no—
350C4—161no—
351C4#386×C8911yescharacteristic
352C4C4×D8×C8911yescharacteristic
353C4#447×C8911yescharacteristic
354C4#463×C8911yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—82no—
359C2—161no—
360C2C89×#187911yescharacteristic
subgroupG/H
centerC4×C89#447
commutatorC2×C8C22×C4×C89
Frattini, cosocleC22×C8C23×C89
socleC2×C89#1879
prime radicalC89×#3414C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C89 ◁ C22×C4×C89 ◁ #405×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#2957C2semidirect
3. C2×#359×C89C2semidirect
4. C89×#2138C2semidirect
5. C89×#1976C2semidirect
6. C89×#1389C2semidirect
7. C89×#1379C2semidirect
12. #510×C89C4semidirect
14. #509×C89C4semidirect
174. #13021C89direct (N × H)
182. #2957C2×C89semidirect
183. C2×#359C2×C89semidirect
184. #2138C2×C89semidirect
185. #1976C2×C89semidirect
186. #1389C2×C89semidirect
187. #1379C2×C89semidirect
188. C89#13021direct (N × H)
193. #510C4×C89semidirect
195. #509C4×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#34142
C89×#29572
C2×#359×C892
C89×#21382
C89×#19762
C89×#13892
C89×#13792
#1302189