Finite Group Databasebeta

← #36707665

Subgroups — #36707665

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19152016C211yesmaximal, characteristic, complemented
2C2×#19152015C211yesmaximal, characteristic, complemented
3#26029153C211yesmaximal, characteristic
4#26022091C214yesmaximal, complemented
5#26022086C214yesmaximal, complemented
6#26022017C211yesmaximal, characteristic, complemented
7#26021879C211yesmaximal, characteristic, complemented
8#26021851C211yesmaximal, characteristic, complemented
9C89×#2117C211yesmaximal, characteristic, complemented
10#19152396C411yescharacteristic
11C2×#15668018C2211yescharacteristic
12#19152016—21no—
13#19152016—21no—
14#19152016C2212yescomplemented
15#19152016C2212yescomplemented
16#19152015—21no—
17#19152015—21no—
18#19152015C2212yescomplemented
19#19152015C2212yescomplemented
20C2×#15667952—21no—
21C2×#15667952C2211yescharacteristic
22C2×#15667951—21no—
23C22×#13083549—21no—
24C22×#13083549C2211yescharacteristic
25C2×C16×D89—21no—
26#19151718C414yes—
27#19151713C2214yes—
28#19151646C411yescharacteristic
29#19151615C411yescharacteristic
30#19151614C411yescharacteristic
31#19151553C2211yescharacteristic
32#19151418C2214yescomplemented
33#19151381C2214yescomplemented
34#19151373C2212yescomplemented
35C16×Dic89C2212yescomplemented
36C2×#131×C89C2211yescharacteristic
37C2×M32×C89C2211yescharacteristic
38C2×M32×C89C2211yescharacteristic
39#350×C89C2214yescomplemented
40C2×#13083574C2×C411yescharacteristic
41#15668043C812yes—
42#15668042C812yes—
43#15668038C812yes—
44#15668037C812yes—
45#15668032C2×C414yes—
46#15668018—21no—
47#15668018—21no—
48#15668018C2×C412yes—
49#15668018C2312yes—
50C2×#13083558C2×C411yescharacteristic
51C2×#13083558C2×C411yescharacteristic
52C2×#13083557—21no—
53C2×C8×D89—21no—
54#15667971C2×C411yescharacteristic
55#15667952—22no—
56#15667952—22no—
57#15667952—21no—
58#15667952C2312yes—
59#15667951—22no—
60#15667951—22no—
61C2×#13083549—22no—
62C2×#13083549—22no—
63C2×#13083549—21no—
64C2×#13083549D411yescharacteristic
65C2×#13083549D411yescharacteristic
66C2×#13083549D411yescharacteristic
67C2×#13083549D411yescharacteristic
68C2×#13083549C2311yescharacteristic
69C2×#13083549C2311yescharacteristic
70C16×D89—22no—
71C16×D89—22no—
72#15667947C2×C414yes—
73#15667939C2×C414yes—
74#15667938C2×C414yes—
75#15667922C2×C412yes—
76C8×Dic89C2312yes—
77C2×#41×C89C2×C411yescharacteristic
78#131×C89—21no—
79#131×C89—21no—
80#131×C89C2×C412yes—
81#131×C89C2312yes—
82C2×M16×C89—21no—
83C2×M16×C89C2×C411yescharacteristic
84C22×C8×C89—21no—
85C22×C8×C89C2311yescharacteristic
86M32×C89—21no—
87M32×C89—21no—
88M32×C89—21no—
89M32×C89C2312yes—
90C2×C16×C89—21no—
91C2×C16×C89C2312yes—
92#13083574—21no—
93#13083574—21no—
94#13083574C2×C812yes—
95#13083574C22×C412yes—
96C2×Dic178C2×C811yescharacteristic
97C2×D356C2×C811yescharacteristic
98C2×C22⋊D89—21no—
99C2×C4×D89—21no—
100#13083564C2×C814yes—
101#13083562C2×C814yes—
102#13083561C2×C814yes—
103#13083559C2×C814yes—
104#13083558—21no—
105#13083558C2×C812yes—
106#13083558C2×C812yes—
107#13083558C22×C412yes—
108#13083557—22no—
109#13083557—22no—
110C2×#10623494—21no—
111C2×#10623494C22×C412yes—
112C8×D89—22no—
113C8×D89—22no—
114C4⋊Dic89—21no—
115C22⋊Dic89—21no—
116C4×Dic89C22×C412yes—
117#13083549—22no—
118#13083549C2×D411yescharacteristic
119#13083549C2×D411yescharacteristic
120#13083549C2×D411yescharacteristic
121#13083549C2×D411yescharacteristic
122#41×C89—21no—
123#41×C89—21no—
124#41×C89C2×C812yes—
125#41×C89C22×C412yes—
126C2×Q8×C89C2×C811yescharacteristic
127C2×D4×C89—21no—
128C2×D4×C89C2×C811yescharacteristic
129C22×C4×C89—21no—
130C22×C4×C89C22×C411yescharacteristic
131M16×C89—21no—
132M16×C89—22no—
133M16×C89—22no—
134M16×C89C22×C412yes—
135C2×C8×C89—21no—
136C2×C8×C89—22no—
137C2×C8×C89—22no—
138C2×C8×C89C22×C411yescharacteristic
139C2×C8×C89C2×D411yescharacteristic
140C2×C8×C89C2×D411yescharacteristic
141C2×C8×C89#4111yescharacteristic
142C2×C8×C89#4111yescharacteristic
143C2×C8×C89C2411yescharacteristic
144C16×C89—22no—
145C16×C89—22no—
146C23×C89—21no—
147C23×C89#13111yescharacteristic
148C22×D89—21no—
149Q8×C89—21no—
150Q8×C89C22×C812yes—
151D4×C89—22no—
152D4×C89—22no—
153D4×C89—21no—
154D4×C89C22×C812yes—
155C2×C4×C89—21no—
156C2×C4×C89—22no—
157C2×C4×C89—22no—
158C2×C4×C89C4×D411yescharacteristic
159C2×C4×C89C4×D411yescharacteristic
160C2×C4×C89C22×C811yescharacteristic
161C2×C4×C89C22×C811yescharacteristic
162C2×C4×C89#13111yescharacteristic
163C2×C4×C89C23×C411yescharacteristic
164C22⋊D89—22no—
165C22⋊D89—22no—
166C2×Dic89—21no—
167C2×Dic89—22no—
168D356—21no—
169D356C22×C812yes—
170C4×D89—22no—
171C4×D89—22no—
172Dic178—21no—
173Dic178C22×C812yes—
174C8×C89—22no—
175C8×C89C4×D411yescharacteristic
176C8×C89C4×D411yescharacteristic
177C8×C89C22×D411yescharacteristic
178C8×C89C2×#4111yescharacteristic
179#10623494—22no—
180#10623494C22×C814yes—
181C22×C89—22no—
182C22×C89—41no—
183C22×C89—42no—
184C22×C89C8×D411yescharacteristic
185C22×C89C8×D411yescharacteristic
186C22×C89C2×#13111yescharacteristic
187C2×D89—22no—
188C2×D89—42no—
189C4×C89—22no—
190C4×C89C8×D411yescharacteristic
191C4×C89C8×D411yescharacteristic
192C4×C89C2×C4×D411yescharacteristic
193C4×C89C23×C811yescharacteristic
194Dic89—22no—
195Dic89—42no—
196#33441—891nomaximal, Sylow
197C2×C89—21no—
198C2×C89—21no—
199C2×C89—21no—
200C2×C89—42no—
201C2×C89C2×C8×D411yescharacteristic
202D89—42no—
203C2×#504—891no—
204C2×#504—891no—
205#2955—891no—
206#2171—894no—
207#2170—894no—
208#2153—891no—
209#2117—891no—
210#2117D8911yescharacteristic, complemented
211#2108—891no—
212C89#3344111yescharacteristic, complemented, Sylow
213#575—891no—
214C2×#131—891no—
215C2×#131C2×D8911yescharacteristic
216#504—892no—
217#504—892no—
218#504—892no—
219#504—892no—
220#504—1781no—
221#504—1781no—
222#504—1781no—
223#504—1781no—
224C2×M32—891no—
225C2×M32—1781no—
226C2×M32—1781no—
227C2×M32C2×D8911yescharacteristic
228C2×M32C2×D8911yescharacteristic
229C22×C16—891no—
230C22×C16—1781no—
231C22×C16—1781no—
232#444—894no—
233#443—894no—
234#430—891no—
235#421—891no—
236#421—891no—
237#405—891no—
238#364—894no—
239#350—894no—
240#350C2×D8914yescomplemented
241#346—892no—
242C4×C16—892no—
243C2×#41—891no—
244C2×#41C4×D8911yescharacteristic
245#137—892no—
246#137—892no—
247AGL(1,ℤ/8)—892no—
248AGL(1,ℤ/8)—892no—
249#135—894no—
250#131—21no—
251#131—21no—
252#131—892no—
253#131—892no—
254#131—1781no—
255#131—1781no—
256#131C4×D8912yes—
257#131C22×D8912yes—
258C2×M16—21no—
259C2×M16—891no—
260C2×M16—891no—
261C2×M16—1781no—
262C2×M16C4×D8911yescharacteristic
263C22×C8—21no—
264C22×C8—1781no—
265C22×C8C22×D8911yescharacteristic
266#117—891no—
267M32—21no—
268M32—21no—
269M32—21no—
270M32—892no—
271M32—1782no—
272M32—1782no—
273M32—1782no—
274M32—1782no—
275M32—1781no—
276M32C22×D8912yes—
277C2×C16—21no—
278C2×C16—891no—
279C2×C16—891no—
280C2×C16—891no—
281C2×C16—891no—
282C2×C16—891no—
283C2×C16—891no—
284C2×C16—1782no—
285C2×C16—1782no—
286C2×C16—1782no—
287C2×C16—1782no—
288C2×C16—1781no—
289C2×C16C22×D8912yes—
290#108—894no—
291#104—894no—
292#104—894no—
293#97—892no—
294C4×C8—892no—
295#41—21no—
296#41—21no—
297#41—892no—
298#41—892no—
299#41—1781no—
300#41—1781no—
301#41C8×D8912yes—
302#41C2×C4×D8912yes—
303C2×Q8—891no—
304C2×Q8C8×D8911yescharacteristic
305C2×D4—21no—
306C2×D4—891no—
307C2×D4—1781no—
308C2×D4C8×D8911yescharacteristic
309C22×C4—21no—
310C22×C4—1781no—
311C22×C4C2×C4×D8911yescharacteristic
312Q16—894no—
313SD16—894no—
314SD16—894no—
315D8—894no—
316M16—21no—
317M16—22no—
318M16—22no—
319M16—892no—
320M16—892no—
321M16—892no—
322M16—1781no—
323M16—1782no—
324M16—1782no—
325M16C2×C4×D8912yes—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—892no—
330C2×C8—1781no—
331C2×C8—1782no—
332C2×C8—1782no—
333C2×C8C2×C4×D8911yescharacteristic
334C2×C8D4×D8911yescharacteristic
335C2×C8D4×D8911yescharacteristic
336C2×C8#1308357311yescharacteristic
337C2×C8#1308357311yescharacteristic
338C2×C8C23×D8911yescharacteristic
339C4⋊C4—1781no—
340C22⋊C4—1781no—
341C42—892no—
342C16—22no—
343C16—22no—
344C16—891no—
345C16—891no—
346C16—891no—
347C16—891no—
348C16—1782no—
349C23—21no—
350C23—1781no—
351C23#1566802011yescharacteristic
352Q8—21no—
353Q8—892no—
354Q8—1781no—
355Q8C2×C8×D8912yes—
356D4—22no—
357D4—22no—
358D4—21no—
359D4—892no—
360D4—1782no—
361D4—1782no—
362D4—1781no—
363D4C2×C8×D8912yes—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—1782no—
368C2×C4—1781no—
369C2×C4—1782no—
370C2×C4—1782no—
371C2×C4#1566797211yescharacteristic
372C2×C4#1566797211yescharacteristic
373C2×C4C2×C8×D8911yescharacteristic
374C2×C4C2×C8×D8911yescharacteristic
375C2×C4#1566802011yescharacteristic
376C2×C4C22×C4×D8911yescharacteristic
377C8—22no—
378C8—894no—
379C8—1782no—
380C8#1566797211yescharacteristic
381C8#1566797211yescharacteristic
382C8C2×D4×D8911yescharacteristic
383C8C2×#1308357311yescharacteristic
384C22—22no—
385C22—41no—
386C22—42no—
387C22—3562no—
388C22—1782no—
389C22#1915166111yescharacteristic
390C22#1915166111yescharacteristic
391C22C2×#1566802011yescharacteristic
392C4—22no—
393C4—3562no—
394C4—1782no—
395C4#1915166111yescharacteristic
396C4#1915166111yescharacteristic
397C4C2×#1566797211yescharacteristic
398C4C22×C8×D8911yescharacteristic
399C2—21no—
400C2—21no—
401C2—21no—
402C2—42no—
403C2—3562no—
404C2C2×#1915166111yescharacteristic
subgroupG/H
centerC8C2×D4×D89
commutatorC4×C89C23×C8
FrattiniC2×C8C23×D89
FittingC89×#2117C2
socleC2×C89C2×C8×D4
prime radical#19152396C4
cosocleC2×C8×C89C24
hypercenter#2117D89
hypocenterC89#33441
LCSG ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
1. C2×#19152016C2semidirect
2. C2×#19152015C2semidirect
4. #26022091C2semidirect
5. #26022086C2semidirect
6. #26022017C2semidirect
7. #26021879C2semidirect
8. #26021851C2semidirect
9. C89×#2117C2semidirect
14. #19152016C22semidirect
15. #19152016C22semidirect
18. #19152015C22semidirect
19. #19152015C22semidirect
32. #19151418C22semidirect
33. #19151381C22semidirect
34. #19151373C22semidirect
35. C16×Dic89C22semidirect
39. #350×C89C22semidirect
210. #2117D89semidirect
212. C89#33441semidirect
240. #350C2×D89semidirect

Maximal subgroups

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

subgroupindex
C2×#191520162
C2×#191520152
#260291532
#260220912
#260220862
#260220172
#260218792
#260218512
C89×#21172
#3344189