Finite Group Databasebeta

← #34258482

Subgroups — #34258482

Order 20224.

666 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773404C211yesmaximal, characteristic
2#23772335C211yesmaximal, characteristic, complemented
3C2×#19098640C211yesmaximal, characteristic
4#23762287C214yesmaximal, complemented
5#23762240C214yesmaximal, complemented
6#23762170C211yesmaximal, characteristic, complemented
7#23761826C211yesmaximal, characteristic
8#23761684C211yesmaximal, characteristic, complemented
9C79×#1481C211yesmaximal, characteristic, complemented
10C22×#11929607—21no—
11C2×C4×D316C2211yescharacteristic
12#19098891C412yes—
13#19098850C412yes—
14#19098765C412yes—
15#19098713C412yes—
16#19098694—21no—
17#19098694C2212yescomplemented
18Q8×C79⋊C8—21no—
19Q8×C79⋊C8C2212yescomplemented
20C4×#11929611C2214yes—
21#19098640—21no—
22#19098640—21no—
23#19098640C2212yes—
24#19098640C2212yes—
25#19098616C2211yescharacteristic
26#19098590—21no—
27#19098587C2211yescharacteristic
28C2×(C2×D79)⋊C8—21no—
29#19098519—21no—
30C2×C4×C79⋊C8C2211yescharacteristic
31#19098313C2214yescomplemented
32#19098292C2214yes—
33#19098285C2212yescomplemented
34C8×C79⋊C8C2212yescomplemented
35#533×C79C2211yescharacteristic
36#431×C79C2211yescharacteristic
37C2×C4⋊C8×C79C2211yescharacteristic
38Q8⋊C8×C79C2214yescomplemented
39C22×D316C2×C411yescharacteristic
40C22×C4×D79—21no—
41C2×#11929611C2×C414yes—
42C2×#11929608—21no—
43C2×#11929607—22no—
44C2×#11929607—22no—
45C2×#11929607—44no—
46C22×C79⋊C8D411yescharacteristic
47C22×C79⋊C8D411yescharacteristic
48C4×D316—21no—
49C4×D316—21no—
50C4×D316C2×C412yes—
51C4×D316C2312yes—
52C2×C4⋊Dic79C2×C411yescharacteristic
53C2×(C2×D79)⋊C4—21no—
54#14579092C2×C414yes—
55C4⋊(C79⋊C8)—21no—
56C4⋊(C79⋊C8)—22no—
57C4⋊(C79⋊C8)—22no—
58C4⋊(C79⋊C8)C2×C412yes—
59C4⋊(C79⋊C8)C2312yes—
60#14579085C2×C414yes—
61#14579081C2×C414yes—
62C22⋊(C79⋊C8)C2×C412yes—
63(C2×D79)⋊C8—22no—
64(C2×D79)⋊C8—22no—
65#14579072—22no—
66C4×C79⋊C8—22no—
67C4×C79⋊C8D411yescharacteristic
68C4×C79⋊C8D411yescharacteristic
69C4×C79⋊C8C2312yes—
70C4×C79⋊C8C2311yescharacteristic
71C4×C79⋊C8C2311yescharacteristic
72C22×C8×C79—21no—
73C4⋊Q8×C79C2×C411yescharacteristic
74C4⋊Q8×C79C2×C411yescharacteristic
75#125×C79—21no—
76C22⋊Q8×C79—21no—
77C22⋊Q8×C79C2×C412yes—
78C4×Q8×C79—21no—
79C4×Q8×C79C2312yes—
80#117×C79—21no—
81C2×C42×C79C2311yescharacteristic
82C4⋊C8×C79—21no—
83C4⋊C8×C79—21no—
84C4⋊C8×C79C2312yes—
85C22⋊C8×C79—21no—
86C4×C8×C79C2312yes—
87C23×D79—21no—
88C2×D316—21no—
89C2×D316—42no—
90C2×D316C22×C412yes—
91C22×Dic79—21no—
92C2×C4×D79—22no—
93C2×C4×D79—22no—
94C2×C4×D79—42no—
95C2×C4×D79—42no—
96#11929611—44no—
97#11929608—42no—
98#11929607—44no—
99#11929607—44no—
100C2×C79⋊C8—22no—
101C2×C79⋊C8—22no—
102C2×C79⋊C8SD1614yescomplemented
103C2×C79⋊C8C22×C414yes—
104C2×C79⋊C8C2×D411yescharacteristic
105C2×C79⋊C8C2×D411yescharacteristic
106C2×C79⋊C8C2×D411yescharacteristic
107C2×C79⋊C8C2×D411yescharacteristic
108C4⋊Dic79—21no—
109C4⋊Dic79C22×C412yes—
110(C2×D79)⋊C4—22no—
111(C2×D79)⋊C4—22no—
112C2×Q8×C79—21no—
113C2×Q8×C79C22×C412yes—
114C22×C4×C79C22×C411yescharacteristic
115C22×C4×C79C2×D411yescharacteristic
116C22×C4×C79#4111yescharacteristic
117C2×C8×C79—22no—
118C2×C8×C79—22no—
119C2×C8×C79—42no—
120C4⋊C4×C79—21no—
121C4⋊C4×C79—21no—
122C4⋊C4×C79—21no—
123C4⋊C4×C79—22no—
124C4⋊C4×C79—22no—
125C4⋊C4×C79C22×C412yes—
126C22⋊C4×C79—22no—
127C42×C79—22no—
128C42×C79C22×C411yescharacteristic
129C42×C79C2×D411yescharacteristic
130C42×C79#4111yescharacteristic
131C42×C79C2411yescharacteristic
132C23×C79C4×D411yescharacteristic
133C22×D79—22no—
134C22×D79—42no—
135C22×D79—42no—
136C22×D79—42no—
137Q8×C79—24no—
138Q8×C79—42no—
139C2×C4×C79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—22no—
142C2×C4×C79—41no—
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79C4×D411yescharacteristic
145C2×C4×C79C4×D411yescharacteristic
146C2×C4×C79C2×SD1612yes—
147C2×C4×C79#13512yes—
148C2×C4×C79C23×C411yescharacteristic
149C2×C4×C79C22×D411yescharacteristic
150C2×C4×C79C2×#4111yescharacteristic
151C2×Dic79—22no—
152C2×Dic79—42no—
153D316—22no—
154D316—22no—
155D316—42no—
156C4×D79—42no—
157C4×D79—42no—
158C4×D79—44no—
159C8×C79—42no—
160C8×C79—42no—
161C79⋊C8—24no—
162C79⋊C8—21no—
163C79⋊C8—21no—
164C79⋊C8—42no—
165C79⋊C8C2×SD1612yes—
166C79⋊C8C2×SD1612yes—
167C22×C79—21no—
168C22×C79—21no—
169C22×C79C4×SD1612yes—
170C22×C79C2×C4×D411yescharacteristic
171C2×D79—42no—
172C2×D79—42no—
173C2×D79—42no—
174C2×D79—84no—
175C4×C79—21no—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—42no—
179C4×C79—42no—
180C4×C79C4×SD1612yes—
181C4×C79#56812yes—
182C4×C79C22×SD1611yescharacteristic
183C4×C79C2×#13511yescharacteristic
184Dic79—42no—
185#17930—791nomaximal, Sylow
186C2×C79—22no—
187C2×C79#293011yescharacteristic
188C2×C79C2×C4×SD1611yescharacteristic
189C2×C79#295511yescharacteristic
190D79—82no—
191#3101—791no—
192#2960—791no—
193C2×#436—791no—
194#1590—794no—
195#1583—794no—
196#1570—791no—
197#1486—791no—
198#1481D7911yescharacteristic, complemented
199#1450—791no—
200C79#1793011yescharacteristic, complemented, Sylow
201C22×M16—1581no—
202#533C2×D7911yescharacteristic
203C2×C4×D4—791no—
204#483—792no—
205#476—792no—
206#461—792no—
207#450—792no—
208#446—792no—
209#446—1581no—
210C8×Q8—792no—
211C8×Q8—1581no—
212C4×SD16—794no—
213#436—792no—
214#436—792no—
215#436—1581no—
216#436—1581no—
217#431—791no—
218#431C2×D7911yescharacteristic
219#424—1581no—
220#423—791no—
221C2×C4⋊C8C2×D7911yescharacteristic
222C2×C22⋊C8—1581no—
223#405—1581no—
224C2×C4×C8—791no—
225#334—794no—
226Q8⋊C8C2×D7914yescomplemented
227D4⋊C8—794no—
228#322—792no—
229C82—792no—
230C22×D4—791no—
231C23×C4—1581no—
232C2×SD16—794no—
233C2×M16—3164no—
234C2×M16—1582no—
235C2×M16—1582no—
236C2×M16—1581no—
237C22×C8—21no—
238C22×C8—791no—
239C22×C8—791no—
240C4⋊Q8C4×D7911yescharacteristic
241C4⋊Q8C4×D7911yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8C4×D7912yes—
245C4×Q8—21no—
246C4×Q8C22×D7912yes—
247C4×D4—792no—
248C4×D4—792no—
249C4×D4—1581no—
250C4×D4—1581no—
251#117—21no—
252C2×C4⋊C4—791no—
253C2×C22⋊C4—1581no—
254C2×C42C22×D7911yescharacteristic
255#106—794no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—792no—
259C4⋊C8—792no—
260C4⋊C8—1582no—
261C4⋊C8—1582no—
262C4⋊C8—1581no—
263C4⋊C8C22×D7912yes—
264#103—794no—
265#102—794no—
266C22⋊C8—21no—
267C22⋊C8—792no—
268C22⋊C8—1582no—
269C22⋊C8—1582no—
270#97—1582no—
271C4×C8—791no—
272C4×C8—791no—
273C4×C8—791no—
274C4×C8—792no—
275C4×C8—791no—
276C4×C8—1582no—
277C4×C8C22×D7912yes—
278C24—1581no—
279C2×Q8—21no—
280C2×Q8C2×C4×D7912yes—
281C2×D4—3162no—
282C2×D4—792no—
283C2×D4—1581no—
284C22×C4—3162no—
285C22×C4—3162no—
286C22×C4—1581no—
287C22×C4—1582no—
288C22×C4—1582no—
289C22×C4C2×C4×D7911yescharacteristic
290C22×C4D4×D7911yescharacteristic
291C22×C4#1192962311yescharacteristic
292SD16—3164no—
293M16—3164no—
294M16—3164no—
295M16—3162no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—794no—
300C2×C8—791no—
301C2×C8—791no—
302C2×C8—794no—
303C2×C8—791no—
304C2×C8—791no—
305C2×C8—1582no—
306C2×C8—1582no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—792no—
313C4⋊C4—1581no—
314C4⋊C4C2×C4×D7912yes—
315C22⋊C4—22no—
316C22⋊C4—1582no—
317C22⋊C4—1582no—
318C42—22no—
319C42C2×C4×D7911yescharacteristic
320C42D4×D7911yescharacteristic
321C42#1192962311yescharacteristic
322C42C23×D7911yescharacteristic
323C23—3162no—
324C23—3162no—
325C23—3162no—
326C23—1582no—
327C23(C22⋊D79)⋊C411yescharacteristic
328Q8—24no—
329Q8—42no—
330D4—3162no—
331D4—1582no—
332D4—1582no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—3162no—
338C2×C4—3164no—
339C2×C4—3162no—
340C2×C4—3162no—
341C2×C4—1582no—
342C2×C4(C22⋊D79)⋊C411yescharacteristic
343C2×C4(C22⋊D79)⋊C411yescharacteristic
344C2×C4(C22⋊D79)⋊C411yescharacteristic
345C2×C4SD16×D7912yes—
346C2×C4#1457918412yes—
347C2×C4C22×C4×D7911yescharacteristic
348C2×C4C2×D4×D7911yescharacteristic
349C2×C4C2×#1192962311yescharacteristic
350C8—42no—
351C8—42no—
352C8—3162no—
353C8—792no—
354C8—792no—
355C8—1584no—
356C8—1581no—
357C8—1581no—
358C22—21no—
359C22—21no—
360C22—3162no—
361C22—3162no—
362C22—3162no—
363C22—6324no—
364C22#1909864812yes—
365C22C2×(C22⋊D79)⋊C411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—42no—
371C4—3162no—
372C4#1909864812yes—
373C4#1909932812yes—
374C4C2×SD16×D7911yescharacteristic
375C4C2×#1457918411yescharacteristic
376C2—22no—
377C2—6322no—
378C2#2377210611yescharacteristic
379C2C2×#1909864811yescharacteristic
380C2#2377229911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D79
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1481C2
socleC22×C79C2×C4×D4
prime radicalC22×D316C2×C4
cosocleC42×C79C24
hypercenter#1481D79
hypocenterC79#17930
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
2. #23772335C2semidirect
4. #23762287C2semidirect
5. #23762240C2semidirect
6. #23762170C2semidirect
8. #23761684C2semidirect
9. C79×#1481C2semidirect
17. #19098694C22semidirect
19. Q8×C79⋊C8C22semidirect
31. #19098313C22semidirect
33. #19098285C22semidirect
34. C8×C79⋊C8C22semidirect
38. Q8⋊C8×C79C22semidirect
102. C2×C79⋊C8SD16semidirect
198. #1481D79semidirect
200. C79#17930semidirect
226. Q8⋊C8C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237734042
#237723352
C2×#190986402
#237622872
#237622402
#237621702
#237618262
#237616842
C79×#14812
#1793079