Finite Group Databasebeta

← #34258481

Subgroups — #34258481

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#23773409C211yesmaximal, characteristic
2#23772334C211yesmaximal, characteristic, complemented
3C2×#19098642C211yesmaximal, characteristic
4#23762292C214yesmaximal, complemented
5#23762245C214yesmaximal, complemented
6#23762171C211yesmaximal, characteristic, complemented
7#23761826C211yesmaximal, characteristic
8#23761804C211yesmaximal, characteristic, complemented
9C79×#1450C211yesmaximal, characteristic, complemented
10C22×#11929607—21no—
11#19099120C2211yescharacteristic
12C2×C4×D316C2211yescharacteristic
13#19098894C412yes—
14#19098854C412yes—
15#19098769C412yes—
16#19098716C412yes—
17#19098693—21no—
18#19098693C2212yescomplemented
19C8×Dic158—21no—
20C8×Dic158C2212yescomplemented
21C4×#11929613C2214yes—
22#19098642—21no—
23#19098642—21no—
24#19098642C2212yes—
25#19098642C2212yes—
26#19098589—21no—
27#19098587C2211yescharacteristic
28C2×C4⋊(C79⋊C8)C2211yescharacteristic
29C2×(C2×D79)⋊C8—21no—
30#19098518—21no—
31#19098314C2214yescomplemented
32#19098295C2214yescomplemented
33#19098292C2214yes—
34#431×C79C2211yescharacteristic
35#431×C79C2211yescharacteristic
36C2×C4×C8×C79C2211yescharacteristic
37#322×C79C2212yescomplemented
38C82×C79C2212yescomplemented
39C22×D316C2×C411yescharacteristic
40C22×C4×D79—21no—
41C2×#11929613C2×C414yes—
42C2×#11929608—21no—
43C2×#11929607—22no—
44C2×#11929607—22no—
45C2×#11929607—44no—
46C22×C79⋊C8—21no—
47#14579161C2×C411yescharacteristic
48#14579161C2×C411yescharacteristic
49#14579153—21no—
50#14579140—21no—
51#14579140C2×C412yes—
52C4×Dic158—21no—
53C4×Dic158C2312yes—
54C4×D316—21no—
55C4×D316—21no—
56C4×D316C2×C412yes—
57C4×D316C2312yes—
58#14579120—21no—
59C2×C4⋊Dic79C2×C411yescharacteristic
60C2×(C2×D79)⋊C4—21no—
61#14579093C2×C414yes—
62C4⋊(C79⋊C8)—21no—
63C4⋊(C79⋊C8)—21no—
64C4⋊(C79⋊C8)C2312yes—
65C4⋊(C79⋊C8)C2312yes—
66Dic79⋊C8—22no—
67Dic79⋊C8—22no—
68#14579086C2×C414yes—
69#14579083C2×C414yes—
70(C2×D79)⋊C8—22no—
71(C2×D79)⋊C8—22no—
72#14579071—22no—
73C8×Dic79—22no—
74C22×C8×C79D411yescharacteristic
75C22×C8×C79D411yescharacteristic
76C2×C42×C79C2311yescharacteristic
77C4⋊C8×C79—21no—
78C4⋊C8×C79C2×C412yes—
79C22⋊C8×C79—21no—
80C22⋊C8×C79C2×C412yes—
81C4×C8×C79D411yescharacteristic
82C4×C8×C79D411yescharacteristic
83C4×C8×C79C2312yes—
84C4×C8×C79C2312yes—
85C4×C8×C79C2311yescharacteristic
86C4×C8×C79C2311yescharacteristic
87C23×D79—21no—
88C2×Dic158—21no—
89C2×Dic158C22×C412yes—
90C2×D316—21no—
91C2×D316—42no—
92C2×D316C22×C412yes—
93C22×Dic79—21no—
94C2×C4×D79—22no—
95C2×C4×D79—22no—
96C2×C4×D79—42no—
97C2×C4×D79—42no—
98#11929613—44no—
99#11929608—42no—
100#11929607—44no—
101#11929607—44no—
102C2×C79⋊C8—22no—
103C2×C79⋊C8—22no—
104C2×C79⋊C8—42no—
105C4⋊Dic79—21no—
106C4⋊Dic79—21no—
107C4⋊Dic79—21no—
108C4⋊Dic79—21no—
109C4⋊Dic79C22×C412yes—
110C4⋊Dic79C22×C412yes—
111Dic79⋊C4—22no—
112Dic79⋊C4—22no—
113C22⋊Dic79—22no—
114(C2×D79)⋊C4—22no—
115(C2×D79)⋊C4—22no—
116C4×Dic79—22no—
117C22×C4×C79C22×C411yescharacteristic
118C22×C4×C79C2×D411yescharacteristic
119C22×C4×C79#4111yescharacteristic
120C2×C8×C79—22no—
121C2×C8×C79—22no—
122C2×C8×C79SD1614yescomplemented
123C2×C8×C79C22×C414yes—
124C2×C8×C79C2×D411yescharacteristic
125C2×C8×C79C2×D411yescharacteristic
126C2×C8×C79C2×D411yescharacteristic
127C2×C8×C79C2×D411yescharacteristic
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—
137C2×C4×C79—22no—
138C2×C4×C79—41no—
139C2×C4×C79C4×D411yescharacteristic
140C2×C4×C79C4×D411yescharacteristic
141C2×C4×C79C4×D411yescharacteristic
142C2×C4×C79C2×SD1612yes—
143C2×C4×C79#13512yes—
144C2×C4×C79C23×C411yescharacteristic
145C2×C4×C79C22×D411yescharacteristic
146C2×C4×C79C2×#4111yescharacteristic
147C2×Dic79—22no—
148C2×Dic79—22no—
149C2×Dic79—22no—
150C2×Dic79—42no—
151D316—22no—
152D316—22no—
153D316—42no—
154C4×D79—42no—
155C4×D79—42no—
156C4×D79—44no—
157Dic158—24no—
158Dic158—42no—
159C8×C79—24no—
160C8×C79—21no—
161C8×C79—21no—
162C8×C79—42no—
163C8×C79C2×SD1612yes—
164C8×C79C2×SD1612yes—
165C79⋊C8—42no—
166C79⋊C8—42no—
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×C79C4×SD1612yes—
179C4×C79#56812yes—
180C4×C79C22×SD1611yescharacteristic
181C4×C79C2×#13511yescharacteristic
182Dic79—42no—
183Dic79—42no—
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#1481—791no—
199#1450D7911yescharacteristic, complemented
200C79#1793011yescharacteristic, complemented, Sylow
201C22×M16—1581no—
202#533—791no—
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#431C2×D7911yescharacteristic
218#431C2×D7911yescharacteristic
219#424—1581no—
220#423—791no—
221C2×C4⋊C8—791no—
222C2×C22⋊C8—1581no—
223#405—1581no—
224C2×C4×C8C2×D7911yescharacteristic
225#334—794no—
226Q8⋊C8—794no—
227D4⋊C8—794no—
228#322C2×D7912yescomplemented
229C82C2×D7912yescomplemented
230C22×D4—791no—
231C23×C4—1581no—
232C2×SD16—794no—
233C2×M16—3164no—
234C2×M16—1581no—
235C2×M16—1582no—
236C2×M16—1582no—
237C22×C8—1581no—
238C22×C8D31611yescharacteristic
239C22×C8D31611yescharacteristic
240C4⋊Q8—791no—
241C4⋊Q8—791no—
242#125—1581no—
243C22⋊Q8—792no—
244C22⋊Q8—1581no—
245C4×Q8—792no—
246C4×Q8—1581no—
247C4×D4—792no—
248C4×D4—792no—
249C4×D4—1581no—
250C4×D4—1581no—
251#117—1581no—
252C2×C4⋊C4—791no—
253C2×C22⋊C4—1581no—
254C2×C42C22×D7911yescharacteristic
255#106—794no—
256C4⋊C8—21no—
257C4⋊C8—792no—
258C4⋊C8—792no—
259C4⋊C8—1582no—
260C4⋊C8—1582no—
261C4⋊C8—1581no—
262C4⋊C8—1581no—
263C4⋊C8C4×D7912yes—
264#103—794no—
265#102—794no—
266C22⋊C8—21no—
267C22⋊C8—1582no—
268C22⋊C8—1582no—
269C22⋊C8C4×D7912yes—
270#97—1582no—
271C4×C8—1582no—
272C4×C8D31611yescharacteristic
273C4×C8D31611yescharacteristic
274C4×C8C22×D7912yes—
275C4×C8C22×D7912yes—
276C4×C8C22×D7911yescharacteristic
277C4×C8C22×D7911yescharacteristic
278C24—1581no—
279C2×Q8—792no—
280C2×Q8—1581no—
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×C4C2×D31611yescharacteristic
291C22×C4#1192962411yescharacteristic
292SD16—3164no—
293M16—3162no—
294M16—3164no—
295M16—3164no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—3162no—
299C2×C8—1582no—
300C2×C8—1582no—
301C2×C8#1192961314yescomplemented
302C2×C8C2×C4×D7914yes—
303C2×C8C2×D31611yescharacteristic
304C2×C8C2×D31611yescharacteristic
305C2×C8C2×D31611yescharacteristic
306C2×C8C2×D31611yescharacteristic
307C4⋊C4—792no—
308C4⋊C4—792no—
309C4⋊C4—1582no—
310C4⋊C4—1582no—
311C4⋊C4—1581no—
312C4⋊C4—1581no—
313C4⋊C4—1581no—
314C4⋊C4—1581no—
315C22⋊C4—1582no—
316C22⋊C4—1582no—
317C22⋊C4—1582no—
318C42—1582no—
319C42C2×C4×D7911yescharacteristic
320C42C2×D31611yescharacteristic
321C42#1192962411yescharacteristic
322C42C23×D7911yescharacteristic
323C23—3162no—
324C23—3162no—
325C23—3162no—
326C23—1582no—
327C23C4×D31611yescharacteristic
328Q8—3162no—
329Q8—1584no—
330D4—3162no—
331D4—1582no—
332D4—1582no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—3162no—
336C2×C4—3164no—
337C2×C4—3162no—
338C2×C4—3162no—
339C2×C4—1582no—
340C2×C4—1582no—
341C2×C4—1582no—
342C2×C4C4×D31611yescharacteristic
343C2×C4C4×D31611yescharacteristic
344C2×C4C4×D31611yescharacteristic
345C2×C4C2×#1192961312yes—
346C2×C4#1457918312yes—
347C2×C4C22×C4×D7911yescharacteristic
348C2×C4C22×D31611yescharacteristic
349C2×C4C2×#1192962411yescharacteristic
350C8—24no—
351C8—21no—
352C8—21no—
353C8—42no—
354C8—3162no—
355C8—3162no—
356C8C2×#1192961312yes—
357C8C2×#1192961312yes—
358C22—21no—
359C22—21no—
360C22—3162no—
361C22—3162no—
362C22—3162no—
363C22—6324no—
364C22C4×#1192961312yes—
365C22C2×C4×D31611yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—3162no—
370C4—3162no—
371C4—3162no—
372C4C4×#1192961312yes—
373C4#1909932812yes—
374C4C22×#1192961311yescharacteristic
375C4C2×#1457918311yescharacteristic
376C2—22no—
377C2—6322no—
378C2#2377210511yescharacteristic
379C2C2×C4×#1192961311yescharacteristic
380C2#2377230411yescharacteristic
subgroupG/H
centerC2×C4C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC42C23×D79
FittingC79×#1450C2
socleC22×C79C2×C4×D4
prime radicalC22×D316C2×C4
cosocleC42×C79C24
hypercenter#1450D79
hypocenterC79#17930
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #1450

Complements

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

NH ≅ G/Ntype
2. #23772334C2semidirect
4. #23762292C2semidirect
5. #23762245C2semidirect
6. #23762171C2semidirect
8. #23761804C2semidirect
9. C79×#1450C2semidirect
18. #19098693C22semidirect
20. C8×Dic158C22semidirect
31. #19098314C22semidirect
32. #19098295C22semidirect
37. #322×C79C22semidirect
38. C82×C79C22semidirect
122. C2×C8×C79SD16semidirect
199. #1450D79semidirect
200. C79#17930semidirect
228. #322C2×D79semidirect
229. C82C2×D79semidirect
301. C2×C8#11929613semidirect

Maximal subgroups

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

subgroupindex
#237734092
#237723342
C2×#190986422
#237622922
#237622452
#237621712
#237618262
#237618042
C79×#14502
#1793079