Finite Group Databasebeta

← #34187622

Subgroups — #34187622

Order 20224.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C8×C4⋊Dic79C211yesmaximal, characteristic, complemented
2#23762148C211yesmaximal, characteristic, complemented
3#23761899C211yesmaximal, characteristic, complemented
4#23761895C211yesmaximal, characteristic
5#23761803C211yesmaximal, characteristic, complemented
6#23761799C211yesmaximal, characteristic
7C79×#3131C211yesmaximal, characteristic
8#19098616C2211yescharacteristic
9#19098616C2211yescharacteristic
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×C8×Dic79—21no—
12C4×C4⋊Dic79C2211yescharacteristic
13#19098317—21no—
14#19098316—21no—
15#19098314—21no—
16#19098300—22no—
17#19098297—22no—
18Q8⋊(C79⋊C8)—22no—
19D4⋊(C79⋊C8)—22no—
20#533×C79C2211yescharacteristic
21#532×C79C2211yescharacteristic
22#488×C79—21no—
23#488×C79—21no—
24#487×C79C411yescharacteristic, complemented
25#487×C79C411yescharacteristic, complemented
26#486×C79C411yescharacteristic, complemented
27#486×C79C411yescharacteristic, complemented
28#416×C79—21no—
29#416×C79—21no—
30C2×C4×C8×C79C2211yescharacteristic
31C22×C79⋊C8—21no—
32C2×C4⋊Dic79—21no—
33C2×C4×Dic79—21no—
34C4⋊(C79⋊C8)—21no—
35C4⋊(C79⋊C8)—21no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)D412yes—
38C22⋊(C79⋊C8)—21no—
39C22⋊(C79⋊C8)—21no—
40C4×C79⋊C8—21no—
41C4×C79⋊C8—21no—
42C8×Dic79—42no—
43C79⋊#95D412yes—
44C2×#41×C79—21no—
45C22×C8×C79D411yescharacteristic
46C22×C8×C79D411yescharacteristic
47C4⋊Q8×C79C2×C411yescharacteristic
48C4⋊Q8×C79C2×C411yescharacteristic
49C4⋊Q8×C79C2×C411yescharacteristic
50#127×C79C2×C411yescharacteristic
51#125×C79—21no—
52#124×C79—21no—
53C22⋊Q8×C79—21no—
54C22⋊Q8×C79—21no—
55#121×C79—21no—
56#121×C79—21no—
57C4×Q8×C79—21no—
58C4×Q8×C79—21no—
59C4×D4×C79—21no—
60C4×D4×C79—21no—
61#117×C79—21no—
62C2×C42×C79C2311yescharacteristic
63#103×C79—21no—
64#103×C79—21no—
65#103×C79—21no—
66#103×C79—21no—
67#102×C79—21no—
68#102×C79—21no—
69#102×C79—21no—
70#102×C79—21no—
71C4×C8×C79C2×C411yescharacteristic
72C4×C8×C79C2×C411yescharacteristic
73C4×C8×C79D411yescharacteristic
74C4×C8×C79D411yescharacteristic
75C22×Dic79—22no—
76C2×C79⋊C8—22no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8—41no—
79C2×C79⋊C8—41no—
80C2×C79⋊C8—41no—
81C4⋊Dic79—24no—
82C4×Dic79—41no—
83C4×Dic79—41no—
84#41×C79—41no—
85#41×C79—41no—
86#41×C79—41no—
87#41×C79—41no—
88C2×Q8×C79—21no—
89C2×Q8×C79—21no—
90C2×Q8×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79—21no—
93C2×D4×C79—21no—
94C2×D4×C79—41no—
95C22×C4×C79—41no—
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79#4111yescharacteristic
99C2×C8×C79—21no—
100C2×C8×C79—21no—
101C2×C8×C79—21no—
102C2×C8×C79—21no—
103C2×C8×C79C22⋊C411yescharacteristic
104C2×C8×C79C22⋊C411yescharacteristic
105C2×C8×C79C22⋊C411yescharacteristic
106C2×C8×C79C22⋊C411yescharacteristic
107C4⋊C4×C79—21no—
108C4⋊C4×C79—21no—
109C4⋊C4×C79—21no—
110C4⋊C4×C79—21no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79—41no—
115C22⋊C4×C79—41no—
116C22⋊C4×C79—41no—
117C22⋊C4×C79—41no—
118C42×C79—41no—
119C42×C79C22×C411yescharacteristic
120C42×C79C2×D411yescharacteristic
121C42×C79C2×D411yescharacteristic
122C42×C79#4111yescharacteristic
123C23×C79—41no—
124C23×C79#12111yescharacteristic
125Q8×C79—41no—
126Q8×C79—41no—
127Q8×C79—42no—
128Q8×C79—81no—
129D4×C79—41no—
130D4×C79—41no—
131D4×C79—42no—
132D4×C79—81no—
133D4×C79—81no—
134D4×C79—81no—
135C2×C4×C79—22no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—81no—
141C2×C4×C79—81no—
142C2×C4×C79C2×C22⋊C411yescharacteristic
143C2×C4×C79C4×D411yescharacteristic
144C2×C4×C79C4×D411yescharacteristic
145C2×C4×C79#12111yescharacteristic
146C2×C4×C79#12411yescharacteristic
147C2×C4×C79#12711yescharacteristic
148C2×C4×C79#13511yescharacteristic
149C2×C4×C79#13511yescharacteristic
150C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C79#13711yescharacteristic
152C2×Dic79—24no—
153C2×Dic79—42no—
154C2×Dic79—42no—
155C8×C79—21no—
156C8×C79—21no—
157C8×C79—21no—
158C8×C79—21no—
159C79⋊C8—42no—
160C79⋊C8—81no—
161C79⋊C8—81no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—81no—
165C22×C79—81no—
166C22×C79—81no—
167C22×C79#39011yescharacteristic
168C22×C79#46311yescharacteristic
169C22×C79#46411yescharacteristic
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—81no—
174C4×C79—81no—
175C4×C79—81no—
176C4×C79#41611yescharacteristic
177C4×C79#41711yescharacteristic
178C4×C79#44312yes—
179C4×C79#46311yescharacteristic
180C4×C79#46411yescharacteristic
181Dic79—44no—
182#9751—791nomaximal, Sylow
183C2×C79—21no—
184C2×C79—21no—
185C2×C79—161no—
186C2×C79#197411yescharacteristic
187C2×C79#197611yescharacteristic
188C2×C79#198311yescharacteristic
189#3131D7911yescharacteristic
190C8×C4⋊C4—791no—
191#1564—791no—
192#1503—791no—
193#1502—791no—
194#1481—791no—
195#1480—791no—
196C79#975111yescharacteristic, complemented, Sylow
197#533C2×D7911yescharacteristic
198#532C2×D7911yescharacteristic
199#488—21no—
200#488—21no—
201#487Dic7911yescharacteristic, complemented
202#487Dic7911yescharacteristic, complemented
203#486Dic7911yescharacteristic, complemented
204#486Dic7911yescharacteristic, complemented
205#431—791no—
206#431—791no—
207C2×C4⋊C8—791no—
208#416—21no—
209#416—21no—
210C2×C4×C8—1581no—
211C2×C4×C8C2×D7911yescharacteristic
212C4×C4⋊C4—791no—
213#336—1581no—
214#335—1581no—
215#334—1581no—
216#328—1582no—
217#327—1582no—
218Q8⋊C8—1582no—
219D4⋊C8—1582no—
220C2×#41—21no—
221C22×C8—1581no—
222C22×C8C22⋊D7911yescharacteristic
223C22×C8C22⋊D7911yescharacteristic
224C4⋊Q8C2×Dic7911yescharacteristic
225C4⋊Q8C2×Dic7911yescharacteristic
226C4⋊Q8C2×Dic7911yescharacteristic
227#127C2×Dic7911yescharacteristic
228#125—21no—
229#124—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—21no—
233#121—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117—21no—
239C2×C4⋊C4—1581no—
240C2×C42—1581no—
241C2×C42C22×D7911yescharacteristic
242C4⋊C8—792no—
243C4⋊C8—1581no—
244C4⋊C8—1581no—
245C4⋊C8—1581no—
246#103—21no—
247#103—21no—
248#103—21no—
249#103—21no—
250#102—21no—
251#102—21no—
252#102—21no—
253#102—21no—
254C22⋊C8—1581no—
255C22⋊C8—1581no—
256C4×C8—3162no—
257C4×C8—1581no—
258C4×C8—1581no—
259C4×C8C2×Dic7911yescharacteristic
260C4×C8C2×Dic7911yescharacteristic
261C4×C8C22⋊D7911yescharacteristic
262C4×C8C22⋊D7911yescharacteristic
263#95—792no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C22×C4—41no—
276C22×C4—1582no—
277C22×C4D4×D7911yescharacteristic
278C22×C4C2×C22⋊D7911yescharacteristic
279C22×C4#1192962311yescharacteristic
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—3161no—
285C2×C8—3161no—
286C2×C8—3161no—
287C2×C8—3161no—
288C2×C8—1582no—
289C2×C8C22⋊Dic7911yescharacteristic
290C2×C8C22⋊Dic7911yescharacteristic
291C2×C8C22⋊Dic7911yescharacteristic
292C2×C8C22⋊Dic7911yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—1584no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C42—41no—
306C42—3161no—
307C42—3161no—
308C42C22×Dic7911yescharacteristic
309C42D4×D7911yescharacteristic
310C42C2×C22⋊D7911yescharacteristic
311C42#1192962311yescharacteristic
312C23—41no—
313C23#1457913211yescharacteristic
314Q8—42no—
315Q8—41no—
316Q8—41no—
317Q8—81no—
318D4—42no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—3162no—
332C2×C4—3162no—
333C2×C4—1584no—
334C2×C4C2×C22⋊Dic7911yescharacteristic
335C2×C4D4×Dic7911yescharacteristic
336C2×C4D4×Dic7911yescharacteristic
337C2×C4#1457913211yescharacteristic
338C2×C4#1457914811yescharacteristic
339C2×C4#1457915711yescharacteristic
340C2×C4#1457918111yescharacteristic
341C2×C4#1457918111yescharacteristic
342C2×C4#1457918611yescharacteristic
343C2×C4#1457919111yescharacteristic
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C8—3162no—
349C8—6321no—
350C8—6321no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1909846611yescharacteristic
357C22#1909877511yescharacteristic
358C22#1909878211yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—3164no—
366C4#1909855911yescharacteristic
367C4#1909856411yescharacteristic
368C4#1909868012yes—
369C4#1909877511yescharacteristic
370C4#1909878211yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2#2376428911yescharacteristic
375C2#2376429911yescharacteristic
376C2#2376433311yescharacteristic
subgroupG/H
centerC2×C4#14579132
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3131C2
socleC22×C79#390
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#3131D79
hypocenterC79#9751
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3131

Complements

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

NH ≅ G/Ntype
1. C8×C4⋊Dic79C2semidirect
2. #23762148C2semidirect
3. #23761899C2semidirect
5. #23761803C2semidirect
24. #487×C79C4semidirect
25. #487×C79C4semidirect
26. #486×C79C4semidirect
27. #486×C79C4semidirect
196. C79#9751semidirect
201. #487Dic79semidirect
202. #487Dic79semidirect
203. #486Dic79semidirect
204. #486Dic79semidirect

Maximal subgroups

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

subgroupindex
C8×C4⋊Dic792
#237621482
#237618992
#237618952
#237618032
#237617992
C79×#31312
#975179