Finite Group Databasebeta

← #34187420

Subgroups — #34187420

Order 20224.

468 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#23764345C211yesmaximal, characteristic, complemented
2#23764320C211yesmaximal, characteristic
3C4×#14579093C211yesmaximal, characteristic, complemented
4#23761899C211yesmaximal, characteristic, complemented
5#23761895C211yesmaximal, characteristic
6C8×#11929608C211yesmaximal, characteristic, complemented
7C79×#3167C211yesmaximal, characteristic
8#19098616C2211yescharacteristic
9C2×#14579093—21no—
10C2×#14579088—22no—
11C4×#11929608C2211yescharacteristic
12C4×C4⋊Dic79C2211yescharacteristic
13C4×C4⋊Dic79C2211yescharacteristic
14#19098326—21no—
15#19098322—22no—
16#19098300—22no—
17#19098297—22no—
18#19098285—21no—
19C8×C79⋊C8—21no—
20#533×C79C2211yescharacteristic
21#532×C79C2211yescharacteristic
22#498×C79C411yescharacteristic, complemented
23#492×C79C411yescharacteristic, complemented
24#487×C79C411yescharacteristic, complemented
25#486×C79C411yescharacteristic, complemented
26#465×C79—21no—
27#465×C79—21no—
28C4×SD16×C79—21no—
29C4×SD16×C79—21no—
30C2×C4×C8×C79C2211yescharacteristic
31C2×#11929608D412yes—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34C2×C4×Dic79—21no—
35C2×C4×Dic79—21no—
36#14579093—24no—
37C4⋊(C79⋊C8)—21no—
38#14579088—44no—
39C22⋊(C79⋊C8)D412yes—
40#14579072—21no—
41C4×C79⋊C8—21no—
42C4×C79⋊C8—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C2×#41×C79—21no—
46C2×SD16×C79—21no—
47C2×SD16×C79—21no—
48C22×C8×C79D411yescharacteristic
49C22×C8×C79D411yescharacteristic
50C4⋊Q8×C79C2×C411yescharacteristic
51C4⋊Q8×C79C2×C411yescharacteristic
52C4⋊Q8×C79C2×C411yescharacteristic
53#127×C79C2×C411yescharacteristic
54#125×C79—21no—
55#124×C79—21no—
56C22⋊Q8×C79—21no—
57C22⋊Q8×C79—21no—
58#121×C79—21no—
59#121×C79—21no—
60C4×Q8×C79—21no—
61C4×Q8×C79—21no—
62C4×D4×C79—21no—
63C4×D4×C79—21no—
64#117×C79—21no—
65C2×C42×C79C2311yescharacteristic
66#106×C79—21no—
67#106×C79—21no—
68#103×C79—21no—
69#103×C79—21no—
70#102×C79—21no—
71#102×C79—21no—
72C4×C8×C79C2×C411yescharacteristic
73C4×C8×C79C2×C411yescharacteristic
74C4×C8×C79D411yescharacteristic
75C4×C8×C79D411yescharacteristic
76C22×Dic79—41no—
77C22×Dic79—41no—
78#11929608—22no—
79#11929608SD1614yescomplemented
80C2×C79⋊C8—22no—
81C2×C79⋊C8—22no—
82C4⋊Dic79—22no—
83C4⋊Dic79—22no—
84C4⋊Dic79—41no—
85C4⋊Dic79—41no—
86C4×Dic79—42no—
87C4×Dic79—42no—
88#41×C79—42no—
89#41×C79—42no—
90C2×Q8×C79—21no—
91C2×Q8×C79—21no—
92C2×Q8×C79—21no—
93C2×D4×C79—21no—
94C2×D4×C79—21no—
95C2×D4×C79—21no—
96C2×D4×C79—41no—
97C22×C4×C79—41no—
98C22×C4×C79C2×D411yescharacteristic
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79C2×D411yescharacteristic
101SD16×C79—42no—
102SD16×C79—42no—
103C2×C8×C79—22no—
104C2×C8×C79—41no—
105C2×C8×C79C22⋊C411yescharacteristic
106C2×C8×C79C22⋊C411yescharacteristic
107C2×C8×C79C22⋊C411yescharacteristic
108C2×C8×C79C22⋊C411yescharacteristic
109C4⋊C4×C79—21no—
110C4⋊C4×C79—21no—
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—41no—
116C4⋊C4×C79—41no—
117C22⋊C4×C79—41no—
118C22⋊C4×C79—41no—
119C22⋊C4×C79—41no—
120C42×C79—41no—
121C42×C79C22×C411yescharacteristic
122C42×C79C2×D411yescharacteristic
123C42×C79#4111yescharacteristic
124C42×C79#4111yescharacteristic
125C23×C79—41no—
126C23×C79#12711yescharacteristic
127Q8×C79—41no—
128Q8×C79—41no—
129Q8×C79—42no—
130Q8×C79—81no—
131D4×C79—41no—
132D4×C79—41no—
133D4×C79—42no—
134D4×C79—81no—
135D4×C79—81no—
136D4×C79—81no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—82no—
144C2×C4×C79C2×C22⋊C411yescharacteristic
145C2×C4×C79C4×D411yescharacteristic
146C2×C4×C79C4×D411yescharacteristic
147C2×C4×C79#12111yescharacteristic
148C2×C4×C79#12111yescharacteristic
149C2×C4×C79#12411yescharacteristic
150C2×C4×C79C2×SD1612yes—
151C2×C4×C79#13512yes—
152C2×Dic79—42no—
153C2×Dic79—42no—
154C2×Dic79—81no—
155C2×Dic79—81no—
156C2×Dic79—81no—
157C2×Dic79—81no—
158C8×C79—21no—
159C8×C79—21no—
160C8×C79—21no—
161C8×C79#10412yescomplemented
162C79⋊C8—24no—
163C79⋊C8—42no—
164C22×C79—21no—
165C22×C79—21no—
166C22×C79—81no—
167C22×C79—81no—
168C22×C79—81no—
169C22×C79#39011yescharacteristic
170C22×C79#49211yescharacteristic
171C22×C79#49511yescharacteristic
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—21no—
175C4×C79—81no—
176C4×C79—81no—
177C4×C79—81no—
178C4×C79C2×#10411yescharacteristic
179C4×C79#42111yescharacteristic
180C4×C79C4×SD1612yes—
181C4×C79#46512yes—
182Dic79—82no—
183Dic79—82no—
184#9717—791nomaximal, Sylow
185C2×C79—21no—
186C2×C79—21no—
187C2×C79—161no—
188C2×C79#196811yescharacteristic
189C2×C79#197611yescharacteristic
190C2×C79#198011yescharacteristic
191#3167D7911yescharacteristic
192#1986—791no—
193#1980—791no—
194C4×#106—791no—
195#1503—791no—
196#1502—791no—
197C8×M16—791no—
198C79#971711yescharacteristic, complemented, Sylow
199#533C2×D7911yescharacteristic
200#532C2×D7911yescharacteristic
201#498Dic7911yescharacteristic, complemented
202#492Dic7911yescharacteristic, complemented
203#487Dic7911yescharacteristic, complemented
204#486Dic7911yescharacteristic, complemented
205#465—21no—
206#465—21no—
207C4×SD16—21no—
208C4×SD16—21no—
209#431—791no—
210C2×#106—1581no—
211C2×#104—1582no—
212C4×M16—791no—
213C2×C4×C8C2×D7911yescharacteristic
214C4×C4⋊C4—791no—
215C4×C4⋊C4—791no—
216#340—1581no—
217#339—1582no—
218#328—1582no—
219#327—1582no—
220#322—1581no—
221C82—1581no—
222C2×#41—21no—
223C2×SD16—21no—
224C2×SD16—21no—
225C2×M16—792no—
226C22×C8C22⋊D7911yescharacteristic
227C22×C8C22⋊D7911yescharacteristic
228C4⋊Q8C2×Dic7911yescharacteristic
229C4⋊Q8C2×Dic7911yescharacteristic
230C4⋊Q8C2×Dic7911yescharacteristic
231#127C2×Dic7911yescharacteristic
232#125—21no—
233#124—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236#121—21no—
237#121—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—21no—
241C4×D4—21no—
242#117—21no—
243C2×C4⋊C4—1581no—
244C2×C4⋊C4—1581no—
245C2×C42—1581no—
246C2×C42—1581no—
247C2×C42C22×D7911yescharacteristic
248#106—21no—
249#106—21no—
250#106—1584no—
251C4⋊C8—1581no—
252#104—3164no—
253#103—21no—
254#103—21no—
255#102—21no—
256#102—21no—
257C22⋊C8—792no—
258#97—1581no—
259C4×C8—1581no—
260C4×C8—1581no—
261C4×C8C2×Dic7911yescharacteristic
262C4×C8C2×Dic7911yescharacteristic
263C4×C8C22⋊D7911yescharacteristic
264C4×C8C22⋊D7911yescharacteristic
265#95—1581no—
266#95—1581no—
267#41—42no—
268#41—42no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C22×C4—41no—
277C22×C4—3161no—
278C22×C4—3161no—
279C22×C4D4×D7911yescharacteristic
280C22×C4D4×D7911yescharacteristic
281C22×C4C2×C22⋊D7911yescharacteristic
282SD16—42no—
283SD16—42no—
284M16—794no—
285M16—1582no—
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—1582no—
289C2×C8—1582no—
290C2×C8C22⋊Dic7911yescharacteristic
291C2×C8C22⋊Dic7911yescharacteristic
292C2×C8C22⋊Dic7911yescharacteristic
293C2×C8C22⋊Dic7911yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—3161no—
303C4⋊C4—3161no—
304C4⋊C4—1582no—
305C4⋊C4—1582no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C42—41no—
310C42—3162no—
311C42—3162no—
312C42C22×Dic7911yescharacteristic
313C42C2×C22⋊D7911yescharacteristic
314C42#1192962311yescharacteristic
315C42#1192962311yescharacteristic
316C23—41no—
317C23#1457915711yescharacteristic
318Q8—42no—
319Q8—41no—
320Q8—41no—
321Q8—81no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—82no—
335C2×C4—3162no—
336C2×C4—3162no—
337C2×C4—6321no—
338C2×C4—6321no—
339C2×C4—6321no—
340C2×C4—6321no—
341C2×C4C2×C22⋊Dic7911yescharacteristic
342C2×C4D4×Dic7911yescharacteristic
343C2×C4D4×Dic7911yescharacteristic
344C2×C4#1457913211yescharacteristic
345C2×C4#1457913211yescharacteristic
346C2×C4#1457914811yescharacteristic
347C2×C4SD16×D7912yes—
348C2×C4#1457918212yes—
349C8—21no—
350C8—21no—
351C8—21no—
352C8—3162no—
353C8—1584no—
354C8#1457908812yescomplemented
355C22—21no—
356C22—21no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1909846611yescharacteristic
361C22#1909893411yescharacteristic
362C22#1909894411yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—6322no—
370C4—6322no—
371C4C2×#1457908811yescharacteristic
372C4#1909858011yescharacteristic
373C4SD16×Dic7912yes—
374C4#1909878912yes—
375C2—21no—
376C2—21no—
377C2—161no—
378C2#2376425611yescharacteristic
379C2#2376429911yescharacteristic
380C2#2376432011yescharacteristic
subgroupG/H
centerC2×C4#14579132
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3167C2
socleC22×C79#390
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#3167D79
hypocenterC79#9717
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #23764345C2semidirect
3. C4×#14579093C2semidirect
4. #23761899C2semidirect
6. C8×#11929608C2semidirect
22. #498×C79C4semidirect
23. #492×C79C4semidirect
24. #487×C79C4semidirect
25. #486×C79C4semidirect
79. #11929608SD16semidirect
161. C8×C79#104semidirect
198. C79#9717semidirect
201. #498Dic79semidirect
202. #492Dic79semidirect
203. #487Dic79semidirect
204. #486Dic79semidirect
354. C8#14579088semidirect

Maximal subgroups

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

subgroupindex
#237643452
#237643202
C4×#145790932
#237618992
#237618952
C8×#119296082
C79×#31672
#971779