Finite Group Databasebeta

← #34187649

Subgroups — #34187649

Order 20224.

400 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773725C211yesmaximal, characteristic
2#23763530C211yesmaximal, characteristic, complemented
3#23761964C211yesmaximal, characteristic, complemented
4#23761961C211yesmaximal, characteristic
5#23761959C211yesmaximal, characteristic
6#23761684C211yesmaximal, characteristic, complemented
7C79×#1518C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098920—21no—
10#19098920—21no—
11#19098915C411yescharacteristic, complemented
12#19098915C411yescharacteristic, complemented
13#19098911C411yescharacteristic, complemented
14#19098911C411yescharacteristic, complemented
15#19098616C2211yescharacteristic
16#19098557—21no—
17#19098557—21no—
18C2×C4×C79⋊C8C2211yescharacteristic
19#19098429C2211yescharacteristic
20#19098325—21no—
21#19098325—21no—
22#19098300—22no—
23#19098298—22no—
24#19098296—22no—
25#19098285—21no—
26C8×C79⋊C8—21no—
27#533×C79C2211yescharacteristic
28#431×C79C2211yescharacteristic
29#382×C79C2211yescharacteristic
30#328×C79—22no—
31C2×#11929624—21no—
32C22×C79⋊C8D411yescharacteristic
33C22×C79⋊C8D411yescharacteristic
34#14579161C2×C411yescharacteristic
35#14579158C2×C411yescharacteristic
36#14579149—21no—
37#14579135—21no—
38#14579135—21no—
39C4×D316—21no—
40C4×D316—21no—
41C2×C4⋊Dic79—21no—
42C4⋊(C79⋊C8)—21no—
43#14579084—21no—
44#14579084—21no—
45#14579084—21no—
46#14579084—21no—
47#14579081—21no—
48#14579081—21no—
49#14579081—21no—
50#14579081—21no—
51C22⋊(C79⋊C8)D412yes—
52C4×C79⋊C8—21no—
53C4×C79⋊C8C2×C411yescharacteristic
54C4×C79⋊C8C2×C411yescharacteristic
55C4×C79⋊C8D411yescharacteristic
56C4×C79⋊C8D411yescharacteristic
57C79⋊#95—21no—
58C79⋊#95—21no—
59C4⋊Q8×C79C2×C411yescharacteristic
60C4⋊Q8×C79C2×C411yescharacteristic
61#125×C79—21no—
62C22⋊Q8×C79—21no—
63C22⋊Q8×C79—21no—
64C4×Q8×C79—21no—
65C4×Q8×C79—21no—
66#117×C79—21no—
67C2×C4⋊C4×C79—21no—
68C2×C42×C79C2311yescharacteristic
69C4⋊C8×C79—21no—
70C22⋊C8×C79D412yes—
71C4×C8×C79—21no—
72#95×C79—21no—
73#95×C79—21no—
74#11929624—41no—
75#11929624—41no—
76#11929624—41no—
77#11929624—41no—
78C2×Dic158—21no—
79C2×D316—21no—
80C2×D316—21no—
81C2×D316—21no—
82C2×C22⋊D79—41no—
83C22×Dic79—41no—
84C2×C4×D79—41no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8—21no—
87C2×C79⋊C8—21no—
88C2×C79⋊C8—21no—
89C2×C79⋊C8—21no—
90C2×C79⋊C8C22⋊C411yescharacteristic
91C2×C79⋊C8C22⋊C411yescharacteristic
92C2×C79⋊C8C22⋊C411yescharacteristic
93C2×C79⋊C8C22⋊C411yescharacteristic
94C4⋊Dic79—21no—
95C4⋊Dic79—21no—
96C4⋊Dic79—41no—
97C4⋊Dic79—41no—
98(C2×D79)⋊C4—41no—
99(C2×D79)⋊C4—41no—
100C2×Q8×C79—21no—
101C2×Q8×C79—21no—
102C22×C4×C79—41no—
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C2×C8×C79—22no—
107C4⋊C4×C79—21no—
108C4⋊C4×C79—21no—
109C4⋊C4×C79—21no—
110C4⋊C4×C79—21no—
111C4⋊C4×C79—41no—
112C4⋊C4×C79—41no—
113C4⋊C4×C79—41no—
114C4⋊C4×C79—41no—
115C22⋊C4×C79—41no—
116C42×C79—41no—
117C42×C79C22×C411yescharacteristic
118C42×C79C2×D411yescharacteristic
119C42×C79#4111yescharacteristic
120C42×C79#4111yescharacteristic
121C23×C79#12711yescharacteristic
122C22×D79—41no—
123Q8×C79—81no—
124Q8×C79—81no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—81no—
130C2×C4×C79—81no—
131C2×C4×C79—81no—
132C2×C4×C79C2×C22⋊C411yescharacteristic
133C2×C4×C79C4×D411yescharacteristic
134C2×C4×C79C4×D411yescharacteristic
135C2×C4×C79#12111yescharacteristic
136C2×C4×C79#12111yescharacteristic
137C2×C4×C79#12411yescharacteristic
138C2×C4×C79#13511yescharacteristic
139C2×C4×C79#13511yescharacteristic
140C2×C4×C79#13511yescharacteristic
141C2×C4×C79#13511yescharacteristic
142C22⋊D79—81no—
143C22⋊D79—81no—
144C2×Dic79—41no—
145C2×Dic79—81no—
146C2×Dic79—81no—
147C2×Dic79—81no—
148D316—41no—
149D316—41no—
150D316—81no—
151D316—81no—
152C4×D79—81no—
153C4×D79—81no—
154Dic158—41no—
155Dic158—41no—
156C8×C79—42no—
157C79⋊C8—21no—
158C79⋊C8—21no—
159C79⋊C8—21no—
160C79⋊C8—21no—
161C79⋊C8—42no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79#39011yescharacteristic
165C22×C79#49511yescharacteristic
166C22×C79#49511yescharacteristic
167C2×D79—81no—
168C2×D79—81no—
169C2×D79—81no—
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—81no—
175C4×C79—81no—
176C4×C79—161no—
177C4×C79#41611yescharacteristic
178C4×C79#41611yescharacteristic
179C4×C79#48811yescharacteristic
180C4×C79#48811yescharacteristic
181Dic79—81no—
182Dic79—161no—
183#9757—791nomaximal, Sylow
184C2×C79—21no—
185C2×C79—21no—
186C2×C79#195811yescharacteristic
187C2×C79#197611yescharacteristic
188C2×C79#197611yescharacteristic
189D79—161no—
190#3131—791no—
191#1835—791no—
192#1518—791no—
193#1518D7911yescharacteristic, complemented
194#1517—791no—
195#1517—791no—
196#1450—791no—
197C79#975711yescharacteristic, complemented, Sylow
198#533C2×D7911yescharacteristic
199#532—791no—
200#488—1581no—
201#488—1581no—
202#487—791no—
203#487—791no—
204#486—791no—
205#486—791no—
206#431—791no—
207#431C2×D7911yescharacteristic
208#416—1581no—
209#416—1581no—
210C2×C4×C8—791no—
211#382—791no—
212#382C2×D7911yescharacteristic
213#340—1581no—
214#340—1581no—
215#328—22no—
216#328—1582no—
217#327—1582no—
218#327—1582no—
219#322—1581no—
220C82—1581no—
221C2×#41—1581no—
222C22×C8—791no—
223C22×C8—791no—
224C4⋊Q8—791no—
225C4⋊Q8C4×D7911yescharacteristic
226C4⋊Q8C4×D7911yescharacteristic
227#127—791no—
228#125—21no—
229#124—1581no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232#121—1581no—
233#121—1581no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—1581no—
237C4×D4—1581no—
238#117—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—1581no—
241C2×C42C22×D7911yescharacteristic
242C4⋊C8—21no—
243C4⋊C8—1581no—
244#103—1581no—
245#103—1581no—
246#103—1581no—
247#103—1581no—
248#102—1581no—
249#102—1581no—
250#102—1581no—
251#102—1581no—
252C22⋊C8—792no—
253C22⋊C8C22⋊D7912yes—
254C4×C8—21no—
255C4×C8—791no—
256C4×C8—791no—
257C4×C8—791no—
258C4×C8—791no—
259C4×C8—1581no—
260#95—21no—
261#95—21no—
262#95—1581no—
263#95—1581no—
264#41—3161no—
265#41—3161no—
266#41—3161no—
267#41—3161no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—1581no—
271C2×D4—3161no—
272C2×D4—1581no—
273C2×D4—1581no—
274C2×D4—1581no—
275C22×C4—41no—
276C22×C4—3161no—
277C22×C4—3161no—
278C22×C4D4×D7911yescharacteristic
279C22×C4D4×D7911yescharacteristic
280C22×C4C2×C22⋊D7911yescharacteristic
281C2×C8—22no—
282C2×C8—791no—
283C2×C8—791no—
284C2×C8—791no—
285C2×C8—791no—
286C2×C8—1582no—
287C2×C8—1581no—
288C2×C8—1581no—
289C2×C8—1581no—
290C2×C8—1581no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—3161no—
300C4⋊C4—3161no—
301C4⋊C4—1581no—
302C4⋊C4—1581no—
303C22⋊C4—41no—
304C22⋊C4—3161no—
305C22⋊C4—3161no—
306C42—41no—
307C42C2×C4×D7911yescharacteristic
308C42D4×D7911yescharacteristic
309C42#1192962311yescharacteristic
310C42#1192962411yescharacteristic
311C23—3161no—
312C23#1457915711yescharacteristic
313Q8—81no—
314Q8—81no—
315Q8—3161no—
316Q8—3161no—
317D4—3161no—
318D4—3161no—
319D4—6321no—
320D4—6321no—
321D4—6321no—
322D4—6321no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—3161no—
331C2×C4—6321no—
332C2×C4—6321no—
333C2×C4—6321no—
334C2×C4—6321no—
335C2×C4—6321no—
336C2×C4C22⋊C4×D7911yescharacteristic
337C2×C4(C22⋊D79)⋊C411yescharacteristic
338C2×C4C4×C22⋊D7911yescharacteristic
339C2×C4#1457913311yescharacteristic
340C2×C4#1457913411yescharacteristic
341C2×C4#1457914711yescharacteristic
342C2×C4#1457918211yescharacteristic
343C2×C4#1457918211yescharacteristic
344C2×C4#1457918411yescharacteristic
345C2×C4#1457918411yescharacteristic
346C8—42no—
347C8—3162no—
348C8—1581no—
349C8—1581no—
350C8—1581no—
351C8—1581no—
352C22—21no—
353C22—21no—
354C22—6321no—
355C22—6321no—
356C22—6321no—
357C22#1909846311yescharacteristic
358C22#1909894411yescharacteristic
359C22#1909894811yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—81no—
366C4—161no—
367C4—6321no—
368C4—12641no—
369C4#1909855811yescharacteristic
370C4#1909855811yescharacteristic
371C4#1909891911yescharacteristic
372C4#1909891911yescharacteristic
373C2—21no—
374C2—21no—
375C2—12641no—
376C2#2376420811yescharacteristic
377C2#2376429511yescharacteristic
378C2#2376429711yescharacteristic
subgroupG/H
centerC22#19098463
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1518C2
socleC22×C79#390
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#1518D79
hypocenterC79#9757
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1518

Complements

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

NH ≅ G/Ntype
2. #23763530C2semidirect
3. #23761964C2semidirect
6. #23761684C2semidirect
7. C79×#1518C2semidirect
11. #19098915C4semidirect
12. #19098915C4semidirect
13. #19098911C4semidirect
14. #19098911C4semidirect
193. #1518D79semidirect
197. C79#9757semidirect

Maximal subgroups

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

subgroupindex
#237737252
#237635302
#237619642
#237619612
#237619592
#237616842
C79×#15182
#975779