Finite Group Databasebeta

← #34186657

Subgroups — #34186657

Order 20224.

454 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764326C211yesmaximal, characteristic
2#23764205C211yesmaximal, characteristic
3#23764161C211yesmaximal, characteristic, complemented
4#23763270C211yesmaximal, characteristic, complemented
5C2×#19098306C211yesmaximal, characteristic
6#23761234C211yesmaximal, characteristic, complemented
7C79×#1296C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9C2×#14579092—21no—
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×#14579083—21no—
12C2×#14579083—21no—
13C2×#14579081—21no—
14C2×#14579081—21no—
15C2×#14579071—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17C4×C4⋊Dic79C2211yescharacteristic
18#19098325—21no—
19#19098317—21no—
20#19098306C414yescomplemented
21C2×C4⋊C8×C79C2211yescharacteristic
22C2×C4⋊C8×C79C2211yescharacteristic
23#384×C79C2211yescharacteristic
24C22×D316—21no—
25C22×D316—21no—
26C22×D316—21no—
27C22×C79⋊C8—21no—
28C22×C79⋊C8D411yescharacteristic
29C22×C79⋊C8D411yescharacteristic
30#14579158—21no—
31#14579158—21no—
32#14579158—21no—
33#14579158C2×C412yes—
34C2×C4⋊Dic79D411yescharacteristic
35C2×C4⋊Dic79D411yescharacteristic
36C2×C4×Dic79—21no—
37#14579092—42no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)C2×C412yes—
40#14579083—42no—
41#14579083—42no—
42#14579081—42no—
43#14579081—42no—
44#14579071—42no—
45C4×C79⋊C8—21no—
46C4×C79⋊C8—21no—
47C79⋊#95—21no—
48C22×C8×C79—21no—
49C22×C8×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79—21no—
52C2×C4⋊C4×C79—21no—
53C2×C42×C79C2311yescharacteristic
54C4⋊C8×C79—21no—
55C4⋊C8×C79—21no—
56C4⋊C8×C79—21no—
57C4⋊C8×C79C2×C412yes—
58C23×D79—41no—
59C2×D316—22no—
60C2×D316—22no—
61C2×D316—22no—
62C2×D316—41no—
63C2×D316—41no—
64C2×D316—41no—
65C2×D316—41no—
66C2×D316—41no—
67C2×D316—41no—
68C2×D316—41no—
69C2×D316—41no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C2×D316—81no—
74C2×D316—81no—
75C22×Dic79—21no—
76C22×Dic79—21no—
77C2×C79⋊C8—22no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8—21no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8SD1612yes—
86C4⋊Dic79—21no—
87C4⋊Dic79—21no—
88C4⋊Dic79—21no—
89C4⋊Dic79—21no—
90C4×Dic79—41no—
91C4×Dic79—41no—
92C22×C4×C79—41no—
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79#4111yescharacteristic
96C2×C8×C79—22no—
97C2×C8×C79—41no—
98C2×C8×C79—41no—
99C2×C8×C79—41no—
100C2×C8×C79—41no—
101C2×C8×C79—41no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79—41no—
105C4⋊C4×C79—41no—
106C4⋊C4×C79—81no—
107C42×C79C22×C411yescharacteristic
108C42×C79#4111yescharacteristic
109C42×C79#4111yescharacteristic
110C42×C79#4111yescharacteristic
111C23×C79#12111yescharacteristic
112C22×D79—42no—
113C22×D79—81no—
114C22×D79—81no—
115C22×D79—81no—
116C22×D79—81no—
117C22×D79—81no—
118C22×D79—81no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—81no—
122C2×C4×C79—81no—
123C2×C4×C79—81no—
124C2×C4×C79#11711yescharacteristic
125C2×C4×C79C4×D411yescharacteristic
126C2×C4×C79C4×D411yescharacteristic
127C2×C4×C79#12311yescharacteristic
128C2×C4×C79#12411yescharacteristic
129C2×C4×C79#12611yescharacteristic
130C2×C4×C79C2×SD1611yescharacteristic
131C2×C4×C79#13511yescharacteristic
132C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C79#13711yescharacteristic
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—41no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144D316—42no—
145D316—42no—
146D316—42no—
147D316—42no—
148D316—82no—
149D316—82no—
150D316—81no—
151D316—81no—
152D316—81no—
153D316—81no—
154C8×C79—42no—
155C8×C79—81no—
156C79⋊C8—22no—
157C79⋊C8—42no—
158C79⋊C8—41no—
159C22×C79#38811yescharacteristic
160C22×C79#46011yescharacteristic
161C22×C79#46311yescharacteristic
162C22×C79#46811yescharacteristic
163C22×C79#46911yescharacteristic
164C22×C79#47111yescharacteristic
165C22×C79#47211yescharacteristic
166C2×D79—82no—
167C2×D79—82no—
168C2×D79—82no—
169C2×D79—161no—
170C2×D79—161no—
171C2×D79—161no—
172C2×D79—161no—
173C4×C79—21no—
174C4×C79—21no—
175C4×C79—161no—
176C4×C79C4×SD1611yescharacteristic
177C4×C79#44011yescharacteristic
178C4×C79#44211yescharacteristic
179C4×C79#44211yescharacteristic
180C4×C79#48111yescharacteristic
181C4×C79#48211yescharacteristic
182C4×C79#48911yescharacteristic
183C4×C79#49011yescharacteristic
184Dic79—81no—
185Dic79—81no—
186#9572—791nomaximal, Sylow
187C2×C79#168611yescharacteristic
188C2×C79#168811yescharacteristic
189C2×C79#169711yescharacteristic
190C2×C79#170111yescharacteristic
191C2×C79#192611yescharacteristic
192C2×C79#193111yescharacteristic
193C2×C79#193611yescharacteristic
194D79—162no—
195#1982—791no—
196#1957—791no—
197#1948—791no—
198#1770—791no—
199C2×#331—791no—
200#1296D7911yescharacteristic, complemented
201#1277—791no—
202C79#957211yescharacteristic, complemented, Sylow
203C2×#127—791no—
204C2×#106—1581no—
205C2×C4⋊C8—791no—
206C2×C4⋊C8C2×D7911yescharacteristic
207C2×C4⋊C8C2×D7911yescharacteristic
208C2×#102—1581no—
209C2×#102—1581no—
210C2×#102—1581no—
211C2×#102—1581no—
212C2×#97—1581no—
213C2×C4×C8—791no—
214#384C2×D7911yescharacteristic
215C4×C4⋊C4—791no—
216#340—1581no—
217#336—1581no—
218#331—794no—
219C22×D4—1581no—
220C22×D4—1581no—
221C22×D4—1581no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—791no—
225C22×C8—791no—
226C22×C8—1581no—
227#127—792no—
228#127—1581no—
229#127—1581no—
230#127—1581no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—791no—
235C2×C4⋊C4—791no—
236C2×C42—1581no—
237C2×C42C22×D7911yescharacteristic
238#106—3162no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—792no—
243C4⋊C8—1581no—
244C4⋊C8C4×D7912yes—
245#102—3162no—
246#102—3162no—
247#102—3162no—
248#102—3162no—
249#97—3162no—
250C4×C8—1581no—
251C4×C8—1581no—
252#95—1581no—
253C24—3161no—
254C2×D4—3161no—
255C2×D4—3161no—
256C2×D4—3161no—
257C2×D4—3161no—
258C2×D4—3161no—
259C2×D4—3161no—
260C2×D4—3161no—
261C2×D4—3161no—
262C2×D4—3161no—
263C2×D4—3161no—
264C2×D4—3161no—
265C2×D4—6321no—
266C2×D4—6321no—
267C2×D4—1582no—
268C2×D4—1582no—
269C2×D4—1582no—
270C22×C4—41no—
271C22×C4—1581no—
272C22×C4—1581no—
273C22×C4D4×D7911yescharacteristic
274C22×C4D4×D7911yescharacteristic
275C22×C4#1192962411yescharacteristic
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—3161no—
283C2×C8—3161no—
284C2×C8—792no—
285C2×C8—1582no—
286C2×C8—1581no—
287C2×C8—1581no—
288C2×C8—1581no—
289C2×C8—1581no—
290C2×C8—1581no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—1581no—
297C4⋊C4—1581no—
298C4⋊C4—1581no—
299C4⋊C4—1581no—
300C42—3161no—
301C42—3161no—
302C42C2×C4×D7911yescharacteristic
303C42#1192962311yescharacteristic
304C42#1192962411yescharacteristic
305C42#1192962511yescharacteristic
306C23—3162no—
307C23—6321no—
308C23—6321no—
309C23—6321no—
310C23—6321no—
311C23—6321no—
312C23—6321no—
313C23#1457913411yescharacteristic
314D4—3162no—
315D4—3162no—
316D4—3162no—
317D4—3162no—
318D4—6322no—
319D4—6322no—
320D4—6321no—
321D4—6321no—
322D4—6321no—
323D4—6321no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—3161no—
330C2×C4—3161no—
331C2×C4—3161no—
332C2×C4—3161no—
333C2×C4—3161no—
334C2×C4—3161no—
335C2×C4#1457911811yescharacteristic
336C2×C4(C22⋊D79)⋊C411yescharacteristic
337C2×C4#1457912311yescharacteristic
338C2×C4#1457914611yescharacteristic
339C2×C4#1457914711yescharacteristic
340C2×C4#1457915611yescharacteristic
341C2×C4SD16×D7911yescharacteristic
342C2×C4#1457918411yescharacteristic
343C2×C4#1457918511yescharacteristic
344C2×C4#1457918511yescharacteristic
345C2×C4#1457918811yescharacteristic
346C2×C4#1457918811yescharacteristic
347C2×C4#1457918811yescharacteristic
348C2×C4#1457919311yescharacteristic
349C8—42no—
350C8—81no—
351C8—3161no—
352C8—3162no—
353C8—1582no—
354C22—6322no—
355C22—6322no—
356C22—6322no—
357C22—12641no—
358C22—12641no—
359C22—12641no—
360C22—12641no—
361C22#1909845211yescharacteristic
362C22#1909876211yescharacteristic
363C22#1909878111yescharacteristic
364C22#1909881211yescharacteristic
365C22#1909881511yescharacteristic
366C22#1909882911yescharacteristic
367C22#1909883211yescharacteristic
368C4—21no—
369C4—21no—
370C4—161no—
371C4—6321no—
372C4—6321no—
373C4#1909864811yescharacteristic
374C4#1909866211yescharacteristic
375C4#1909867211yescharacteristic
376C4#1909867411yescharacteristic
377C4#1909888311yescharacteristic
378C4#1909889011yescharacteristic
379C4#1909892411yescharacteristic
380C4#1909892611yescharacteristic
381C2—12642no—
382C2#2376287511yescharacteristic
383C2#2376288311yescharacteristic
384C2#2376293611yescharacteristic
385C2#2376296411yescharacteristic
386C2#2376402511yescharacteristic
387C2#2376406011yescharacteristic
388C2#2376409511yescharacteristic
subgroupG/H
centerC23#14579134
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1296C2
socleC23×C79#121
prime radicalC2×#14579158C22
cosocleC2×C42×C79C23
hypercenter#1296D79
hypocenterC79#9572
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
3. #23764161C2semidirect
4. #23763270C2semidirect
6. #23761234C2semidirect
7. C79×#1296C2semidirect
20. #19098306C4semidirect
200. #1296D79semidirect
202. C79#9572semidirect

Maximal subgroups

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

subgroupindex
#237643262
#237642052
#237641612
#237632702
C2×#190983062
#237612342
C79×#12962
#957279