Finite Group Databasebeta

← #34204284

Subgroups — #34204284

Order 20224.

434 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23765041C212yesmaximal
2#23763450C211yesmaximal, characteristic, complemented
3#23763446C211yesmaximal, characteristic, complemented
4#23761529C212yesmaximal, complemented
5C79×#2929C211yesmaximal, characteristic
6C2×#14579097—21no—
7#19098601—21no—
8#19098547C2212yes—
9#19098537C2212yes—
10#19098353—22no—
11#19098350—22no—
12C79⋊#338—21no—
13#19098320—21no—
14C2×#131×C79C2211yescharacteristic
15C23×C8×C79C411yescharacteristic, complemented
16#525×C79C2211yescharacteristic
17#435×C79—21no—
18#435×C79—21no—
19C8×D4×C79—21no—
20C8×D4×C79—21no—
21#424×C79C411yescharacteristic, complemented
22#408×C79C2211yescharacteristic
23#407×C79C411yescharacteristic, complemented
24#405×C79C411yescharacteristic, complemented
25C2×#11929608—22no—
26#14579120—22no—
27#14579097—41no—
28#14579097—41no—
29#14579095—41no—
30#14579093—41no—
31#14579080—22no—
32#14579078—22no—
33#14579076—22no—
34#14579076—22no—
35C2×#41×C79C2311yescharacteristic
36C23×C4×C79C2×C411yescharacteristic
37#131×C79—41no—
38#131×C79—41no—
39C2×M16×C79C2×C411yescharacteristic
40C2×M16×C79C2×C411yescharacteristic
41C2×M16×C79D411yescharacteristic
42C2×M16×C79D411yescharacteristic
43C22×C8×C79—21no—
44C22×C8×C79—21no—
45C22×C8×C79—41no—
46C22×C8×C79—41no—
47C22×C8×C79C2×C411yescharacteristic
48C22×C8×C79C2×C411yescharacteristic
49C22×C8×C79D411yescharacteristic
50C22×C8×C79D411yescharacteristic
51#123×C79D411yescharacteristic
52#123×C79Q811yescharacteristic
53C22⋊Q8×C79—21no—
54#121×C79—21no—
55#120×C79D411yescharacteristic
56#120×C79Q811yescharacteristic
57C4×D4×C79—21no—
58C4×D4×C79—21no—
59#117×C79C2×C411yescharacteristic
60C4⋊C8×C79—21no—
61C4⋊C8×C79—21no—
62C22⋊C8×C79—21no—
63C22⋊C8×C79—21no—
64C22⋊C8×C79—21no—
65C22⋊C8×C79—21no—
66#97×C79—21no—
67C4×C8×C79—21no—
68#11929608—42no—
69#11929608—42no—
70C2×C79⋊C8—42no—
71C4⋊Dic79—42no—
72C22⋊Dic79—42no—
73C4×Dic79—42no—
74C24×C79C4⋊C411yescharacteristic
75#41×C79—41no—
76#41×C79—41no—
77C2×Q8×C79#4111yescharacteristic
78C2×D4×C79—21no—
79C2×D4×C79—21no—
80C2×D4×C79C2×D411yescharacteristic
81C2×D4×C79C2×Q811yescharacteristic
82C2×D4×C79#4111yescharacteristic
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—41no—
86C22×C4×C79—41no—
87C22×C4×C79C4⋊C411yescharacteristic
88C22×C4×C79C22×C411yescharacteristic
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91M16×C79—41no—
92M16×C79—41no—
93M16×C79—41no—
94M16×C79—41no—
95C2×C8×C79—21no—
96C2×C8×C79—21no—
97C2×C8×C79—21no—
98C2×C8×C79—21no—
99C2×C8×C79—21no—
100C2×C8×C79—21no—
101C2×C8×C79—21no—
102C2×C8×C79—21no—
103C2×C8×C79—41no—
104C2×C8×C79—42no—
105C2×C8×C79—41no—
106C2×C8×C79—41no—
107C2×C8×C79C22⋊C411yescharacteristic
108C2×C8×C79C22⋊C411yescharacteristic
109C2×C8×C79C22⋊C411yescharacteristic
110C2×C8×C79C22⋊C411yescharacteristic
111C4⋊C4×C79—21no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C22⋊C4×C79—21no—
115C22⋊C4×C79—21no—
116C22⋊C4×C79—21no—
117C22⋊C4×C79—21no—
118C22⋊C4×C79C4⋊C411yescharacteristic
119C22⋊C4×C79C4⋊C411yescharacteristic
120C42×C79—21no—
121C23×C79—21no—
122C23×C79—41no—
123C23×C79—42no—
124C23×C79C2×C4⋊C411yescharacteristic
125C23×C79#12111yescharacteristic
126C23×C79C22⋊Q811yescharacteristic
127Q8×C79—41no—
128D4×C79—41no—
129D4×C79—41no—
130D4×C79—41no—
131D4×C79—42no—
132D4×C79—42no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—42no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79C2×C22⋊C411yescharacteristic
147C2×C4×C79#11711yescharacteristic
148C2×C4×C79#12111yescharacteristic
149C2×C4×C79C22⋊Q811yescharacteristic
150C2×Dic79—82no—
151C8×C79—21no—
152C8×C79—21no—
153C8×C79—21no—
154C8×C79—41no—
155C8×C79—41no—
156C8×C79#9911yescharacteristic
157C8×C79#9911yescharacteristic
158C79⋊C8—82no—
159C22×C79—22no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—21no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—41no—
166C22×C79—41no—
167C22×C79—42no—
168C22×C79—42no—
169C22×C79—41no—
170C22×C79#38011yescharacteristic
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—41no—
175C4×C79—41no—
176C4×C79—41no—
177C4×C79C2×#9911yescharacteristic
178C4×C79#41011yescharacteristic
179Dic79—162no—
180#12805—791nomaximal, Sylow
181C2×C79—21no—
182C2×C79—41no—
183C2×C79—41no—
184C2×C79—42no—
185C2×C79—41no—
186C2×C79#179911yescharacteristic
187#2929D7911yescharacteristic
188#2120—792no—
189#1813—791no—
190#1812—791no—
191#1395—792no—
192C79#1280511yescharacteristic, complemented, Sylow
193C2×#131C2×D7911yescharacteristic
194C23×C8Dic7911yescharacteristic, complemented
195#525C2×D7911yescharacteristic
196#435—21no—
197#435—21no—
198C8×D4—21no—
199C8×D4—21no—
200C2×#108—1581no—
201#427—1581no—
202#424Dic7911yescharacteristic, complemented
203#413—792no—
204#410—792no—
205#408C2×D7911yescharacteristic
206#407Dic7911yescharacteristic, complemented
207#405Dic7911yescharacteristic, complemented
208#352—1582no—
209AΣL(1,𝔽16)—1582no—
210#338—1581no—
211H(3,ℤ/4)—1581no—
212C2×#41C22×D7911yescharacteristic
213C23×C4C2×Dic7911yescharacteristic
214#131—41no—
215#131—41no—
216C2×M16—1582no—
217C2×M16C2×Dic7911yescharacteristic
218C2×M16C2×Dic7911yescharacteristic
219C2×M16C22⋊D7911yescharacteristic
220C2×M16C22⋊D7911yescharacteristic
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—41no—
224C22×C8—41no—
225C22×C8C2×Dic7911yescharacteristic
226C22×C8C2×Dic7911yescharacteristic
227C22×C8C22⋊D7911yescharacteristic
228C22×C8C22⋊D7911yescharacteristic
229#123Dic15811yescharacteristic
230#123D31611yescharacteristic
231C22⋊Q8—21no—
232#121—21no—
233#120Dic15811yescharacteristic
234#120D31611yescharacteristic
235C4×D4—21no—
236C4×D4—21no—
237#117—1582no—
238#117C2×Dic7911yescharacteristic
239#108—3161no—
240#108—3161no—
241#107—3161no—
242#106—3161no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245#101—1582no—
246#100—1582no—
247#99—1582no—
248#99—1582no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253#97—21no—
254C4×C8—21no—
255C24C4⋊Dic7911yescharacteristic
256#41—41no—
257#41—41no—
258C2×Q8#1192962411yescharacteristic
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C2×D31611yescharacteristic
262C2×D4C2×Dic15811yescharacteristic
263C2×D4#1192962411yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4C4⋊Dic7911yescharacteristic
269C22×C4C22×Dic7911yescharacteristic
270C22×C4C2×C22⋊D7911yescharacteristic
271C22×C4C2×C22⋊D7911yescharacteristic
272M16—41no—
273M16—41no—
274M16—41no—
275M16—41no—
276M16—3162no—
277M16—3162no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—3162no—
291C2×C8C22⋊Dic7911yescharacteristic
292C2×C8C22⋊Dic7911yescharacteristic
293C2×C8C22⋊Dic7911yescharacteristic
294C2×C8C22⋊Dic7911yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—3162no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—3162no—
304C22⋊C4C4⋊Dic7911yescharacteristic
305C22⋊C4C4⋊Dic7911yescharacteristic
306C42—21no—
307C42—3162no—
308C23—21no—
309C23—41no—
310C23—42no—
311C23C2×C4⋊Dic7911yescharacteristic
312C23#1457913511yescharacteristic
313C23#1457914011yescharacteristic
314Q8—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—42no—
319D4—42no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—6322no—
334C2×C4C2×C22⋊Dic7911yescharacteristic
335C2×C4#1457912011yescharacteristic
336C2×C4#1457913511yescharacteristic
337C2×C4#1457914011yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—41no—
342C8—41no—
343C8—6322no—
344C8#1457907611yescharacteristic
345C8#1457907611yescharacteristic
346C22—22no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22#1909842311yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—12642no—
365C4C2×#1457907611yescharacteristic
366C4#1909853711yescharacteristic
367C2—21no—
368C2—41no—
369C2—42no—
370C2—41no—
371C2—41no—
372C2#2376338811yescharacteristic
subgroupG/H
centerC4#19098537
commutatorC22×C4×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#2929C2
socleC2×C79#1799
prime radical#525×C79C22
cosocleC2×#41×C79C23
hypercenter#2929D79
hypocenterC79#12805
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #2929

Complements

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

NH ≅ G/Ntype
2. #23763450C2semidirect
3. #23763446C2semidirect
4. #23761529C2semidirect
15. C23×C8×C79C4semidirect
21. #424×C79C4semidirect
23. #407×C79C4semidirect
24. #405×C79C4semidirect
192. C79#12805semidirect
194. C23×C8Dic79semidirect
202. #424Dic79semidirect
206. #407Dic79semidirect
207. #405Dic79semidirect

Maximal subgroups

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

subgroupindex
#237650412
#237634502
#237634462
#237615292
C79×#29292
#1280579