Finite Group Databasebeta

← #34204289

Subgroups — #34204289

Order 20224.

420 conjugacy classes of subgroups.

The subgroup lattice has 372 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×#2930C211yesmaximal, 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
15C22×M16×C79C411yescharacteristic, complemented
16#525×C79C2211yescharacteristic
17#436×C79—21no—
18#436×C79—21no—
19#435×C79—21no—
20#435×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×C79—21no—
40C2×M16×C79—21no—
41C2×M16×C79—41no—
42C2×M16×C79—41no—
43C2×M16×C79C2×C411yescharacteristic
44C2×M16×C79C2×C411yescharacteristic
45C2×M16×C79D411yescharacteristic
46C2×M16×C79D411yescharacteristic
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—
95M16×C79—42no—
96M16×C79—41no—
97M16×C79—41no—
98C2×C8×C79—21no—
99C2×C8×C79—21no—
100C2×C8×C79—21no—
101C2×C8×C79—21no—
102C2×C8×C79—21no—
103C2×C8×C79—21no—
104C2×C8×C79—21no—
105C2×C8×C79—21no—
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—81no—
124C23×C79C2×C4⋊C411yescharacteristic
125C23×C79#12111yescharacteristic
126C23×C79C22⋊Q811yescharacteristic
127Q8×C79—41no—
128D4×C79—41no—
129D4×C79—41no—
130D4×C79—41no—
131D4×C79—81no—
132D4×C79—81no—
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—21no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—41no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—41no—
166C22×C79—41no—
167C22×C79—41no—
168C22×C79—82no—
169C22×C79#38011yescharacteristic
170C4×C79—21no—
171C4×C79—21no—
172C4×C79—21no—
173C4×C79—41no—
174C4×C79—41no—
175C4×C79—41no—
176C4×C79C2×#9911yescharacteristic
177C4×C79#41011yescharacteristic
178Dic79—162no—
179#12806—791nomaximal, Sylow
180C2×C79—21no—
181C2×C79—41no—
182C2×C79—41no—
183C2×C79—41no—
184C2×C79—81no—
185C2×C79#179911yescharacteristic
186#2930D7911yescharacteristic
187#2120—792no—
188#1813—791no—
189#1812—791no—
190#1395—792no—
191C79#1280611yescharacteristic, complemented, Sylow
192C2×#131C2×D7911yescharacteristic
193C22×M16Dic7911yescharacteristic, complemented
194#525C2×D7911yescharacteristic
195#436—21no—
196#436—21no—
197#435—21no—
198#435—21no—
199C2×#108—1581no—
200#427—1581no—
201#424Dic7911yescharacteristic, complemented
202#413—792no—
203#410—792no—
204#408C2×D7911yescharacteristic
205#407Dic7911yescharacteristic, complemented
206#405Dic7911yescharacteristic, complemented
207#352—1582no—
208AΣL(1,𝔽16)—1582no—
209#338—1581no—
210H(3,ℤ/4)—1581no—
211C2×#41C22×D7911yescharacteristic
212C23×C4C2×Dic7911yescharacteristic
213#131—41no—
214#131—41no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—41no—
218C2×M16—41no—
219C2×M16—1582no—
220C2×M16C2×Dic7911yescharacteristic
221C2×M16C2×Dic7911yescharacteristic
222C2×M16C22⋊D7911yescharacteristic
223C2×M16C22⋊D7911yescharacteristic
224C22×C8C2×Dic7911yescharacteristic
225C22×C8C2×Dic7911yescharacteristic
226C22×C8C22⋊D7911yescharacteristic
227C22×C8C22⋊D7911yescharacteristic
228#123Dic15811yescharacteristic
229#123D31611yescharacteristic
230C22⋊Q8—21no—
231#121—21no—
232#120Dic15811yescharacteristic
233#120D31611yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236#117—1582no—
237#117C2×Dic7911yescharacteristic
238#108—3161no—
239#108—3161no—
240#107—3161no—
241#106—3161no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244#101—1582no—
245#100—1582no—
246#99—1582no—
247#99—1582no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252#97—21no—
253C4×C8—21no—
254C24C4⋊Dic7911yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8#1192962411yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4C2×D31611yescharacteristic
261C2×D4C2×Dic15811yescharacteristic
262C2×D4#1192962411yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C4⋊Dic7911yescharacteristic
268C22×C4C22×Dic7911yescharacteristic
269C22×C4C2×C22⋊D7911yescharacteristic
270C22×C4C2×C22⋊D7911yescharacteristic
271M16—41no—
272M16—41no—
273M16—41no—
274M16—42no—
275M16—41no—
276M16—41no—
277M16—41no—
278M16—3162no—
279M16—3162no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—3162no—
290C2×C8C22⋊Dic7911yescharacteristic
291C2×C8C22⋊Dic7911yescharacteristic
292C2×C8C22⋊Dic7911yescharacteristic
293C2×C8C22⋊Dic7911yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—3162no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—3162no—
303C22⋊C4C4⋊Dic7911yescharacteristic
304C22⋊C4C4⋊Dic7911yescharacteristic
305C42—21no—
306C42—3162no—
307C23—21no—
308C23—41no—
309C23—81no—
310C23C2×C4⋊Dic7911yescharacteristic
311C23#1457913511yescharacteristic
312C23#1457914011yescharacteristic
313Q8—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—81no—
318D4—81no—
319C2×C4—21no—
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—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—6322no—
333C2×C4C2×C22⋊Dic7911yescharacteristic
334C2×C4#1457912011yescharacteristic
335C2×C4#1457913511yescharacteristic
336C2×C4#1457914011yescharacteristic
337C8—21no—
338C8—21no—
339C8—21no—
340C8—41no—
341C8—41no—
342C8—6322no—
343C8#1457907611yescharacteristic
344C8#1457907611yescharacteristic
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—82no—
355C22#1909842311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—12642no—
363C4C2×#1457907611yescharacteristic
364C4#1909853711yescharacteristic
365C2—21no—
366C2—41no—
367C2—41no—
368C2—41no—
369C2—81no—
370C2#2376338811yescharacteristic
subgroupG/H
centerC4#19098537
commutatorC22×C4×C79C22×C4
FrattiniC2×#41C22×D79
FittingC79×#2930C2
socleC2×C79#1799
prime radical#525×C79C22
cosocleC2×#41×C79C23
hypercenter#2930D79
hypocenterC79#12806
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ #2930

Complements

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

NH ≅ G/Ntype
2. #23763450C2semidirect
3. #23763446C2semidirect
4. #23761529C2semidirect
15. C22×M16×C79C4semidirect
21. #424×C79C4semidirect
23. #407×C79C4semidirect
24. #405×C79C4semidirect
191. C79#12806semidirect
193. C22×M16Dic79semidirect
201. #424Dic79semidirect
205. #407Dic79semidirect
206. #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×#29302
#1280679