Finite Group Databasebeta

← #34183820

Subgroups — #34183820

Order 20224.

560 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#23772429C211yesmaximal, characteristic
2#23761718C211yesmaximal, characteristic
3C2×#19098289C212yesmaximal, complemented
4#23761248C212yesmaximal, complemented
5C79×#2322C211yesmaximal, characteristic
6#19098635—24no—
7#19098620—21no—
8#19098527—21no—
9C2×C22⋊(C79⋊C8)C2212yes—
10C2×C22⋊(C79⋊C8)C2212yes—
11#19098289—24no—
12#19098289—21no—
13#19098287—21no—
14C2×#123×C79C411yescharacteristic
15C2×C4×D4×C79C2211yescharacteristic
16C22×C4⋊C4×C79C2211yescharacteristic
17#389×C79—21no—
18#388×C79—21no—
19#387×C79—21no—
20#386×C79C411yescharacteristic
21#385×C79—21no—
22#385×C79C411yescharacteristic
23#381×C79C411yescharacteristic
24C4×C22⋊C4×C79C2211yescharacteristic
25C2×#11929608—22no—
26C22×C79⋊C8—22no—
27C22×C79⋊C8—22no—
28C4⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)D414yes—
34#14579072—22no—
35C22×D4×C79C2×C411yescharacteristic
36C23×C4×C79C2×C411yescharacteristic
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79D411yescharacteristic
39C23×C4×C79C2311yescharacteristic
40#123×C79—22no—
41#123×C79—22no—
42C4×D4×C79—22no—
43C4×D4×C79—22no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—22no—
46C2×C4⋊C4×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—22no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79D411yescharacteristic
51C2×C22⋊C4×C79—21no—
52C2×C22⋊C4×C79—21no—
53C2×C22⋊C4×C79C2×C411yescharacteristic
54C2×C22⋊C4×C79C2×C411yescharacteristic
55C2×C22⋊C4×C79C2×C411yescharacteristic
56C2×C22⋊C4×C79D411yescharacteristic
57C2×C42×C79—21no—
58C2×C42×C79D411yescharacteristic
59#95×C79—21no—
60#95×C79—21no—
61#95×C79—21no—
62#95×C79—21no—
63#95×C79—21no—
64#11929608—82no—
65C2×C79⋊C8—24no—
66C2×C79⋊C8—42no—
67C2×C79⋊C8—82no—
68C2×C79⋊C8—82no—
69C24×C79C22⋊C411yescharacteristic
70C24×C79C22×C411yescharacteristic
71C2×D4×C79—22no—
72C2×D4×C79—21no—
73C2×D4×C79—21no—
74C2×D4×C79—22no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—22no—
78C22×C4×C79—21no—
79C22×C4×C79—22no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—22no—
85C22×C4×C79—81no—
86C22×C4×C79—81no—
87C22×C4×C79—81no—
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79M1612yescomplemented
92C22×C4×C79C2×D412yes—
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79#4112yes—
96C4⋊C4×C79—24no—
97C4⋊C4×C79—22no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—42no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—81no—
102C22⋊C4×C79—24no—
103C22⋊C4×C79—22no—
104C22⋊C4×C79—21no—
105C22⋊C4×C79—22no—
106C22⋊C4×C79—22no—
107C22⋊C4×C79—81no—
108C22⋊C4×C79M1612yescomplemented
109C42×C79—21no—
110C42×C79—21no—
111C42×C79—81no—
112C23×C79—21no—
113C23×C79—21no—
114C23×C79—21no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79C2×C22⋊C411yescharacteristic
120C23×C79C4×D412yes—
121C23×C79C2×M1611yescharacteristic
122C23×C79C2×M1611yescharacteristic
123C23×C79#13112yes—
124D4×C79—42no—
125D4×C79—42no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—22no—
130C2×C4×C79—22no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—24no—
134C2×C4×C79—42no—
135C2×C4×C79—42no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—42no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—42no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—81no—
153C2×C4×C79—82no—
154C2×C4×C79#12011yescharacteristic
155C2×C4×C79#12112yes—
156C2×C4×C79#12311yescharacteristic
157C79⋊C8—44no—
158C79⋊C8—82no—
159C22×C79—21no—
160C22×C79—22no—
161C22×C79—21no—
162C22×C79—22no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—42no—
166C22×C79—42no—
167C22×C79—41no—
168C22×C79—41no—
169C22×C79—41no—
170C22×C79—81no—
171C22×C79#38611yescharacteristic
172C22×C79#40711yescharacteristic
173C22×C79#40811yescharacteristic
174C22×C79#43512yes—
175C22×C79#43612yes—
176C4×C79—21no—
177C4×C79—21no—
178C4×C79—22no—
179C4×C79—42no—
180C4×C79—42no—
181C4×C79—41no—
182C4×C79—41no—
183C4×C79—41no—
184C4×C79—81no—
185C4×C79—81no—
186#9110—791nomaximal, Sylow
187C2×C79—21no—
188C2×C79—21no—
189C2×C79—22no—
190C2×C79—22no—
191C2×C79—81no—
192C2×C79#187011yescharacteristic
193C2×C79#191311yescharacteristic
194C2×C79#191511yescharacteristic
195#2973—791no—
196#2322D7911yescharacteristic
197#1459—791no—
198C2×#324—792no—
199#1282—792no—
200C79#911011yescharacteristic, complemented, Sylow
201C2×#123Dic7911yescharacteristic
202C2×C4×D4C2×D7911yescharacteristic
203C22×C4⋊C4C2×D7911yescharacteristic
204#435—1584no—
205#432—1581no—
206#407—1581no—
207C2×C22⋊C8—792no—
208C2×C22⋊C8—792no—
209#389—21no—
210#388—21no—
211#387—21no—
212#386Dic7911yescharacteristic
213#385—21no—
214#385Dic7911yescharacteristic
215#381Dic7911yescharacteristic
216C4×C22⋊C4C2×D7911yescharacteristic
217#324—1584no—
218#324—1581no—
219#323—1581no—
220C22×D4C2×Dic7911yescharacteristic
221C23×C4C2×Dic7911yescharacteristic
222C23×C4C2×Dic7911yescharacteristic
223C23×C4C22⋊D7911yescharacteristic
224C23×C4C22×D7911yescharacteristic
225C2×M16—1582no—
226C22×C8—1582no—
227C22×C8—1582no—
228#123—22no—
229#123—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C22⋊D7911yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4C2×Dic7911yescharacteristic
242C2×C22⋊C4C2×Dic7911yescharacteristic
243C2×C22⋊C4C2×Dic7911yescharacteristic
244C2×C22⋊C4C22⋊D7911yescharacteristic
245C2×C42—21no—
246C2×C42C22⋊D7911yescharacteristic
247C4⋊C8—1582no—
248C22⋊C8—794no—
249C22⋊C8—1582no—
250C22⋊C8—1582no—
251C22⋊C8—1582no—
252C22⋊C8—1582no—
253#97—1582no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259C24C22⋊Dic7911yescharacteristic
260C24C22×Dic7911yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4C22⋊Dic7911yescharacteristic
279C22×C4C22⋊Dic7911yescharacteristic
280C22×C4C22⋊Dic7911yescharacteristic
281C22×C4#1192960812yescomplemented
282C22×C4D4×D7912yes—
283C22×C4C2×C22⋊D7911yescharacteristic
284C22×C4C2×C22⋊D7911yescharacteristic
285C22×C4#1192962312yes—
286M16—6322no—
287C2×C8—3162no—
288C2×C8—6322no—
289C2×C8—6322no—
290C2×C8—1584no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—81no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—21no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—81no—
303C22⋊C4#1192960812yescomplemented
304C42—21no—
305C42—21no—
306C42—81no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C2×C22⋊Dic7911yescharacteristic
315C23D4×Dic7912yes—
316C23C2×#1192960811yescharacteristic
317C23C2×#1192960811yescharacteristic
318C23#1457916812yes—
319D4—42no—
320D4—42no—
321C2×C4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—82no—
349C2×C4#1457912911yescharacteristic
350C2×C4#1457913312yes—
351C2×C4#1457914211yescharacteristic
352C8—3164no—
353C8—6322no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1909844311yescharacteristic
367C22#1909852711yescharacteristic
368C22#1909853011yescharacteristic
369C22#1909863512yes—
370C22#1909864112yes—
371C4—21no—
372C4—22no—
373C4—21no—
374C4—42no—
375C4—41no—
376C4—41no—
377C4—42no—
378C4—41no—
379C4—81no—
380C4—81no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#2376369811yescharacteristic
387C2#2376394311yescharacteristic
388C2#2376395311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2322C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#2322D79
hypocenterC79#9110
LCSG ▷ C24×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2322

Complements

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

NH ≅ G/Ntype
3. C2×#19098289C2semidirect
4. #23761248C2semidirect
91. C22×C4×C79M16semidirect
108. C22⋊C4×C79M16semidirect
200. C79#9110semidirect
281. C22×C4#11929608semidirect
303. C22⋊C4#11929608semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237617182
C2×#190982892
#237612482
C79×#23222
#911079