Finite Group Databasebeta

← #34182970

Subgroups — #34182970

Order 20224.

600 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772429C211yesmaximal, characteristic
2#23763468C212yesmaximal, complemented
3#23761718C211yesmaximal, characteristic
4#23761248C212yesmaximal, complemented
5C4×#123×C79C211yesmaximal, characteristic
6#19098635—24no—
7#19098620—21no—
8#19098527—21no—
9C2×C22⋊(C79⋊C8)C2212yes—
10C2×C22⋊(C79⋊C8)C2212yes—
11#19098318—24no—
12#19098289—21no—
13#19098287—21no—
14C2×#123×C79C411yescharacteristic
15C2×C4×D4×C79C2211yescharacteristic
16C22×C42×C79C2211yescharacteristic
17#388×C79—21no—
18#387×C79—21no—
19#386×C79C411yescharacteristic
20#385×C79C411yescharacteristic
21#381×C79C411yescharacteristic
22C4×C4⋊C4×C79—21no—
23C4×C22⋊C4×C79—21no—
24C4×C22⋊C4×C79C2211yescharacteristic
25C2×#11929608—22no—
26C22×C79⋊C8—22no—
27C22×C79⋊C8D414yes—
28C4⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)—22no—
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—21no—
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C22⋊C4×C79—21no—
48C2×C22⋊C4×C79—21no—
49C2×C22⋊C4×C79C2×C411yescharacteristic
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C42×C79—22no—
54C2×C42×C79—21no—
55C2×C42×C79—21no—
56C2×C42×C79—21no—
57C2×C42×C79—22no—
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—44no—
67C2×C79⋊C8—44no—
68C2×C79⋊C8—42no—
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—21no—
78C22×C4×C79—22no—
79C22×C4×C79—22no—
80C22×C4×C79—21no—
81C22×C4×C79—22no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—42no—
86C22×C4×C79—42no—
87C22×C4×C79—42no—
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—42no—
99C22⋊C4×C79—24no—
100C22⋊C4×C79—22no—
101C22⋊C4×C79—21no—
102C22⋊C4×C79—22no—
103C22⋊C4×C79—22no—
104C22⋊C4×C79—42no—
105C22⋊C4×C79M1612yescomplemented
106C42×C79—21no—
107C42×C79—21no—
108C42×C79—42no—
109C42×C79—41no—
110C42×C79—42no—
111C42×C79—41no—
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—24no—
127C2×C4×C79—22no—
128C2×C4×C79—22no—
129C2×C4×C79—24no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—22no—
134C2×C4×C79—22no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—24no—
138C2×C4×C79—41no—
139C2×C4×C79—42no—
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—42no—
148C2×C4×C79—42no—
149C2×C4×C79—41no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—81no—
155C2×C4×C79#12011yescharacteristic
156C2×C4×C79#12112yes—
157C2×C4×C79#12311yescharacteristic
158C79⋊C8—44no—
159C79⋊C8—82no—
160C22×C79—21no—
161C22×C79—22no—
162C22×C79—21no—
163C22×C79—22no—
164C22×C79—41no—
165C22×C79—41no—
166C22×C79—42no—
167C22×C79—42no—
168C22×C79—41no—
169C22×C79—41no—
170C22×C79—41no—
171C22×C79—81no—
172C22×C79#38611yescharacteristic
173C22×C79#40711yescharacteristic
174C22×C79#40811yescharacteristic
175C22×C79#43512yes—
176C22×C79#43612yes—
177C4×C79—21no—
178C4×C79—22no—
179C4×C79—42no—
180C4×C79—42no—
181C4×C79—41no—
182C4×C79—41no—
183C4×C79—42no—
184C4×C79—41no—
185C4×C79—81no—
186C4×C79#45812yes—
187#8968—791nomaximal, Sylow
188C2×C79—21no—
189C2×C79—21no—
190C2×C79—22no—
191C2×C79—22no—
192C2×C79—81no—
193C2×C79#187011yescharacteristic
194C2×C79#189011yescharacteristic
195C2×C79#189111yescharacteristic
196#2973—791no—
197C4×#123D7911yescharacteristic
198#1817—792no—
199#1459—791no—
200#1282—792no—
201C79#896811yescharacteristic, complemented, Sylow
202C2×#123Dic7911yescharacteristic
203C2×C4×D4C2×D7911yescharacteristic
204C22×C42C2×D7911yescharacteristic
205#435—1584no—
206#432—1581no—
207#407—1581no—
208C2×C22⋊C8—792no—
209C2×C22⋊C8—792no—
210#388—21no—
211#387—21no—
212#386Dic7911yescharacteristic
213#385Dic7911yescharacteristic
214#381Dic7911yescharacteristic
215C4×C4⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4C2×D7911yescharacteristic
218#336—1584no—
219#324—1581no—
220#323—1581no—
221C22×D4C2×Dic7911yescharacteristic
222C23×C4C2×Dic7911yescharacteristic
223C23×C4C2×Dic7911yescharacteristic
224C23×C4C22⋊D7911yescharacteristic
225C23×C4C22×D7911yescharacteristic
226C2×M16—1582no—
227C22×C8—794no—
228C22×C8—1582no—
229#123—22no—
230#123—22no—
231C4×D4—22no—
232C4×D4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22⋊D7911yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4C2×Dic7911yescharacteristic
239C2×C22⋊C4C2×Dic7911yescharacteristic
240C2×C22⋊C4C2×Dic7911yescharacteristic
241C2×C22⋊C4C22⋊D7911yescharacteristic
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42C22⋊D7911yescharacteristic
248C4⋊C8—1582no—
249C22⋊C8—1582no—
250C22⋊C8—1582no—
251C22⋊C8—1582no—
252C22⋊C8—1582no—
253C22⋊C8—1582no—
254#97—1582no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24C22⋊Dic7911yescharacteristic
261C24C22×Dic7911yescharacteristic
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4C22⋊Dic7911yescharacteristic
280C22×C4C22⋊Dic7911yescharacteristic
281C22×C4C22⋊Dic7911yescharacteristic
282C22×C4#1192960812yescomplemented
283C22×C4D4×D7912yes—
284C22×C4C2×C22⋊D7911yescharacteristic
285C22×C4C2×C22⋊D7911yescharacteristic
286C22×C4#1192962312yes—
287M16—6322no—
288C2×C8—3164no—
289C2×C8—3164no—
290C2×C8—3162no—
291C2×C8—1584no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C22⋊C4—24no—
296C22⋊C4—22no—
297C22⋊C4—21no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—42no—
301C22⋊C4#1192960812yescomplemented
302C42—21no—
303C42—21no—
304C42—42no—
305C42—41no—
306C42—42no—
307C42—41no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23C2×C22⋊Dic7911yescharacteristic
316C23D4×Dic7912yes—
317C23C2×#1192960811yescharacteristic
318C23C2×#1192960811yescharacteristic
319C23#1457916812yes—
320D4—42no—
321D4—42no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4#1457912911yescharacteristic
352C2×C4#1457913312yes—
353C2×C4#1457914211yescharacteristic
354C8—3164no—
355C8—6322no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22#1909844311yescharacteristic
369C22#1909852711yescharacteristic
370C22#1909853011yescharacteristic
371C22#1909863512yes—
372C22#1909864112yes—
373C4—22no—
374C4—21no—
375C4—42no—
376C4—41no—
377C4—42no—
378C4—41no—
379C4—42no—
380C4—41no—
381C4—81no—
382C4#1909875212yes—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—22no—
387C2—81no—
388C2#2376369811yescharacteristic
389C2#2376381411yescharacteristic
390C2#2376381911yescharacteristic
subgroupG/H
centerC2×C4#14579129
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC4×#123×C79C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC23×C4×C79C23
hypercenterC4×#123D79
hypocenterC79#8968
LCSG ▷ C24×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
2. #23763468C2semidirect
4. #23761248C2semidirect
91. C22×C4×C79M16semidirect
105. C22⋊C4×C79M16semidirect
201. C79#8968semidirect
282. C22×C4#11929608semidirect
301. C22⋊C4#11929608semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237634682
#237617182
#237612482
C4×#123×C792
#896879