Finite Group Databasebeta

← #34217418

Subgroups — #34217418

Order 20224.

1558 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23761649C214yesmaximal, complemented
2#23761641C212yesmaximal
3#23761637C214yesmaximal, complemented
4#23761632C212yesmaximal, complemented
5#23761627C212yesmaximal, complemented
6C2×#390×C79C211yesmaximal, characteristic
7C22×C22⋊Dic79C2212yes—
8C22×C22⋊Dic79C2214yes—
9C22×C22⋊Dic79C2212yes—
10C22×C4×Dic79C2212yes—
11C2×C79⋊#95C2214yes—
12C2×C79⋊#95C2214yes—
13C2×C79⋊#95C2212yes—
14C2×C79⋊#95C2212yes—
15C2×C79⋊#95C2214yes—
16C2×C79⋊#95C2212yes—
17C23×D4×C79C2211yescharacteristic
18C22×C4⋊C4×C79C2211yescharacteristic
19C22×C22⋊C4×C79C2214yes—
20C22×C42×C79C2211yescharacteristic
21#390×C79C418yescomplemented
22C23×Dic79C2314yes—
23C23×Dic79C2314yes—
24C2×C22⋊Dic79—28no—
25C2×C22⋊Dic79—28no—
26C2×C22⋊Dic79—42no—
27C2×C22⋊Dic79—44no—
28C2×C22⋊Dic79—44no—
29C2×C22⋊Dic79—42no—
30C2×C22⋊Dic79—44no—
31C2×C4×Dic79—44no—
32C2×C4×Dic79—42no—
33C79⋊#95—28no—
34C79⋊#95—28no—
35C79⋊#95—44no—
36C79⋊#95—42no—
37C79⋊#95—42no—
38C79⋊#95—42no—
39C25×C79C2312yes—
40C22×D4×C79—24no—
41C22×D4×C79—42no—
42C22×D4×C79—41no—
43C22×D4×C79—41no—
44C22×D4×C79C2×C414yes—
45C23×C4×C79C2312yes—
46C23×C4×C79C2312yes—
47C23×C4×C79C2311yescharacteristic
48C2×C4⋊C4×C79—41no—
49C2×C4⋊C4×C79—41no—
50C2×C4⋊C4×C79C2×C414yes—
51C2×C22⋊C4×C79—44no—
52C2×C22⋊C4×C79—44no—
53C2×C22⋊C4×C79C2×C4116yes—
54C2×C42×C79—42no—
55C2×C42×C79C2×C414yes—
56C22×Dic79—24no—
57C22×Dic79—24no—
58C22×Dic79—24no—
59C22×Dic79—24no—
60C22×Dic79—24no—
61C22×Dic79—24no—
62C22×Dic79—24no—
63C22×Dic79—24no—
64C22×Dic79—24no—
65C22×Dic79—24no—
66C22×Dic79—24no—
67C22×Dic79—24no—
68C22×Dic79—24no—
69C22×Dic79#4118yes—
70C22⋊Dic79—48no—
71C22⋊Dic79—416no—
72C22⋊Dic79—48no—
73C4×Dic79—48no—
74C24×C79—24no—
75C24×C79—24no—
76C24×C79—24no—
77C24×C79—24no—
78C24×C79—24no—
79C24×C79—24no—
80C24×C79—22no—
81C24×C79—22no—
82C24×C79C22×C414yes—
83C24×C79C2411yescharacteristic
84C2×D4×C79—28no—
85C2×D4×C79—44no—
86C2×D4×C79—44no—
87C2×D4×C79—48no—
88C2×D4×C79—48no—
89C22×C4×C79—22no—
90C22×C4×C79—21no—
91C22×C4×C79—21no—
92C22×C4×C79—24no—
93C22×C4×C79—22no—
94C22×C4×C79—22no—
95C22×C4×C79—24no—
96C22×C4×C79—24no—
97C22×C4×C79—24no—
98C22×C4×C79—22no—
99C22×C4×C79—22no—
100C22×C4×C79C22×C414yes—
101C22×C4×C79C22×C414yes—
102C22×C4×C79C22×C412yes—
103C22×C4×C79#4114yes—
104C4⋊C4×C79—44no—
105C22⋊C4×C79—416no—
106C42×C79—44no—
107C23×C79—24no—
108C23×C79—28no—
109C23×C79—28no—
110C23×C79—24no—
111C23×C79—24no—
112C23×C79—44no—
113C23×C79—44no—
114C23×C79—44no—
115C23×C79—44no—
116C23×C79—44no—
117C23×C79—44no—
118C23×C79—42no—
119C23×C79—42no—
120C23×C79—44no—
121C23×C79—44no—
122C23×C79—44no—
123C23×C79—44no—
124C23×C79—44no—
125C23×C79—42no—
126C23×C79—44no—
127C23×C79—42no—
128C23×C79C23×C411yescharacteristic
129C23×C79C2×#4114yes—
130C23×C79C2×#4112yes—
131C23×C79H(5,𝔽2)12yes—
132C23×C79H(5,𝔽2)11yescharacteristic
133C23×C79H(5,𝔽2)11yescharacteristic
134C23×C79H(5,𝔽2)14yes—
135D4×C79—416no—
136C2×C4×C79—22no—
137C2×C4×C79—22no—
138C2×C4×C79—24no—
139C2×C4×C79—24no—
140C2×C4×C79—24no—
141C2×C4×C79—24no—
142C2×C4×C79—24no—
143C2×C4×C79—28no—
144C2×C4×C79—24no—
145C2×C4×C79—44no—
146C2×C4×C79—44no—
147C2×C4×C79—42no—
148C2×C4×C79—42no—
149C2×C4×C79—44no—
150C2×C4×C79#11714yes—
151C2×Dic79—28no—
152C2×Dic79—28no—
153C2×Dic79—28no—
154C2×Dic79—44no—
155C2×Dic79—44no—
156C2×Dic79—44no—
157C2×Dic79—44no—
158C2×Dic79—44no—
159C2×Dic79—44no—
160C2×Dic79—44no—
161C2×Dic79—44no—
162C2×Dic79—44no—
163C2×Dic79—44no—
164C2×Dic79—44no—
165C22×C79—44no—
166C22×C79—48no—
167C22×C79—48no—
168C22×C79—44no—
169C22×C79—44no—
170C22×C79—84no—
171C22×C79—84no—
172C22×C79—84no—
173C22×C79—84no—
174C22×C79C2×#11711yescharacteristic
175C22×C79C4×#4112yes—
176C22×C79#51811yescharacteristic
177C22×C79#51811yescharacteristic
178C22×C79#51812yes—
179C22×C79#53814yes—
180C22×C79#53812yes—
181C22×C79#53812yes—
182C22×C79#54014yes—
183C22×C79#55112yes—
184C22×C79#55314yes—
185C22×C79#55512yes—
186C22×C79#55914yes—
187C22×C79#56012yes—
188C22×C79#56012yes—
189C4×C79—24no—
190C4×C79—44no—
191C4×C79—44no—
192Dic79—48no—
193Dic79—84no—
194#14503—791nomaximal, Sylow
195C2×C79—84no—
196C2×C79#231211yescharacteristic
197C2×C79#233312yes—
198C2×C79#235712yes—
199C2×C79#237211yescharacteristic
200C2×C79#238011yescharacteristic
201C2×C79#279714yes—
202C2×C79#280314yes—
203C2×#390D7911yescharacteristic
204#1440—794no—
205#1438—792no—
206#1437—794no—
207#1436—792no—
208#1434—792no—
209C79#1450311yescharacteristic, complemented, Sylow
210C23×D4C2×D7911yescharacteristic
211C22×C4⋊C4C2×D7911yescharacteristic
212C22×C22⋊C4—792no—
213C22×C22⋊C4—794no—
214C22×C22⋊C4—792no—
215C22×C22⋊C4C2×D7914yes—
216C22×C42—792no—
217C22×C42C2×D7911yescharacteristic
218#390Dic7918yescomplemented
219C2×#95—792no—
220C2×#95—792no—
221C2×#95—794no—
222C2×#95—792no—
223C2×#95—794no—
224C2×#95—794no—
225C25C22×D7912yes—
226C22×D4—24no—
227C22×D4—42no—
228C22×D4—41no—
229C22×D4—41no—
230C22×D4C2×Dic7914yes—
231C23×C4—794no—
232C23×C4—794no—
233C23×C4C22×D7912yes—
234C23×C4C22×D7912yes—
235C23×C4C22×D7911yescharacteristic
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4C2×Dic7914yes—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—3162no—
242C2×C22⋊C4—3164no—
243C2×C22⋊C4—3164no—
244C2×C22⋊C4—3164no—
245C2×C22⋊C4—3162no—
246C2×C22⋊C4—1588no—
247C2×C22⋊C4—1588no—
248C2×C22⋊C4C2×Dic79116yes—
249C2×C42—42no—
250C2×C42—3164no—
251C2×C42—3162no—
252C2×C42C2×Dic7914yes—
253#95—3162no—
254#95—3162no—
255#95—3164no—
256#95—3162no—
257#95—1588no—
258#95—1588no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—22no—
266C24—22no—
267C24C22×Dic7914yes—
268C24C23×D7911yescharacteristic
269C2×D4—28no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—48no—
273C2×D4—48no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—798no—
286C22×C4—1584no—
287C22×C4—1584no—
288C22×C4—1584no—
289C22×C4—1584no—
290C22×C4—1584no—
291C22×C4—1584no—
292C22×C4—1584no—
293C22×C4—1584no—
294C22×C4—1584no—
295C22×C4—1584no—
296C22×C4—1584no—
297C22×C4—1584no—
298C22×C4—1584no—
299C22×C4C22×Dic7914yes—
300C22×C4C22×Dic7914yes—
301C22×C4C22×Dic7912yes—
302C22×C4#1192962414yes—
303C4⋊C4—44no—
304C22⋊C4—416no—
305C22⋊C4—3168no—
306C22⋊C4—31616no—
307C22⋊C4—3168no—
308C42—44no—
309C42—3168no—
310C23—24no—
311C23—28no—
312C23—28no—
313C23—24no—
314C23—24no—
315C23—44no—
316C23—44no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—42no—
322C23—42no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—42no—
329C23—44no—
330C23—42no—
331C23C23×Dic7911yescharacteristic
332C23#41×D7914yes—
333C23C2×#1192962412yes—
334C23#1457920612yes—
335C23#1457920614yes—
336C23#1457920611yescharacteristic
337C23#1457920611yescharacteristic
338D4—416no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—28no—
347C2×C4—24no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—3164no—
354C2×C4—3164no—
355C2×C4—3164no—
356C2×C4—3164no—
357C2×C4—3164no—
358C2×C4—3164no—
359C2×C4—3164no—
360C2×C4—3164no—
361C2×C4—3164no—
362C2×C4—3164no—
363C2×C4—3164no—
364C2×C4—1588no—
365C2×C4—1588no—
366C2×C4—1588no—
367C2×C4#1457912014yes—
368C22—44no—
369C22—48no—
370C22—48no—
371C22—44no—
372C22—44no—
373C22—84no—
374C22—84no—
375C22—84no—
376C22—84no—
377C22C2×#1457912011yescharacteristic
378C22#41×Dic7912yes—
379C22#1909903812yes—
380C22#1909903811yescharacteristic
381C22#1909903811yescharacteristic
382C22#1909914514yes—
383C22#1909915112yes—
384C22#1909915112yes—
385C22#1909916314yes—
386C22#1909924312yes—
387C22#1909926314yes—
388C22#1909927512yes—
389C22#1909929614yes—
390C22#1909930012yes—
391C22#1909930012yes—
392C4—24no—
393C4—44no—
394C4—44no—
395C4—3168no—
396C4—6324no—
397C2—84no—
398C2#2376599811yescharacteristic
399C2#2376616712yes—
400C2#2376637312yes—
401C2#2376651211yescharacteristic
402C2#2376657711yescharacteristic
403C2#2377125414yes—
404C2#2377132914yes—
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC2×#390×C79C2
socle, cosocleC24×C79C24
prime radicalC23×D4×C79C22
hypercenterC2×#390D79
hypocenterC79#14503
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. #23761649C2semidirect
3. #23761637C2semidirect
4. #23761632C2semidirect
5. #23761627C2semidirect
21. #390×C79C4semidirect
209. C79#14503semidirect
218. #390Dic79semidirect

Maximal subgroups

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

subgroupindex
#237616492
#237616412
#237616372
#237616322
#237616272
C2×#390×C792
#1450379