Finite Group Databasebeta

← #34203227

Subgroups — #34203227

Order 20224.

768 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098537C212yesmaximal
2#23761521C214yesmaximal, complemented
3C79×#2879C211yesmaximal, characteristic
4C2×#14579120—22no—
5#19098537—22no—
6#19098537—22no—
7#19098537C414yescomplemented
8#19098537C2214yes—
9C2×#14579076—22no—
10C2×#14579076C414yescomplemented
11C22×#41×C79C2211yescharacteristic
12C22×M16×C79C411yescharacteristic, complemented
13C22×M16×C79C411yescharacteristic, complemented
14#408×C79—22no—
15#408×C79C412yescomplemented
16#408×C79C2212yes—
17#407×C79—22no—
18#14579120—24no—
19#14579120—24no—
20#14579120—42no—
21#14579120—42no—
22C2×C4⋊Dic79—42no—
23C2×C22⋊Dic79—24no—
24C2×C4×Dic79—42no—
25#14579076—24no—
26#14579076—24no—
27#14579076—24no—
28#14579076C2×C418yescomplemented
29C2×#41×C79—42no—
30C2×#41×C79—42no—
31C2×#41×C79C2×C412yes—
32C2×#41×C79C2×C411yescharacteristic
33C2×#41×C79D412yes—
34C2×#41×C79D411yescharacteristic
35C2×#41×C79Q811yescharacteristic
36C2×#41×C79C2311yescharacteristic
37C22×Q8×C79Q811yescharacteristic
38C22×D4×C79C2×C412yes—
39C22×D4×C79D411yescharacteristic
40C23×C4×C79C2×C411yescharacteristic
41C23×C4×C79D412yes—
42C2×M16×C79—21no—
43C2×M16×C79—21no—
44C2×M16×C79—21no—
45C2×M16×C79—42no—
46C2×M16×C79—42no—
47C2×M16×C79C2×C412yescomplemented
48C22×C8×C79—21no—
49C22×C8×C79C2×C412yescomplemented
50C22⋊C8×C79—24no—
51C22⋊C8×C79—24no—
52C22×Dic79—44no—
53C4⋊Dic79—44no—
54C4⋊Dic79—44no—
55C22⋊Dic79—28no—
56C22⋊Dic79—44no—
57C4×Dic79—44no—
58C4×Dic79—44no—
59C24×C79C4211yescharacteristic
60C24×C79C22⋊C412yes—
61#41×C79—42no—
62#41×C79—42no—
63#41×C79—41no—
64#41×C79—41no—
65#41×C79—42no—
66#41×C79—42no—
67#41×C79—42no—
68#41×C79—42no—
69#41×C79—41no—
70#41×C79—41no—
71C2×Q8×C79—42no—
72C2×Q8×C79C4⋊C412yes—
73C2×Q8×C79C4⋊C411yescharacteristic
74C2×Q8×C79C2×Q811yescharacteristic
75C2×D4×C79—42no—
76C2×D4×C79—42no—
77C2×D4×C79—42no—
78C2×D4×C79C4212yes—
79C2×D4×C79C22⋊C412yes—
80C2×D4×C79C22⋊C412yes—
81C2×D4×C79C4⋊C412yes—
82C2×D4×C79C4⋊C411yescharacteristic
83C2×D4×C79C22×C412yes—
84C2×D4×C79C2×D411yescharacteristic
85C22×C4×C79—22no—
86C22×C4×C79—22no—
87C22×C4×C79—22no—
88C22×C4×C79—22no—
89C22×C4×C79—22no—
90C22×C4×C79—22no—
91C22×C4×C79C4211yescharacteristic
92C22×C4×C79C22⋊C412yes—
93C22×C4×C79C22⋊C412yes—
94C22×C4×C79C22⋊C411yescharacteristic
95C22×C4×C79C22⋊C411yescharacteristic
96C22×C4×C79C4⋊C412yes—
97C22×C4×C79C4⋊C412yes—
98C22×C4×C79C4⋊C411yescharacteristic
99C22×C4×C79C4⋊C411yescharacteristic
100C22×C4×C79C22×C411yescharacteristic
101C22×C4×C79C2×D412yes—
102M16×C79—42no—
103M16×C79—42no—
104M16×C79—42no—
105M16×C79—42no—
106C2×C8×C79—22no—
107C2×C8×C79—22no—
108C2×C8×C79—42no—
109C2×C8×C79—42no—
110C23×C79—22no—
111C23×C79—22no—
112C23×C79—22no—
113C23×C79—82no—
114C23×C79—81no—
115C23×C79#9512yes—
116C23×C79#9511yescharacteristic
117C23×C79#9511yescharacteristic
118C23×C79C2×C4211yescharacteristic
119C23×C79C2×C22⋊C412yes—
120Q8×C79—42no—
121Q8×C79—41no—
122Q8×C79—41no—
123D4×C79—42no—
124D4×C79—42no—
125D4×C79—42no—
126D4×C79—42no—
127D4×C79—41no—
128D4×C79—42no—
129D4×C79—41no—
130C2×C4×C79—22no—
131C2×C4×C79—22no—
132C2×C4×C79—22no—
133C2×C4×C79—22no—
134C2×C4×C79—22no—
135C2×C4×C79—22no—
136C2×C4×C79—22no—
137C2×C4×C79—22no—
138C2×C4×C79—22no—
139C2×C4×C79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—22no—
146C2×C4×C79—22no—
147C2×C4×C79#9511yescharacteristic
148C2×C4×C79#9512yes—
149C2×C4×C79#9511yescharacteristic
150C2×C4×C79#9912yes—
151C2×C4×C79#9912yes—
152C2×C4×C79C2×C22⋊C411yescharacteristic
153C2×C4×C79C2×C4⋊C412yes—
154C2×C4×C79C2×C4⋊C411yescharacteristic
155C2×Dic79—48no—
156C2×Dic79—88no—
157C8×C79—42no—
158C8×C79—42no—
159C22×C79—22no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—22no—
163C22×C79—41no—
164C22×C79—42no—
165C22×C79—41no—
166C22×C79—41no—
167C22×C79—42no—
168C22×C79—44no—
169C22×C79—82no—
170C22×C79—82no—
171C22×C79#34212yes—
172C22×C79C2×#9511yescharacteristic
173C4×C79—22no—
174C4×C79—21no—
175C4×C79—22no—
176C4×C79—21no—
177C4×C79#34212yes—
178C4×C79C2×#9911yescharacteristic
179C4×C79C2×#9911yescharacteristic
180Dic79—88no—
181#12577—791nomaximal, Sylow
182C2×C79—21no—
183C2×C79—22no—
184C2×C79—42no—
185C2×C79—42no—
186C2×C79—41no—
187C2×C79—41no—
188C2×C79C2×#34211yescharacteristic
189C2×#410—792no—
190#2879D7911yescharacteristic
191#1391—794no—
192C79#1257711yescharacteristic, complemented, Sylow
193C22×#41C2×D7911yescharacteristic
194C22×M16Dic7911yescharacteristic, complemented
195C22×M16Dic7911yescharacteristic, complemented
196C2×#117—1582no—
197#410—794no—
198#410—794no—
199#410—1582no—
200#410—1582no—
201C2×#99—794no—
202C2×#99—1582no—
203#408—22no—
204#408Dic7912yescomplemented
205#408C2×D7912yes—
206#407—22no—
207C2×#41—42no—
208C2×#41—42no—
209C2×#41Dic15811yescharacteristic
210C2×#41C4×D7912yes—
211C2×#41D31611yescharacteristic
212C2×#41C2×Dic7911yescharacteristic
213C2×#41C22⋊D7912yes—
214C2×#41C22×D7911yescharacteristic
215C22×Q8Dic15811yescharacteristic
216C22×D4C4×D7912yes—
217C22×D4D31611yescharacteristic
218C23×C4C2×Dic7911yescharacteristic
219C23×C4C22⋊D7912yes—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16—21no—
223C2×M16—42no—
224C2×M16—42no—
225C2×M16C2×Dic7912yescomplemented
226C22×C8—21no—
227C22×C8C2×Dic7912yescomplemented
228#117—3162no—
229#117—3162no—
230#117—1584no—
231#117—1584no—
232C2×C4⋊C4—3162no—
233C2×C22⋊C4—1584no—
234C2×C42—3162no—
235#99—798no—
236#99—1584no—
237#99—1584no—
238#99—1584no—
239C22⋊C8—24no—
240C22⋊C8—24no—
241C24C4×Dic7911yescharacteristic
242C24(C2×D79)⋊C412yes—
243#41—42no—
244#41—42no—
245#41—41no—
246#41—41no—
247#41—42no—
248#41—42no—
249#41—42no—
250#41—42no—
251#41—41no—
252#41—41no—
253C2×Q8—42no—
254C2×Q8Dic79⋊C412yes—
255C2×Q8C4⋊Dic7911yescharacteristic
256C2×Q8C2×Dic15811yescharacteristic
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4C4×Dic7912yes—
261C2×D4(C2×D79)⋊C412yes—
262C2×D4(C2×D79)⋊C412yes—
263C2×D4Dic79⋊C412yes—
264C2×D4C4⋊Dic7911yescharacteristic
265C2×D4C2×C4×D7912yes—
266C2×D4C2×D31611yescharacteristic
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—3164no—
274C22×C4C4×Dic7911yescharacteristic
275C22×C4(C2×D79)⋊C412yes—
276C22×C4C22⋊Dic7912yes—
277C22×C4C22⋊Dic7911yescharacteristic
278C22×C4C22⋊Dic7911yescharacteristic
279C22×C4Dic79⋊C412yes—
280C22×C4Dic79⋊C412yes—
281C22×C4C4⋊Dic7911yescharacteristic
282C22×C4C4⋊Dic7911yescharacteristic
283C22×C4C22×Dic7911yescharacteristic
284C22×C4C2×C22⋊D7912yes—
285M16—42no—
286M16—42no—
287M16—42no—
288M16—42no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—3164no—
294C4⋊C4—3164no—
295C22⋊C4—3164no—
296C22⋊C4—1588no—
297C42—3164no—
298C42—3164no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—82no—
303C23—81no—
304C23C79⋊#9512yes—
305C23C79⋊#9511yescharacteristic
306C23C79⋊#9511yescharacteristic
307C23C2×C4×Dic7911yescharacteristic
308C23C2×(C2×D79)⋊C412yes—
309Q8—42no—
310Q8—41no—
311Q8—41no—
312D4—42no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—42no—
318D4—41no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—3168no—
337C2×C4—6328no—
338C2×C4C79⋊#9511yescharacteristic
339C2×C4C79⋊#9512yes—
340C2×C4C79⋊#9511yescharacteristic
341C2×C4#1457907612yes—
342C2×C4#1457907612yes—
343C2×C4C2×C22⋊Dic7911yescharacteristic
344C2×C4C2×Dic79⋊C412yes—
345C2×C4C2×C4⋊Dic7911yescharacteristic
346C8—42no—
347C8—42no—
348C22—22no—
349C22—22no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—44no—
358C22—82no—
359C22—82no—
360C22#1909833012yes—
361C22C2×C79⋊#9511yescharacteristic
362C4—22no—
363C4—21no—
364C4—22no—
365C4—21no—
366C4—6328no—
367C4#1909833012yes—
368C4C2×#1457907611yescharacteristic
369C4C2×#1457907611yescharacteristic
370C2—21no—
371C2—22no—
372C2—42no—
373C2—42no—
374C2—41no—
375C2—41no—
376C2C2×#1909833011yescharacteristic
subgroupG/H
centerC4C2×#14579076
commutatorC23×C79C2×C42
FrattiniC2×#41C22×D79
FittingC79×#2879C2
socleC2×C79C2×#342
prime radicalC22×#41×C79C22
cosocleC2×#41×C79C23
hypercenter#2879D79
hypocenterC79#12577
LCSG ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ #2879

Complements

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

NH ≅ G/Ntype
2. #23761521C2semidirect
7. #19098537C4semidirect
10. C2×#14579076C4semidirect
12. C22×M16×C79C4semidirect
13. C22×M16×C79C4semidirect
15. #408×C79C4semidirect
28. #14579076C2×C4semidirect
47. C2×M16×C79C2×C4semidirect
49. C22×C8×C79C2×C4semidirect
192. C79#12577semidirect
194. C22×M16Dic79semidirect
195. C22×M16Dic79semidirect
204. #408Dic79semidirect
225. C2×M16C2×Dic79semidirect
227. C22×C8C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190985372
#237615212
C79×#28792
#1257779