Finite Group Databasebeta

← #34181243

Subgroups — #34181243

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772094C211yesmaximal, characteristic
2#23763438C211yesmaximal, characteristic, complemented
3#23761941C212yesmaximal
4#23761306C212yesmaximal, complemented
5C79×#1808C211yesmaximal, characteristic, complemented
6C2×#14579167C411yescharacteristic, complemented
7#19099074C2211yescharacteristic
8#19098634—21no—
9#19098634—21no—
10C8×C22⋊D79—21no—
11C8×C22⋊D79—21no—
12C2×#14579097—21no—
13#19098589C411yescharacteristic, complemented
14#19098529C411yescharacteristic, complemented
15#19098527C2212yes—
16#19098527C2211yescharacteristic
17#19098518C411yescharacteristic, complemented
18#19098328—21no—
19#19098299—22no—
20#19098296—22no—
21C23×C8×C79C2211yescharacteristic
22C2×#107×C79—21no—
23#380×C79C2212yes—
24#340×C79—21no—
25C2×#11929624C2×C411yescharacteristic
26#14579167—42no—
27C2×#11929608—22no—
28C2×#11929608C2×C411yescharacteristic
29C2×#11929608C2×C411yescharacteristic
30C2×#11929607—21no—
31C2×C8×D79—21no—
32#14579144—21no—
33#14579140D411yescharacteristic
34#14579140Q811yescharacteristic
35#14579135D411yescharacteristic
36#14579135Q811yescharacteristic
37#14579130—21no—
38C4×C22⋊D79—21no—
39C4×C22⋊D79—21no—
40#14579120C2×C411yescharacteristic
41#14579097—42no—
42Dic79⋊C8—21no—
43Dic79⋊C8—21no—
44C22⋊(C79⋊C8)—21no—
45C22⋊(C79⋊C8)—21no—
46C22⋊(C79⋊C8)—22no—
47C22⋊(C79⋊C8)—22no—
48(C2×D79)⋊C8—21no—
49(C2×D79)⋊C8—21no—
50#14579071—21no—
51C8×Dic79—21no—
52C23×C4×C79C2311yescharacteristic
53C22×C8×C79—42no—
54C22×C8×C79C2×C411yescharacteristic
55C22×C8×C79C2×C411yescharacteristic
56C22×C8×C79D411yescharacteristic
57C22×C8×C79D411yescharacteristic
58C22×C8×C79D411yescharacteristic
59C22×C8×C79D411yescharacteristic
60C2×C4⋊C4×C79—22no—
61C2×C22⋊C4×C79—22no—
62#107×C79—42no—
63#95×C79—22no—
64#11929624—41no—
65#11929624—41no—
66C2×Dic158C4⋊C411yescharacteristic
67C2×D316C4⋊C411yescharacteristic
68C2×C22⋊D79—21no—
69C2×C22⋊D79—21no—
70C2×C22⋊D79—21no—
71C2×C4×D79—21no—
72#11929608—41no—
73#11929608—41no—
74#11929608—42no—
75#11929608—42no—
76#11929607—42no—
77C2×C79⋊C8—21no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—42no—
80C8×D79—42no—
81C4⋊Dic79C4⋊C411yescharacteristic
82C4⋊Dic79C4⋊C411yescharacteristic
83Dic79⋊C4—21no—
84Dic79⋊C4—21no—
85C22⋊Dic79—21no—
86C22⋊Dic79—21no—
87C22⋊Dic79—21no—
88(C2×D79)⋊C4—21no—
89(C2×D79)⋊C4—21no—
90C4×Dic79—21no—
91C24×C79#4111yescharacteristic
92C22×C4×C79—41no—
93C22×C4×C79—41no—
94C22×C4×C79—42no—
95C22×C4×C79C22×C411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98C22×C4×C79C2×D411yescharacteristic
99C22×C4×C79C2×Q811yescharacteristic
100C22×C4×C79#4111yescharacteristic
101C2×C8×C79—22no—
102C2×C8×C79—22no—
103C2×C8×C79—22no—
104C2×C8×C79—41no—
105C2×C8×C79—41no—
106C2×C8×C79—41no—
107C2×C8×C79C22⋊C411yescharacteristic
108C2×C8×C79C22⋊C411yescharacteristic
109C2×C8×C79C22⋊C411yescharacteristic
110C2×C8×C79C22⋊C411yescharacteristic
111C4⋊C4×C79—42no—
112C4⋊C4×C79—42no—
113C22⋊C4×C79—82no—
114C23×C79—21no—
115C23×C79—21no—
116C23×C79—41no—
117C23×C79—41no—
118C23×C79#11711yescharacteristic
119C23×C79#12111yescharacteristic
120C23×C79C22⋊Q811yescharacteristic
121C22×D79—21no—
122C2×C4×C79—21no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—82no—
132C2×C4×C79—82no—
133C2×C4×C79—82no—
134C2×C4×C79C2×C22⋊C411yescharacteristic
135C2×C4×C79C2×C4⋊C411yescharacteristic
136C2×C4×C79#12111yescharacteristic
137C2×C4×C79C22⋊Q811yescharacteristic
138C22⋊D79—42no—
139C22⋊D79—42no—
140C22⋊D79—41no—
141C22⋊D79—41no—
142C2×Dic79—21no—
143C2×Dic79—21no—
144C2×Dic79—21no—
145D316—41no—
146C4×D79—41no—
147C4×D79—41no—
148Dic158—41no—
149C8×C79—21no—
150C8×C79—21no—
151C8×C79—21no—
152C8×C79#9912yes—
153C79⋊C8—41no—
154C79⋊C8—41no—
155C79⋊C8—82no—
156C22×C79—21no—
157C22×C79—21no—
158C22×C79—21no—
159C22×C79—21no—
160C22×C79—21no—
161C22×C79—21no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—42no—
165C22×C79—42no—
166C22×C79#38011yescharacteristic
167C22×C79#47211yescharacteristic
168C22×C79#47311yescharacteristic
169C2×D79—41no—
170C2×D79—41no—
171C2×D79—41no—
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—21no—
175C4×C79—162no—
176C4×C79C2×#9911yescharacteristic
177C4×C79#41011yescharacteristic
178Dic79—41no—
179Dic79—41no—
180Dic79—41no—
181#8627—791nomaximal, Sylow
182C2×C79—21no—
183C2×C79—21no—
184C2×C79—41no—
185C2×C79—41no—
186C2×C79#179911yescharacteristic
187C2×C79#181411yescharacteristic
188C2×C79#181611yescharacteristic
189D79—81no—
190#2929—791no—
191#1810—791no—
192#1808D7911yescharacteristic, complemented
193#1513—792no—
194#1305—792no—
195C79#862711yescharacteristic, complemented, Sylow
196C2×#131—791no—
197C23×C8C2×D7911yescharacteristic
198#525—791no—
199#435—1581no—
200#435—1581no—
201C8×D4—1581no—
202C8×D4—1581no—
203C2×#108—1581no—
204C2×#107—21no—
205#424—791no—
206#408—791no—
207#407—791no—
208#407—792no—
209#405—791no—
210#380C2×D7912yes—
211#341—1581no—
212#340—21no—
213#328—1582no—
214#327—1582no—
215C2×#41—791no—
216C23×C4C22×D7911yescharacteristic
217#131—3162no—
218C2×M16—791no—
219C2×M16—791no—
220C2×M16—1582no—
221C2×M16—1581no—
222C22×C8—42no—
223C22×C8—1581no—
224C22×C8C4×D7911yescharacteristic
225C22×C8C4×D7911yescharacteristic
226C22×C8D31611yescharacteristic
227C22×C8D31611yescharacteristic
228C22×C8C22⋊D7911yescharacteristic
229C22×C8C22⋊D7911yescharacteristic
230#123—1581no—
231C22⋊Q8—791no—
232C22⋊Q8—791no—
233#121—791no—
234#121—791no—
235#120—1581no—
236C4×D4—1581no—
237C4×D4—1581no—
238#117—791no—
239C2×C4⋊C4—22no—
240C2×C22⋊C4—22no—
241#108—3162no—
242#107—42no—
243C4⋊C8—1581no—
244C4⋊C8—1581no—
245C22⋊C8—1582no—
246C22⋊C8—1582no—
247C22⋊C8—1581no—
248C22⋊C8—1581no—
249C22⋊C8—1581no—
250C22⋊C8—1581no—
251#97—1581no—
252C4×C8—1581no—
253#95—22no—
254C24#1192962311yescharacteristic
255#41—3161no—
256#41—3161no—
257C2×Q8—791no—
258C2×D4—791no—
259C2×D4—1581no—
260C2×D4—1581no—
261C2×D4—1581no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—1581no—
266C22×C4C2×C4×D7911yescharacteristic
267C22×C4D4×D7911yescharacteristic
268C22×C4C2×C22⋊D7911yescharacteristic
269C22×C4C2×D31611yescharacteristic
270C22×C4Q8×D7911yescharacteristic
271C22×C4#1192962511yescharacteristic
272M16—3162no—
273M16—3161no—
274M16—3161no—
275M16—3162no—
276M16—3162no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—3162no—
284C2×C8—3162no—
285C2×C8—1581no—
286C2×C8—1581no—
287C2×C8(C2×D79)⋊C411yescharacteristic
288C2×C8(C2×D79)⋊C411yescharacteristic
289C2×C8(C2×D79)⋊C411yescharacteristic
290C2×C8(C2×D79)⋊C411yescharacteristic
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—791no—
294C4⋊C4—791no—
295C4⋊C4—1581no—
296C4⋊C4—1581no—
297C22⋊C4—82no—
298C22⋊C4—1581no—
299C22⋊C4—1581no—
300C22⋊C4—1581no—
301C22⋊C4—1581no—
302C22⋊C4—1581no—
303C42—1581no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—1581no—
309C23#1457911911yescharacteristic
310C23#1457913211yescharacteristic
311C23#1457914111yescharacteristic
312Q8—3161no—
313D4—3161no—
314D4—3162no—
315D4—3162no—
316D4—3161no—
317D4—3161no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—3161no—
331C2×C4—3161no—
332C2×C4—1581no—
333C2×C4—1581no—
334C2×C4—1581no—
335C2×C4C2×(C2×D79)⋊C411yescharacteristic
336C2×C4C4⋊C4×D7911yescharacteristic
337C2×C4#1457913611yescharacteristic
338C2×C4#1457913711yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—3161no—
343C8—3161no—
344C8—6322no—
345C8#1457907512yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—3161no—
357C22—3161no—
358C22—3161no—
359C22#1909842011yescharacteristic
360C22#1909883011yescharacteristic
361C22#1909883511yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—162no—
366C4—3161no—
367C4—3161no—
368C4—3161no—
369C4C2×#1457907511yescharacteristic
370C4#1909853611yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—6321no—
376C2#2376338511yescharacteristic
377C2#2376345111yescharacteristic
378C2#2376346511yescharacteristic
subgroupG/H
centerC22#19098420
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1808C2
socleC22×C79#380
prime radical#19099074C22
cosocleC23×C4×C79C23
hypercenter#1808D79
hypocenterC79#8627
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
2. #23763438C2semidirect
4. #23761306C2semidirect
5. C79×#1808C2semidirect
6. C2×#14579167C4semidirect
13. #19098589C4semidirect
14. #19098529C4semidirect
17. #19098518C4semidirect
192. #1808D79semidirect
195. C79#8627semidirect

Maximal subgroups

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

subgroupindex
#237720942
#237634382
#237619412
#237613062
C79×#18082
#862779