Finite Group Databasebeta

← #34182266

Subgroups — #34182266

Order 20224.

612 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#23766018C211yesmaximal, characteristic
2#23763593C212yesmaximal
3C2×#19098289C211yesmaximal, characteristic
4#23761279C212yesmaximal
5C79×#1441C211yesmaximal, characteristic
6C2×#14579120—21no—
7C2×#14579092—22no—
8C2×C22⋊(C79⋊C8)C2212yes—
9#19098462—22no—
10#19098462—22no—
11#19098423—21no—
12#19098423C2212yes—
13#19098325—22no—
14#19098289C414yescomplemented
15C22×C4⋊C4×C79C2211yescharacteristic
16C22×C4⋊C4×C79C2211yescharacteristic
17C2×#95×C79—21no—
18C2×#95×C79—21no—
19C2×#95×C79C2211yescharacteristic
20C22×C79⋊C8—22no—
21#14579120—42no—
22#14579120—42no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79D412yes—
28C2×C4⋊Dic79D412yes—
29C2×C22⋊Dic79—22no—
30C2×C4×Dic79—22no—
31#14579092—44no—
32C22⋊(C79⋊C8)—22no—
33C22⋊(C79⋊C8)C2×C414yes—
34C79⋊#95—22no—
35C79⋊#95—22no—
36C23×C4×C79—21no—
37C23×C4×C79—21no—
38C23×C4×C79D411yescharacteristic
39C23×C4×C79D411yescharacteristic
40C23×C4×C79C2311yescharacteristic
41C2×C4⋊C4×C79—22no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—22no—
44C2×C4⋊C4×C79—42no—
45C2×C4⋊C4×C79C2×C412yes—
46C2×C4⋊C4×C79Q812yes—
47#95×C79—22no—
48#95×C79—42no—
49#95×C79—42no—
50C22×Dic79—22no—
51C22×Dic79—22no—
52C2×C79⋊C8—24no—
53C2×C79⋊C8—42no—
54C2×C79⋊C8—42no—
55C4⋊Dic79—42no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—42no—
59C4⋊Dic79—81no—
60C4⋊Dic79—81no—
61C22⋊Dic79—42no—
62C22⋊Dic79—42no—
63C4×Dic79—44no—
64C24×C79C2×D411yescharacteristic
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—22no—
72C22×C4×C79—22no—
73C22×C4×C79—41no—
74C22×C4×C79—41no—
75C22×C4×C79—41no—
76C22×C4×C79—41no—
77C22×C4×C79—41no—
78C22×C4×C79—41no—
79C22×C4×C79—41no—
80C22×C4×C79—41no—
81C22×C4×C79—41no—
82C22×C4×C79—41no—
83C22×C4×C79—41no—
84C22×C4×C79—41no—
85C22×C4×C79—41no—
86C22×C4×C79—41no—
87C22×C4×C79—41no—
88C22×C4×C79—41no—
89C22×C4×C79C4⋊C412yes—
90C22×C4×C79C4⋊C412yes—
91C22×C4×C79SD1612yes—
92C22×C4×C79SD1612yes—
93C22×C4×C79C22×C411yescharacteristic
94C22×C4×C79C2×D412yes—
95C22×C4×C79C2×Q811yescharacteristic
96C22×C4×C79#4112yes—
97C4⋊C4×C79—24no—
98C4⋊C4×C79—24no—
99C4⋊C4×C79—24no—
100C4⋊C4×C79—42no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—21no—
104C23×C79—21no—
105C23×C79C2×C4⋊C411yescharacteristic
106C23×C79#12011yescharacteristic
107C23×C79#12311yescharacteristic
108C23×C79C2×SD1611yescharacteristic
109C23×C79C2×SD1611yescharacteristic
110C23×C79AGL(1,ℤ/8)11yescharacteristic
111C23×C79#13711yescharacteristic
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—22no—
115C2×C4×C79—22no—
116C2×C4×C79—22no—
117C2×C4×C79—22no—
118C2×C4×C79—22no—
119C2×C4×C79—22no—
120C2×C4×C79—22no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—42no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—42no—
130C2×C4×C79—42no—
131C2×C4×C79—42no—
132C2×C4×C79—42no—
133C2×C4×C79—81no—
134C2×C4×C79—81no—
135C2×C4×C79—81no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79—81no—
140C2×C4×C79—81no—
141C2×C4×C79—81no—
142C2×C4×C79#10614yescomplemented
143C2×C4×C79C4×D412yes—
144C2×C4×C79C22⋊Q812yes—
145C2×Dic79—42no—
146C2×Dic79—42no—
147C2×Dic79—42no—
148C2×Dic79—42no—
149C2×Dic79—42no—
150C2×Dic79—42no—
151C79⋊C8—44no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—21no—
155C22×C79—21no—
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#38511yescharacteristic
165C22×C79C2×#10611yescharacteristic
166C22×C79#42811yescharacteristic
167C22×C79#44811yescharacteristic
168C22×C79#45011yescharacteristic
169C22×C79#45411yescharacteristic
170C22×C79#45611yescharacteristic
171C22×C79B(4,𝔽2)11yescharacteristic
172C22×C79#45811yescharacteristic
173C22×C79#48111yescharacteristic
174C22×C79#48311yescharacteristic
175C4×C79—24no—
176C4×C79—22no—
177C4×C79—22no—
178C4×C79—42no—
179C4×C79—82no—
180C4×C79—82no—
181Dic79—82no—
182Dic79—82no—
183#8857—791nomaximal, Sylow
184C2×C79—21no—
185C2×C79—21no—
186C2×C79—21no—
187C2×C79—21no—
188C2×C79#159911yescharacteristic
189C2×C79#160411yescharacteristic
190C2×C79#161111yescharacteristic
191C2×C79#161611yescharacteristic
192C2×C79#185311yescharacteristic
193C2×C79#185711yescharacteristic
194C2×C79#186711yescharacteristic
195#2316—791no—
196#1853—792no—
197C2×#324—791no—
198#1441D7911yescharacteristic
199#1294—792no—
200C79#885711yescharacteristic, complemented, Sylow
201C2×#117—1581no—
202C22×C4⋊C4C2×D7911yescharacteristic
203C22×C4⋊C4C2×D7911yescharacteristic
204C2×#106—1582no—
205C2×C22⋊C8—792no—
206#389—1582no—
207#389—1582no—
208#380—792no—
209#380—1581no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95C2×D7911yescharacteristic
213#340—1582no—
214#324—794no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C22⋊D7911yescharacteristic
218C23×C4C22⋊D7911yescharacteristic
219C23×C4C22×D7911yescharacteristic
220C22×C8—1582no—
221#117—3162no—
222#117—3162no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—792no—
228C2×C4⋊C4—792no—
229C2×C4⋊C4—1581no—
230C2×C4⋊C4—1581no—
231C2×C4⋊C4—1581no—
232C2×C4⋊C4—1581no—
233C2×C4⋊C4Dic15812yes—
234C2×C4⋊C4C4×D7912yes—
235C2×C22⋊C4—1582no—
236C2×C42—1582no—
237#106—3164no—
238C22⋊C8—794no—
239C22⋊C8—1582no—
240#95—22no—
241#95—42no—
242#95—42no—
243#95—1582no—
244#95—1582no—
245C24C2×C22⋊D7911yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—1582no—
271C22×C4—1582no—
272C22×C4Dic79⋊C412yes—
273C22×C4Dic79⋊C412yes—
274C22×C4#1192961112yes—
275C22×C4#1192961212yes—
276C22×C4C2×C4×D7911yescharacteristic
277C22×C4D4×D7912yes—
278C22×C4C2×Dic15811yescharacteristic
279C22×C4#1192962312yes—
280C2×C8—3162no—
281C2×C8—3162no—
282C2×C8—1584no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—42no—
287C4⋊C4—3162no—
288C4⋊C4—3162no—
289C4⋊C4—3162no—
290C4⋊C4—3162no—
291C4⋊C4—6321no—
292C4⋊C4—6321no—
293C22⋊C4—3162no—
294C22⋊C4—3162no—
295C42—3164no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23C2×Dic79⋊C411yescharacteristic
301C23#1457912911yescharacteristic
302C23#1457914211yescharacteristic
303C23C2×#1192961111yescharacteristic
304C23C2×#1192961211yescharacteristic
305C23#1457918611yescharacteristic
306C23#1457919111yescharacteristic
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—3162no—
338C2×C4—3162no—
339C2×C4—3162no—
340C2×C4—3162no—
341C2×C4—3162no—
342C2×C4—3162no—
343C2×C4#1457909214yescomplemented
344C2×C4(C22⋊D79)⋊C412yes—
345C2×C4#1457913812yes—
346C8—3164no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22#1909843511yescharacteristic
360C22C2×#1457909211yescharacteristic
361C22#1909860611yescharacteristic
362C22#1909870111yescharacteristic
363C22#1909871311yescharacteristic
364C22#1909873511yescharacteristic
365C22#1909874511yescharacteristic
366C22#1909874911yescharacteristic
367C22#1909875211yescharacteristic
368C22#1909887911yescharacteristic
369C22#1909889111yescharacteristic
370C4—24no—
371C4—22no—
372C4—22no—
373C4—42no—
374C4—82no—
375C4—82no—
376C4—6322no—
377C4—6322no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—21no—
382C2#2376233911yescharacteristic
383C2#2376236811yescharacteristic
384C2#2376240711yescharacteristic
385C2#2376243611yescharacteristic
386C2#2376359211yescharacteristic
387C2#2376361411yescharacteristic
388C2#2376368011yescharacteristic
subgroupG/H
centerC23#14579129
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1441C2
socleC23×C79#120
prime radicalC24×C79C2×D4
cosocleC23×C4×C79C23
hypercenter#1441D79
hypocenterC79#8857
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1441

Complements

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

NH ≅ G/Ntype
14. #19098289C4semidirect
142. C2×C4×C79#106semidirect
200. C79#8857semidirect
343. C2×C4#14579092semidirect

Maximal subgroups

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

subgroupindex
#237660182
#237635932
C2×#190982892
#237612792
C79×#14412
#885779