Finite Group Databasebeta

← #23769739

Subgroups — #23769739

Order 10112.

568 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14579144C211yesmaximal, characteristic
2C2×#14579143C211yesmaximal, characteristic
3C2×#14579137C211yesmaximal, characteristic, complemented
4#19098496C211yesmaximal, characteristic, complemented
5#19098493C211yesmaximal, characteristic, complemented
6#19098491C211yesmaximal, characteristic, complemented
7#19098489C211yesmaximal, characteristic, complemented
8#19098482C211yesmaximal, characteristic, complemented
9#19098478C211yesmaximal, characteristic
10#19098456C211yesmaximal, characteristic, complemented
11#19098453C211yesmaximal, characteristic, complemented
12#19098436C211yesmaximal, characteristic, complemented
13#19098434C211yesmaximal, characteristic, complemented
14#19098432C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C79C211yesmaximal, characteristic, complemented
16C22×C22⋊D79C2211yescharacteristic
17C22×C4×D79C2211yescharacteristic
18#14579144—24no—
19#14579143—24no—
20#14579137—24no—
21C2×C4⋊Dic79C2211yescharacteristic
22C2×C4⋊Dic79C2211yescharacteristic
23C2×Dic79⋊C4C2211yescharacteristic
24C2×Dic79⋊C4C2211yescharacteristic
25C2×Dic79⋊C4C2211yescharacteristic
26C2×Dic79⋊C4C2211yescharacteristic
27C2×Dic79⋊C4C2211yescharacteristic
28C2×Dic79⋊C4C2211yescharacteristic
29C2×Dic79⋊C4C2211yescharacteristic
30C2×C22⋊Dic79C2211yescharacteristic
31C2×C22⋊Dic79C2211yescharacteristic
32C2×C22⋊Dic79C2211yescharacteristic
33C2×(C2×D79)⋊C4C2211yescharacteristic
34C2×(C2×D79)⋊C4C2211yescharacteristic
35C2×(C2×D79)⋊C4C2211yescharacteristic
36C2×(C2×D79)⋊C4C2211yescharacteristic
37C2×(C2×D79)⋊C4C2211yescharacteristic
38C2×C4×Dic79C2211yescharacteristic
39C79⋊#95C2211yescharacteristic
40C79⋊#95C2211yescharacteristic
41C79⋊#95C2211yescharacteristic
42C79⋊#95C2211yescharacteristic
43C79⋊#95C2211yescharacteristic
44C79⋊#95C2211yescharacteristic
45C79⋊#95C2211yescharacteristic
46C79⋊#95C2211yescharacteristic
47C22×Q8×C79C2211yescharacteristic
48C23×C4×C79C2211yescharacteristic
49C22⋊Q8×C79—24no—
50C2×C4⋊C4×C79C2211yescharacteristic
51C2×C4⋊C4×C79C2211yescharacteristic
52C2×C4⋊C4×C79C2211yescharacteristic
53C2×C22⋊C4×C79C2211yescharacteristic
54C2×C22⋊C4×C79C2211yescharacteristic
55C23×D79C2311yescharacteristic
56C2×C22⋊D79—22no—
57C2×C22⋊D79—22no—
58C2×C22⋊D79—41no—
59C22×Dic79C2311yescharacteristic
60C22×Dic79C2311yescharacteristic
61C22×Dic79C2311yescharacteristic
62C22×Dic79C2311yescharacteristic
63C22×Dic79C2311yescharacteristic
64C22×Dic79C2311yescharacteristic
65C22×Dic79C2311yescharacteristic
66C2×C4×D79—22no—
67C2×C4×D79—22no—
68C2×C4×D79—41no—
69C4⋊Dic79—22no—
70C4⋊Dic79—22no—
71Dic79⋊C4—22no—
72Dic79⋊C4—22no—
73Dic79⋊C4—22no—
74Dic79⋊C4—22no—
75Dic79⋊C4—22no—
76Dic79⋊C4—41no—
77Dic79⋊C4—41no—
78C22⋊Dic79—22no—
79C22⋊Dic79—22no—
80C22⋊Dic79—22no—
81(C2×D79)⋊C4—22no—
82(C2×D79)⋊C4—22no—
83(C2×D79)⋊C4—22no—
84(C2×D79)⋊C4—22no—
85(C2×D79)⋊C4D414yes—
86C4×Dic79—22no—
87C24×C79C2311yescharacteristic
88C2×Q8×C79—22no—
89C2×Q8×C79—22no—
90C2×Q8×C79—41no—
91C22×C4×C79—22no—
92C22×C4×C79—22no—
93C22×C4×C79—41no—
94C22×C4×C79C2311yescharacteristic
95C22×C4×C79C2311yescharacteristic
96C22×C4×C79C2311yescharacteristic
97C22×C4×C79C2311yescharacteristic
98C22×C4×C79C2311yescharacteristic
99C22×C4×C79C2311yescharacteristic
100C4⋊C4×C79—22no—
101C4⋊C4×C79—22no—
102C4⋊C4×C79—22no—
103C22⋊C4×C79—22no—
104C22⋊C4×C79D414yes—
105C23×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—21no—
111C23×C79C2×D412yes—
112C23×C79C2411yescharacteristic
113C22×D79—21no—
114C22×D79—21no—
115C22×D79—21no—
116C22×D79—21no—
117C22×D79—21no—
118C22×D79—21no—
119C22×D79C2×D412yes—
120Q8×C79—44no—
121C2×C4×C79—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—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—21no—
136C2×C4×C79—44no—
137C2×C4×C79C2×D412yes—
138C2×C4×C79C2×D412yes—
139C2×C4×C79C2×D412yes—
140C22⋊D79—44no—
141C2×Dic79—21no—
142C2×Dic79—21no—
143C2×Dic79—21no—
144C2×Dic79—21no—
145C2×Dic79—21no—
146C2×Dic79—21no—
147C2×Dic79—21no—
148C2×Dic79—21no—
149C2×Dic79—21no—
150C2×Dic79—21no—
151C2×Dic79—21no—
152C2×Dic79—21no—
153C2×Dic79—21no—
154C2×Dic79—21no—
155C2×Dic79—21no—
156C2×Dic79—21no—
157C2×Dic79—21no—
158C2×Dic79—21no—
159C2×Dic79—21no—
160C2×Dic79C2×D412yes—
161C2×Dic79#4112yes—
162C4×D79—44no—
163C22×C79—22no—
164C22×C79—22no—
165C22×C79—22no—
166C22×C79—41no—
167C22×C79—41no—
168C22×C79—41no—
169C22×C79—41no—
170C22×C79C22×D411yescharacteristic
171C22×C79C22×D411yescharacteristic
172C22×C79C2×#4111yescharacteristic
173C22×C79H(5,𝔽2)11yescharacteristic
174C22×C792−1+411yescharacteristic
175C22×C792−1+411yescharacteristic
176C22×C792−1+411yescharacteristic
177C2×D79—22no—
178C2×D79—22no—
179C2×D79—22no—
180C2×D79—41no—
181C2×D79—41no—
182C2×D79—41no—
183C2×D79—41no—
184C4×C79—22no—
185C4×C79—22no—
186C4×C79—22no—
187C4×C79—41no—
188C4×C79—41no—
189C4×C79—41no—
190Dic79—22no—
191Dic79—22no—
192Dic79—41no—
193Dic79—41no—
194Dic79—41no—
195Dic79—41no—
196Dic79—41no—
197C2×C79—42no—
198C2×C79#53611yescharacteristic
199C2×C79#53611yescharacteristic
200C2×C79#53911yescharacteristic
201C2×C79D4211yescharacteristic
202C2×C79#54711yescharacteristic
203C2×C79#54711yescharacteristic
204C2×C79#56311yescharacteristic
205D79—42no—
206#2680—791nomaximal, Sylow
207C79#268011yescharacteristic, complemented, Sylow
208C2×#123—791no—
209C2×#123—791no—
210C2×C22⋊Q8—791no—
211C2×C22⋊Q8D7911yescharacteristic, complemented
212#398—791no—
213#398—791no—
214#397—791no—
215#397—791no—
216#395—791no—
217#394—791no—
218#388—791no—
219#388—791no—
220#385—791no—
221#385—791no—
222#384—791no—
223C22×Q8C2×D7911yescharacteristic
224C22×D4—791no—
225C23×C4—791no—
226C23×C4C2×D7911yescharacteristic
227#123—1584no—
228#123—1584no—
229C22⋊Q8—24no—
230C22⋊Q8—1584no—
231C2×C4⋊C4—791no—
232C2×C4⋊C4—791no—
233C2×C4⋊C4—791no—
234C2×C4⋊C4—791no—
235C2×C4⋊C4—791no—
236C2×C4⋊C4—791no—
237C2×C4⋊C4—791no—
238C2×C4⋊C4—791no—
239C2×C4⋊C4—791no—
240C2×C4⋊C4C2×D7911yescharacteristic
241C2×C4⋊C4C2×D7911yescharacteristic
242C2×C4⋊C4C2×D7911yescharacteristic
243C2×C22⋊C4—791no—
244C2×C22⋊C4—791no—
245C2×C22⋊C4—791no—
246C2×C22⋊C4—791no—
247C2×C22⋊C4—791no—
248C2×C22⋊C4—791no—
249C2×C22⋊C4—791no—
250C2×C22⋊C4—791no—
251C2×C22⋊C4C2×D7911yescharacteristic
252C2×C22⋊C4C2×D7911yescharacteristic
253C2×C42—791no—
254#95—791no—
255#95—791no—
256#95—791no—
257#95—791no—
258#95—791no—
259#95—791no—
260#95—791no—
261#95—791no—
262C24—791no—
263C24C22×D7911yescharacteristic
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×D4—3161no—
268C2×D4—1582no—
269C2×D4—1582no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—41no—
273C22×C4—3161no—
274C22×C4—791no—
275C22×C4—791no—
276C22×C4—791no—
277C22×C4—791no—
278C22×C4—791no—
279C22×C4—791no—
280C22×C4—791no—
281C22×C4—1582no—
282C22×C4—1582no—
283C22×C4C22×D7911yescharacteristic
284C22×C4C22×D7911yescharacteristic
285C22×C4C22×D7911yescharacteristic
286C22×C4C22×D7911yescharacteristic
287C22×C4C22×D7911yescharacteristic
288C22×C4C22×D7911yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—3161no—
293C4⋊C4—3161no—
294C4⋊C4—1582no—
295C4⋊C4—1582no—
296C4⋊C4—1582no—
297C4⋊C4—1582no—
298C4⋊C4—1582no—
299C4⋊C4—1582no—
300C4⋊C4—1582no—
301C22⋊C4—22no—
302C22⋊C4—794no—
303C22⋊C4—1582no—
304C22⋊C4—1582no—
305C22⋊C4—1582no—
306C22⋊C4—1582no—
307C22⋊C4—1582no—
308C22⋊C4—1582no—
309C22⋊C4—1582no—
310C22⋊C4C22⋊D7914yes—
311C42—1582no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—792no—
319C23—1581no—
320C23—1581no—
321C23—1581no—
322C23—1581no—
323C23—1581no—
324C23—1581no—
325C23C2×C22⋊D7912yes—
326C23C23×D7911yescharacteristic
327Q8—44no—
328D4—3164no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—44no—
345C2×C4—3164no—
346C2×C4—792no—
347C2×C4—792no—
348C2×C4—1581no—
349C2×C4—1581no—
350C2×C4—1581no—
351C2×C4—1581no—
352C2×C4—1581no—
353C2×C4—1581no—
354C2×C4—1581no—
355C2×C4—1581no—
356C2×C4—1581no—
357C2×C4—1581no—
358C2×C4—1581no—
359C2×C4—1581no—
360C2×C4—1581no—
361C2×C4—1581no—
362C2×C4—1581no—
363C2×C4—1581no—
364C2×C4—1581no—
365C2×C4—1581no—
366C2×C4—1581no—
367C2×C4D4×D7912yes—
368C2×C4C2×C22⋊D7912yes—
369C2×C4C2×C22⋊D7912yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3161no—
378C22—3161no—
379C22—3161no—
380C22—3161no—
381C22—1582no—
382C22—1582no—
383C22—1582no—
384C22C2×D4×D7911yescharacteristic
385C22C22×C22⋊D7911yescharacteristic
386C22#41×D7911yescharacteristic
387C22#1457920611yescharacteristic
388C22#1457920811yescharacteristic
389C22#1457920911yescharacteristic
390C22#1457920911yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—3161no—
398C4—3161no—
399C4—3161no—
400C4—3161no—
401C4—3161no—
402C4—1582no—
403C4—1582no—
404C2—42no—
405C2—3162no—
406C2#1909913611yescharacteristic
407C2#1909913711yescharacteristic
408C2#1909915911yescharacteristic
409C2D4×C22⋊D7911yescharacteristic
410C2#1909921211yescharacteristic
411C2#1909921511yescharacteristic
412C2#1909931411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D79
commutator, socle, cosocleC23×C79C24
FittingC2×C22⋊Q8×C79C2
prime radicalC22×C22⋊D79C22
hypercenterC2×C22⋊Q8D79
hypocenterC79#2680
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#14579137C2semidirect
4. #19098496C2semidirect
5. #19098493C2semidirect
6. #19098491C2semidirect
7. #19098489C2semidirect
8. #19098482C2semidirect
10. #19098456C2semidirect
11. #19098453C2semidirect
12. #19098436C2semidirect
13. #19098434C2semidirect
14. #19098432C2semidirect
15. C2×C22⋊Q8×C79C2semidirect
207. C79#2680semidirect
211. C2×C22⋊Q8D79semidirect