Finite Group Databasebeta

← #23770313

Subgroups — #23770313

Order 10112.

498 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14579156C211yesmaximal, characteristic, complemented
2#19098498C211yesmaximal, characteristic, complemented
3#19098489C211yesmaximal, characteristic, complemented
4#19098488C211yesmaximal, characteristic, complemented
5#19098482C211yesmaximal, characteristic, complemented
6#19098479C211yesmaximal, characteristic
7#19098474C211yesmaximal, characteristic, complemented
8#19098463C211yesmaximal, characteristic
9#19098459C211yesmaximal, characteristic, complemented
10#19098451C211yesmaximal, characteristic, complemented
11#19098449C211yesmaximal, characteristic, complemented
12#19098442C211yesmaximal, characteristic
13#19098436C211yesmaximal, characteristic, complemented
14#19098428C211yesmaximal, characteristic, complemented
15C2×C22⋊Q8×C79C211yesmaximal, characteristic, complemented
16C22×C22⋊D79C2211yescharacteristic
17#14579156—24no—
18C2×C4⋊Dic79C2211yescharacteristic
19C2×C4⋊Dic79C2211yescharacteristic
20C2×Dic79⋊C4C2211yescharacteristic
21C2×Dic79⋊C4C2211yescharacteristic
22C2×Dic79⋊C4C2211yescharacteristic
23C2×Dic79⋊C4C2211yescharacteristic
24C2×C22⋊Dic79C2211yescharacteristic
25C2×C22⋊Dic79C2211yescharacteristic
26C2×C22⋊Dic79C2211yescharacteristic
27C2×(C2×D79)⋊C4C2211yescharacteristic
28C2×(C2×D79)⋊C4C2211yescharacteristic
29C2×(C2×D79)⋊C4C2211yescharacteristic
30C2×(C2×D79)⋊C4C2211yescharacteristic
31C2×(C2×D79)⋊C4C2211yescharacteristic
32C2×(C2×D79)⋊C4C2211yescharacteristic
33C2×C4×Dic79C2211yescharacteristic
34C2×C4×Dic79C2211yescharacteristic
35C79⋊#95C2211yescharacteristic
36C79⋊#95C2211yescharacteristic
37C79⋊#95C2211yescharacteristic
38C79⋊#95C2211yescharacteristic
39C79⋊#95C2211yescharacteristic
40C79⋊#95C2211yescharacteristic
41C79⋊#95C2211yescharacteristic
42C79⋊#95C2211yescharacteristic
43C79⋊#95C2211yescharacteristic
44C79⋊#95C2211yescharacteristic
45C22×Q8×C79C2211yescharacteristic
46C23×C4×C79C2211yescharacteristic
47C22⋊Q8×C79—24no—
48C2×C4⋊C4×C79C2211yescharacteristic
49C2×C4⋊C4×C79C2211yescharacteristic
50C2×C4⋊C4×C79C2211yescharacteristic
51C2×C22⋊C4×C79C2211yescharacteristic
52C2×C22⋊C4×C79C2211yescharacteristic
53C23×D79C2311yescharacteristic
54C2×C22⋊D79—22no—
55C2×C22⋊D79—22no—
56C2×C22⋊D79—41no—
57C22×Dic79C2311yescharacteristic
58C22×Dic79C2311yescharacteristic
59C22×Dic79C2311yescharacteristic
60C22×Dic79C2311yescharacteristic
61C22×Dic79C2311yescharacteristic
62C22×Dic79C2311yescharacteristic
63C22×Dic79C2311yescharacteristic
64C4⋊Dic79—22no—
65C4⋊Dic79—22no—
66Dic79⋊C4—22no—
67Dic79⋊C4—41no—
68Dic79⋊C4—41no—
69Dic79⋊C4—41no—
70C22⋊Dic79—22no—
71C22⋊Dic79—22no—
72C22⋊Dic79—41no—
73(C2×D79)⋊C4—22no—
74(C2×D79)⋊C4—22no—
75(C2×D79)⋊C4—22no—
76(C2×D79)⋊C4—41no—
77(C2×D79)⋊C4—41no—
78(C2×D79)⋊C4—41no—
79C4×Dic79—22no—
80C4×Dic79—22no—
81C24×C79C2311yescharacteristic
82C2×Q8×C79—22no—
83C2×Q8×C79—41no—
84C2×Q8×C79—41no—
85C22×C4×C79—22no—
86C22×C4×C79—41no—
87C22×C4×C79—41no—
88C22×C4×C79C2311yescharacteristic
89C22×C4×C79C2311yescharacteristic
90C22×C4×C79C2311yescharacteristic
91C22×C4×C79C2311yescharacteristic
92C22×C4×C79C2311yescharacteristic
93C22×C4×C79C2311yescharacteristic
94C4⋊C4×C79—22no—
95C4⋊C4×C79—22no—
96C4⋊C4×C79D414yescomplemented
97C22⋊C4×C79—22no—
98C22⋊C4×C79—22no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—21no—
102C23×C79—21no—
103C23×C79—21no—
104C23×C79—21no—
105C23×C79#4112yes—
106C23×C79C2411yescharacteristic
107C22×D79—21no—
108C22×D79—21no—
109C22×D79—21no—
110C22×D79—21no—
111C22×D79—21no—
112C22×D79—21no—
113C22×D79—21no—
114Q8×C79—44no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
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—44no—
131C2×C4×C79C2×D412yes—
132C2×C4×C79C2×D412yes—
133C2×C4×C79C2×D412yes—
134C22⋊D79—44no—
135C2×Dic79—21no—
136C2×Dic79—21no—
137C2×Dic79—21no—
138C2×Dic79—21no—
139C2×Dic79—21no—
140C2×Dic79—21no—
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#4112yes—
156C22×C79—22no—
157C22×C79—22no—
158C22×C79—22no—
159C22×C79—41no—
160C22×C79—41no—
161C22×C79—41no—
162C22×C79—41no—
163C22×C79C22×D411yescharacteristic
164C22×C79C2×#4111yescharacteristic
165C22×C79C2×#4111yescharacteristic
166C22×C79H(5,𝔽2)11yescharacteristic
167C22×C79H(5,𝔽2)11yescharacteristic
168C22×C792−1+411yescharacteristic
169C22×C792−1+411yescharacteristic
170C2×D79—41no—
171C2×D79—41no—
172C2×D79—41no—
173C2×D79—41no—
174C2×D79—41no—
175C2×D79—41no—
176C2×D79—41no—
177C4×C79—22no—
178C4×C79—22no—
179C4×C79—22no—
180C4×C79—41no—
181C4×C79—41no—
182C4×C79—41no—
183Dic79—22no—
184Dic79—41no—
185Dic79—41no—
186Dic79—41no—
187Dic79—41no—
188Dic79—41no—
189Dic79—41no—
190C2×C79—42no—
191C2×C79#53511yescharacteristic
192C2×C79#53911yescharacteristic
193C2×C79#54211yescharacteristic
194C2×C79#54711yescharacteristic
195C2×C79#54811yescharacteristic
196C2×C79#55111yescharacteristic
197C2×C79#56311yescharacteristic
198D79—81no—
199#2723—791nomaximal, Sylow
200C79#272311yescharacteristic, complemented, Sylow
201C2×#126—791no—
202C2×C22⋊Q8D7911yescharacteristic, complemented
203#399—791no—
204#397—791no—
205#397—791no—
206#395—791no—
207#394—791no—
208#393—791no—
209#390—791no—
210#389—791no—
211#388—791no—
212#387—791no—
213#386—791no—
214#385—791no—
215#382—791no—
216C22×Q8C2×D7911yescharacteristic
217C22×D4—791no—
218C23×C4C2×D7911yescharacteristic
219#126—1584no—
220C22⋊Q8—24no—
221C2×C4⋊C4—791no—
222C2×C4⋊C4—791no—
223C2×C4⋊C4—791no—
224C2×C4⋊C4—791no—
225C2×C4⋊C4—791no—
226C2×C4⋊C4—791no—
227C2×C4⋊C4C2×D7911yescharacteristic
228C2×C4⋊C4C2×D7911yescharacteristic
229C2×C4⋊C4C2×D7911yescharacteristic
230C2×C22⋊C4—791no—
231C2×C22⋊C4—791no—
232C2×C22⋊C4—791no—
233C2×C22⋊C4—791no—
234C2×C22⋊C4—791no—
235C2×C22⋊C4—791no—
236C2×C22⋊C4—791no—
237C2×C22⋊C4—791no—
238C2×C22⋊C4—791no—
239C2×C22⋊C4C2×D7911yescharacteristic
240C2×C22⋊C4C2×D7911yescharacteristic
241C2×C42—791no—
242C2×C42—791no—
243#95—791no—
244#95—791no—
245#95—791no—
246#95—791no—
247#95—791no—
248#95—791no—
249#95—791no—
250#95—791no—
251#95—791no—
252#95—791no—
253C24—791no—
254C24C22×D7911yescharacteristic
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×D4—3161no—
259C2×D4—1582no—
260C2×D4—1582no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—791no—
265C22×C4—791no—
266C22×C4—791no—
267C22×C4—791no—
268C22×C4—791no—
269C22×C4—791no—
270C22×C4—791no—
271C22×C4C22×D7911yescharacteristic
272C22×C4C22×D7911yescharacteristic
273C22×C4C22×D7911yescharacteristic
274C22×C4C22×D7911yescharacteristic
275C22×C4C22×D7911yescharacteristic
276C22×C4C22×D7911yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—3161no—
280C4⋊C4—3161no—
281C4⋊C4—3161no—
282C4⋊C4—1582no—
283C4⋊C4—1582no—
284C4⋊C4—1582no—
285C4⋊C4C22⋊D7914yescomplemented
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—3161no—
289C22⋊C4—3161no—
290C22⋊C4—3161no—
291C22⋊C4—3161no—
292C22⋊C4—1582no—
293C22⋊C4—1582no—
294C22⋊C4—1582no—
295C22⋊C4—1582no—
296C22⋊C4—1582no—
297C42—1582no—
298C42—1582no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—1581no—
306C23—1581no—
307C23—1581no—
308C23—1581no—
309C23—1581no—
310C23—1581no—
311C23—1581no—
312C23#1192962512yes—
313C23C23×D7911yescharacteristic
314Q8—44no—
315D4—3164no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—792no—
333C2×C4—1581no—
334C2×C4—1581no—
335C2×C4—1581no—
336C2×C4—1581no—
337C2×C4—1581no—
338C2×C4—1581no—
339C2×C4—1581no—
340C2×C4—1581no—
341C2×C4—1581no—
342C2×C4—1581no—
343C2×C4—1581no—
344C2×C4—1581no—
345C2×C4—1581no—
346C2×C4—1581no—
347C2×C4—1581no—
348C2×C4—1581no—
349C2×C4—1581no—
350C2×C4—1581no—
351C2×C4—1581no—
352C2×C4—1581no—
353C2×C4C2×C22⋊D7912yes—
354C2×C4C2×C22⋊D7912yes—
355C2×C4C2×C22⋊D7912yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3161no—
364C22—3161no—
365C22—3161no—
366C22—3161no—
367C22—3161no—
368C22—3161no—
369C22—3161no—
370C22C22×C22⋊D7911yescharacteristic
371C22#41×D7911yescharacteristic
372C22C2×#1192962511yescharacteristic
373C22#1457920611yescharacteristic
374C22#1457920611yescharacteristic
375C22#1457920811yescharacteristic
376C22#1457920911yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3161no—
384C4—3161no—
385C4—3161no—
386C4—3161no—
387C4—3161no—
388C4—3161no—
389C4—1582no—
390C2—42no—
391C2—6321no—
392C2#1909913011yescharacteristic
393C2#1909916211yescharacteristic
394C2#1909918311yescharacteristic
395C2#1909921211yescharacteristic
396C2#1909922411yescharacteristic
397C2#1909924011yescharacteristic
398C2#1909931411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D79
commutator, socle, cosocleC23×C79C24
FittingC2×C22⋊Q8×C79C2
prime radicalC22×C22⋊D79C22
hypercenterC2×C22⋊Q8D79
hypocenterC79#2723
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#14579156C2semidirect
2. #19098498C2semidirect
3. #19098489C2semidirect
4. #19098488C2semidirect
5. #19098482C2semidirect
7. #19098474C2semidirect
9. #19098459C2semidirect
10. #19098451C2semidirect
11. #19098449C2semidirect
13. #19098436C2semidirect
14. #19098428C2semidirect
15. C2×C22⋊Q8×C79C2semidirect
96. C4⋊C4×C79D4semidirect
200. C79#2723semidirect
202. C2×C22⋊Q8D79semidirect
285. C4⋊C4C22⋊D79semidirect