Finite Group Databasebeta

← #23777476

Subgroups — #23777476

Order 10112.

1104 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19099295C213yesmaximal, complemented
2Q8×Dic158C211yesmaximal, characteristic, complemented
3#19099271C213yesmaximal, complemented
4#19099232C211yesmaximal, characteristic, complemented
5#19099224C211yesmaximal, characteristic, complemented
6#19099217C213yesmaximal, complemented
7#19099213C213yesmaximal, complemented
8#19099186C213yesmaximal, complemented
9#19099182C213yesmaximal, complemented
10#19099108C213yesmaximal, complemented
11#19099093C213yesmaximal, complemented
12#19099046C211yesmaximal, characteristic, complemented
13#19099045C211yesmaximal, characteristic, complemented
14C2×Q8×Dic79C211yesmaximal, characteristic, complemented
15#548×C79C211yesmaximal, characteristic, complemented
16C2×#11929625C2211yescharacteristic
17C2×#11929623C2216yes—
18#14579161C2213yes—
19#14579160C2213yes—
20#14579160C2213yes—
21#14579157C2213yes—
22#14579156C2213yes—
23#14579154C2213yes—
24#14579151C2213yes—
25#14579150C2213yes—
26#14579147C2216yes—
27#14579147C2213yes—
28#14579145C2216yes—
29#14579145C2213yes—
30#14579143C2216yes—
31#14579142C2213yes—
32#14579141C2216yes—
33#14579138C2216yes—
34#14579138C2211yescharacteristic
35#14579137C2213yes—
36#14579135C2213yes—
37#14579135C2213yes—
38#14579133C2213yes—
39#14579132C2213yes—
40Q8×Dic79C2213yes—
41Q8×Dic79C2211yescharacteristic
42Q8×Dic79C2211yescharacteristic
43Q8×Dic79C2213yes—
44Q8×Dic79C2211yescharacteristic
45C4×Dic158C2213yes—
46C4×D316C2213yes—
47D4×Dic79C2213yes—
48D4×Dic79C2213yes—
49#14579123C2213yes—
50C4×C22⋊D79C2213yes—
51C4×C22⋊D79C2213yes—
52(C22⋊D79)⋊C4C2216yes—
53#14579120C2213yes—
54#14579118C2213yes—
55C2×C4⋊Dic79C2213yes—
56C2×C4⋊Dic79C2213yes—
57C4⋊C4×D79C2213yes—
58C2×C4×Dic79C2213yes—
59C2×#41×C79C2211yescharacteristic
60C22×Q8×C79C2211yescharacteristic
61#124×C79C2213yes—
62C22⋊Q8×C79C2213yes—
63#121×C79C2213yes—
64C4×Q8×C79C2211yescharacteristic
65C4×D4×C79C2213yes—
66#11929625—24no—
67#11929623—412no—
68C2×Dic158C2311yescharacteristic
69C2×Dic158C2316yes—
70C2×D316C2313yes—
71C2×C22⋊D79C2313yes—
72C2×C22⋊D79C2313yes—
73C2×C22⋊D79C2316yes—
74C22×Dic79C2313yes—
75C22×Dic79C2311yescharacteristic
76C22×Dic79C2316yes—
77C2×C4×D79C2313yes—
78C2×C4×D79C2316yes—
79C4⋊Dic79C2313yes—
80C4⋊Dic79C2313yes—
81C4⋊Dic79C2313yes—
82C4⋊Dic79C2313yes—
83C4⋊Dic79C2316yes—
84C4⋊Dic79C2311yescharacteristic
85C4⋊Dic79C2313yes—
86Dic79⋊C4C2313yes—
87Dic79⋊C4C2311yescharacteristic
88Dic79⋊C4C2313yes—
89Dic79⋊C4C2311yescharacteristic
90Dic79⋊C4C2313yes—
91Dic79⋊C4C2313yes—
92C22⋊Dic79C2313yes—
93C22⋊Dic79C2313yes—
94C22⋊Dic79C2311yescharacteristic
95C22⋊Dic79C2313yes—
96(C2×D79)⋊C4C2313yes—
97(C2×D79)⋊C4C2313yes—
98(C2×D79)⋊C4C2316yes—
99(C2×D79)⋊C4C2313yes—
100(C2×D79)⋊C4C2313yes—
101C4×Dic79C2313yes—
102C4×Dic79C2313yes—
103C4×Dic79C2313yes—
104C4×Dic79C2313yes—
105C4×Dic79C2313yes—
106#41×C79—24no—
107C2×Q8×C79—24no—
108C2×Q8×C79C2313yes—
109C2×Q8×C79C2311yescharacteristic
110C2×D4×C79C2313yes—
111C2×D4×C79C2313yes—
112C22×C4×C79C2313yes—
113C22×C4×C79C2313yes—
114C4⋊C4×C79C2313yes—
115C4⋊C4×C79C2313yes—
116C22⋊C4×C79C2311yescharacteristic
117C22⋊C4×C79C2313yes—
118C22⋊C4×C79C2316yes—
119C42×C79C2313yes—
120C23×C79C2413yes—
121C23×C79C2411yescharacteristic
122C22×D79C2413yes—
123Q8×C79—26no—
124Q8×C79#4114yescomplemented
125D4×C79—26no—
126D4×C79—26no—
127C2×C4×C79—26no—
128C2×C4×C79—26no—
129C2×C4×C79C2413yes—
130C2×C4×C79C2411yescharacteristic
131C2×C4×C79C2411yescharacteristic
132C2×C4×C79C2413yes—
133C2×C4×C79C2413yes—
134C22⋊D79—26no—
135C22⋊D79—26no—
136C22⋊D79—46no—
137C2×Dic79—26no—
138C2×Dic79—22no—
139C2×Dic79—46no—
140C2×Dic79C2411yescharacteristic
141C2×Dic79C2411yescharacteristic
142C2×Dic79C2413yes—
143C2×Dic79C2413yes—
144C2×Dic79C2413yes—
145C2×Dic79C2411yescharacteristic
146C2×Dic79C2411yescharacteristic
147D316—26no—
148C4×D79—26no—
149C4×D79—46no—
150Dic158—22no—
151Dic158—46no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—23no—
155C22×C79—23no—
156C22×C79—23no—
157C22×C79H(5,𝔽2)12yes—
158C22×C79C2511yescharacteristic
159C2×D79—23no—
160C2×D79—23no—
161C2×D79—23no—
162C4×C79—21no—
163C4×C79—23no—
164C4×C79—23no—
165C4×C79—21no—
166C4×C79C2×#4116yes—
167Dic79—21no—
168Dic79—23no—
169Dic79—23no—
170Dic79—23no—
171Dic79—21no—
172Dic79—21no—
173Dic792−1+412yes—
174C2×C79—22no—
175C2×C79—43no—
176C2×C79C22×#4111yescharacteristic
177C2×C79C2×H(5,𝔽2)11yescharacteristic
178C2×C79C2×2−1+411yescharacteristic
179D79—43no—
180#3522—791nomaximal, Sylow
181C79#352211yescharacteristic, complemented, Sylow
182#559—793no—
183Q82—791no—
184#554—793no—
185#550—791no—
186#548—791no—
187#548D7911yescharacteristic, complemented
188#547—793no—
189#547—793no—
190#542—793no—
191#542—793no—
192#532—793no—
193#529—793no—
194#520—791no—
195#520—791no—
196C2×C4×Q8—791no—
197C2×#41—791no—
198C2×#41—796no—
199C2×#41C2×D7911yescharacteristic
200C22×Q8C2×D7911yescharacteristic
201C4⋊Q8—793no—
202C4⋊Q8—793no—
203C4⋊Q8—793no—
204#127—793no—
205#126—793no—
206#126—793no—
207#125—793no—
208#124—793no—
209#124—793no—
210#124—796no—
211#124C2×D7913yes—
212#123—793no—
213#123—793no—
214#123—796no—
215#123—796no—
216C22⋊Q8—793no—
217C22⋊Q8—796no—
218C22⋊Q8—791no—
219C22⋊Q8—796no—
220C22⋊Q8C2×D7913yes—
221#121—793no—
222#121—793no—
223#121—793no—
224#121—793no—
225#121C2×D7913yes—
226C4×Q8—791no—
227C4×Q8—793no—
228C4×Q8—791no—
229C4×Q8—793no—
230C4×Q8—791no—
231C4×Q8—793no—
232C4×Q8C2×D7911yescharacteristic
233C4×D4—793no—
234C4×D4—793no—
235C4×D4—793no—
236C4×D4—793no—
237C4×D4—793no—
238C4×D4—796no—
239C4×D4—793no—
240C4×D4C2×D7913yes—
241#117—793no—
242#117—793no—
243C2×C4⋊C4—793no—
244C2×C4⋊C4—793no—
245C2×C4⋊C4—793no—
246C2×C42—793no—
247#41—24no—
248#41—31612no—
249#41—1584no—
250C2×Q8—24no—
251C2×Q8—796no—
252C2×Q8—791no—
253C2×Q8C22×D7913yes—
254C2×Q8C22×D7911yescharacteristic
255C2×D4—796no—
256C2×D4—793no—
257C2×D4—793no—
258C2×D4—793no—
259C2×D4C22×D7913yes—
260C2×D4C22×D7913yes—
261C22×C4—796no—
262C22×C4—796no—
263C22×C4—791no—
264C22×C4—793no—
265C22×C4—793no—
266C22×C4C22×D7913yes—
267C22×C4C22×D7913yes—
268C4⋊C4—793no—
269C4⋊C4—791no—
270C4⋊C4—791no—
271C4⋊C4—793no—
272C4⋊C4—793no—
273C4⋊C4—793no—
274C4⋊C4—793no—
275C4⋊C4—793no—
276C4⋊C4—793no—
277C4⋊C4—793no—
278C4⋊C4—796no—
279C4⋊C4—791no—
280C4⋊C4—793no—
281C4⋊C4C22×D7913yes—
282C4⋊C4C22×D7913yes—
283C22⋊C4—793no—
284C22⋊C4—793no—
285C22⋊C4—791no—
286C22⋊C4—793no—
287C22⋊C4—793no—
288C22⋊C4—793no—
289C22⋊C4—793no—
290C22⋊C4—796no—
291C22⋊C4—793no—
292C22⋊C4C22×D7916yes—
293C22⋊C4C22×D7911yescharacteristic
294C22⋊C4C22×D7913yes—
295C42—793no—
296C42—793no—
297C42—793no—
298C42—793no—
299C42—793no—
300C42C22×D7913yes—
301C23—793no—
302C23C23×D7913yes—
303C23C23×D7911yescharacteristic
304Q8—26no—
305Q8—3166no—
306Q8—1582no—
307Q8#1192962314yescomplemented
308D4—26no—
309D4—26no—
310D4—3166no—
311D4—1586no—
312D4—1586no—
313D4—1586no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—3166no—
317C2×C4—3166no—
318C2×C4—791no—
319C2×C4—791no—
320C2×C4—791no—
321C2×C4—793no—
322C2×C4—793no—
323C2×C4—793no—
324C2×C4—791no—
325C2×C4—1582no—
326C2×C4—1586no—
327C2×C4—1586no—
328C2×C4C23×D7913yes—
329C2×C4C23×D7911yescharacteristic
330C2×C4C23×D7913yes—
331C2×C4C23×D7911yescharacteristic
332C2×C4C23×D7913yes—
333C22—21no—
334C22—21no—
335C22—23no—
336C22—23no—
337C22—23no—
338C22—1583no—
339C22—1583no—
340C22—1583no—
341C22#1457920712yes—
342C22C24×D7911yescharacteristic
343C4—23no—
344C4—21no—
345C4—21no—
346C4—23no—
347C4—792no—
348C4—1583no—
349C4—1583no—
350C4—1583no—
351C4—1581no—
352C4—1581no—
353C4—1581no—
354C4C2×#1192962316yes—
355C2—22no—
356C2—43no—
357C2—3163no—
358C2C22×#1192962311yescharacteristic
359C2C2×#1457920711yescharacteristic
360C22−1+4×D7911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#548×C79C2
hypercenter#548D79
hypocenterC79#3522
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #19099295C2semidirect
2. Q8×Dic158C2semidirect
3. #19099271C2semidirect
4. #19099232C2semidirect
5. #19099224C2semidirect
6. #19099217C2semidirect
7. #19099213C2semidirect
8. #19099186C2semidirect
9. #19099182C2semidirect
10. #19099108C2semidirect
11. #19099093C2semidirect
12. #19099046C2semidirect
13. #19099045C2semidirect
14. C2×Q8×Dic79C2semidirect
15. #548×C79C2semidirect
124. Q8×C79#41semidirect
181. C79#3522semidirect
187. #548D79semidirect
307. Q8#11929623semidirect