Finite Group Databasebeta

← #23777502

Subgroups — #23777502

Order 10112.

1030 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19099295C213yesmaximal, complemented
2#19099289C211yesmaximal, characteristic, complemented
3#19099277C213yesmaximal, complemented
4#19099259C213yesmaximal, complemented
5#19099255C213yesmaximal, complemented
6#19099232C211yesmaximal, characteristic, complemented
7#19099224C211yesmaximal, characteristic, complemented
8#19099186C213yesmaximal, complemented
9#19099166C213yesmaximal, complemented
10#19099095C213yesmaximal, complemented
11#19099092C213yesmaximal, complemented
12#19099046C211yesmaximal, characteristic, complemented
13#19099041C211yesmaximal, characteristic, complemented
14C4×#11929625C211yesmaximal, characteristic, complemented
15#548×C79C211yesmaximal, characteristic, complemented
16C2×#11929625C2211yescharacteristic
17#14579160C2213yes—
18#14579157C2213yes—
19#14579156C2216yes—
20#14579154C2213yes—
21#14579154C2211yescharacteristic
22#14579153C2213yes—
23#14579152C2213yes—
24#14579152C2213yes—
25#14579150C2213yes—
26#14579148C2213yes—
27#14579147C2216yes—
28#14579145C2216yes—
29#14579143C2216yes—
30#14579143C2213yes—
31#14579143C2213yes—
32#14579141C2213yes—
33#14579137C2213yes—
34#14579135C2213yes—
35#14579135C2213yes—
36#14579133C2216yes—
37#14579132C2213yes—
38#14579132C2213yes—
39Q8×Dic79C2213yes—
40Q8×Dic79C2211yescharacteristic
41Q8×Dic79C2211yescharacteristic
42Q8×Dic79C2213yes—
43Q8×Dic79C2211yescharacteristic
44C4×Dic158C2213yes—
45C4×D316C2213yes—
46D4×Dic79C2213yes—
47D4×Dic79C2213yes—
48#14579123C2213yes—
49C4×C22⋊D79C2213yes—
50C4×C22⋊D79C2213yes—
51(C22⋊D79)⋊C4C2216yes—
52#14579120C2213yes—
53#14579120C2213yes—
54#14579120C2213yes—
55#14579119C2213yes—
56#14579117C2216yes—
57C2×C4⋊Dic79C2213yes—
58C42×D79C2213yes—
59C2×#41×C79C2211yescharacteristic
60C22×Q8×C79C2211yescharacteristic
61#124×C79C2213yes—
62C22⋊Q8×C79C2213yes—
63#121×C79C2213yes—
64C4×Q8×C79C2211yescharacteristic
65C4×D4×C79C2213yes—
66#11929625—24no—
67C2×Dic158C2313yes—
68C2×D316C2313yes—
69C2×C22⋊D79C2316yes—
70C2×C22⋊D79C2313yes—
71C2×C22⋊D79C2313yes—
72C22×Dic79C2316yes—
73C2×C4×D79C2313yes—
74C2×C4×D79C2313yes—
75C2×C4×D79C2313yes—
76C4⋊Dic79C2313yes—
77C4⋊Dic79C2313yes—
78C4⋊Dic79C2313yes—
79C4⋊Dic79C2311yescharacteristic
80C4⋊Dic79C2313yes—
81C4⋊Dic79C2313yes—
82C4⋊Dic79C2313yes—
83C4⋊Dic79C2313yes—
84Dic79⋊C4C2313yes—
85Dic79⋊C4C2316yes—
86Dic79⋊C4C2313yes—
87Dic79⋊C4C2311yescharacteristic
88Dic79⋊C4C2311yescharacteristic
89C22⋊Dic79C2313yes—
90C22⋊Dic79C2311yescharacteristic
91C22⋊Dic79C2313yes—
92C22⋊Dic79C2313yes—
93C22⋊Dic79C2313yes—
94C22⋊Dic79C2311yescharacteristic
95(C2×D79)⋊C4C2316yes—
96(C2×D79)⋊C4C2316yes—
97(C2×D79)⋊C4C2313yes—
98(C2×D79)⋊C4C2313yes—
99C4×Dic79C2311yescharacteristic
100C4×Dic79C2313yes—
101C4×Dic79C2313yes—
102C4×Dic79C2316yes—
103C4×Dic79C2313yes—
104C4×Dic79C2313yes—
105#41×C79—24no—
106C2×Q8×C79—24no—
107C2×Q8×C79C2313yes—
108C2×Q8×C79C2311yescharacteristic
109C2×D4×C79C2313yes—
110C2×D4×C79C2313yes—
111C22×C4×C79C2313yes—
112C22×C4×C79C2313yes—
113C4⋊C4×C79C2313yes—
114C4⋊C4×C79C2313yes—
115C22⋊C4×C79C2311yescharacteristic
116C22⋊C4×C79C2313yes—
117C22⋊C4×C79C2316yes—
118C42×C79C2313yes—
119C23×C79C2411yescharacteristic
120C23×C79C2413yes—
121C22×D79C2413yes—
122Q8×C79—26no—
123Q8×C79#4114yes—
124D4×C79—26no—
125D4×C79—43no—
126C2×C4×C79—26no—
127C2×C4×C79—26no—
128C2×C4×C79C2413yes—
129C2×C4×C79C2413yes—
130C2×C4×C79C2411yescharacteristic
131C2×C4×C79C2411yescharacteristic
132C2×C4×C79C2413yes—
133C22⋊D79—43no—
134C22⋊D79—46no—
135C22⋊D79—43no—
136C2×Dic79—46no—
137C2×Dic79C2413yes—
138C2×Dic79C2411yescharacteristic
139C2×Dic79C2411yescharacteristic
140C2×Dic79C2411yescharacteristic
141C2×Dic79C2413yes—
142C2×Dic79C2411yescharacteristic
143C2×Dic79C2413yes—
144D316—26no—
145C4×D79—26no—
146C4×D79—43no—
147C4×D79—43no—
148Dic158—43no—
149C22×C79—21no—
150C22×C79—23no—
151C22×C79—21no—
152C22×C79—23no—
153C22×C79—23no—
154C22×C79—21no—
155C22×C79C2511yescharacteristic
156C2×D79—23no—
157C2×D79—23no—
158C2×D79—23no—
159C4×C79—21no—
160C4×C79—23no—
161C4×C79—23no—
162C4×C79—21no—
163C4×C79C2×#4116yes—
164Dic79—21no—
165Dic79—21no—
166Dic79—23no—
167Dic79—23no—
168Dic79—21no—
169Dic79—21no—
170Dic79—23no—
171C2×C79—41no—
172C2×C79—43no—
173C2×C79C22×#4111yescharacteristic
174C2×C79#58511yescharacteristic
175C2×C79#58511yescharacteristic
176D79—43no—
177#3524—791nomaximal, Sylow
178C79#352411yescharacteristic, complemented, Sylow
179#559—793no—
180#557—791no—
181#555—793no—
182#553—793no—
183#552—793no—
184#550—791no—
185#548—791no—
186#548D7911yescharacteristic, complemented
187#542—793no—
188#540—793no—
189#529—793no—
190#529—793no—
191#520—791no—
192#519—791no—
193C4×#41—791no—
194C2×#41—791no—
195C2×#41C2×D7911yescharacteristic
196C22×Q8C2×D7911yescharacteristic
197C4⋊Q8—793no—
198#127—793no—
199#126—796no—
200#126—791no—
201#126—793no—
202#125—793no—
203#125—793no—
204#125—793no—
205#124—793no—
206#124—793no—
207#124—796no—
208#124C2×D7913yes—
209#123—796no—
210#123—793no—
211#123—793no—
212#123—796no—
213C22⋊Q8—793no—
214C22⋊Q8—793no—
215C22⋊Q8C2×D7913yes—
216#121—793no—
217#121—793no—
218#121—796no—
219#121—793no—
220#121—793no—
221#121C2×D7913yes—
222C4×Q8—791no—
223C4×Q8—793no—
224C4×Q8—791no—
225C4×Q8—793no—
226C4×Q8—791no—
227C4×Q8—793no—
228C4×Q8C2×D7911yescharacteristic
229C4×D4—793no—
230C4×D4—793no—
231C4×D4—793no—
232C4×D4—793no—
233C4×D4—793no—
234C4×D4—796no—
235C4×D4—793no—
236C4×D4C2×D7913yes—
237#117—793no—
238#117—793no—
239#117—793no—
240#117—793no—
241#117—796no—
242C2×C4⋊C4—793no—
243C2×C42—793no—
244#41—24no—
245#41—1584no—
246C2×Q8—24no—
247C2×Q8—793no—
248C2×Q8C22×D7913yes—
249C2×Q8C22×D7911yescharacteristic
250C2×D4—793no—
251C2×D4—796no—
252C2×D4—793no—
253C2×D4—793no—
254C2×D4C22×D7913yes—
255C2×D4C22×D7913yes—
256C22×C4—796no—
257C22×C4—793no—
258C22×C4—793no—
259C22×C4—793no—
260C22×C4C22×D7913yes—
261C22×C4C22×D7913yes—
262C4⋊C4—793no—
263C4⋊C4—796no—
264C4⋊C4—791no—
265C4⋊C4—793no—
266C4⋊C4—791no—
267C4⋊C4—793no—
268C4⋊C4—793no—
269C4⋊C4—793no—
270C4⋊C4—791no—
271C4⋊C4—793no—
272C4⋊C4—793no—
273C4⋊C4—793no—
274C4⋊C4—793no—
275C4⋊C4C22×D7913yes—
276C4⋊C4C22×D7913yes—
277C22⋊C4—791no—
278C22⋊C4—793no—
279C22⋊C4—793no—
280C22⋊C4—793no—
281C22⋊C4—793no—
282C22⋊C4—791no—
283C22⋊C4—796no—
284C22⋊C4—793no—
285C22⋊C4—796no—
286C22⋊C4—793no—
287C22⋊C4C22×D7916yes—
288C22⋊C4C22×D7913yes—
289C22⋊C4C22×D7911yescharacteristic
290C42—791no—
291C42—793no—
292C42—793no—
293C42—796no—
294C42—793no—
295C42—793no—
296C42C22×D7913yes—
297C23—793no—
298C23C23×D7911yescharacteristic
299C23C23×D7913yes—
300Q8—26no—
301Q8—3163no—
302Q8#1192962314yes—
303D4—26no—
304D4—43no—
305D4—3166no—
306D4—3163no—
307D4—3163no—
308D4—1586no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—3163no—
312C2×C4—3163no—
313C2×C4—3166no—
314C2×C4—791no—
315C2×C4—793no—
316C2×C4—791no—
317C2×C4—791no—
318C2×C4—793no—
319C2×C4—791no—
320C2×C4—793no—
321C2×C4—1586no—
322C2×C4C23×D7913yes—
323C2×C4C23×D7911yescharacteristic
324C2×C4C23×D7913yes—
325C2×C4C23×D7911yescharacteristic
326C2×C4C23×D7913yes—
327C22—21no—
328C22—23no—
329C22—23no—
330C22—21no—
331C22—21no—
332C22—23no—
333C22—1583no—
334C22—1583no—
335C22—1583no—
336C22C24×D7911yescharacteristic
337C4—23no—
338C4—21no—
339C4—21no—
340C4—23no—
341C4—1581no—
342C4—1583no—
343C4—1581no—
344C4—1583no—
345C4—1583no—
346C4—1581no—
347C4—1581no—
348C4C2×#1192962316yes—
349C2—41no—
350C2—43no—
351C2—3163no—
352C2C22×#1192962311yescharacteristic
353C2#1909939911yescharacteristic
354C2#1909940011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#548×C79C2
hypercenter#548D79
hypocenterC79#3524
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #19099295C2semidirect
2. #19099289C2semidirect
3. #19099277C2semidirect
4. #19099259C2semidirect
5. #19099255C2semidirect
6. #19099232C2semidirect
7. #19099224C2semidirect
8. #19099186C2semidirect
9. #19099166C2semidirect
10. #19099095C2semidirect
11. #19099092C2semidirect
12. #19099046C2semidirect
13. #19099041C2semidirect
14. C4×#11929625C2semidirect
15. #548×C79C2semidirect
178. C79#3524semidirect
186. #548D79semidirect