Finite Group Databasebeta

← #23773125

Subgroups — #23773125

Order 10112.

466 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19099361C211yesmaximal, characteristic, complemented
2C2×#14579187C211yesmaximal, characteristic, complemented
3#19099330C211yesmaximal, characteristic, complemented
4#19098831C214yesmaximal, complemented
5#19098824C214yesmaximal, complemented
6#19098611C211yesmaximal, characteristic, complemented
7#19098545C211yesmaximal, characteristic, complemented
8C2×#14579078C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C79C211yesmaximal, characteristic, complemented
10C2×#11929624C2211yescharacteristic
11#14579192—21no—
12#14579191C2212yescomplemented
13#14579187—21no—
14#14579187—22no—
15#14579187—22no—
16#14579186C2212yescomplemented
17#14579181—22no—
18C2×#11929612C2212yescomplemented
19C2×#11929609C2212yescomplemented
20#14579168—22no—
21#14579168C2214yescomplemented
22C2×#11929608—22no—
23C2×#11929608C2211yescharacteristic
24C2×#11929608C2211yescharacteristic
25C2×#11929608C2211yescharacteristic
26#14579096C2214yescomplemented
27#14579079C2212yescomplemented
28#14579078C2214yescomplemented
29#14579077C2212yescomplemented
30C2×#41×C79C2211yescharacteristic
31C22×D4×C79C2211yescharacteristic
32AGL(1,ℤ/8)×C79—22no—
33AGL(1,ℤ/8)×C79C2214yes—
34C2×SD16×C79—21no—
35C2×D8×C79—21no—
36C2×M16×C79C2211yescharacteristic
37#11929624—21no—
38#11929624—21no—
39#11929624—21no—
40#11929624—21no—
41C2×Dic158C2311yescharacteristic
42C2×D316C2311yescharacteristic
43C2×C22⋊D79—21no—
44C2×C4×D79—21no—
45#11929614—22no—
46#11929612—24no—
47#11929612—22no—
48#11929611—22no—
49#11929609—24no—
50#11929609—22no—
51#11929608—24no—
52#11929608—21no—
53#11929608—21no—
54#11929608D412yes—
55#11929608D412yes—
56#11929608C2312yes—
57#11929608C2312yes—
58C2×C79⋊C8—24no—
59C2×C79⋊C8—21no—
60C2×C79⋊C8—21no—
61C2×C79⋊C8C2312yes—
62C24×C79—21no—
63#41×C79—21no—
64#41×C79—21no—
65#41×C79D412yes—
66#41×C79C2312yes—
67C2×Q8×C79D411yescharacteristic
68C2×D4×C79—21no—
69C2×D4×C79—21no—
70C2×D4×C79—21no—
71C2×D4×C79—21no—
72C2×D4×C79—21no—
73C2×D4×C79—21no—
74C2×D4×C79D411yescharacteristic
75C2×D4×C79C2312yes—
76C22×C4×C79—21no—
77C22×C4×C79C2311yescharacteristic
78SD16×C79—24no—
79D8×C79—24no—
80M16×C79—21no—
81M16×C79C2312yes—
82C2×C8×C79—21no—
83C23×C79—21no—
84C23×C79—22no—
85C23×C79—41no—
86C23×C79—41no—
87C23×C79—82no—
88C23×C79#4111yescharacteristic
89C22×D79—21no—
90Q8×C79—21no—
91Q8×C79C2×D412yes—
92D4×C79—22no—
93D4×C79—22no—
94D4×C79—21no—
95D4×C79—22no—
96D4×C79—22no—
97D4×C79—41no—
98D4×C79—41no—
99D4×C79C2×D412yes—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—22no—
103C2×C4×C79C2×D411yescharacteristic
104C2×C4×C79C2×D411yescharacteristic
105C2×C4×C79C2×D411yescharacteristic
106C2×C4×C79C2×D411yescharacteristic
107C2×C4×C79#4111yescharacteristic
108C2×C4×C79C2411yescharacteristic
109C22⋊D79—41no—
110C22⋊D79—41no—
111C2×Dic79—21no—
112D316—21no—
113D316—21no—
114C4×D79—41no—
115C4×D79—41no—
116Dic158—21no—
117Dic158—21no—
118C8×C79—22no—
119C79⋊C8—22no—
120C79⋊C8—22no—
121C79⋊C8C2×D414yes—
122C22×C79—22no—
123C22×C79—41no—
124C22×C79—42no—
125C22×C79—81no—
126C22×C79—82no—
127C22×C79—82no—
128C22×C79—82no—
129C22×C79#12111yescharacteristic
130C22×C79#12111yescharacteristic
131C22×C79C2×#4111yescharacteristic
132C2×D79—41no—
133C2×D79—81no—
134C4×C79—22no—
135C4×C79#12111yescharacteristic
136C4×C79#12111yescharacteristic
137C4×C79C22×D411yescharacteristic
138C4×C79C22×D411yescharacteristic
139Dic79—41no—
140C2×C79—21no—
141C2×C79—21no—
142C2×C79—21no—
143C2×C79—42no—
144C2×C79—82no—
145C2×C79C2×#12111yescharacteristic
146D79—81no—
147#3068—791nomaximal, Sylow
148C79#306811yescharacteristic, complemented, Sylow
149#575—791no—
150C2×AGL(1,ℤ/8)—791no—
151C2×AGL(1,ℤ/8)D7911yescharacteristic, complemented
152#568—791no—
153#472—794no—
154#471—794no—
155#430—791no—
156#413—791no—
157C2×#100—791no—
158C2×#41—791no—
159C2×#41C2×D7911yescharacteristic
160C22×D4C2×D7911yescharacteristic
161#137—792no—
162#137—1581no—
163AGL(1,ℤ/8)—22no—
164AGL(1,ℤ/8)—792no—
165AGL(1,ℤ/8)—1582no—
166AGL(1,ℤ/8)—1582no—
167AGL(1,ℤ/8)—1581no—
168AGL(1,ℤ/8)C2×D7914yes—
169#135—1582no—
170C2×SD16—21no—
171C2×SD16—792no—
172C2×D8—21no—
173C2×D8—792no—
174#131—794no—
175#131—1582no—
176C2×M16—791no—
177C2×M16—791no—
178C2×M16—791no—
179C2×M16—1582no—
180C2×M16C2×D7911yescharacteristic
181#108—794no—
182#101—792no—
183#100—794no—
184#100—792no—
185C24—21no—
186#41—21no—
187#41—21no—
188#41—1581no—
189#41—1581no—
190#41—1581no—
191#41—1581no—
192#41C22⋊D7912yes—
193#41C22×D7912yes—
194C2×Q8—791no—
195C2×Q8C22⋊D7911yescharacteristic
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—791no—
203C2×D4—1581no—
204C2×D4C22⋊D7911yescharacteristic
205C2×D4C22×D7912yes—
206C22×C4—21no—
207C22×C4—1581no—
208C22×C4C22×D7911yescharacteristic
209Q16—1582no—
210SD16—24no—
211SD16—1584no—
212SD16—1582no—
213SD16—1582no—
214D8—24no—
215D8—1584no—
216D8—1582no—
217M16—21no—
218M16—792no—
219M16—792no—
220M16—792no—
221M16—792no—
222M16—1584no—
223M16—1581no—
224M16—1581no—
225M16C22×D7912yes—
226C2×C8—21no—
227C2×C8—792no—
228C2×C8—1584no—
229C2×C8—1581no—
230C2×C8—1581no—
231C23—21no—
232C23—22no—
233C23—41no—
234C23—41no—
235C23—82no—
236C23—1581no—
237C23#1192962311yescharacteristic
238Q8—21no—
239Q8—1581no—
240Q8—1581no—
241Q8C2×C22⋊D7912yes—
242D4—22no—
243D4—22no—
244D4—21no—
245D4—22no—
246D4—22no—
247D4—41no—
248D4—41no—
249D4—3161no—
250D4—3161no—
251D4—1581no—
252D4—1581no—
253D4C2×C22⋊D7912yes—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—3161no—
258C2×C4—3161no—
259C2×C4—1581no—
260C2×C4D4×D7911yescharacteristic
261C2×C4D4×D7911yescharacteristic
262C2×C4C2×C22⋊D7911yescharacteristic
263C2×C4C2×C22⋊D7911yescharacteristic
264C2×C4#1192962311yescharacteristic
265C2×C4C23×D7911yescharacteristic
266C8—22no—
267C8—794no—
268C8—1582no—
269C8—1582no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—3161no—
278C22—6321no—
279C22#1457913311yescharacteristic
280C22#1457913311yescharacteristic
281C22C2×#1192962311yescharacteristic
282C4—22no—
283C4—3161no—
284C4#1457913311yescharacteristic
285C4#1457913311yescharacteristic
286C4C2×D4×D7911yescharacteristic
287C4C22×C22⋊D7911yescharacteristic
288C2—21no—
289C2—21no—
290C2—21no—
291C2—42no—
292C2—82no—
293C2—6321no—
294C2C2×#1457913311yescharacteristic
subgroupG/H
centerC2C2×#14579133
commutator, cosocleC2×C4×C79C24
FrattiniC2×C4C23×D79
FittingC2×AGL(1,ℤ/8)×C79C2
socleC2×C79C2×#121
hypercenterC2×AGL(1,ℤ/8)D79
hypocenterC79#3068
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #19099361C2semidirect
2. C2×#14579187C2semidirect
3. #19099330C2semidirect
4. #19098831C2semidirect
5. #19098824C2semidirect
6. #19098611C2semidirect
7. #19098545C2semidirect
8. C2×#14579078C2semidirect
9. C2×AGL(1,ℤ/8)×C79C2semidirect
12. #14579191C22semidirect
16. #14579186C22semidirect
18. C2×#11929612C22semidirect
19. C2×#11929609C22semidirect
21. #14579168C22semidirect
26. #14579096C22semidirect
27. #14579079C22semidirect
28. #14579078C22semidirect
29. #14579077C22semidirect
148. C79#3068semidirect
151. C2×AGL(1,ℤ/8)D79semidirect