Finite Group Databasebeta

← #34444922

Subgroups — #34444922

Order 20224.

812 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×D16×D79C212yesmaximal, complemented
2#23775075C212yesmaximal, complemented
3#23765483C212yesmaximal, complemented
4#23765432C212yesmaximal, complemented
5#23765310C214yesmaximal, complemented
6C4×D1264C211yesmaximal, characteristic, complemented
7C16×D316C211yesmaximal, characteristic, complemented
8C79×#2246C211yesmaximal, characteristic, complemented
9C2×D8×D79C2214yescomplemented
10#19099198—22no—
11C2×D1264C2211yescharacteristic
12C2×#14579102C2214yescomplemented
13D16×D79—28no—
14C2×C16×D79C2212yescomplemented
15#19098796C2214yescomplemented
16#19098755—22no—
17#19098695—24no—
18C4×D632C2212yescomplemented
19C8×D316C2211yescharacteristic
20#19098389C2211yescharacteristic, complemented
21C4⋊(C79⋊C16)C2211yescharacteristic, complemented
22#19098365C2212yescomplemented
23#19098364C2214yescomplemented
24(C2×D79)⋊C16C2212yescomplemented
25C2×D16×C79C2212yescomplemented
26C2×D16×C79C2212yescomplemented
27#493×C79C2212yescomplemented
28C4×C16×C79C2211yescharacteristic, complemented
29C2×D4×D79—24no—
30C2×D4×D79—42no—
31C2×D632C2312yes—
32C2×#11929609—24no—
33D8×D79—28no—
34D8×D79—44no—
35C2×C8×D79C2312yes—
36#14579132—24no—
37#14579132—42no—
38#14579129—44no—
39C4×D316—22no—
40C4×D316D411yescharacteristic
41D1264D414yescomplemented
42#14579102—28no—
43C2×C79⋊C16C2312yes—
44C16×D79—24no—
45#14579095C2312yes—
46C4⋊(C79⋊C8)D411yescharacteristic
47#14579083—22no—
48#14579082—24no—
49(C2×D79)⋊C8D412yes—
50C2×D8×C79—22no—
51C2×D8×C79C2314yes—
52#127×C79—22no—
53D16×C79—24no—
54D16×C79—42no—
55C2×C16×C79C2311yescharacteristic
56C2×C16×C79C2311yescharacteristic
57C4×C8×C79C2311yescharacteristic
58C23×D79—44no—
59C2×D316—22no—
60C2×D316D811yescharacteristic
61C2×C22⋊D79—44no—
62C2×C22⋊D79—44no—
63D4×D79—48no—
64D4×D79—84no—
65D4×D79—88no—
66C2×C4×D79—42no—
67C2×C4×D79C2×D412yes—
68D632—22no—
69D632C2×D414yescomplemented
70#11929609—48no—
71C2×C79⋊C8C2×D412yes—
72C8×D79—22no—
73C8×D79—22no—
74C4⋊Dic79—22no—
75C4⋊Dic79D811yescharacteristic
76C22⋊Dic79—44no—
77(C2×D79)⋊C4—42no—
78(C2×D79)⋊C4D812yes—
79C79⋊C16—22no—
80C2×D4×C79—24no—
81C2×D4×C79—42no—
82C2×D4×C79—42no—
83D8×C79—24no—
84D8×C79—24no—
85D8×C79—82no—
86C2×C8×C79C2×D411yescharacteristic
87C2×C8×C79C2411yescharacteristic
88C42×C79C2×D411yescharacteristic
89C16×C79—21no—
90C16×C79C2×D412yescomplemented
91C23×C79—44no—
92C22×D79—42no—
93C22×D79—88no—
94C22×D79—88no—
95C22×D79—164no—
96C22×D79C2×D812yes—
97D4×C79—44no—
98D4×C79—44no—
99D4×C79—84no—
100D4×C79—162no—
101C2×C4×C79C2×D811yescharacteristic
102C2×C4×C79C2×D811yescharacteristic
103C2×C4×C79C22×D411yescharacteristic
104C22⋊D79—88no—
105C22⋊D79—88no—
106C2×Dic79—42no—
107C2×Dic79C2×D812yes—
108D316—24no—
109D316—42no—
110D316D1614yescomplemented
111C4×D79—22no—
112C4×D79—22no—
113C4×D79—84no—
114C8×C79—21no—
115C8×C79C22×D411yescharacteristic
116C8×C79H(5,𝔽2)11yescharacteristic
117C79⋊C8—22no—
118C22×C79—84no—
119C22×C79—84no—
120C22×C79—84no—
121C22×C79C22×D811yescharacteristic
122C2×D79—22no—
123C2×D79—22no—
124C2×D79—44no—
125C2×D79—82no—
126C2×D79—82no—
127C2×D79—1616no—
128C2×D79C2×D1614yescomplemented
129C4×C79—21no—
130C4×C79C2×D1612yescomplemented
131C4×C79D4211yescharacteristic
132C4×C79#57611yescharacteristic
133Dic79—22no—
134Dic79—82no—
135D4×D16—791nomaximal, Sylow
136C2×C79—164no—
137C2×C79D4×D811yescharacteristic
138C2×C79C22×D1611yescharacteristic
139C2×C79#341511yescharacteristic
140D79—24no—
141D79—84no—
142C22×D16—792no—
143D4×D8—792no—
144#2246D7911yescharacteristic, complemented
145#2215—792no—
146#2206—792no—
147#2184—794no—
148C4×D16—791no—
149D4×C16—791no—
150C79D4×D1611yescharacteristic, complemented, Sylow
151C22×D8—794no—
152D42—1582no—
153C2×D16—794no—
154C2×D16—791no—
155C2×D16—1588no—
156C2×D16C2×D7912yescomplemented
157C2×D16C2×D7912yescomplemented
158C22×C16—792no—
159#493C2×D7912yescomplemented
160#466—794no—
161#459—1582no—
162#447—1584no—
163C4×D8—792no—
164C8×D4—791no—
165#366—791no—
166C4⋊C16—791no—
167#357—794no—
168#357—792no—
169C22⋊C16—792no—
170C4×C16C2×D7911yescharacteristic, complemented
171C22×D4—3162no—
172C22×D4—1584no—
173C2×D8—22no—
174C2×D8—3164no—
175C2×D8—792no—
176C2×D8—1584no—
177C2×D8—1588no—
178C2×D8C22×D7914yes—
179C22×C8—792no—
180#127—22no—
181#121—3162no—
182#121—1584no—
183#120—3164no—
184C4×D4—791no—
185C4×D4—1582no—
186D16—24no—
187D16—42no—
188D16—794no—
189D16—1588no—
190C2×C16—792no—
191C2×C16—1584no—
192C2×C16C22×D7911yescharacteristic
193C2×C16C22×D7911yescharacteristic
194#107—792no—
195C4⋊C8—791no—
196#102—1584no—
197#102—1582no—
198C22⋊C8—792no—
199C4×C8C22×D7911yescharacteristic
200C24—3164no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—3164no—
205C2×D4—3164no—
206C2×D4—3168no—
207C2×D4—791no—
208C2×D4—6328no—
209C2×D4—6324no—
210C2×D4—1582no—
211C22×C4—3162no—
212C22×C4—792no—
213D8—24no—
214D8—24no—
215D8—82no—
216D8—3168no—
217D8—794no—
218D8—1582no—
219C2×C8—792no—
220C2×C8—1582no—
221C2×C8—1582no—
222C2×C8D4×D7911yescharacteristic
223C2×C8C23×D7911yescharacteristic
224C4⋊C4—791no—
225C4⋊C4—1582no—
226C22⋊C4—3164no—
227C22⋊C4—3162no—
228C22⋊C4—792no—
229C42D4×D7911yescharacteristic
230C16—21no—
231C16—1582no—
232C16D4×D7912yescomplemented
233C23—44no—
234C23—3162no—
235C23—792no—
236C23—6328no—
237C23—6328no—
238C23—12644no—
239D4—44no—
240D4—44no—
241D4—84no—
242D4—162no—
243D4—3162no—
244D4—794no—
245D4—6328no—
246D4—6328no—
247D4—1584no—
248C2×C4—3162no—
249C2×C4—792no—
250C2×C4—6324no—
251C2×C4—1582no—
252C2×C4—1582no—
253C2×C4D8×D7911yescharacteristic
254C2×C4D8×D7911yescharacteristic
255C2×C4C2×D4×D7911yescharacteristic
256C8—21no—
257C8—1582no—
258C8C2×D4×D7911yescharacteristic
259C8#1457920611yescharacteristic
260C22—84no—
261C22—84no—
262C22—84no—
263C22—3164no—
264C22—794no—
265C22—6322no—
266C22—6322no—
267C22—1582no—
268C22—1582no—
269C22—126416no—
270C22C2×D8×D7911yescharacteristic
271C4—21no—
272C4—6322no—
273C4—1582no—
274C4D16×D7912yescomplemented
275C4#1909919811yescharacteristic
276C4#1909936411yescharacteristic
277C2—164no—
278C2—6324no—
279C2—1584no—
280C2#2377507511yescharacteristic
281C2C2×D16×D7911yescharacteristic
282C2#2377653711yescharacteristic
subgroupG/H
centerC22C2×D8×D79
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingC79×#2246C2
socleC22×C79C22×D8
hypercenter#2246D79
hypocenterC79D4×D16
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2246

Complements

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

NH ≅ G/Ntype
1. C2×D16×D79C2semidirect
2. #23775075C2semidirect
3. #23765483C2semidirect
4. #23765432C2semidirect
5. #23765310C2semidirect
6. C4×D1264C2semidirect
7. C16×D316C2semidirect
8. C79×#2246C2semidirect
9. C2×D8×D79C22semidirect
12. C2×#14579102C22semidirect
14. C2×C16×D79C22semidirect
15. #19098796C22semidirect
18. C4×D632C22semidirect
20. #19098389C22semidirect
21. C4⋊(C79⋊C16)C22semidirect
22. #19098365C22semidirect
23. #19098364C22semidirect
24. (C2×D79)⋊C16C22semidirect
25. C2×D16×C79C22semidirect
26. C2×D16×C79C22semidirect
27. #493×C79C22semidirect
28. C4×C16×C79C22semidirect
41. D1264D4semidirect
69. D632C2×D4semidirect
90. C16×C79C2×D4semidirect
110. D316D16semidirect
128. C2×D79C2×D16semidirect
130. C4×C79C2×D16semidirect
144. #2246D79semidirect
150. C79D4×D16semidirect
156. C2×D16C2×D79semidirect
157. C2×D16C2×D79semidirect
159. #493C2×D79semidirect
170. C4×C16C2×D79semidirect
232. C16D4×D79semidirect
274. C4D16×D79semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D16×D792
#237750752
#237654832
#237654322
#237653102
C4×D12642
C16×D3162
C79×#22462
D4×D1679