Finite Group Databasebeta

← #34177059

Subgroups — #34177059

Order 20224.

492 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 83 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#23770330C211yesmaximal, characteristic
2#23764725C212yesmaximal, complemented
3#23761224C212yesmaximal, complemented
4#23761221C211yesmaximal, characteristic
5C79×#2013C211yesmaximal, characteristic
6C2×#14579142—21no—
7C2×#14579094—22no—
8#19098493—21no—
9#19098488—22no—
10#19098461C2212yes—
11#19098438—22no—
12#19098325—22no—
13#19098318C2212yes—
14C22×SD16×C79—21no—
15C2×C4⋊Q8×C79C2211yescharacteristic
16C2×#103×C79—21no—
17#390×C79C2211yescharacteristic
18#336×C79C2211yescharacteristic
19C23×Dic79—21no—
20C22×C79⋊C8—22no—
21#14579142—42no—
22#14579142—42no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79D412yes—
25C2×C4⋊Dic79D412yes—
26C2×Dic79⋊C4—22no—
27C2×C22⋊Dic79—22no—
28C2×C22⋊Dic79—21no—
29#14579094—44no—
30C79⋊#95—22no—
31C79⋊#95—21no—
32C79⋊#95—22no—
33C22×Q8×C79—21no—
34C22×D4×C79D411yescharacteristic
35C2×SD16×C79—42no—
36C2×SD16×C79—42no—
37C2×SD16×C79—42no—
38C22×C8×C79—21no—
39C4⋊Q8×C79—42no—
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79D411yescharacteristic
42C2×C22⋊C4×C79—22no—
43C2×C42×C79C2311yescharacteristic
44#103×C79—42no—
45C22×Dic79—22no—
46C22×Dic79—22no—
47C22×Dic79—42no—
48C22×Dic79—81no—
49C22×Dic79—81no—
50C2×C79⋊C8—24no—
51C2×C79⋊C8—42no—
52C2×C79⋊C8—42no—
53C4⋊Dic79—22no—
54C4⋊Dic79—22no—
55C4⋊Dic79—42no—
56C4⋊Dic79—81no—
57Dic79⋊C4—42no—
58Dic79⋊C4—42no—
59C22⋊Dic79—42no—
60C22⋊Dic79—44no—
61C24×C79—21no—
62C2×Q8×C79—22no—
63C2×Q8×C79—41no—
64C2×Q8×C79—41no—
65C2×Q8×C79—41no—
66C2×Q8×C79—41no—
67C2×Q8×C79—41no—
68C2×D4×C79—21no—
69C2×D4×C79—41no—
70C2×D4×C79—41no—
71C2×D4×C79D812yes—
72C22×C4×C79—21no—
73C22×C4×C79—41no—
74C22×C4×C79C2×D411yescharacteristic
75C22×C4×C79C2×D412yes—
76SD16×C79—48no—
77C2×C8×C79—22no—
78C2×C8×C79—41no—
79C2×C8×C79—41no—
80C4⋊C4×C79—24no—
81C4⋊C4×C79—42no—
82C4⋊C4×C79—41no—
83C22⋊C4×C79—42no—
84C22⋊C4×C79—42no—
85C42×C79—41no—
86C23×C79—22no—
87C23×C79—42no—
88C23×C79—41no—
89C23×C79—42no—
90C23×C79—41no—
91C23×C79#12011yescharacteristic
92Q8×C79—42no—
93Q8×C79—42no—
94Q8×C79—81no—
95Q8×C79—81no—
96D4×C79—42no—
97D4×C79—81no—
98C2×C4×C79—21no—
99C2×C4×C79—22no—
100C2×C4×C79—22no—
101C2×C4×C79—21no—
102C2×C4×C79—42no—
103C2×C4×C79—41no—
104C2×C4×C79—42no—
105C2×C4×C79—81no—
106C2×C4×C79—81no—
107C2×C4×C79C2×D811yescharacteristic
108C2×C4×C79#13512yes—
109C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
110C2×C4×C79#13712yes—
111C2×Dic79—42no—
112C2×Dic79—42no—
113C2×Dic79—42no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—84no—
118C8×C79—42no—
119C79⋊C8—44no—
120C22×C79—42no—
121C22×C79—44no—
122C22×C79—82no—
123C22×C79—81no—
124C22×C79—81no—
125C22×C79#44711yescharacteristic
126C22×C79#45212yes—
127C22×C79#45311yescharacteristic
128C22×C79#45612yes—
129C22×C79B(4,𝔽2)11yescharacteristic
130C4×C79—21no—
131C4×C79—21no—
132C4×C79—22no—
133C4×C79—22no—
134C4×C79—82no—
135C4×C79—81no—
136Dic79—82no—
137Dic79—82no—
138#7751—791nomaximal, Sylow
139C2×C79—82no—
140C2×C79#159511yescharacteristic
141C2×C79#160312yes—
142C2×C79#163011yescharacteristic
143C2×C79#164712yes—
144C2×C79#166011yescharacteristic
145#2724—791no—
146#2057—792no—
147#2013D7911yescharacteristic
148#1271—792no—
149#1270—791no—
150C79#775111yescharacteristic, complemented, Sylow
151C22×SD16—21no—
152C2×C4⋊Q8C2×D7911yescharacteristic
153C2×#123—1581no—
154C2×#107—1582no—
155C2×#103—21no—
156#398—1581no—
157#397—1582no—
158#390C2×D7911yescharacteristic
159#389—792no—
160#385—1582no—
161#340—1582no—
162#336—792no—
163#336C2×D7911yescharacteristic
164C22×Q8—21no—
165C22×D4C22⋊D7911yescharacteristic
166C23×C4—1581no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×SD16—42no—
170C22×C8—21no—
171C22×C8—1582no—
172C4⋊Q8—42no—
173#123—3162no—
174#123—3162no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—792no—
177C2×C4⋊C4—792no—
178C2×C4⋊C4—1582no—
179C2×C4⋊C4—1581no—
180C2×C4⋊C4C22⋊D7911yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1582no—
183C2×C22⋊C4—1581no—
184C2×C42C22×D7911yescharacteristic
185#107—3164no—
186#103—42no—
187#95—1582no—
188#95—1581no—
189#95—1582no—
190C24—21no—
191C2×Q8—22no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4#1192960912yes—
201C22×C4—21no—
202C22×C4—41no—
203C22×C4—3162no—
204C22×C4—6321no—
205C22×C4—6321no—
206C22×C4—1582no—
207C22×C4—1582no—
208C22×C4D4×D7912yes—
209C22×C4C2×C22⋊D7911yescharacteristic
210SD16—48no—
211C2×C8—22no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—3162no—
215C2×C8—3162no—
216C2×C8—1584no—
217C4⋊C4—24no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—3162no—
221C4⋊C4—3162no—
222C4⋊C4—3162no—
223C4⋊C4—6321no—
224C4⋊C4—1582no—
225C4⋊C4—1582no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—3162no—
229C22⋊C4—3164no—
230C42—41no—
231C23—22no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23—41no—
236C23#1457912911yescharacteristic
237Q8—42no—
238Q8—42no—
239Q8—81no—
240Q8—81no—
241D4—42no—
242D4—81no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—3162no—
253C2×C4—3162no—
254C2×C4—3162no—
255C2×C4—3162no—
256C2×C4—3162no—
257C2×C4—3162no—
258C2×C4—6324no—
259C2×C4C2×#1192960911yescharacteristic
260C2×C4#1457918212yes—
261C2×C4#1457918611yescharacteristic
262C2×C4#1457919012yes—
263C8—42no—
264C8—3164no—
265C22—42no—
266C22—44no—
267C22—81no—
268C22—81no—
269C22—82no—
270C22#1909869611yescharacteristic
271C22#1909872212yes—
272C22#1909873011yescharacteristic
273C22#1909874412yes—
274C22#1909874911yescharacteristic
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—81no—
280C4—82no—
281C4—6322no—
282C4—6322no—
283C2—82no—
284C2#2376231711yescharacteristic
285C2#2376236012yes—
286C2#2376251911yescharacteristic
287C2#2376263312yes—
288C2#2376270911yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2013C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2013D79
hypocenterC79#7751
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #23764725C2semidirect
3. #23761224C2semidirect
150. C79#7751semidirect

Maximal subgroups

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

subgroupindex
#237703302
#237647252
#237612242
#237612212
C79×#20132
#775179