Finite Group Databasebeta

← #34177366

Subgroups — #34177366

Order 20224.

478 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 87 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#23769772C211yesmaximal, characteristic
2#23764872C212yesmaximal, complemented
3#23761222C212yesmaximal, complemented
4#23761221C211yesmaximal, characteristic
5C79×#2093C211yesmaximal, characteristic
6C2×#14579138—22no—
7C2×#14579094—22no—
8#19098486—22no—
9#19098483—21no—
10#19098461C2212yes—
11#19098443—21no—
12#19098325—22no—
13#19098318C2212yes—
14C2×#107×C79—21no—
15C2×#102×C79—21no—
16#390×C79C2211yescharacteristic
17#384×C79C2211yescharacteristic
18#336×C79C2211yescharacteristic
19C22×Dic158—21no—
20C23×Dic79—21no—
21C22×C79⋊C8—22no—
22#14579138—44no—
23#14579138—44no—
24C2×C4⋊Dic79D412yes—
25C2×C4⋊Dic79D412yes—
26C2×Dic79⋊C4—22no—
27C2×Dic79⋊C4—22no—
28C2×C22⋊Dic79—22no—
29C2×C22⋊Dic79—21no—
30#14579094—44no—
31C79⋊#95—21no—
32C79⋊#95—22no—
33C22×D4×C79D411yescharacteristic
34C22×C8×C79—21no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—22no—
37C2×C4⋊C4×C79D411yescharacteristic
38C2×C22⋊C4×C79—22no—
39C2×C42×C79C2311yescharacteristic
40#107×C79—42no—
41#102×C79—42no—
42C2×Dic158—44no—
43C2×Dic158—81no—
44C22×Dic79—22no—
45C22×Dic79—22no—
46C22×Dic79—44no—
47C22×Dic79—81no—
48C2×C79⋊C8—24no—
49C2×C79⋊C8—42no—
50C2×C79⋊C8—42no—
51C4⋊Dic79—22no—
52C4⋊Dic79—42no—
53C4⋊Dic79D814yes—
54Dic79⋊C4—44no—
55Dic79⋊C4—42no—
56Dic79⋊C4—42no—
57C22⋊Dic79—44no—
58C22⋊Dic79—81no—
59C24×C79—21no—
60C2×D4×C79—21no—
61C2×D4×C79—41no—
62C2×D4×C79—41no—
63C2×D4×C79Q1612yes—
64C22×C4×C79—21no—
65C22×C4×C79—41no—
66C22×C4×C79C2×D412yes—
67C22×C4×C79C2×D411yescharacteristic
68C2×C8×C79—22no—
69C2×C8×C79—41no—
70C2×C8×C79—41no—
71C4⋊C4×C79—41no—
72C4⋊C4×C79—41no—
73C4⋊C4×C79—41no—
74C4⋊C4×C79—42no—
75C4⋊C4×C79—42no—
76C22⋊C4×C79—42no—
77C22⋊C4×C79—42no—
78C42×C79—41no—
79C23×C79—22no—
80C23×C79—42no—
81C23×C79—41no—
82C23×C79—42no—
83C23×C79—41no—
84C23×C79#12011yescharacteristic
85D4×C79—42no—
86D4×C79—81no—
87C2×C4×C79—21no—
88C2×C4×C79—22no—
89C2×C4×C79—22no—
90C2×C4×C79—21no—
91C2×C4×C79—42no—
92C2×C4×C79—41no—
93C2×C4×C79—82no—
94C2×C4×C79—81no—
95C2×C4×C79C2×D812yes—
96C2×C4×C79C2×Q1611yescharacteristic
97C2×C4×C79AGL(1,ℤ/8)12yes—
98C2×C4×C79#13711yescharacteristic
99C2×Dic79—24no—
100C2×Dic79—42no—
101C2×Dic79—42no—
102C2×Dic79—42no—
103C2×Dic79—42no—
104C2×Dic79—42no—
105C2×Dic79—84no—
106Dic158—84no—
107C8×C79—42no—
108C79⋊C8—44no—
109C22×C79—42no—
110C22×C79—44no—
111C22×C79—82no—
112C22×C79—81no—
113C22×C79—81no—
114C22×C79#44712yes—
115C22×C79#45111yescharacteristic
116C22×C79#45311yescharacteristic
117C22×C79#45612yes—
118C22×C79#45811yescharacteristic
119C4×C79—22no—
120C4×C79—22no—
121C4×C79—21no—
122C4×C79—21no—
123C4×C79—81no—
124C4×C79—161no—
125Dic79—44no—
126Dic79—82no—
127#7797—791nomaximal, Sylow
128C2×C79—82no—
129C2×C79#161012yes—
130C2×C79#161411yescharacteristic
131C2×C79#161911yescharacteristic
132C2×C79#164912yes—
133C2×C79#166011yescharacteristic
134#2683—791no—
135#2093D7911yescharacteristic
136#2083—792no—
137#1271—792no—
138#1270—791no—
139C79#779711yescharacteristic, complemented, Sylow
140C2×C22⋊Q8—1582no—
141C2×#107—21no—
142C2×#107—1582no—
143C2×#102—21no—
144#396—1582no—
145#395—1581no—
146#390C2×D7911yescharacteristic
147#389—792no—
148#386—1581no—
149#384C2×D7911yescharacteristic
150#340—1582no—
151#336—792no—
152#336C2×D7911yescharacteristic
153C22×Q8—1581no—
154C22×D4C22⋊D7911yescharacteristic
155C23×C4—1581no—
156C22×C8—21no—
157C22×C8—1582no—
158C22⋊Q8—3164no—
159C22⋊Q8—3164no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—792no—
163C2×C4⋊C4—792no—
164C2×C4⋊C4—1582no—
165C2×C4⋊C4—1582no—
166C2×C4⋊C4C22⋊D7911yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—1582no—
169C2×C22⋊C4—1581no—
170C2×C42C22×D7911yescharacteristic
171#107—42no—
172#107—3164no—
173#102—42no—
174#95—1581no—
175#95—1582no—
176C24—21no—
177C2×Q8—3164no—
178C2×Q8—6321no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1192961412yes—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—3164no—
186C22×C4—6321no—
187C22×C4—1582no—
188C22×C4—1582no—
189C22×C4D4×D7912yes—
190C22×C4C2×C22⋊D7911yescharacteristic
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—3162no—
195C2×C8—3162no—
196C2×C8—1584no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—3162no—
203C4⋊C4—3164no—
204C4⋊C4—3162no—
205C4⋊C4—3162no—
206C4⋊C4—794no—
207C4⋊C4—1582no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—3164no—
211C22⋊C4—6321no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1457912911yescharacteristic
219Q8—6324no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—3162no—
231C2×C4—3162no—
232C2×C4—3162no—
233C2×C4—3162no—
234C2×C4—3162no—
235C2×C4—6324no—
236C2×C4—1584no—
237C2×C4D8×D7912yes—
238C2×C4C2×#1192961411yescharacteristic
239C2×C4#1457918512yes—
240C2×C4#1457919111yescharacteristic
241C8—42no—
242C8—3164no—
243C22—44no—
244C22—42no—
245C22—81no—
246C22—81no—
247C22—82no—
248C22#1909869512yes—
249C22#1909871811yescharacteristic
250C22#1909873011yescharacteristic
251C22#1909874412yes—
252C22#1909875211yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—81no—
258C4—161no—
259C4—3164no—
260C4—6322no—
261C2—82no—
262C2#2376240112yes—
263C2#2376242411yescharacteristic
264C2#2376245211yescharacteristic
265C2#2376264812yes—
266C2#2376270911yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2093C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2093D79
hypocenterC79#7797
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
2. #23764872C2semidirect
3. #23761222C2semidirect
139. C79#7797semidirect

Maximal subgroups

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

subgroupindex
#237697722
#237648722
#237612222
#237612212
C79×#20932
#779779