Finite Group Databasebeta

← #34207960

Subgroups — #34207960

Order 20224.

384 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 81 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#23771388C211yesmaximal, characteristic
2#23763617C212yesmaximal
3#23763614C211yesmaximal, characteristic
4#23761526C212yesmaximal
5C79×#1394C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098605—22no—
8#19098603—21no—
9C2×#14579080C2211yescharacteristic
10#19098502—21no—
11#19098495—22no—
12#19098470C2212yes—
13#19098461—21no—
14#19098450—22no—
15#19098324—22no—
16#19098323—21no—
17C2×#101×C79C2212yes—
18#391×C79C2211yescharacteristic
19C2×#11929608—21no—
20Q8×Dic79—41no—
21Q8×Dic79—41no—
22#14579120—21no—
23#14579120—22no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79—21no—
27C2×C4⋊Dic79D412yes—
28C2×Dic79⋊C4—22no—
29C2×C4×Dic79D412yes—
30#14579095—42no—
31#14579094—41no—
32#14579093—42no—
33#14579092—41no—
34#14579080—21no—
35#14579080Q812yes—
36C79⋊#95—21no—
37C79⋊#95—22no—
38C79⋊#95—22no—
39C79⋊#95—22no—
40C22×Q8×C79C2311yescharacteristic
41C2×M16×C79—22no—
42C2×C4⋊C4×C79D411yescharacteristic
43C2×C42×C79D411yescharacteristic
44#101×C79—22no—
45#101×C79Q814yes—
46#95×C79—22no—
47C22×Dic79—22no—
48C22×Dic79—22no—
49#11929608—22no—
50#11929608—41no—
51C2×C79⋊C8—41no—
52C4⋊Dic79—21no—
53C4⋊Dic79—22no—
54C4⋊Dic79—21no—
55C4⋊Dic79—22no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—81no—
59C4⋊Dic79—81no—
60Dic79⋊C4—44no—
61C22⋊Dic79—41no—
62C4×Dic79—42no—
63C4×Dic79—42no—
64C4×Dic79—41no—
65C2×Q8×C79—41no—
66C2×Q8×C79—41no—
67C2×Q8×C79C2×Q811yescharacteristic
68C2×Q8×C79C2×Q812yes—
69C2×Q8×C79#4111yescharacteristic
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79C2×D412yes—
73C22×C4×C79C2×D411yescharacteristic
74M16×C79—24no—
75M16×C79—42no—
76C2×C8×C79—42no—
77C4⋊C4×C79—41no—
78C42×C79—41no—
79C23×C79#12011yescharacteristic
80Q8×C79—41no—
81Q8×C79—42no—
82Q8×C79—41no—
83C2×C4×C79—22no—
84C2×C4×C79—21no—
85C2×C4×C79—22no—
86C2×C4×C79—21no—
87C2×C4×C79—22no—
88C2×C4×C79—41no—
89C2×C4×C79—41no—
90C2×C4×C79—42no—
91C2×C4×C79—41no—
92C2×C4×C79C22⋊Q812yes—
93C2×C4×C79C22⋊Q811yescharacteristic
94C2×C4×C79C4⋊Q812yes—
95C2×C4×C79C4⋊Q811yescharacteristic
96C2×Dic79—42no—
97C2×Dic79—42no—
98C2×Dic79—42no—
99C2×Dic79—42no—
100C2×Dic79—41no—
101C2×Dic79—41no—
102C2×Dic79—42no—
103C2×Dic79—42no—
104C8×C79—44no—
105C79⋊C8—42no—
106C22×C79—21no—
107C22×C79—21no—
108C22×C79#39511yescharacteristic
109C22×C79B(4,𝔽2)11yescharacteristic
110C22×C79#45811yescharacteristic
111C4×C79—21no—
112C4×C79—22no—
113C4×C79—22no—
114C4×C79—21no—
115C4×C79—42no—
116C4×C79—81no—
117Dic79—82no—
118Dic79—81no—
119Dic79—81no—
120Dic79—82no—
121#13465—791nomaximal, Sylow
122C2×C79—21no—
123C2×C79—21no—
124C2×C79#206211yescharacteristic
125C2×C79#220311yescharacteristic
126C2×C79#220411yescharacteristic
127#2809—791no—
128#1857—792no—
129#1857—791no—
130#1394—792no—
131#1394D7911yescharacteristic
132C79#1346511yescharacteristic, complemented, Sylow
133#519—791no—
134#428—1582no—
135#428—1581no—
136C2×#101—791no—
137C2×#101C2×D7912yes—
138#400—1581no—
139#398—1582no—
140#391—792no—
141#391C2×D7911yescharacteristic
142#389—1581no—
143#387—1582no—
144#339—1582no—
145#339—1581no—
146C22×Q8C22×D7911yescharacteristic
147C2×M16—22no—
148C2×M16—1581no—
149C4×Q8—3161no—
150C4×Q8—3161no—
151#117—1582no—
152#117—1581no—
153C2×C4⋊C4—792no—
154C2×C4⋊C4—1582no—
155C2×C4⋊C4—1581no—
156C2×C4⋊C4—1581no—
157C2×C4⋊C4—1581no—
158C2×C4⋊C4D31611yescharacteristic
159C2×C42—792no—
160C2×C42D31611yescharacteristic
161#107—3161no—
162#107—3162no—
163#106—3161no—
164#106—3162no—
165#101—22no—
166#101—792no—
167#101—1581no—
168#101Dic15814yes—
169#95—22no—
170#95—1581no—
171#95—1582no—
172#95—1582no—
173#95—1582no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8Q8×D7911yescharacteristic
177C2×Q8C2×Dic15812yes—
178C2×Q8#1192962311yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—1582no—
182C22×C4—1582no—
183C22×C4D4×D7912yes—
184C22×C4C2×D31611yescharacteristic
185M16—24no—
186M16—42no—
187M16—3161no—
188M16—1582no—
189C2×C8—42no—
190C2×C8—3161no—
191C4⋊C4—41no—
192C4⋊C4—3162no—
193C4⋊C4—3164no—
194C4⋊C4—3162no—
195C4⋊C4—6321no—
196C4⋊C4—6321no—
197C4⋊C4—1581no—
198C4⋊C4—1582no—
199C4⋊C4—1581no—
200C4⋊C4—1582no—
201C22⋊C4—3161no—
202C42—41no—
203C42—3162no—
204C42—3161no—
205C42—3162no—
206C23#1457913111yescharacteristic
207Q8—41no—
208Q8—42no—
209Q8—41no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—3162no—
220C2×C4—3162no—
221C2×C4—3162no—
222C2×C4—3162no—
223C2×C4—3161no—
224C2×C4—3161no—
225C2×C4—3162no—
226C2×C4—3162no—
227C2×C4#1457913812yes—
228C2×C4#1457914111yescharacteristic
229C2×C4#1457916012yes—
230C2×C4#1457916111yescharacteristic
231C8—44no—
232C8—3162no—
233C22—21no—
234C22—21no—
235C22#1909848411yescharacteristic
236C22#1909875111yescharacteristic
237C22#1909875411yescharacteristic
238C4—22no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—81no—
244C4—6322no—
245C4—6321no—
246C4—6322no—
247C4—6321no—
248C2—21no—
249C2—21no—
250C2#2376475411yescharacteristic
251C2#2376542311yescharacteristic
252C2#2376542811yescharacteristic
subgroupG/H
centerC22#19098751
commutator, cosocleC22×Q8×C79C23
FrattiniC22×Q8C22×D79
FittingC79×#1394C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#1394D79
hypocenterC79#13465
LCSG ▷ C22×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
132. C79#13465semidirect

Maximal subgroups

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

subgroupindex
#237713882
#237636172
#237636142
#237615262
C79×#13942
#1346579