Finite Group Databasebeta

← #34207956

Subgroups — #34207956

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#23763614C212yesmaximal
2#23763614C211yesmaximal, characteristic
3#23761527C211yesmaximal, characteristic
4#23761527C212yesmaximal
5C79×#2809C211yesmaximal, characteristic
6#19099041C2211yescharacteristic
7#19098603—22no—
8#19098603—21no—
9C2×#14579080C2212yes—
10C2×#14579080C2211yescharacteristic
11#19098323—22no—
12#19098323—21no—
13#400×C79—21no—
14#398×C79—22no—
15#391×C79C2211yescharacteristic
16#391×C79C2212yes—
17#389×C79—21no—
18#387×C79—22no—
19C2×#11929608—22no—
20C2×#11929608—21no—
21Q8×Dic79—41no—
22Q8×Dic79—41no—
23#14579120—22no—
24#14579120—21no—
25#14579094—42no—
26#14579094—41no—
27#14579092—42no—
28#14579092—41no—
29#14579080—22no—
30#14579080—21no—
31#14579080Q814yes—
32#14579080Q812yes—
33C22×Q8×C79C2311yescharacteristic
34C2×C4⋊C4×C79—22no—
35C2×C4⋊C4×C79—21no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79D412yes—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C42×C79D412yes—
41C2×C42×C79D411yescharacteristic
42#95×C79—21no—
43#95×C79—22no—
44#95×C79—22no—
45#95×C79—22no—
46#95×C79—22no—
47#11929608—24no—
48#11929608—22no—
49#11929608—41no—
50#11929608—42no—
51C2×C79⋊C8—42no—
52C2×C79⋊C8—41no—
53C4⋊Dic79—22no—
54C4⋊Dic79—21no—
55C4⋊Dic79—21no—
56C4⋊Dic79—22no—
57C22⋊Dic79—41no—
58C4×Dic79—42no—
59C4×Dic79—41no—
60C2×Q8×C79—41no—
61C2×Q8×C79—41no—
62C2×Q8×C79C2×Q812yes—
63C2×Q8×C79C2×Q811yescharacteristic
64C2×Q8×C79#4111yescharacteristic
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—22no—
68C22×C4×C79—22no—
69C22×C4×C79C2×D412yes—
70C22×C4×C79C2×D411yescharacteristic
71C4⋊C4×C79—44no—
72C4⋊C4×C79—42no—
73C4⋊C4×C79—42no—
74C4⋊C4×C79—41no—
75C4⋊C4×C79—81no—
76C4⋊C4×C79—81no—
77C42×C79—41no—
78C42×C79—42no—
79C23×C79#12011yescharacteristic
80Q8×C79—42no—
81Q8×C79—41no—
82Q8×C79—41no—
83C2×C4×C79—22no—
84C2×C4×C79—21no—
85C2×C4×C79—21no—
86C2×C4×C79—22no—
87C2×C4×C79—22no—
88C2×C4×C79—41no—
89C2×C4×C79—41no—
90C2×C4×C79—42no—
91C2×C4×C79—42no—
92C2×C4×C79—42no—
93C2×C4×C79—42no—
94C2×C4×C79—42no—
95C2×C4×C79—41no—
96C2×C4×C79—42no—
97C2×C4×C79—42no—
98C2×C4×C79C22⋊Q812yes—
99C2×C4×C79C22⋊Q811yescharacteristic
100C2×C4×C79C4⋊Q812yes—
101C2×C4×C79C4⋊Q811yescharacteristic
102C2×Dic79—41no—
103C2×Dic79—41no—
104C79⋊C8—44no—
105C79⋊C8—42no—
106C22×C79—21no—
107C22×C79—21no—
108C22×C79#39511yescharacteristic
109C22×C79B(4,𝔽2)11yescharacteristic
110C22×C79#45811yescharacteristic
111C4×C79—22no—
112C4×C79—21no—
113C4×C79—22no—
114C4×C79—21no—
115C4×C79—42no—
116C4×C79—82no—
117C4×C79—81no—
118C4×C79—82no—
119Dic79—81no—
120Dic79—81no—
121#13465—791nomaximal, Sylow
122C2×C79—21no—
123C2×C79—21no—
124C2×C79#206211yescharacteristic
125C2×C79#220311yescharacteristic
126C2×C79#220411yescharacteristic
127#2809D7911yescharacteristic
128#1857—792no—
129#1857—791no—
130#1394—791no—
131#1394—792no—
132C79#1346511yescharacteristic, complemented, Sylow
133#519—791no—
134#428—1582no—
135#428—1581no—
136C2×#101—792no—
137C2×#101—791no—
138#400—21no—
139#398—22no—
140#391C2×D7911yescharacteristic
141#391C2×D7912yes—
142#389—21no—
143#387—22no—
144#339—1582no—
145#339—1581no—
146C22×Q8C22×D7911yescharacteristic
147C2×M16—1582no—
148C2×M16—1581no—
149C4×Q8—3161no—
150C4×Q8—3161no—
151#117—1582no—
152#117—1581no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D7912yes—
158C2×C4⋊C4C22⋊D7911yescharacteristic
159C2×C42C22⋊D7912yes—
160C2×C42C22⋊D7911yescharacteristic
161#107—3162no—
162#107—3161no—
163#106—3162no—
164#106—3161no—
165#101—794no—
166#101—792no—
167#101—1582no—
168#101—1581no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8Q8×D7912yes—
177C2×Q8Q8×D7911yescharacteristic
178C2×Q8#1192962511yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4C2×C22⋊D7912yes—
184C22×C4C2×C22⋊D7911yescharacteristic
185M16—3161no—
186M16—3162no—
187M16—1584no—
188M16—1582no—
189C2×C8—3162no—
190C2×C8—3161no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—1582no—
198C4⋊C4—1581no—
199C4⋊C4—1581no—
200C4⋊C4—1582no—
201C22⋊C4—3161no—
202C42—41no—
203C42—42no—
204C42—3162no—
205C42—3161no—
206C23#1457913011yescharacteristic
207Q8—42no—
208Q8—41no—
209Q8—41no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—3161no—
226C2×C4—3161no—
227C2×C4#1457913712yes—
228C2×C4#1457913711yescharacteristic
229C2×C4#1457915912yes—
230C2×C4#1457915911yescharacteristic
231C8—3164no—
232C8—3162no—
233C22—21no—
234C22—21no—
235C22#1909848211yescharacteristic
236C22#1909875011yescharacteristic
237C22#1909875311yescharacteristic
238C4—22no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—82no—
244C4—81no—
245C4—82no—
246C4—6321no—
247C4—6321no—
248C2—21no—
249C2—21no—
250C2#2376475211yescharacteristic
251C2#2376542111yescharacteristic
252C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC22×Q8×C79C23
FrattiniC22×Q8C22×D79
FittingC79×#2809C2
socleC22×C79B(4,𝔽2)
prime radicalC23×C79#120
hypercenter#2809D79
hypocenterC79#13465
LCSG ▷ C22×Q8×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2809

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
#237636142
#237615272
C79×#28092
#1346579