Finite Group Databasebeta

← #34201065

Subgroups — #34201065

Order 20224.

388 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 59 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#23769065C211yesmaximal, characteristic
2#23762425C212yesmaximal
3#23761423C212yesmaximal
4#23761330C211yesmaximal, characteristic
5C79×#2030C211yesmaximal, characteristic
6#19098896—22no—
7#19098499C2212yes—
8#19098475—21no—
9#19098467—22no—
10#19098450—22no—
11C4×C22⋊Dic79—21no—
12#19098300—22no—
13#19098287C2212yes—
14#543×C79C2211yescharacteristic
15#381×C79—21no—
16#381×C79C2211yescharacteristic
17#342×C79—21no—
18#342×C79C2211yescharacteristic
19C22×Dic158—21no—
20C2×C4⋊Dic79D412yes—
21C2×Dic79⋊C4—22no—
22C2×C22⋊Dic79—22no—
23C2×C4×Dic79—21no—
24C2×C4×Dic79—21no—
25#14579094—42no—
26#14579085—42no—
27C22⋊(C79⋊C8)—22no—
28C79⋊#95—22no—
29C79⋊#95—21no—
30C79⋊#95—22no—
31C79⋊#95D412yes—
32C23×C4×C79D411yescharacteristic
33C22⋊Q8×C79—22no—
34C22⋊Q8×C79—42no—
35C2×C22⋊C4×C79—21no—
36C2×C22⋊C4×C79—21no—
37C2×C22⋊C4×C79D411yescharacteristic
38C2×C22⋊C4×C79C2311yescharacteristic
39#95×C79—22no—
40#95×C79—42no—
41C2×Dic158—44no—
42C2×Dic158—81no—
43C22×Dic79—22no—
44C22×Dic79—22no—
45C2×C79⋊C8—42no—
46C4⋊Dic79—42no—
47Dic79⋊C4—44no—
48C22⋊Dic79—44no—
49C4×Dic79—81no—
50C4×Dic79—81no—
51C24×C79C2×D411yescharacteristic
52C2×Q8×C79—22no—
53C22×C4×C79—21no—
54C22×C4×C79—21no—
55C22×C4×C79—21no—
56C22×C4×C79—22no—
57C22×C4×C79—41no—
58C22×C4×C79—41no—
59C22×C4×C79—41no—
60C22×C4×C79Q1612yes—
61C22×C4×C79C2×D412yes—
62C4⋊C4×C79—22no—
63C4⋊C4×C79—42no—
64C22⋊C4×C79—41no—
65C22⋊C4×C79—41no—
66C22⋊C4×C79—41no—
67C22⋊C4×C79—41no—
68C22⋊C4×C79—41no—
69C22⋊C4×C79—41no—
70C23×C79—21no—
71C23×C79—21no—
72C23×C79—42no—
73C23×C79#12011yescharacteristic
74C23×C79C2×Q1611yescharacteristic
75C23×C79#13711yescharacteristic
76Q8×C79—82no—
77C2×C4×C79—22no—
78C2×C4×C79—22no—
79C2×C4×C79—41no—
80C2×C4×C79—41no—
81C2×C4×C79—42no—
82C2×C4×C79—42no—
83C2×C4×C79—42no—
84C2×C4×C79—41no—
85C2×C4×C79—42no—
86C2×C4×C79—81no—
87C2×C4×C79—82no—
88C2×C4×C79—81no—
89C2×C4×C79—82no—
90C2×Dic79—24no—
91C2×Dic79—42no—
92C2×Dic79—42no—
93C2×Dic79—42no—
94C2×Dic79—42no—
95C2×Dic79—42no—
96Dic158—84no—
97C79⋊C8—82no—
98C22×C79—22no—
99C22×C79—21no—
100C22×C79—21no—
101C22×C79—41no—
102C22×C79—41no—
103C22×C79—42no—
104C22×C79—81no—
105C22×C79#45111yescharacteristic
106C22×C79#45511yescharacteristic
107C22×C79B(4,𝔽2)11yescharacteristic
108C4×C79—42no—
109C4×C79—42no—
110C4×C79—81no—
111C4×C79—82no—
112C4×C79—161no—
113Dic79—44no—
114Dic79—82no—
115#12216—791nomaximal, Sylow
116C2×C79—21no—
117C2×C79—21no—
118C2×C79—41no—
119C2×C79—41no—
120C2×C79#160211yescharacteristic
121C2×C79#219812yes—
122#2607—791no—
123#2030D7911yescharacteristic
124#1614—792no—
125#1352—792no—
126#1317—791no—
127C79#1221611yescharacteristic, complemented, Sylow
128#543C2×D7911yescharacteristic
129#484—1582no—
130#399—792no—
131#393—1581no—
132#391—1582no—
133#387—1582no—
134#381—21no—
135#381C2×D7911yescharacteristic
136C4×C22⋊C4—1581no—
137#342—21no—
138#342C2×D7911yescharacteristic
139#328—1582no—
140#323—792no—
141C22×Q8—1581no—
142C23×C4C22⋊D7911yescharacteristic
143C22⋊Q8—22no—
144C22⋊Q8—42no—
145C2×C4⋊C4—792no—
146C2×C4⋊C4—1582no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—1582no—
150C2×C22⋊C4C22⋊D7911yescharacteristic
151C2×C22⋊C4C22×D7911yescharacteristic
152C2×C42—1581no—
153C2×C42—1581no—
154#107—3162no—
155#103—3162no—
156C22⋊C8—1582no—
157#95—22no—
158#95—42no—
159#95—792no—
160#95—1582no—
161#95—1581no—
162#95—1582no—
163C24C2×C22⋊D7911yescharacteristic
164C2×Q8—22no—
165C2×Q8—3164no—
166C2×Q8—6321no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—1582no—
175C22×C4—1582no—
176C22×C4#1192961412yes—
177C22×C4D4×D7912yes—
178C2×C8—3162no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—3162no—
182C4⋊C4—3164no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—3164no—
190C42—6321no—
191C42—6321no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23#1457912911yescharacteristic
196C23C2×#1192961411yescharacteristic
197C23#1457919111yescharacteristic
198Q8—82no—
199Q8—6324no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—81no—
210C2×C4—82no—
211C2×C4—81no—
212C2×C4—82no—
213C2×C4—3162no—
214C2×C4—3162no—
215C2×C4—3162no—
216C2×C4—3162no—
217C2×C4—3162no—
218C2×C4—1584no—
219C8—6322no—
220C22—22no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—81no—
227C22#1909871811yescharacteristic
228C22#1909874011yescharacteristic
229C22#1909874911yescharacteristic
230C4—42no—
231C4—42no—
232C4—81no—
233C4—82no—
234C4—161no—
235C4—3164no—
236C4—6322no—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2#2376235611yescharacteristic
242C2#2376539412yes—
subgroupG/H
centerC22#19098749
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2030C2
socleC22×C79B(4,𝔽2)
prime radicalC24×C79C2×D4
hypercenter#2030D79
hypocenterC79#12216
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2030

Complements

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

NH ≅ G/Ntype
127. C79#12216semidirect

Maximal subgroups

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

subgroupindex
#237690652
#237624252
#237614232
#237613302
C79×#20302
#1221679