Finite Group Databasebeta

← #34175641

Subgroups — #34175641

Order 20224.

302 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 39 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#23763472C211yesmaximal, characteristic
2#23761328C211yesmaximal, characteristic
3C79×#1316C211yesmaximal, characteristic
4C22×C22⋊Dic79C411yescharacteristic
5#19098419C411yescharacteristic
6#19098329—21no—
7#19098329—21no—
8#19098287—21no—
9#379×C79C2211yescharacteristic
10#323×C79—21no—
11C23×Dic79—21no—
12C2×C22⋊Dic79—21no—
13C2×C22⋊Dic79—21no—
14C2×C22⋊Dic79—41no—
15C2×C22⋊Dic79—41no—
16C2×C22⋊Dic79—41no—
17C2×C22⋊Dic79C812yescomplemented
18C22⋊(C79⋊C8)—41no—
19C25×C79C2×C411yescharacteristic
20C2×C22⋊C4×C79—41no—
21C2×C22⋊C4×C79D411yescharacteristic
22C2×C22⋊C4×C79D411yescharacteristic
23C22⋊C8×C79—41no—
24C22×Dic79—22no—
25C22×Dic79—41no—
26C22×Dic79—41no—
27C22×Dic79—41no—
28C22×Dic79—82no—
29C2×C79⋊C8—81no—
30C22⋊Dic79—42no—
31C22⋊Dic79—41no—
32C22⋊Dic79—41no—
33C22⋊Dic79—41no—
34C22⋊Dic79—41no—
35C22⋊Dic79—82no—
36C24×C79—21no—
37C24×C79—21no—
38C24×C79—41no—
39C24×C79—41no—
40C24×C79—82no—
41C24×C79C22⋊C411yescharacteristic
42C24×C79C2×C811yescharacteristic
43C24×C79M1611yescharacteristic
44C22×C4×C79—21no—
45C22×C4×C79—21no—
46C2×C8×C79—81no—
47C22⋊C4×C79—41no—
48C22⋊C4×C79—41no—
49C22⋊C4×C79—82no—
50C23×C79—21no—
51C23×C79—21no—
52C23×C79—21no—
53C23×C79—21no—
54C23×C79—42no—
55C23×C79—41no—
56C23×C79—42no—
57C23×C79—42no—
58C23×C79—42no—
59C23×C79—41no—
60C23×C79—41no—
61C23×C79—41no—
62C23×C79—81no—
63C23×C79—81no—
64C23×C79—81no—
65C23×C79—81no—
66C23×C79—82no—
67C23×C79—82no—
68C23×C79—82no—
69C23×C79—82no—
70C23×C79C22⋊C811yescharacteristic
71C23×C79#9911yescharacteristic
72C23×C79#10011yescharacteristic
73C2×C4×C79—41no—
74C2×C4×C79—41no—
75C2×C4×C79—81no—
76C2×C4×C79—81no—
77C2×Dic79—42no—
78C2×Dic79—82no—
79C2×Dic79—82no—
80C2×Dic79—82no—
81C2×Dic79—81no—
82C2×Dic79—82no—
83C8×C79—161no—
84C79⋊C8—161no—
85C22×C79—22no—
86C22×C79—21no—
87C22×C79—21no—
88C22×C79—41no—
89C22×C79—41no—
90C22×C79—41no—
91C22×C79—41no—
92C22×C79—41no—
93C22×C79—41no—
94C22×C79—42no—
95C22×C79—42no—
96C22×C79—41no—
97C22×C79—41no—
98C22×C79—81no—
99C22×C79—81no—
100C22×C79—82no—
101C22×C79—82no—
102C22×C79—84no—
103C22×C79—81no—
104C22×C79—81no—
105C22×C79#32311yescharacteristic
106C22×C79#35211yescharacteristic
107C22×C79#35211yescharacteristic
108C4×C79—81no—
109C4×C79—81no—
110Dic79—82no—
111Dic79—161no—
112#7349—791nomaximal, Sylow
113C2×C79—21no—
114C2×C79—21no—
115C2×C79—21no—
116C2×C79—41no—
117C2×C79—42no—
118C2×C79—41no—
119C2×C79—81no—
120C2×C79#131711yescharacteristic
121#1819—791no—
122#1316—791no—
123#1316D7911yescharacteristic
124C79#734911yescharacteristic, complemented, Sylow
125C22×C22⋊C4—791no—
126#379—791no—
127#379C2×D7911yescharacteristic
128#342—1581no—
129#342—1581no—
130#323—21no—
131#323—1581no—
132C25C4×D7911yescharacteristic
133C23×C4—1581no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—3161no—
136C2×C22⋊C4—3161no—
137C2×C22⋊C4—3161no—
138C2×C22⋊C4—792no—
139C2×C22⋊C4—1581no—
140C2×C22⋊C4—1581no—
141C2×C22⋊C4D31611yescharacteristic
142C2×C22⋊C4C22⋊D7911yescharacteristic
143C22⋊C8—41no—
144C22⋊C8—3161no—
145C24—21no—
146C24—21no—
147C24—41no—
148C24—41no—
149C24—82no—
150C24(C2×D79)⋊C411yescharacteristic
151C24C8×D7911yescharacteristic
152C24#1192960711yescharacteristic
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—3161no—
156C22×C4—3161no—
157C22×C4—3161no—
158C22×C4—6322no—
159C22×C4—1582no—
160C2×C8—81no—
161C2×C8—6321no—
162C22⋊C4—41no—
163C22⋊C4—41no—
164C22⋊C4—82no—
165C22⋊C4—3161no—
166C22⋊C4—3161no—
167C22⋊C4—3162no—
168C22⋊C4—3161no—
169C22⋊C4—3161no—
170C22⋊C4—6322no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—42no—
176C23—41no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23—81no—
187C23—82no—
188C23—82no—
189C23—82no—
190C23—82no—
191C23(C2×D79)⋊C811yescharacteristic
192C23#1457907511yescharacteristic
193C23#1457907711yescharacteristic
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—3162no—
199C2×C4—6322no—
200C2×C4—6322no—
201C2×C4—6322no—
202C2×C4—6321no—
203C2×C4—6322no—
204C8—161no—
205C8—12641no—
206C22—22no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—42no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—81no—
221C22—82no—
222C22—82no—
223C22—84no—
224C22—81no—
225C22—81no—
226C22#1909828611yescharacteristic
227C22#1909835211yescharacteristic
228C22#1909835411yescharacteristic
229C4—81no—
230C4—81no—
231C4—6322no—
232C4—12641no—
233C2—21no—
234C2—21no—
235C2—21no—
236C2—41no—
237C2—42no—
238C2—41no—
239C2—81no—
240C2#2376132911yescharacteristic
subgroupG/H
centerC2#23761329
commutatorC24×C79C2×C8
Frattini#379C2×D79
FittingC79×#1316C2
socleC2×C79#1317
prime radicalC25×C79C2×C4
cosocle#379×C79C22
hypercenter#1316D79
hypocenterC79#7349
LCSG ▷ C24×C79 ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ #1316

Complements

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

NH ≅ G/Ntype
17. C2×C22⋊Dic79C8semidirect
124. C79#7349semidirect

Maximal subgroups

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

subgroupindex
#237634722
#237613282
C79×#13162
#734979