Finite Group Databasebeta

← #34177539

Subgroups — #34177539

Order 20224.

418 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 83 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#23770330C211yesmaximal, characteristic
2#23764964C212yesmaximal, complemented
3#23764933C211yesmaximal, characteristic
4#23761222C212yesmaximal, complemented
5C79×#1270C211yesmaximal, characteristic
6C2×#14579142—21no—
7C2×#14579092—21no—
8C2×#14579082—21no—
9#19098493—21no—
10#19098488—22no—
11#19098461C2212yes—
12#19098438—22no—
13#19098433C2211yescharacteristic
14#19098326—22no—
15#19098326—22no—
16#19098318C2211yescharacteristic
17#390×C79C2211yescharacteristic
18#336×C79C2212yes—
19C23×Dic79—21no—
20C22×C79⋊C8—21no—
21#14579142—42no—
22#14579142—42no—
23C2×C4⋊Dic79—21no—
24C2×C4⋊Dic79—22no—
25C2×C4⋊Dic79—21no—
26C2×C4⋊Dic79D412yes—
27C2×C4⋊Dic79D412yes—
28C2×Dic79⋊C4—22no—
29C2×C22⋊Dic79—22no—
30C2×C22⋊Dic79—21no—
31#14579092—42no—
32#14579082—42no—
33C79⋊#95—22no—
34C79⋊#95—21no—
35C79⋊#95—22no—
36C22×D4×C79D411yescharacteristic
37C22×C8×C79—22no—
38C2×C4⋊C4×C79D411yescharacteristic
39C2×C22⋊C4×C79—22no—
40C2×C42×C79C2311yescharacteristic
41C22×Dic79—22no—
42C22×Dic79—22no—
43C22×Dic79—42no—
44C22×Dic79—41no—
45C22×Dic79—81no—
46C22×Dic79—81no—
47C2×C79⋊C8—22no—
48C2×C79⋊C8—41no—
49C2×C79⋊C8—41no—
50C4⋊Dic79—22no—
51C4⋊Dic79—22no—
52C4⋊Dic79—42no—
53C4⋊Dic79—41no—
54C4⋊Dic79—41no—
55C4⋊Dic79—44no—
56C4⋊Dic79—81no—
57Dic79⋊C4—42no—
58Dic79⋊C4—42no—
59C22⋊Dic79—42no—
60C22⋊Dic79—44no—
61C24×C79—21no—
62C2×D4×C79—21no—
63C2×D4×C79—41no—
64C2×D4×C79—41no—
65C2×D4×C79SD1612yes—
66C22×C4×C79—21no—
67C22×C4×C79C2×D411yescharacteristic
68C22×C4×C79C2×D412yes—
69C2×C8×C79—42no—
70C2×C8×C79—42no—
71C2×C8×C79—42no—
72C4⋊C4×C79—41no—
73C22⋊C4×C79—42no—
74C22⋊C4×C79—42no—
75C42×C79—41no—
76C23×C79—22no—
77C23×C79—42no—
78C23×C79—41no—
79C23×C79—42no—
80C23×C79—41no—
81C23×C79#12011yescharacteristic
82D4×C79—42no—
83D4×C79—81no—
84C2×C4×C79—21no—
85C2×C4×C79—22no—
86C2×C4×C79—22no—
87C2×C4×C79—21no—
88C2×C4×C79—42no—
89C2×C4×C79—41no—
90C2×C4×C79C2×SD1611yescharacteristic
91C2×C4×C79#13512yes—
92C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
93C2×C4×C79#13712yes—
94C2×Dic79—42no—
95C2×Dic79—42no—
96C2×Dic79—42no—
97C2×Dic79—42no—
98C2×Dic79—42no—
99C2×Dic79—42no—
100C2×Dic79—84no—
101C2×Dic79—81no—
102C2×Dic79—82no—
103C8×C79—82no—
104C79⋊C8—42no—
105C22×C79—42no—
106C22×C79—44no—
107C22×C79—82no—
108C22×C79—81no—
109C22×C79—81no—
110C22×C79#45011yescharacteristic
111C22×C79#45212yes—
112C22×C79#45311yescharacteristic
113C22×C79#45412yes—
114C22×C79#45811yescharacteristic
115C4×C79—21no—
116C4×C79—21no—
117C4×C79—22no—
118C4×C79—22no—
119C4×C79—81no—
120Dic79—82no—
121Dic79—82no—
122Dic79—161no—
123#7824—791nomaximal, Sylow
124C2×C79—82no—
125C2×C79#161311yescharacteristic
126C2×C79#161512yes—
127C2×C79#163011yescharacteristic
128C2×C79#163512yes—
129C2×C79#165711yescharacteristic
130#2724—791no—
131#2100—792no—
132#2094—791no—
133#1271—792no—
134#1270D7911yescharacteristic
135C79#782411yescharacteristic, complemented, Sylow
136C2×#123—1581no—
137C2×#106—1581no—
138C2×#102—1581no—
139#398—1581no—
140#397—1582no—
141#390C2×D7911yescharacteristic
142#389—792no—
143#385—1582no—
144#384—791no—
145#340—1582no—
146#340—1582no—
147#336—791no—
148#336C2×D7912yes—
149C22×D4D31611yescharacteristic
150C23×C4—1581no—
151C22×C8—22no—
152C22×C8—1581no—
153#123—3162no—
154#123—3162no—
155C2×C4⋊C4—792no—
156C2×C4⋊C4—792no—
157C2×C4⋊C4—1582no—
158C2×C4⋊C4—1581no—
159C2×C4⋊C4—1582no—
160C2×C4⋊C4—1581no—
161C2×C4⋊C4D31611yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—1582no—
164C2×C22⋊C4—1581no—
165C2×C42C22×D7911yescharacteristic
166#106—3162no—
167#102—3162no—
168#95—1582no—
169#95—1581no—
170#95—1582no—
171C24—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4#1192961312yes—
176C22×C4—21no—
177C22×C4—3162no—
178C22×C4—3161no—
179C22×C4—6321no—
180C22×C4—6321no—
181C22×C4—1582no—
182C22×C4—1582no—
183C22×C4D4×D7912yes—
184C22×C4C2×D31611yescharacteristic
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—3161no—
189C2×C8—3161no—
190C2×C8—1582no—
191C4⋊C4—41no—
192C4⋊C4—3162no—
193C4⋊C4—3161no—
194C4⋊C4—3161no—
195C4⋊C4—3164no—
196C4⋊C4—3162no—
197C4⋊C4—3162no—
198C4⋊C4—6321no—
199C4⋊C4—1582no—
200C4⋊C4—1582no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—3162no—
204C22⋊C4—3164no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1457913111yescharacteristic
212D4—42no—
213D4—81no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—3162no—
221C2×C4—3162no—
222C2×C4—3162no—
223C2×C4—3162no—
224C2×C4—3162no—
225C2×C4—3162no—
226C2×C4—6324no—
227C2×C4—6321no—
228C2×C4—6322no—
229C2×C4C2×#1192961311yescharacteristic
230C2×C4#1457918012yes—
231C2×C4#1457918911yescharacteristic
232C2×C4#1457919012yes—
233C8—82no—
234C8—3162no—
235C22—44no—
236C22—42no—
237C22—82no—
238C22—81no—
239C22—81no—
240C22#1909871611yescharacteristic
241C22#1909872612yes—
242C22#1909873311yescharacteristic
243C22#1909873712yes—
244C22#1909875411yescharacteristic
245C4—22no—
246C4—22no—
247C4—21no—
248C4—21no—
249C4—81no—
250C4—6322no—
251C4—6322no—
252C4—12641no—
253C2—82no—
254C2#2376242211yescharacteristic
255C2#2376243212yes—
256C2#2376252211yescharacteristic
257C2#2376255312yes—
258C2#2376269911yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1270C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#1270D79
hypocenterC79#7824
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #23764964C2semidirect
4. #23761222C2semidirect
135. C79#7824semidirect

Maximal subgroups

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

subgroupindex
#237703302
#237649642
#237649332
#237612222
C79×#12702
#782479