Finite Group Databasebeta

← #34177513

Subgroups — #34177513

Order 20224.

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

normal subgroups only

This group has 434 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#23764928C211yesmaximal, characteristic
3#23764878C212yesmaximal, complemented
4#23761222C212yesmaximal, complemented
5C79×#1270C211yesmaximal, characteristic
6C2×#14579142—21no—
7C2×#14579094—21no—
8C2×#14579093—22no—
9C2×#14579082—21no—
10#19098493—21no—
11#19098488—22no—
12#19098461C2212yes—
13#19098438—22no—
14#19098433C2211yescharacteristic
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#14579094—42no—
32#14579093—44no—
33#14579082—42no—
34C79⋊#95—22no—
35C79⋊#95—21no—
36C79⋊#95—22no—
37C22×D4×C79D411yescharacteristic
38C22×C8×C79—22no—
39C2×C4⋊C4×C79D411yescharacteristic
40C2×C22⋊C4×C79—22no—
41C2×C42×C79C2311yescharacteristic
42C22×Dic79—22no—
43C22×Dic79—22no—
44C22×Dic79—42no—
45C22×Dic79—41no—
46C22×Dic79—81no—
47C22×Dic79—81no—
48C2×C79⋊C8—22no—
49C2×C79⋊C8—41no—
50C2×C79⋊C8—41no—
51C4⋊Dic79—22no—
52C4⋊Dic79—22no—
53C4⋊Dic79—42no—
54C4⋊Dic79—41no—
55C4⋊Dic79—41no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79—81no—
59Dic79⋊C4—42no—
60Dic79⋊C4—42no—
61C22⋊Dic79—42no—
62C22⋊Dic79—44no—
63C24×C79—21no—
64C2×D4×C79—21no—
65C2×D4×C79—41no—
66C2×D4×C79—41no—
67C2×D4×C79SD1612yes—
68C22×C4×C79—21no—
69C22×C4×C79C2×D411yescharacteristic
70C22×C4×C79C2×D412yes—
71C2×C8×C79—24no—
72C2×C8×C79—42no—
73C2×C8×C79—42no—
74C4⋊C4×C79—41no—
75C22⋊C4×C79—42no—
76C22⋊C4×C79—42no—
77C42×C79—41no—
78C23×C79—22no—
79C23×C79—42no—
80C23×C79—41no—
81C23×C79—42no—
82C23×C79—41no—
83C23×C79#12011yescharacteristic
84D4×C79—42no—
85D4×C79—81no—
86C2×C4×C79—21no—
87C2×C4×C79—22no—
88C2×C4×C79—22no—
89C2×C4×C79—21no—
90C2×C4×C79—42no—
91C2×C4×C79—41no—
92C2×C4×C79C2×SD1611yescharacteristic
93C2×C4×C79#13512yes—
94C2×C4×C79AGL(1,ℤ/8)12yes—
95C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
96C2×Dic79—42no—
97C2×Dic79—42no—
98C2×Dic79—42no—
99C2×Dic79—42no—
100C2×Dic79—42no—
101C2×Dic79—42no—
102C2×Dic79—84no—
103C2×Dic79—82no—
104C2×Dic79—81no—
105C8×C79—44no—
106C79⋊C8—42no—
107C22×C79—42no—
108C22×C79—44no—
109C22×C79—82no—
110C22×C79—81no—
111C22×C79—81no—
112C22×C79#44912yes—
113C22×C79#45011yescharacteristic
114C22×C79#45311yescharacteristic
115C22×C79#45612yes—
116C22×C79#45811yescharacteristic
117C4×C79—22no—
118C4×C79—22no—
119C4×C79—21no—
120C4×C79—21no—
121C4×C79—81no—
122Dic79—82no—
123Dic79—82no—
124Dic79—161no—
125#7819—791nomaximal, Sylow
126C2×C79—82no—
127C2×C79#160912yes—
128C2×C79#161311yescharacteristic
129C2×C79#162011yescharacteristic
130C2×C79#165412yes—
131C2×C79#166011yescharacteristic
132#2724—791no—
133#2093—791no—
134#2084—792no—
135#1271—792no—
136#1270D7911yescharacteristic
137C79#781911yescharacteristic, complemented, Sylow
138C2×#123—1581no—
139C2×#107—1581no—
140C2×#106—1582no—
141C2×#102—1581no—
142#398—1581no—
143#397—1582no—
144#390C2×D7911yescharacteristic
145#389—792no—
146#385—1582no—
147#384—791no—
148#340—1582no—
149#336—791no—
150#336C2×D7912yes—
151C22×D4D31611yescharacteristic
152C23×C4—1581no—
153C22×C8—22no—
154C22×C8—1581no—
155#123—3162no—
156#123—3162no—
157C2×C4⋊C4—792no—
158C2×C4⋊C4—792no—
159C2×C4⋊C4—1582no—
160C2×C4⋊C4—1581no—
161C2×C4⋊C4—1582no—
162C2×C4⋊C4—1581no—
163C2×C4⋊C4D31611yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—1582no—
166C2×C22⋊C4—1581no—
167C2×C42C22×D7911yescharacteristic
168#107—3162no—
169#106—3164no—
170#102—3162no—
171#95—1582no—
172#95—1581no—
173#95—1582no—
174C24—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4#1192961312yes—
179C22×C4—21no—
180C22×C4—3162no—
181C22×C4—3161no—
182C22×C4—6321no—
183C22×C4—6321no—
184C22×C4—1582no—
185C22×C4—1582no—
186C22×C4D4×D7912yes—
187C22×C4C2×D31611yescharacteristic
188C2×C8—24no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—3161no—
192C2×C8—3161no—
193C2×C8—1582no—
194C4⋊C4—41no—
195C4⋊C4—3162no—
196C4⋊C4—3161no—
197C4⋊C4—3161no—
198C4⋊C4—3162no—
199C4⋊C4—3162no—
200C4⋊C4—3162no—
201C4⋊C4—3162no—
202C4⋊C4—6321no—
203C4⋊C4—1582no—
204C4⋊C4—1582no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—3162no—
208C22⋊C4—3164no—
209C42—41no—
210C23—22no—
211C23—42no—
212C23—41no—
213C23—42no—
214C23—41no—
215C23#1457913111yescharacteristic
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—3162no—
225C2×C4—3162no—
226C2×C4—3162no—
227C2×C4—3162no—
228C2×C4—3162no—
229C2×C4—3162no—
230C2×C4—6324no—
231C2×C4—6322no—
232C2×C4—6321no—
233C2×C4C2×#1192961311yescharacteristic
234C2×C4#1457918212yes—
235C2×C4#1457918512yes—
236C2×C4#1457918911yescharacteristic
237C8—44no—
238C8—3162no—
239C22—42no—
240C22—44no—
241C22—81no—
242C22—81no—
243C22—82no—
244C22#1909870912yes—
245C22#1909871611yescharacteristic
246C22#1909873311yescharacteristic
247C22#1909874712yes—
248C22#1909875411yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—81no—
254C4—6322no—
255C4—6322no—
256C4—12641no—
257C2—82no—
258C2#2376239812yes—
259C2#2376242211yescharacteristic
260C2#2376246011yescharacteristic
261C2#2376268712yes—
262C2#2376271211yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1270C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#1270D79
hypocenterC79#7819
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
3. #23764878C2semidirect
4. #23761222C2semidirect
137. C79#7819semidirect

Maximal subgroups

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

subgroupindex
#237703302
#237649282
#237648782
#237612222
C79×#12702
#781979