Finite Group Databasebeta

← #34177369

Subgroups — #34177369

Order 20224.

478 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 87 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#23769772C211yesmaximal, characteristic
2#23764928C211yesmaximal, characteristic
3#23764873C212yesmaximal, complemented
4#23761224C212yesmaximal, complemented
5C79×#1270C211yesmaximal, characteristic
6C2×#14579138—22no—
7C2×#14579095—22no—
8C2×#14579094—21no—
9C2×#14579082—21no—
10#19098486—22no—
11#19098483—21no—
12#19098461C2212yes—
13#19098443—21no—
14#19098433C2211yescharacteristic
15#19098326—22no—
16#19098318C2211yescharacteristic
17#390×C79C2211yescharacteristic
18#336×C79C2212yes—
19C22×Dic158—21no—
20C23×Dic79—21no—
21C22×C79⋊C8—21no—
22#14579138—44no—
23#14579138—44no—
24C2×C4⋊Dic79—21no—
25C2×C4⋊Dic79—22no—
26C2×C4⋊Dic79D412yes—
27C2×C4⋊Dic79D412yes—
28C2×Dic79⋊C4—22no—
29C2×Dic79⋊C4—22no—
30C2×C22⋊Dic79—22no—
31C2×C22⋊Dic79—21no—
32#14579095—44no—
33#14579094—42no—
34#14579082—42no—
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
42C2×Dic158—44no—
43C2×Dic158—81no—
44C22×Dic79—22no—
45C22×Dic79—22no—
46C22×Dic79—44no—
47C22×Dic79—41no—
48C22×Dic79—81no—
49C2×C79⋊C8—22no—
50C2×C79⋊C8—41no—
51C2×C79⋊C8—41no—
52C4⋊Dic79—22no—
53C4⋊Dic79—42no—
54C4⋊Dic79—41no—
55C4⋊Dic79—41no—
56C4⋊Dic79—42no—
57C4⋊Dic79—42no—
58C4⋊Dic79D814yes—
59Dic79⋊C4—44no—
60Dic79⋊C4—42no—
61Dic79⋊C4—42no—
62C22⋊Dic79—44no—
63C22⋊Dic79—81no—
64C24×C79—21no—
65C2×D4×C79—21no—
66C2×D4×C79—41no—
67C2×D4×C79—41no—
68C2×D4×C79Q1612yes—
69C22×C4×C79—21no—
70C22×C4×C79C2×D412yes—
71C22×C4×C79C2×D411yescharacteristic
72C2×C8×C79—24no—
73C2×C8×C79—42no—
74C2×C8×C79—42no—
75C4⋊C4×C79—41no—
76C22⋊C4×C79—42no—
77C22⋊C4×C79—42no—
78C42×C79—41no—
79C23×C79—22no—
80C23×C79—42no—
81C23×C79—41no—
82C23×C79—42no—
83C23×C79—41no—
84C23×C79#12011yescharacteristic
85D4×C79—42no—
86D4×C79—81no—
87C2×C4×C79—21no—
88C2×C4×C79—22no—
89C2×C4×C79—22no—
90C2×C4×C79—21no—
91C2×C4×C79—42no—
92C2×C4×C79—41no—
93C2×C4×C79C2×D812yes—
94C2×C4×C79C2×Q1611yescharacteristic
95C2×C4×C79AGL(1,ℤ/8)12yes—
96C2×C4×C79#13711yescharacteristic
97C2×Dic79—24no—
98C2×Dic79—42no—
99C2×Dic79—42no—
100C2×Dic79—42no—
101C2×Dic79—42no—
102C2×Dic79—42no—
103C2×Dic79—84no—
104C2×Dic79—82no—
105C2×Dic79—81no—
106Dic158—84no—
107C8×C79—44no—
108C79⋊C8—42no—
109C22×C79—42no—
110C22×C79—44no—
111C22×C79—82no—
112C22×C79—81no—
113C22×C79—81no—
114C22×C79#44712yes—
115C22×C79#45111yescharacteristic
116C22×C79#45311yescharacteristic
117C22×C79#45612yes—
118C22×C79#45811yescharacteristic
119C4×C79—22no—
120C4×C79—22no—
121C4×C79—21no—
122C4×C79—21no—
123C4×C79—81no—
124Dic79—44no—
125Dic79—82no—
126Dic79—161no—
127#7797—791nomaximal, Sylow
128C2×C79—82no—
129C2×C79#161012yes—
130C2×C79#161411yescharacteristic
131C2×C79#161911yescharacteristic
132C2×C79#164912yes—
133C2×C79#166011yescharacteristic
134#2683—791no—
135#2093—791no—
136#2083—792no—
137#1271—792no—
138#1270D7911yescharacteristic
139C79#779711yescharacteristic, complemented, Sylow
140C2×C22⋊Q8—1582no—
141C2×#107—1582no—
142C2×#107—1581no—
143C2×#102—1581no—
144#396—1582no—
145#395—1581no—
146#390C2×D7911yescharacteristic
147#389—792no—
148#386—1581no—
149#384—791no—
150#340—1582no—
151#336—791no—
152#336C2×D7912yes—
153C22×Q8—1581no—
154C22×D4D31611yescharacteristic
155C23×C4—1581no—
156C22×C8—22no—
157C22×C8—1581no—
158C22⋊Q8—3164no—
159C22⋊Q8—3164no—
160C2×C4⋊C4—792no—
161C2×C4⋊C4—792no—
162C2×C4⋊C4—1582no—
163C2×C4⋊C4—1582no—
164C2×C4⋊C4—1581no—
165C2×C4⋊C4—1582no—
166C2×C4⋊C4D31611yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—1582no—
169C2×C22⋊C4—1581no—
170C2×C42C22×D7911yescharacteristic
171#107—3162no—
172#107—3164no—
173#102—3162no—
174#95—1581no—
175#95—1582no—
176C24—21no—
177C2×Q8—3164no—
178C2×Q8—6321no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4Dic31612yes—
183C22×C4—21no—
184C22×C4—3164no—
185C22×C4—3161no—
186C22×C4—6321no—
187C22×C4—1582no—
188C22×C4—1582no—
189C22×C4D4×D7912yes—
190C22×C4C2×D31611yescharacteristic
191C2×C8—24no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—3161no—
195C2×C8—3161no—
196C2×C8—1582no—
197C4⋊C4—41no—
198C4⋊C4—3162no—
199C4⋊C4—3161no—
200C4⋊C4—3161no—
201C4⋊C4—3164no—
202C4⋊C4—3162no—
203C4⋊C4—3162no—
204C4⋊C4—3162no—
205C4⋊C4—3162no—
206C4⋊C4—794no—
207C4⋊C4—1582no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—3164no—
211C22⋊C4—6321no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1457913111yescharacteristic
219Q8—6324no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—3162no—
229C2×C4—3162no—
230C2×C4—3162no—
231C2×C4—3162no—
232C2×C4—3162no—
233C2×C4—6324no—
234C2×C4—6322no—
235C2×C4—6321no—
236C2×C4—1584no—
237C2×C4D8×D7912yes—
238C2×C4C2×Dic31611yescharacteristic
239C2×C4#1457918812yes—
240C2×C4#1457919411yescharacteristic
241C8—44no—
242C8—3162no—
243C22—44no—
244C22—42no—
245C22—81no—
246C22—81no—
247C22—82no—
248C22#1909869812yes—
249C22#1909872111yescharacteristic
250C22#1909873311yescharacteristic
251C22#1909874712yes—
252C22#1909875411yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—81no—
258C4—3164no—
259C4—6322no—
260C4—12641no—
261C2—82no—
262C2#2376240412yes—
263C2#2376242711yescharacteristic
264C2#2376245511yescharacteristic
265C2#2376265212yes—
266C2#2376271211yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1270C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#1270D79
hypocenterC79#7797
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. #23764873C2semidirect
4. #23761224C2semidirect
139. C79#7797semidirect

Maximal subgroups

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

subgroupindex
#237697722
#237649282
#237648732
#237612242
C79×#12702
#779779