Finite Group Databasebeta

← #34177061

Subgroups — #34177061

Order 20224.

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

normal subgroups only

This group has 492 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#23764728C212yesmaximal, complemented
2#23764482C211yesmaximal, characteristic
3#23761223C212yesmaximal, complemented
4#23761221C211yesmaximal, characteristic
5C79×#2724C211yesmaximal, characteristic
6C22×#11929612—21no—
7C2×#14579161C2211yescharacteristic
8C2×#14579094—22no—
9C2×#14579084—21no—
10#19098325—22no—
11#19098318C2211yescharacteristic
12#19098318C2212yes—
13C2×#123×C79—21no—
14#398×C79—21no—
15#397×C79—22no—
16#390×C79C2211yescharacteristic
17#389×C79C2212yes—
18#385×C79—22no—
19C22×Dic158—21no—
20C2×#11929612—42no—
21C2×#11929612—42no—
22C2×#11929612—42no—
23C22×C79⋊C8—21no—
24C22×C79⋊C8—22no—
25#14579161—42no—
26C2×C4⋊Dic79—22no—
27#14579094—44no—
28#14579084—42no—
29C22×D4×C79D411yescharacteristic
30C23×C4×C79—21no—
31#123×C79—42no—
32#123×C79—42no—
33C2×C4⋊C4×C79—22no—
34C2×C4⋊C4×C79—21no—
35C2×C4⋊C4×C79D412yes—
36C2×C4⋊C4×C79D411yescharacteristic
37C2×C4⋊C4×C79D412yes—
38C2×C22⋊C4×C79—22no—
39C2×C22⋊C4×C79—21no—
40C2×C22⋊C4×C79—22no—
41C2×C42×C79C2311yescharacteristic
42#95×C79—22no—
43#95×C79—21no—
44#95×C79—22no—
45C2×Dic158—22no—
46C2×Dic158—41no—
47C2×Dic158—41no—
48C2×Dic158—41no—
49C2×Dic158—41no—
50C2×Dic158—41no—
51C22×Dic79—41no—
52#11929612—48no—
53C2×C79⋊C8—22no—
54C2×C79⋊C8—24no—
55C2×C79⋊C8—42no—
56C2×C79⋊C8—41no—
57C2×C79⋊C8—41no—
58C2×C79⋊C8—42no—
59C4⋊Dic79—24no—
60C4⋊Dic79—42no—
61C24×C79—21no—
62C2×D4×C79—21no—
63C2×D4×C79—41no—
64C2×D4×C79—41no—
65C2×D4×C79D812yes—
66C22×C4×C79—22no—
67C22×C4×C79—21no—
68C22×C4×C79—22no—
69C22×C4×C79—42no—
70C22×C4×C79—81no—
71C22×C4×C79—81no—
72C22×C4×C79C2×D411yescharacteristic
73C22×C4×C79C2×D412yes—
74C4⋊C4×C79—22no—
75C4⋊C4×C79—22no—
76C4⋊C4×C79—41no—
77C4⋊C4×C79—42no—
78C4⋊C4×C79—42no—
79C4⋊C4×C79—42no—
80C4⋊C4×C79—81no—
81C22⋊C4×C79—42no—
82C22⋊C4×C79—44no—
83C22⋊C4×C79—42no—
84C22⋊C4×C79—42no—
85C42×C79—41no—
86C23×C79—22no—
87C23×C79—42no—
88C23×C79—41no—
89C23×C79—42no—
90C23×C79—41no—
91C23×C79#12011yescharacteristic
92D4×C79—42no—
93D4×C79—81no—
94C2×C4×C79—21no—
95C2×C4×C79—22no—
96C2×C4×C79—22no—
97C2×C4×C79—21no—
98C2×C4×C79—42no—
99C2×C4×C79—42no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—41no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—84no—
107C2×C4×C79C2×D811yescharacteristic
108C2×C4×C79#13512yes—
109C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
110C2×C4×C79#13712yes—
111C2×Dic79—42no—
112C2×Dic79—81no—
113C2×Dic79—81no—
114Dic158—42no—
115Dic158—42no—
116Dic158—81no—
117Dic158—81no—
118C79⋊C8—44no—
119C79⋊C8—42no—
120C22×C79—42no—
121C22×C79—44no—
122C22×C79—82no—
123C22×C79—81no—
124C22×C79—81no—
125C22×C79#44711yescharacteristic
126C22×C79#45212yes—
127C22×C79#45311yescharacteristic
128C22×C79#45612yes—
129C22×C79B(4,𝔽2)11yescharacteristic
130C4×C79—21no—
131C4×C79—21no—
132C4×C79—22no—
133C4×C79—22no—
134C4×C79—82no—
135C4×C79—81no—
136C4×C79—82no—
137Dic79—82no—
138#7751—791nomaximal, Sylow
139C2×C79—82no—
140C2×C79#159511yescharacteristic
141C2×C79#160312yes—
142C2×C79#163011yescharacteristic
143C2×C79#164712yes—
144C2×C79#166011yescharacteristic
145#2724D7911yescharacteristic
146#2057—792no—
147#2013—791no—
148#1271—792no—
149#1270—791no—
150C79#775111yescharacteristic, complemented, Sylow
151C22×SD16—1581no—
152C2×C4⋊Q8—791no—
153C2×#123—21no—
154C2×#107—1582no—
155C2×#103—1581no—
156#398—21no—
157#397—22no—
158#390C2×D7911yescharacteristic
159#389C2×D7912yes—
160#385—22no—
161#340—1582no—
162#336—792no—
163#336—791no—
164C22×Q8—1581no—
165C22×D4C22⋊D7911yescharacteristic
166C23×C4—21no—
167C2×SD16—3162no—
168C2×SD16—3162no—
169C2×SD16—3162no—
170C22×C8—1581no—
171C22×C8—1582no—
172C4⋊Q8—3162no—
173#123—42no—
174#123—42no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—1582no—
178C2×C4⋊C4C22⋊D7912yes—
179C2×C4⋊C4C22⋊D7912yes—
180C2×C4⋊C4C22⋊D7911yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C42C22×D7911yescharacteristic
185#107—3164no—
186#103—3162no—
187#95—22no—
188#95—21no—
189#95—22no—
190C24—21no—
191C2×Q8—3161no—
192C2×Q8—3161no—
193C2×Q8—3161no—
194C2×Q8—3161no—
195C2×Q8—3161no—
196C2×Q8—1582no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4#1192960912yes—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—3161no—
208C22×C4C2×C22⋊D7911yescharacteristic
209C22×C4C2×C22⋊D7912yes—
210SD16—3168no—
211C2×C8—3162no—
212C2×C8—3161no—
213C2×C8—3161no—
214C2×C8—3162no—
215C2×C8—1582no—
216C2×C8—1584no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4—3162no—
225C4⋊C4—1584no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C42—41no—
231C23—22no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23—41no—
236C23#1457913011yescharacteristic
237Q8—3162no—
238Q8—3162no—
239Q8—6321no—
240Q8—6321no—
241D4—42no—
242D4—81no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—84no—
256C2×C4—3162no—
257C2×C4—6321no—
258C2×C4—6321no—
259C2×C4C2×#1192960911yescharacteristic
260C2×C4#1457918112yes—
261C2×C4#1457918711yescharacteristic
262C2×C4#1457919112yes—
263C8—3162no—
264C8—3164no—
265C22—42no—
266C22—44no—
267C22—81no—
268C22—81no—
269C22—82no—
270C22#1909869711yescharacteristic
271C22#1909872512yes—
272C22#1909873111yescharacteristic
273C22#1909874612yes—
274C22#1909875011yescharacteristic
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—81no—
280C4—82no—
281C4—82no—
282C4—6322no—
283C2—82no—
284C2#2376231811yescharacteristic
285C2#2376236312yes—
286C2#2376252111yescharacteristic
287C2#2376263512yes—
288C2#2376271111yescharacteristic
subgroupG/H
centerC23#14579130
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#2724C2
socleC23×C79#120
prime radicalC22×D4×C79D4
hypercenter#2724D79
hypocenterC79#7751
LCSG ▷ C2×C42×C79 ▷ C23×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
1. #23764728C2semidirect
3. #23761223C2semidirect
150. C79#7751semidirect

Maximal subgroups

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

subgroupindex
#237647282
#237644822
#237612232
#237612212
C79×#27242
#775179