Finite Group Databasebeta

← #34201262

Subgroups — #34201262

Order 20224.

558 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 63 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#23764745C211yesmaximal, characteristic, complemented
2#23762339C212yesmaximal, complemented
3#23761403C212yesmaximal, complemented
4#23761346C211yesmaximal, characteristic, complemented
5C79×#2537C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098714—22no—
8#19098603—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13C2×#121×C79—22no—
14C2×C4×D4×C79—21no—
15#394×C79—21no—
16#394×C79C2212yes—
17#384×C79C2211yescharacteristic
18#380×C79—22no—
19C2×#11929612—42no—
20C2×#11929608—21no—
21#14579160—41no—
22#14579152—41no—
23#14579140—22no—
24#14579120—21no—
25#14579094—41no—
26#14579092—41no—
27#14579082—42no—
28C22⋊(C79⋊C8)—22no—
29C22×D4×C79—21no—
30C22×D4×C79—21no—
31C22×D4×C79D412yes—
32C23×C4×C79—22no—
33#121×C79—48no—
34C4×D4×C79—42no—
35C4×D4×C79—42no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—22no—
38C2×C4⋊C4×C79D411yescharacteristic
39C2×C4⋊C4×C79C2311yescharacteristic
40C2×C22⋊C4×C79—22no—
41C2×C22⋊C4×C79—22no—
42C2×C22⋊C4×C79—22no—
43C2×C42×C79D411yescharacteristic
44#95×C79—21no—
45#95×C79D412yes—
46C2×Dic158—22no—
47#11929612—84no—
48#11929608—22no—
49#11929608—41no—
50C2×C79⋊C8—41no—
51C2×C79⋊C8—42no—
52C4⋊Dic79—22no—
53C4⋊Dic79—21no—
54C4⋊Dic79—21no—
55Dic79⋊C4—42no—
56C22⋊Dic79—41no—
57C4×Dic79—41no—
58C24×C79—22no—
59C2×D4×C79—24no—
60C2×D4×C79—42no—
61C2×D4×C79—42no—
62C2×D4×C79—42no—
63C2×D4×C79—44no—
64C2×D4×C79—44no—
65C2×D4×C79—81no—
66C22×C4×C79—21no—
67C22×C4×C79—22no—
68C22×C4×C79—21no—
69C22×C4×C79—44no—
70C22×C4×C79—44no—
71C22×C4×C79—44no—
72C22×C4×C79C2×D411yescharacteristic
73C22×C4×C79C2×D412yes—
74C4⋊C4×C79—41no—
75C4⋊C4×C79—42no—
76C4⋊C4×C79—44no—
77C4⋊C4×C79—41no—
78C22⋊C4×C79—44no—
79C22⋊C4×C79—44no—
80C22⋊C4×C79—42no—
81C22⋊C4×C79—42no—
82C42×C79—21no—
83C42×C79SD1612yescomplemented
84C23×C79—24no—
85C23×C79—42no—
86C23×C79—42no—
87C23×C79—42no—
88C23×C79—42no—
89C23×C79—42no—
90C23×C79—42no—
91C23×C79#12011yescharacteristic
92D4×C79—48no—
93D4×C79—44no—
94D4×C79—82no—
95D4×C79—84no—
96C2×C4×C79—22no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—22no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—41no—
108C2×C4×C79—88no—
109C2×C4×C79C2×SD1611yescharacteristic
110C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
111C2×Dic79—41no—
112C2×Dic79—41no—
113Dic158—82no—
114C79⋊C8—42no—
115C79⋊C8—82no—
116C22×C79—21no—
117C22×C79—21no—
118C22×C79—44no—
119C22×C79—44no—
120C22×C79—44no—
121C22×C79—82no—
122C22×C79—82no—
123C22×C79—82no—
124C22×C79—82no—
125C22×C79#45011yescharacteristic
126C22×C79#45611yescharacteristic
127C22×C79B(4,𝔽2)11yescharacteristic
128C4×C79—24no—
129C4×C79—21no—
130C4×C79—21no—
131C4×C79—42no—
132C4×C79—82no—
133C4×C79—81no—
134Dic79—81no—
135Dic79—81no—
136#12246—791nomaximal, Sylow
137C2×C79—21no—
138C2×C79—21no—
139C2×C79—84no—
140C2×C79#160411yescharacteristic
141C2×C79#219611yescharacteristic
142C2×C79#220411yescharacteristic
143#2537D7911yescharacteristic
144#2060—791no—
145#1599—792no—
146#1345—792no—
147#1325—791no—
148C79#1224611yescharacteristic, complemented, Sylow
149#544—791no—
150C2×#121—22no—
151C2×C4×D4—21no—
152#450—1582no—
153#428—1581no—
154#394—21no—
155#394C2×D7912yes—
156#384C2×D7911yescharacteristic
157#380—22no—
158#343—791no—
159#339—1581no—
160#328—1582no—
161#324—792no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22⋊D7912yes—
165C23×C4—22no—
166C2×SD16—3162no—
167C2×M16—1581no—
168C4⋊Q8—3161no—
169#125—3161no—
170C22⋊Q8—1582no—
171#121—48no—
172C4×D4—42no—
173C4×D4—42no—
174#117—1581no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊D7911yescharacteristic
178C2×C4⋊C4C22×D7911yescharacteristic
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42C22⋊D7911yescharacteristic
183#107—3161no—
184#106—3161no—
185#102—3162no—
186C22⋊C8—1582no—
187#95—21no—
188#95C22⋊D7912yes—
189C24—22no—
190C2×Q8—1582no—
191C2×D4—24no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—44no—
204C22×C4C2×C22⋊D7911yescharacteristic
205C22×C4C2×C22⋊D7912yes—
206SD16—6324no—
207M16—3161no—
208M16—1582no—
209C2×C8—3161no—
210C2×C8—3162no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—3162no—
216C4⋊C4—1582no—
217C4⋊C4—1581no—
218C4⋊C4—1581no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—3161no—
224C42—21no—
225C42—3161no—
226C42#1192961212yescomplemented
227C23—24no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#1457913011yescharacteristic
235Q8—6322no—
236D4—48no—
237D4—44no—
238D4—82no—
239D4—84no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—88no—
253C2×C4—3161no—
254C2×C4—3161no—
255C2×C4C2×#1192961211yescharacteristic
256C2×C4#1457918711yescharacteristic
257C8—3162no—
258C8—6322no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#1909871411yescharacteristic
269C22#1909874611yescharacteristic
270C22#1909875011yescharacteristic
271C4—24no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—82no—
276C4—81no—
277C4—6321no—
278C4—6321no—
279C2—21no—
280C2—21no—
281C2—84no—
282C2#2376236911yescharacteristic
283C2#2376538411yescharacteristic
284C2#2376542611yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
Fitting, prime radicalC79×#2537C2
socleC22×C79B(4,𝔽2)
hypercenter#2537D79
hypocenterC79#12246
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. #23764745C2semidirect
2. #23762339C2semidirect
3. #23761403C2semidirect
4. #23761346C2semidirect
83. C42×C79SD16semidirect
148. C79#12246semidirect
226. C42#11929612semidirect

Maximal subgroups

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

subgroupindex
#237647452
#237623392
#237614032
#237613462
C79×#25372
#1224679