Finite Group Databasebeta

← #34201286

Subgroups — #34201286

Order 20224.

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

normal subgroups only

This group has 454 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#23764750C211yesmaximal, characteristic, complemented
2#23762339C212yesmaximal, complemented
3#23761403C212yesmaximal, complemented
4#23761344C211yesmaximal, characteristic, complemented
5C79×#2819C211yesmaximal, characteristic
6#19099194C2211yescharacteristic
7#19098714—22no—
8#19098603—21no—
9#19098332C2211yescharacteristic
10#19098323—21no—
11#19098299—22no—
12#19098289C2212yes—
13C2×#124×C79—21no—
14#394×C79—21no—
15#394×C79C2212yes—
16#393×C79—22no—
17#392×C79—22no—
18#382×C79C2211yescharacteristic
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×Q8×C79—21no—
30C22×D4×C79—21no—
31C22×D4×C79—21no—
32C22×D4×C79D412yes—
33#124×C79—42no—
34#124×C79—42no—
35C2×C4⋊C4×C79D411yescharacteristic
36C2×C4⋊C4×C79C2311yescharacteristic
37C2×C22⋊C4×C79—22no—
38C2×C22⋊C4×C79—22no—
39C2×C22⋊C4×C79—22no—
40C2×C22⋊C4×C79—22no—
41C2×C42×C79D411yescharacteristic
42#95×C79—21no—
43#95×C79—22no—
44#95×C79D412yes—
45C2×Dic158—22no—
46#11929612—84no—
47#11929608—22no—
48#11929608—41no—
49C2×C79⋊C8—41no—
50C2×C79⋊C8—42no—
51C4⋊Dic79—22no—
52C4⋊Dic79—21no—
53C4⋊Dic79—21no—
54Dic79⋊C4—42no—
55C22⋊Dic79—41no—
56C4×Dic79—41no—
57C24×C79—22no—
58C2×Q8×C79—42no—
59C2×Q8×C79—81no—
60C2×Q8×C79—81no—
61C2×D4×C79—42no—
62C2×D4×C79—42no—
63C2×D4×C79—82no—
64C2×D4×C79—81no—
65C2×D4×C79—81no—
66C2×D4×C79—82no—
67C2×D4×C79—81no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—22no—
71C22×C4×C79C2×D411yescharacteristic
72C22×C4×C79C2×D412yes—
73C4⋊C4×C79—41no—
74C4⋊C4×C79—41no—
75C22⋊C4×C79—44no—
76C22⋊C4×C79—42no—
77C22⋊C4×C79—42no—
78C22⋊C4×C79—82no—
79C22⋊C4×C79—82no—
80C42×C79—21no—
81C42×C79SD1612yescomplemented
82C23×C79—42no—
83C23×C79—42no—
84C23×C79—42no—
85C23×C79—42no—
86C23×C79—42no—
87C23×C79—42no—
88C23×C79—42no—
89C23×C79#12011yescharacteristic
90Q8×C79—84no—
91D4×C79—44no—
92D4×C79—82no—
93D4×C79—84no—
94D4×C79—84no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—22no—
98C2×C4×C79—22no—
99C2×C4×C79—41no—
100C2×C4×C79—41no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—41no—
107C2×C4×C79C2×SD1611yescharacteristic
108C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
109C2×Dic79—41no—
110C2×Dic79—41no—
111Dic158—82no—
112C79⋊C8—42no—
113C79⋊C8—82no—
114C22×C79—21no—
115C22×C79—21no—
116C22×C79—82no—
117C22×C79—82no—
118C22×C79—82no—
119C22×C79—82no—
120C22×C79—82no—
121C22×C79—82no—
122C22×C79—82no—
123C22×C79#45011yescharacteristic
124C22×C79#45611yescharacteristic
125C22×C79B(4,𝔽2)11yescharacteristic
126C4×C79—21no—
127C4×C79—21no—
128C4×C79—42no—
129C4×C79—42no—
130C4×C79—82no—
131C4×C79—81no—
132Dic79—81no—
133Dic79—81no—
134#12250—791nomaximal, Sylow
135C2×C79—21no—
136C2×C79—21no—
137C2×C79—162no—
138C2×C79#160411yescharacteristic
139C2×C79#219911yescharacteristic
140C2×C79#220211yescharacteristic
141#2819D7911yescharacteristic
142#2061—791no—
143#1599—792no—
144#1345—792no—
145#1324—791no—
146C79#1225011yescharacteristic, complemented, Sylow
147#544—791no—
148C2×#124—21no—
149#450—1582no—
150#428—1581no—
151#394—21no—
152#394C2×D7912yes—
153#393—22no—
154#392—22no—
155#382C2×D7911yescharacteristic
156#343—791no—
157#339—1581no—
158#328—1582no—
159#324—792no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22⋊D7912yes—
164C2×SD16—3162no—
165C2×M16—1581no—
166C4⋊Q8—3161no—
167#125—3161no—
168#124—42no—
169#124—42no—
170C22⋊Q8—1582no—
171#117—1581no—
172C2×C4⋊C4C22⋊D7911yescharacteristic
173C2×C4⋊C4C22×D7911yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C42C22⋊D7911yescharacteristic
179#107—3161no—
180#106—3161no—
181#102—3162no—
182C22⋊C8—1582no—
183#95—21no—
184#95—22no—
185#95C22⋊D7912yes—
186C24—22no—
187C2×Q8—42no—
188C2×Q8—81no—
189C2×Q8—81no—
190C2×Q8—1582no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—82no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—82no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4C2×C22⋊D7911yescharacteristic
202C22×C4C2×C22⋊D7912yes—
203SD16—6324no—
204M16—3161no—
205M16—1582no—
206C2×C8—3161no—
207C2×C8—3162no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—3162no—
211C4⋊C4—1582no—
212C4⋊C4—1581no—
213C4⋊C4—1581no—
214C22⋊C4—44no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C22⋊C4—3161no—
220C42—21no—
221C42—3161no—
222C42#1192961212yescomplemented
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23#1457913011yescharacteristic
231Q8—84no—
232Q8—6322no—
233D4—44no—
234D4—82no—
235D4—84no—
236D4—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—3161no—
250C2×C4—3161no—
251C2×C4C2×#1192961211yescharacteristic
252C2×C4#1457918711yescharacteristic
253C8—3162no—
254C8—6322no—
255C22—21no—
256C22—21no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22#1909871411yescharacteristic
265C22#1909874611yescharacteristic
266C22#1909875011yescharacteristic
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—82no—
272C4—81no—
273C4—6321no—
274C4—6321no—
275C2—21no—
276C2—21no—
277C2—162no—
278C2#2376236911yescharacteristic
279C2#2376540111yescharacteristic
280C2#2376541711yescharacteristic
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C4⋊C4×C79C23
FrattiniC2×C4⋊C4C22×D79
Fitting, prime radicalC79×#2819C2
socleC22×C79B(4,𝔽2)
hypercenter#2819D79
hypocenterC79#12250
LCSG ▷ C2×C4⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. #23764750C2semidirect
2. #23762339C2semidirect
3. #23761403C2semidirect
4. #23761344C2semidirect
81. C42×C79SD16semidirect
146. C79#12250semidirect
222. C42#11929612semidirect

Maximal subgroups

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

subgroupindex
#237647502
#237623392
#237614032
#237613442
C79×#28192
#1225079