Finite Group Databasebeta

← #34200991

Subgroups — #34200991

Order 20224.

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

normal subgroups only

This group has 532 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#23764564C211yesmaximal, characteristic
2#23762390C212yesmaximal, complemented
3#23761400C212yesmaximal, complemented
4#23761328C211yesmaximal, characteristic
5C79×#2483C211yesmaximal, characteristic
6#19099189C2211yescharacteristic
7#19098893—22no—
8#19098419—21no—
9#19098330C2211yescharacteristic
10#19098329—21no—
11#19098299—22no—
12#19098287C2212yes—
13C22×C22⋊C4×C79—21no—
14#397×C79—21no—
15#397×C79C2212yes—
16#385×C79—22no—
17#381×C79—22no—
18#379×C79C2211yescharacteristic
19#14579140—22no—
20#14579138—42no—
21C2×C22⋊Dic79—21no—
22C2×C22⋊Dic79—21no—
23C2×C22⋊Dic79—42no—
24#14579092—42no—
25#14579084—42no—
26C22⋊(C79⋊C8)—22no—
27C25×C79D411yescharacteristic
28C23×C4×C79—21no—
29C23×C4×C79—21no—
30C2×C4⋊C4×C79—21no—
31C2×C4⋊C4×C79D412yes—
32C2×C22⋊C4×C79—22no—
33C2×C22⋊C4×C79—22no—
34C2×C22⋊C4×C79—22no—
35C2×C22⋊C4×C79—42no—
36C2×C22⋊C4×C79—42no—
37C2×C22⋊C4×C79D411yescharacteristic
38C2×C22⋊C4×C79C2311yescharacteristic
39#95×C79—21no—
40#95×C79—22no—
41#95×C79—22no—
42#95×C79D412yes—
43C2×Dic158—22no—
44C22×Dic79—22no—
45C22×Dic79—41no—
46C2×C79⋊C8—42no—
47C4⋊Dic79—22no—
48Dic79⋊C4—42no—
49C22⋊Dic79—41no—
50C22⋊Dic79—41no—
51C22⋊Dic79—41no—
52C22⋊Dic79—41no—
53C22⋊Dic79—84no—
54C24×C79—21no—
55C24×C79—21no—
56C24×C79—44no—
57C24×C79—41no—
58C24×C79—41no—
59C24×C79SD1612yes—
60C24×C79C2×D411yescharacteristic
61C22×C4×C79—22no—
62C22×C4×C79—21no—
63C22×C4×C79—22no—
64C22×C4×C79—42no—
65C22×C4×C79—42no—
66C22×C4×C79—44no—
67C22×C4×C79—81no—
68C22×C4×C79—81no—
69C22×C4×C79C2×D412yes—
70C4⋊C4×C79—42no—
71C4⋊C4×C79—81no—
72C22⋊C4×C79—41no—
73C22⋊C4×C79—44no—
74C22⋊C4×C79—44no—
75C22⋊C4×C79—42no—
76C22⋊C4×C79—42no—
77C22⋊C4×C79—41no—
78C23×C79—24no—
79C23×C79—22no—
80C23×C79—21no—
81C23×C79—21no—
82C23×C79—42no—
83C23×C79—44no—
84C23×C79—42no—
85C23×C79—42no—
86C23×C79—44no—
87C23×C79—41no—
88C23×C79—41no—
89C23×C79—44no—
90C23×C79—81no—
91C23×C79—84no—
92C23×C79—82no—
93C23×C79#12011yescharacteristic
94C23×C79C2×SD1611yescharacteristic
95C23×C79AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C79—22no—
97C2×C4×C79—42no—
98C2×C4×C79—42no—
99C2×C4×C79—44no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—41no—
103C2×C4×C79—42no—
104C2×C4×C79—42no—
105C2×C4×C79—84no—
106C2×C4×C79—84no—
107C2×Dic79—42no—
108C2×Dic79—82no—
109C2×Dic79—81no—
110C2×Dic79—82no—
111Dic158—82no—
112C79⋊C8—82no—
113C22×C79—22no—
114C22×C79—24no—
115C22×C79—21no—
116C22×C79—21no—
117C22×C79—41no—
118C22×C79—41no—
119C22×C79—42no—
120C22×C79—44no—
121C22×C79—44no—
122C22×C79—42no—
123C22×C79—44no—
124C22×C79—42no—
125C22×C79—81no—
126C22×C79—84no—
127C22×C79—82no—
128C22×C79#45011yescharacteristic
129C22×C79#45511yescharacteristic
130C22×C79B(4,𝔽2)11yescharacteristic
131C4×C79—42no—
132C4×C79—81no—
133C4×C79—82no—
134C4×C79—82no—
135Dic79—82no—
136Dic79—161no—
137#12203—791nomaximal, Sylow
138C2×C79—21no—
139C2×C79—21no—
140C2×C79—41no—
141C2×C79—44no—
142C2×C79—41no—
143C2×C79#160111yescharacteristic
144C2×C79#219712yes—
145#2483D7911yescharacteristic
146#2029—791no—
147#1608—792no—
148#1344—792no—
149#1316—791no—
150C79#1220311yescharacteristic, complemented, Sylow
151#543—791no—
152C22×C22⋊C4—21no—
153#483—1582no—
154#397—21no—
155#397C2×D7912yes—
156#385—22no—
157#381—22no—
158#379—1581no—
159#379C2×D7911yescharacteristic
160#342—791no—
161#342—1581no—
162#328—1582no—
163#323—792no—
164C25C22⋊D7911yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C22⋊Q8—3162no—
168C22⋊Q8—1582no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C22⋊D7912yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—3162no—
177C2×C22⋊C4—1581no—
178C2×C22⋊C4—1581no—
179C2×C22⋊C4C22⋊D7911yescharacteristic
180C2×C22⋊C4C22×D7911yescharacteristic
181#106—3162no—
182#103—3162no—
183C22⋊C8—1582no—
184#95—21no—
185#95—22no—
186#95—22no—
187#95C22⋊D7912yes—
188C24—21no—
189C24—21no—
190C24—41no—
191C24—44no—
192C24—41no—
193C24#1192961212yes—
194C24C2×C22⋊D7911yescharacteristic
195C2×Q8—1582no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—3161no—
205C22×C4—1582no—
206C22×C4C2×C22⋊D7912yes—
207C2×C8—3162no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4—3162no—
211C4⋊C4—1582no—
212C22⋊C4—41no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—3161no—
219C22⋊C4—3161no—
220C22⋊C4—3161no—
221C22⋊C4—3161no—
222C22⋊C4—6324no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—42no—
228C23—44no—
229C23—42no—
230C23—42no—
231C23—44no—
232C23—41no—
233C23—41no—
234C23—44no—
235C23—81no—
236C23—84no—
237C23—82no—
238C23#1457913011yescharacteristic
239C23C2×#1192961211yescharacteristic
240C23#1457918711yescharacteristic
241Q8—6322no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—3162no—
254C2×C4—6322no—
255C2×C4—6321no—
256C2×C4—6322no—
257C8—6322no—
258C22—22no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—44no—
269C22—42no—
270C22—81no—
271C22—84no—
272C22—82no—
273C22#1909871411yescharacteristic
274C22#1909874111yescharacteristic
275C22#1909875011yescharacteristic
276C4—42no—
277C4—82no—
278C4—81no—
279C4—82no—
280C4—6322no—
281C4—12641no—
282C2—21no—
283C2—21no—
284C2—41no—
285C2—44no—
286C2—41no—
287C2#2376235211yescharacteristic
288C2#2376539012yes—
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2483C2
socleC22×C79B(4,𝔽2)
prime radicalC25×C79D4
hypercenter#2483D79
hypocenterC79#12203
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2483

Complements

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

NH ≅ G/Ntype
2. #23762390C2semidirect
3. #23761400C2semidirect
150. C79#12203semidirect

Maximal subgroups

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

subgroupindex
#237645642
#237623902
#237614002
#237613282
C79×#24832
#1220379