Finite Group Databasebeta

← #34201010

Subgroups — #34201010

Order 20224.

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

normal subgroups only

This group has 500 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#23764567C211yesmaximal, characteristic, complemented
2#23762322C212yesmaximal, complemented
3#23761397C212yesmaximal, complemented
4#23761330C211yesmaximal, characteristic, complemented
5C79×#2815C211yesmaximal, characteristic
6#19099189C2211yescharacteristic
7#19098714—22no—
8#19098426—21no—
9#19098330C2211yescharacteristic
10#19098329—21no—
11#19098299—22no—
12#19098287C2212yes—
13C2×#121×C79—21no—
14#397×C79—22no—
15#392×C79—22no—
16#392×C79—21no—
17#392×C79C2212yes—
18#381×C79C2211yescharacteristic
19C2×#11929612—42no—
20#14579140—22no—
21#14579138—42no—
22C2×C22⋊Dic79—21no—
23C2×C22⋊Dic79—21no—
24#14579082—42no—
25C22⋊(C79⋊C8)—22no—
26C79⋊#95—42no—
27C22×D4×C79—21no—
28C22×D4×C79—22no—
29C22×D4×C79—21no—
30C22×D4×C79D412yes—
31C23×C4×C79D411yescharacteristic
32#121×C79—42no—
33#121×C79—42no—
34C2×C4⋊C4×C79—21no—
35C2×C22⋊C4×C79—22no—
36C2×C22⋊C4×C79—22no—
37C2×C22⋊C4×C79—22no—
38C2×C22⋊C4×C79D411yescharacteristic
39C2×C22⋊C4×C79C2311yescharacteristic
40#95×C79—21no—
41#95×C79—22no—
42#95×C79D412yes—
43C2×Dic158—22no—
44C22×Dic79—22no—
45C22×Dic79—41no—
46#11929612—84no—
47C2×C79⋊C8—42no—
48C4⋊Dic79—22no—
49Dic79⋊C4—42no—
50C22⋊Dic79—41no—
51C22⋊Dic79—41no—
52C22⋊Dic79—41no—
53C22⋊Dic79—41no—
54C24×C79—22no—
55C24×C79C2×D411yescharacteristic
56C2×D4×C79—42no—
57C2×D4×C79—42no—
58C2×D4×C79—42no—
59C2×D4×C79—42no—
60C2×D4×C79—82no—
61C2×D4×C79—81no—
62C2×D4×C79—81no—
63C2×D4×C79—82no—
64C2×D4×C79—82no—
65C2×D4×C79—82no—
66C2×D4×C79—81no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—22no—
71C22×C4×C79—41no—
72C22×C4×C79—41no—
73C22×C4×C79SD1612yescomplemented
74C22×C4×C79C2×D412yes—
75C4⋊C4×C79—42no—
76C22⋊C4×C79—41no—
77C22⋊C4×C79—44no—
78C22⋊C4×C79—41no—
79C22⋊C4×C79—82no—
80C22⋊C4×C79—82no—
81C23×C79—21no—
82C23×C79—21no—
83C23×C79—42no—
84C23×C79—42no—
85C23×C79—42no—
86C23×C79—42no—
87C23×C79—42no—
88C23×C79—42no—
89C23×C79—42no—
90C23×C79—42no—
91C23×C79#12011yescharacteristic
92C23×C79C2×SD1611yescharacteristic
93C23×C79AGL(1,ℤ/8)11yescharacteristic
94D4×C79—44no—
95D4×C79—82no—
96D4×C79—84no—
97D4×C79—84no—
98D4×C79—84no—
99D4×C79—84no—
100C2×C4×C79—22no—
101C2×C4×C79—22no—
102C2×C4×C79—41no—
103C2×C4×C79—41no—
104C2×C4×C79—42no—
105C2×C4×C79—42no—
106C2×C4×C79—42no—
107C2×C4×C79—42no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—41no—
111C2×C4×C79—81no—
112C2×Dic79—42no—
113C2×Dic79—82no—
114C2×Dic79—81no—
115C2×Dic79—82no—
116Dic158—82no—
117C79⋊C8—82no—
118C22×C79—22no—
119C22×C79—21no—
120C22×C79—21no—
121C22×C79—41no—
122C22×C79—41no—
123C22×C79—42no—
124C22×C79—81no—
125C22×C79—82no—
126C22×C79—82no—
127C22×C79—82no—
128C22×C79—82no—
129C22×C79—82no—
130C22×C79—82no—
131C22×C79—82no—
132C22×C79#45011yescharacteristic
133C22×C79#45511yescharacteristic
134C22×C79B(4,𝔽2)11yescharacteristic
135C4×C79—42no—
136C4×C79—42no—
137C4×C79—82no—
138C4×C79—81no—
139Dic79—82no—
140Dic79—161no—
141#12206—791nomaximal, Sylow
142C2×C79—21no—
143C2×C79—21no—
144C2×C79—41no—
145C2×C79—41no—
146C2×C79—162no—
147C2×C79#160111yescharacteristic
148C2×C79#220012yes—
149#2815D7911yescharacteristic
150#2030—791no—
151#1596—792no—
152#1343—792no—
153#1317—791no—
154C79#1220611yescharacteristic, complemented, Sylow
155#543—791no—
156C2×#121—21no—
157#450—1582no—
158#397—22no—
159#392—22no—
160#392—21no—
161#392C2×D7912yes—
162#381—1581no—
163#381C2×D7911yescharacteristic
164#342—791no—
165#342—1581no—
166#328—1582no—
167#323—792no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—21no—
171C22×D4C22⋊D7912yes—
172C23×C4C22⋊D7911yescharacteristic
173C2×SD16—3162no—
174C22⋊Q8—3162no—
175C22⋊Q8—1582no—
176#121—42no—
177#121—42no—
178C2×C4⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1581no—
183C2×C22⋊C4—1581no—
184C2×C22⋊C4C22⋊D7911yescharacteristic
185C2×C22⋊C4C22×D7911yescharacteristic
186#102—3162no—
187C22⋊C8—1582no—
188#95—21no—
189#95—22no—
190#95—3162no—
191#95C22⋊D7912yes—
192C24—22no—
193C24C2×C22⋊D7911yescharacteristic
194C2×Q8—1582no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—81no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—3161no—
213C22×C4—1582no—
214C22×C4#1192961212yescomplemented
215C22×C4C2×C22⋊D7912yes—
216SD16—6324no—
217C2×C8—3162no—
218C4⋊C4—42no—
219C4⋊C4—3162no—
220C4⋊C4—1582no—
221C22⋊C4—41no—
222C22⋊C4—44no—
223C22⋊C4—41no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C22⋊C4—3161no—
227C22⋊C4—3161no—
228C22⋊C4—3161no—
229C22⋊C4—3161no—
230C23—21no—
231C23—21no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23#1457913011yescharacteristic
241C23C2×#1192961211yescharacteristic
242C23#1457918711yescharacteristic
243Q8—6322no—
244D4—44no—
245D4—82no—
246D4—84no—
247D4—84no—
248D4—84no—
249D4—84no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—3162no—
263C2×C4—6322no—
264C2×C4—6321no—
265C2×C4—6322no—
266C8—6322no—
267C22—22no—
268C22—21no—
269C22—21no—
270C22—41no—
271C22—41no—
272C22—42no—
273C22—81no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22#1909871411yescharacteristic
282C22#1909874111yescharacteristic
283C22#1909875011yescharacteristic
284C4—42no—
285C4—42no—
286C4—82no—
287C4—81no—
288C4—6322no—
289C4—12641no—
290C2—21no—
291C2—21no—
292C2—41no—
293C2—41no—
294C2—162no—
295C2#2376235211yescharacteristic
296C2#2376540712yes—
subgroupG/H
centerC22#19098750
commutator, cosocleC2×C22⋊C4×C79C23
FrattiniC2×C22⋊C4C22×D79
Fitting, prime radicalC79×#2815C2
socleC22×C79B(4,𝔽2)
hypercenter#2815D79
hypocenterC79#12206
LCSG ▷ C2×C22⋊C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2815

Complements

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

NH ≅ G/Ntype
1. #23764567C2semidirect
2. #23762322C2semidirect
3. #23761397C2semidirect
4. #23761330C2semidirect
73. C22×C4×C79SD16semidirect
154. C79#12206semidirect
214. C22×C4#11929612semidirect

Maximal subgroups

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

subgroupindex
#237645672
#237623222
#237613972
#237613302
C79×#28152
#1220679