Finite Group Databasebeta

← #34184052

Subgroups — #34184052

Order 20224.

648 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 127 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#23771731C211yesmaximal, characteristic
2C2×#19098431C211yesmaximal, characteristic, complemented
3C2×#19098298C212yesmaximal
4#23761280C212yesmaximal, complemented
5C79×#1817C211yesmaximal, characteristic, complemented
6C2×#14579144—21no—
7C2×#14579135C2211yescharacteristic
8#19098465C412yescomplemented
9#19098461C412yescomplemented
10#19098454—22no—
11#19098442—21no—
12#19098431—24no—
13#19098423C2211yescharacteristic
14C2×C79⋊#95—21no—
15C2×C79⋊#95—21no—
16C2×C79⋊#95C2212yes—
17#19098298—24no—
18C22×C42×C79C2211yescharacteristic
19C2×C22⋊C8×C79C2212yes—
20#336×C79—22no—
21C22×D316C2×C411yescharacteristic
22C22×C22⋊D79—21no—
23C23×Dic79—22no—
24#14579144—44no—
25#14579135—22no—
26#14579135—22no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C412yes—
29C2×Dic79⋊C4—22no—
30C2×C22⋊Dic79—21no—
31C2×(C2×D79)⋊C4—22no—
32C2×(C2×D79)⋊C4—22no—
33C2×(C2×D79)⋊C4—21no—
34C79⋊#95—24no—
35C79⋊#95—24no—
36C79⋊#95—22no—
37C79⋊#95—22no—
38C79⋊#95—21no—
39C79⋊#95D414yes—
40C23×C4×C79D411yescharacteristic
41C23×C4×C79D411yescharacteristic
42C23×C4×C79C2311yescharacteristic
43C22×C8×C79—22no—
44C2×C42×C79—22no—
45C2×C42×C79—22no—
46C2×C42×C79C2×C412yes—
47C2×C42×C79D412yes—
48C22⋊C8×C79—22no—
49C22⋊C8×C79—22no—
50C23×D79—21no—
51C2×D316—22no—
52C2×D316—42no—
53C2×C22⋊D79—41no—
54C2×C22⋊D79—41no—
55C2×C22⋊D79—42no—
56C2×C22⋊D79—42no—
57C22×Dic79—22no—
58C22×Dic79—21no—
59C22×Dic79—24no—
60C22×Dic79—44no—
61C22×Dic79—44no—
62C22×Dic79—42no—
63C22×Dic79—42no—
64C4⋊Dic79—22no—
65C4⋊Dic79—42no—
66Dic79⋊C4—44no—
67C22⋊Dic79—42no—
68(C2×D79)⋊C4—44no—
69(C2×D79)⋊C4—44no—
70(C2×D79)⋊C4—42no—
71C24×C79C2×D411yescharacteristic
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—22no—
77C22×C4×C79—22no—
78C22×C4×C79—42no—
79C22×C4×C79—42no—
80C22×C4×C79C22⋊C412yes—
81C22×C4×C79C22⋊C412yes—
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×D412yes—
85C22×C4×C79#4112yes—
86C2×C8×C79—44no—
87C2×C8×C79—42no—
88C42×C79—24no—
89C42×C79—24no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79C2×C22⋊C411yescharacteristic
93C23×C79#12011yescharacteristic
94C23×C79#12311yescharacteristic
95C23×C79#13512yes—
96C23×C79AGL(1,ℤ/8)12yes—
97C22×D79—41no—
98C22×D79—42no—
99C22×D79—41no—
100C22×D79—41no—
101C22×D79—42no—
102C2×C4×C79—22no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—22no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79C4×D412yes—
117C2×C4×C79#12112yes—
118C22⋊D79—82no—
119C22⋊D79—82no—
120C2×Dic79—44no—
121C2×Dic79—44no—
122C2×Dic79—41no—
123C2×Dic79—42no—
124C2×Dic79—41no—
125C2×Dic79—42no—
126C2×Dic79—41no—
127C2×Dic79—42no—
128C2×Dic79—44no—
129C2×Dic79—84no—
130C2×Dic79—84no—
131D316—82no—
132C8×C79—82no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—24no—
138C22×C79—24no—
139C22×C79#38611yescharacteristic
140C22×C79#41012yes—
141C22×C79#41611yescharacteristic
142C22×C79#41811yescharacteristic
143C22×C79#42112yes—
144C22×C79#44912yes—
145C22×C79#48212yes—
146C2×D79—81no—
147C2×D79—81no—
148C2×D79—81no—
149C2×D79—82no—
150C2×D79—82no—
151C4×C79—22no—
152C4×C79—22no—
153C4×C79—42no—
154C4×C79—42no—
155Dic79—84no—
156Dic79—81no—
157Dic79—82no—
158#9157—791nomaximal, Sylow
159C2×C79—22no—
160C2×C79—22no—
161C2×C79#151412yes—
162C2×C79#152112yes—
163C2×C79#187511yescharacteristic
164C2×C79#189711yescharacteristic
165C2×C79#189911yescharacteristic
166D79—161no—
167#2854—791no—
168C2×#383—791no—
169#1817D7911yescharacteristic, complemented
170C2×#327—792no—
171#1294—792no—
172C79#915711yescharacteristic, complemented, Sylow
173C2×#123—1581no—
174C2×#121—791no—
175C22×C42C2×D7911yescharacteristic
176C2×C22⋊C8C2×D7912yes—
177#390—792no—
178#389—792no—
179#388—1582no—
180#386—1581no—
181#383—1584no—
182#380—791no—
183C2×#95—792no—
184C2×#95—1581no—
185C2×#95—1581no—
186#336—22no—
187#327—1584no—
188C22×D4—791no—
189C22×D4—1581no—
190C23×C4—1582no—
191C23×C4D31611yescharacteristic
192C23×C4D31611yescharacteristic
193C23×C4C22×D7911yescharacteristic
194C22×C8—22no—
195#123—3164no—
196#121—1582no—
197#121—1582no—
198C2×C4⋊C4—792no—
199C2×C4⋊C4—791no—
200C2×C4⋊C4—1582no—
201C2×C22⋊C4—1581no—
202C2×C22⋊C4—1582no—
203C2×C22⋊C4—1582no—
204C2×C22⋊C4—1581no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D7912yes—
208C2×C42C22⋊D7912yes—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—794no—
212#95—1584no—
213#95—1584no—
214#95—1582no—
215#95—1582no—
216#95—1581no—
217C24—1581no—
218C24C2×D31611yescharacteristic
219C2×D4—3162no—
220C2×D4—3161no—
221C2×D4—3161no—
222C2×D4—3162no—
223C2×D4—3162no—
224C2×D4—1582no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—3164no—
234C22×C4—3164no—
235C22×C4—3162no—
236C22×C4—3162no—
237C22×C4—1582no—
238C22×C4—1581no—
239C22×C4—1584no—
240C22×C4(C2×D79)⋊C412yes—
241C22×C4(C2×D79)⋊C412yes—
242C22×C4C2×C4×D7911yescharacteristic
243C22×C4D4×D7912yes—
244C22×C4C2×C22⋊D7911yescharacteristic
245C22×C4#1192962312yes—
246C2×C8—44no—
247C2×C8—42no—
248C4⋊C4—3162no—
249C4⋊C4—3164no—
250C4⋊C4—1582no—
251C22⋊C4—3164no—
252C22⋊C4—3164no—
253C22⋊C4—3162no—
254C22⋊C4—3162no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—3161no—
260C23—3162no—
261C23—3161no—
262C23—3161no—
263C23—3162no—
264C23C2×(C2×D79)⋊C411yescharacteristic
265C23#1457913111yescharacteristic
266C23#1457914511yescharacteristic
267C23#1457918312yes—
268C23#1457918912yes—
269D4—6322no—
270D4—6322no—
271D4—6322no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—3164no—
287C2×C4—3164no—
288C2×C4—3161no—
289C2×C4—3164no—
290C2×C4—3162no—
291C2×C4—3161no—
292C2×C4—3162no—
293C2×C4—3161no—
294C2×C4—3162no—
295C2×C4—6324no—
296C2×C4—6324no—
297C2×C4(C22⋊D79)⋊C412yes—
298C2×C4#1457913312yes—
299C8—82no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—6321no—
307C22—6322no—
308C22—6322no—
309C22—6321no—
310C22—6321no—
311C22#1909844011yescharacteristic
312C22#1909853612yes—
313C22#1909856011yescharacteristic
314C22#1909856911yescharacteristic
315C22#1909857912yes—
316C22#1909871012yes—
317C22#1909888812yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—6324no—
323C4—6321no—
324C4—6322no—
325C2—22no—
326C2—22no—
327C2—12641no—
328C2#2376194812yes—
329C2#2376197812yes—
330C2#2376372011yescharacteristic
331C2#2376384611yescharacteristic
332C2#2376385611yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1817C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C22
cosocleC23×C4×C79C23
hypercenter#1817D79
hypocenterC79#9157
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#19098431C2semidirect
4. #23761280C2semidirect
5. C79×#1817C2semidirect
8. #19098465C4semidirect
9. #19098461C4semidirect
169. #1817D79semidirect
172. C79#9157semidirect

Maximal subgroups

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

subgroupindex
#237717312
C2×#190984312
C2×#190982982
#237612802
C79×#18172
#915779