Finite Group Databasebeta

← #34186061

Subgroups — #34186061

Order 20224.

576 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 115 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#23772429C211yesmaximal, characteristic
2#23767708C211yesmaximal, characteristic
3#23763484C212yesmaximal, complemented
4#23761267C212yesmaximal, complemented
5C79×#1799C211yesmaximal, characteristic
6C2×#14579142—21no—
7C22×C4⋊Dic79C2211yescharacteristic
8#19098635—24no—
9#19098620—21no—
10#19098527—21no—
11C2×C22⋊(C79⋊C8)C2212yes—
12#19098493C412yescomplemented
13#19098488C412yescomplemented
14#19098461—21no—
15#19098424—21no—
16#19098423—21no—
17#19098423C2211yescharacteristic
18#19098325—24no—
19C2×C4×D4×C79C2211yescharacteristic
20C2×#99×C79—21no—
21#380×C79C2212yes—
22#342×C79—21no—
23C23×Dic79—21no—
24C2×#11929608—22no—
25C22×C79⋊C8D414yes—
26#14579142—44no—
27C2×C4⋊Dic79—22no—
28C2×C4⋊Dic79—22no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79C2×C412yes—
32C2×Dic79⋊C4—22no—
33C2×C22⋊Dic79—22no—
34C2×C22⋊Dic79—21no—
35C4⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—22no—
38C22⋊(C79⋊C8)—22no—
39#14579072—22no—
40C79⋊#95—22no—
41C79⋊#95—22no—
42C79⋊#95C2×C412yes—
43C22×D4×C79D411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C23×C4×C79C2311yescharacteristic
46C4×D4×C79—22no—
47C4×D4×C79—22no—
48C2×C4⋊C4×C79—22no—
49C2×C4⋊C4×C79C2×C411yescharacteristic
50C2×C22⋊C4×C79—22no—
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54#99×C79—42no—
55#95×C79—22no—
56C22×Dic79—22no—
57C22×Dic79—22no—
58C22×Dic79—42no—
59C22×Dic79—42no—
60C22×Dic79—41no—
61C22×Dic79—41no—
62#11929608—82no—
63C2×C79⋊C8—24no—
64C2×C79⋊C8—44no—
65C4⋊Dic79—24no—
66C4⋊Dic79—42no—
67C4⋊Dic79—41no—
68C4⋊Dic79—41no—
69Dic79⋊C4—44no—
70C22⋊Dic79—44no—
71C22⋊Dic79—42no—
72C24×C79C22⋊C411yescharacteristic
73C24×C79C2×D411yescharacteristic
74C2×D4×C79—22no—
75C2×D4×C79—21no—
76C2×D4×C79—22no—
77C2×D4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—42no—
81C22×C4×C79—42no—
82C22×C4×C79—42no—
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22⋊C411yescharacteristic
86C22×C4×C79C22×C411yescharacteristic
87C22×C4×C79C2×D412yes—
88C22×C4×C79C2×D411yescharacteristic
89C22×C4×C79#4112yes—
90C4⋊C4×C79—22no—
91C4⋊C4×C79—44no—
92C22⋊C4×C79—22no—
93C22⋊C4×C79—22no—
94C22⋊C4×C79—42no—
95C22⋊C4×C79—42no—
96C42×C79—21no—
97C42×C79—21no—
98C23×C79—21no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79—41no—
104C23×C79—41no—
105C23×C79C2×C22⋊C411yescharacteristic
106C23×C79#12112yes—
107C23×C79AGL(1,ℤ/8)12yes—
108C23×C79AGL(1,ℤ/8)11yescharacteristic
109C23×C79#13711yescharacteristic
110D4×C79—42no—
111D4×C79—42no—
112C2×C4×C79—22no—
113C2×C4×C79—24no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—41no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—81no—
125C2×C4×C79—82no—
126C2×C4×C79—82no—
127C2×C4×C79—82no—
128C2×C4×C79C4×D412yes—
129C2×C4×C79#12011yescharacteristic
130C2×C4×C79#12311yescharacteristic
131C2×Dic79—42no—
132C2×Dic79—42no—
133C2×Dic79—42no—
134C2×Dic79—42no—
135C2×Dic79—42no—
136C2×Dic79—42no—
137C2×Dic79—82no—
138C2×Dic79—82no—
139C79⋊C8—44no—
140C22×C79—21no—
141C22×C79—22no—
142C22×C79—21no—
143C22×C79—22no—
144C22×C79—42no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—42no—
148C22×C79—41no—
149C22×C79—41no—
150C22×C79—41no—
151C22×C79—81no—
152C22×C79#38611yescharacteristic
153C22×C79#41711yescharacteristic
154C22×C79#41811yescharacteristic
155C22×C79#46812yes—
156C22×C79#46912yes—
157C4×C79—21no—
158C4×C79—22no—
159C4×C79—41no—
160C4×C79—41no—
161C4×C79—41no—
162C4×C79—162no—
163C4×C79#45812yes—
164Dic79—82no—
165Dic79—82no—
166#9474—791nomaximal, Sylow
167C2×C79—21no—
168C2×C79—22no—
169C2×C79—21no—
170C2×C79—22no—
171C2×C79—81no—
172C2×C79#189011yescharacteristic
173C2×C79#189511yescharacteristic
174C2×C79#190911yescharacteristic
175#2973—791no—
176#2486—791no—
177#1823—792no—
178#1799D7911yescharacteristic
179#1289—792no—
180C79#947411yescharacteristic, complemented, Sylow
181C2×#123—1581no—
182C2×C4×D4C2×D7911yescharacteristic
183C22×C4⋊C4—791no—
184#435—1584no—
185#432—1581no—
186C2×#99—21no—
187#407—1581no—
188C2×C22⋊C8—792no—
189#398—792no—
190#397—792no—
191#389—1581no—
192#381—1581no—
193#380—791no—
194#380—1581no—
195#380C2×D7912yes—
196#342—21no—
197#340—1584no—
198C22×D4C22⋊D7911yescharacteristic
199C23×C4—1581no—
200C23×C4D31611yescharacteristic
201C23×C4C22×D7911yescharacteristic
202C2×M16—1582no—
203C22×C8—794no—
204#123—3164no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—792no—
209C2×C4⋊C4—1582no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4—1581no—
212C2×C4⋊C4—1582no—
213C2×C4⋊C4—1582no—
214C2×C4⋊C4C4×D7911yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—1582no—
217C2×C22⋊C4—1581no—
218C2×C22⋊C4C4×D7911yescharacteristic
219C2×C22⋊C4C22⋊D7911yescharacteristic
220C2×C42D31611yescharacteristic
221C4⋊C8—1582no—
222#99—42no—
223C22⋊C8—1582no—
224C22⋊C8—1582no—
225C22⋊C8—1582no—
226#97—1582no—
227#95—22no—
228#95—792no—
229#95—1582no—
230#95—1582no—
231C24(C2×D79)⋊C411yescharacteristic
232C24C2×C22⋊D7911yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—3162no—
243C22×C4—3162no—
244C22×C4—3161no—
245C22×C4—3161no—
246C22×C4—1582no—
247C22×C4—1582no—
248C22×C4(C2×D79)⋊C411yescharacteristic
249C22×C4(C2×D79)⋊C411yescharacteristic
250C22×C4(C2×D79)⋊C411yescharacteristic
251C22×C4C2×C4×D7911yescharacteristic
252C22×C4D4×D7912yes—
253C22×C4C2×D31611yescharacteristic
254C22×C4#1192962312yes—
255M16—6322no—
256C2×C8—3164no—
257C2×C8—1584no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—3164no—
261C4⋊C4—3162no—
262C4⋊C4—3161no—
263C4⋊C4—3161no—
264C4⋊C4—1584no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—3164no—
270C22⋊C4—3162no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D79)⋊C411yescharacteristic
281C23#1457913312yes—
282C23#1457918612yes—
283C23#1457918711yescharacteristic
284C23#1457919211yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—3162no—
304C2×C4—3162no—
305C2×C4—3162no—
306C2×C4—3162no—
307C2×C4—3162no—
308C2×C4—3162no—
309C2×C4—6322no—
310C2×C4—6322no—
311C2×C4(C22⋊D79)⋊C412yes—
312C2×C4#1457913111yescharacteristic
313C2×C4#1457914511yescharacteristic
314C8—3164no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1909844011yescharacteristic
328C22#1909856211yescharacteristic
329C22#1909856711yescharacteristic
330C22#1909881112yes—
331C22#1909881412yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—162no—
338C4—6322no—
339C4—6322no—
340C4#1909875412yes—
341C2—22no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2376381111yescharacteristic
347C2#2376383611yescharacteristic
348C2#2376392011yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#1799C2
socleC22×C79#386
prime radicalC22×D4×C79D4
cosocleC23×C4×C79C23
hypercenter#1799D79
hypocenterC79#9474
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #23763484C2semidirect
4. #23761267C2semidirect
12. #19098493C4semidirect
13. #19098488C4semidirect
180. C79#9474semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237677082
#237634842
#237612672
C79×#17992
#947479