Finite Group Databasebeta

← #34197675

Subgroups — #34197675

Order 20224.

434 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 87 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#23766024C211yesmaximal, characteristic
2#23761730C211yesmaximal, characteristic
3#23761345C212yesmaximal, complemented
4#23761343C212yesmaximal, complemented
5C79×#1463C211yesmaximal, characteristic
6C2×#14579120C411yescharacteristic
7C2×#14579120C411yescharacteristic
8#19098527—21no—
9#19098459—21no—
10#19098459—21no—
11#19098436—21no—
12#19098436—21no—
13#19098432C411yescharacteristic
14#19098432C2211yescharacteristic
15#19098428C411yescharacteristic
16#19098428C2211yescharacteristic
17#19098289—21no—
18#19098289C2212yes—
19C22×C4⋊C4×C79C2211yescharacteristic
20#407×C79—21no—
21#324×C79—21no—
22#324×C79C2212yes—
23C2×#11929608—41no—
24#14579120—22no—
25#14579120—22no—
26#14579120—41no—
27#14579120—41no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×C4⋊Dic79C2×C411yescharacteristic
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×Dic79⋊C4—21no—
34C2×Dic79⋊C4—21no—
35C2×C22⋊Dic79—21no—
36C2×C22⋊Dic79—21no—
37C2×C4×Dic79—21no—
38C2×C4×Dic79C2×C411yescharacteristic
39C2×C4×Dic79C2×C411yescharacteristic
40C22⋊(C79⋊C8)—22no—
41C22⋊(C79⋊C8)—41no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C23×C4×C79C2×C411yescharacteristic
47C23×C4×C79D411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C2×M16×C79—41no—
50C2×C4⋊C4×C79—41no—
51C2×C4⋊C4×C79—41no—
52C2×C4⋊C4×C79C2×C411yescharacteristic
53C2×C4⋊C4×C79D411yescharacteristic
54C2×C4⋊C4×C79D411yescharacteristic
55C2×C4⋊C4×C79C2311yescharacteristic
56C22⋊C8×C79—22no—
57C22⋊C8×C79—41no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62#11929608—82no—
63C2×C79⋊C8—42no—
64C4⋊Dic79—24no—
65C4⋊Dic79—41no—
66C4⋊Dic79—41no—
67Dic79⋊C4—81no—
68Dic79⋊C4—81no—
69Dic79⋊C4—81no—
70Dic79⋊C4—81no—
71C22⋊Dic79—24no—
72C22⋊Dic79—41no—
73C22⋊Dic79—41no—
74C4×Dic79—21no—
75C4×Dic79—21no—
76C4×Dic79—42no—
77C4×Dic79M1612yescomplemented
78C4×Dic79M1612yescomplemented
79C24×C79C22⋊C411yescharacteristic
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—22no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—81no—
90C22×C4×C79—81no—
91C22×C4×C79C22⋊C411yescharacteristic
92C22×C4×C79C22⋊C411yescharacteristic
93C22×C4×C79C22⋊C411yescharacteristic
94C22×C4×C79C22×C411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97M16×C79—82no—
98C2×C8×C79—42no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C23×C79—21no—
104C23×C79—21no—
105C23×C79—41no—
106C23×C79—41no—
107C23×C79#9911yescharacteristic
108C23×C79#9911yescharacteristic
109C23×C79C2×C22⋊C411yescharacteristic
110C2×C4×C79—22no—
111C2×C4×C79—22no—
112C2×C4×C79—22no—
113C2×C4×C79—22no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—21no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—42no—
123C2×C4×C79—42no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—81no—
129C2×C4×C79—81no—
130C2×C4×C79#10112yes—
131C2×C4×C79C2×M1611yescharacteristic
132C2×C4×C79C2×M1611yescharacteristic
133C2×Dic79—22no—
134C2×Dic79—22no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C2×Dic79—41no—
141C2×Dic79—42no—
142C2×Dic79—41no—
143C2×Dic79—41no—
144C8×C79—82no—
145C79⋊C8—82no—
146C22×C79—21no—
147C22×C79—21no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—21no—
152C22×C79—21no—
153C22×C79—21no—
154C22×C79—42no—
155C22×C79—42no—
156C22×C79#40711yescharacteristic
157C22×C79C2×#9911yescharacteristic
158C22×C79C2×#10111yescharacteristic
159C4×C79—22no—
160C4×C79—21no—
161C4×C79—21no—
162C4×C79—41no—
163C4×C79—41no—
164C4×C79—41no—
165C4×C79—41no—
166Dic79—42no—
167Dic79—42no—
168Dic79—81no—
169Dic79—81no—
170#11514—791nomaximal, Sylow
171C2×C79—21no—
172C2×C79—21no—
173C2×C79—41no—
174C2×C79—41no—
175C2×C79#146311yescharacteristic
176C2×C79#212711yescharacteristic
177C2×C79#213311yescharacteristic
178#2317—791no—
179#1463—791no—
180#1463D7911yescharacteristic
181#1325—792no—
182#1324—792no—
183C79#1151411yescharacteristic, complemented, Sylow
184C2×#117—791no—
185C2×#117—791no—
186C22×C4⋊C4C2×D7911yescharacteristic
187#407—21no—
188#407—1581no—
189#389—1581no—
190#389—1581no—
191#385—1581no—
192#385—1581no—
193#384—791no—
194#384—791no—
195#382—791no—
196#382—791no—
197#324—21no—
198#324—792no—
199#324—1581no—
200#324C2×D7912yes—
201C23×C4C4×D7911yescharacteristic
202C23×C4D31611yescharacteristic
203C23×C4C22⋊D7911yescharacteristic
204C2×M16—41no—
205C2×M16—3161no—
206#117—3161no—
207#117—3161no—
208#117—1582no—
209#117—1582no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—791no—
213C2×C4⋊C4—791no—
214C2×C4⋊C4—1581no—
215C2×C4⋊C4—1581no—
216C2×C4⋊C4—1581no—
217C2×C4⋊C4—1581no—
218C2×C4⋊C4—1581no—
219C2×C4⋊C4C4×D7911yescharacteristic
220C2×C4⋊C4D31611yescharacteristic
221C2×C4⋊C4C22⋊D7911yescharacteristic
222C2×C4⋊C4C22×D7911yescharacteristic
223C2×C22⋊C4—1581no—
224C2×C22⋊C4—1581no—
225C2×C42—791no—
226C2×C42—791no—
227C2×C42—1581no—
228C22⋊C8—22no—
229C22⋊C8—41no—
230C22⋊C8—3161no—
231C22⋊C8—1582no—
232#95—1581no—
233#95—1581no—
234#95—1581no—
235#95—1581no—
236C24(C2×D79)⋊C411yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—1581no—
249C22×C4—1581no—
250C22×C4—1581no—
251C22×C4—1581no—
252C22×C4(C2×D79)⋊C411yescharacteristic
253C22×C4(C2×D79)⋊C411yescharacteristic
254C22×C4(C2×D79)⋊C411yescharacteristic
255C22×C4C2×C4×D7911yescharacteristic
256C22×C4C2×C22⋊D7911yescharacteristic
257C22×C4C2×D31611yescharacteristic
258M16—82no—
259M16—6322no—
260C2×C8—42no—
261C2×C8—3162no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—3161no—
267C4⋊C4—3161no—
268C4⋊C4—6321no—
269C4⋊C4—6321no—
270C4⋊C4—6321no—
271C4⋊C4—6321no—
272C4⋊C4—1584no—
273C22⋊C4—3161no—
274C22⋊C4—3161no—
275C22⋊C4—1584no—
276C42—3162no—
277C42—792no—
278C42—792no—
279C42—1581no—
280C42—1581no—
281C23—21no—
282C23—21no—
283C23—41no—
284C23—41no—
285C23#1457907511yescharacteristic
286C23#1457907511yescharacteristic
287C23C2×(C2×D79)⋊C411yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—3161no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—3161no—
313C2×C4—3161no—
314C2×C4—3162no—
315C2×C4—3161no—
316C2×C4—3161no—
317C2×C4—1582no—
318C2×C4—1582no—
319C2×C4#1457907912yes—
320C2×C4M16×D7911yescharacteristic
321C2×C4M16×D7911yescharacteristic
322C8—82no—
323C8—6322no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—42no—
333C22—42no—
334C22#1909852611yescharacteristic
335C22C2×#1457907511yescharacteristic
336C22C2×#1457907911yescharacteristic
337C4—22no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—3162no—
345C4—3162no—
346C4—6321no—
347C4—6321no—
348C2—21no—
349C2—21no—
350C2—41no—
351C2—41no—
352C2#2376172911yescharacteristic
353C2#2376506711yescharacteristic
354C2#2376508711yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1463C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1463D79
hypocenterC79#11514
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #23761345C2semidirect
4. #23761343C2semidirect
77. C4×Dic79M16semidirect
78. C4×Dic79M16semidirect
183. C79#11514semidirect

Maximal subgroups

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

subgroupindex
#237660242
#237617302
#237613452
#237613432
C79×#14632
#1151479