Finite Group Databasebeta

← #34198738

Subgroups — #34198738

Order 20224.

444 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 83 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#23768977C211yesmaximal, characteristic
2#23761734C211yesmaximal, characteristic
3#23761420C212yesmaximal
4#23761399C212yesmaximal
5C79×#1740C211yesmaximal, characteristic
6C2×C4×Dic158C411yescharacteristic, complemented
7#19098620—21no—
8#19098493C411yescharacteristic, complemented
9#19098488C2211yescharacteristic
10#19098483C411yescharacteristic, complemented
11#19098475C2211yescharacteristic
12#19098451—21no—
13#19098447—21no—
14#19098444—21no—
15#19098436—21no—
16#19098431C411yescharacteristic, complemented
17#19098289—21no—
18#19098287C2212yes—
19C2×#117×C79—21no—
20C2×#117×C79C2211yescharacteristic
21#342×C79—21no—
22#342×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C4×Dic158—41no—
25C4×Dic158—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—21no—
32C2×C4×Dic79—21no—
33C4⋊(C79⋊C8)—41no—
34C22⋊(C79⋊C8)—22no—
35#14579072—41no—
36C79⋊#95—21no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45#117×C79—24no—
46#117×C79—41no—
47#117×C79—41no—
48#117×C79—42no—
49C2×C4⋊C4×C79—41no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C22⋊C4×C79—22no—
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79C2311yescharacteristic
55C2×C42×C79—41no—
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79D411yescharacteristic
58C2×Dic158—41no—
59C2×Dic158—42no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C2×C79⋊C8—42no—
65C4⋊Dic79—41no—
66Dic79⋊C4—42no—
67Dic79⋊C4—41no—
68Dic79⋊C4—41no—
69Dic79⋊C4—42no—
70C22⋊Dic79—42no—
71C22⋊Dic79—41no—
72C22⋊Dic79—41no—
73C4×Dic79—42no—
74C24×C79C2×D411yescharacteristic
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—22no—
80C22×C4×C79—42no—
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22×C411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C4⋊C4×C79—41no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—44no—
90C22⋊C4×C79—24no—
91C22⋊C4×C79—41no—
92C22⋊C4×C79—41no—
93C22⋊C4×C79—42no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79—44no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79C2×C22⋊C411yescharacteristic
104C23×C79AGL(1,ℤ/8)11yescharacteristic
105C23×C79#13711yescharacteristic
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—44no—
123C2×C4×C79—82no—
124C2×C4×C79—82no—
125C2×C4×C79#9911yescharacteristic
126C2×C4×C79#9911yescharacteristic
127C2×C4×C79#10412yes—
128C2×Dic79—22no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140Dic158—42no—
141Dic158—42no—
142C79⋊C8—82no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—21no—
146C22×C79—21no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—41no—
150C22×C79—41no—
151C22×C79—81no—
152C22×C79C2×#9911yescharacteristic
153C22×C79#41711yescharacteristic
154C22×C79C2×#10411yescharacteristic
155C4×C79—21no—
156C4×C79—21no—
157C4×C79—22no—
158C4×C79—41no—
159C4×C79—41no—
160C4×C79—41no—
161C4×C79—41no—
162C4×C79—84no—
163Dic79—42no—
164Dic79—81no—
165Dic79—81no—
166Dic79—81no—
167#11756—791nomaximal, Sylow
168C2×C79—21no—
169C2×C79—21no—
170C2×C79—41no—
171C2×C79—41no—
172C2×C79#150411yescharacteristic
173C2×C79#212011yescharacteristic
174C2×C79#212911yescharacteristic
175#2599—791no—
176#1740D7911yescharacteristic
177#1464—791no—
178#1351—792no—
179#1344—792no—
180C79#1175611yescharacteristic, complemented, Sylow
181C2×C4×Q8—791no—
182C2×#117—21no—
183C2×#117C2×D7911yescharacteristic
184#432—1581no—
185#398—791no—
186#397—791no—
187#395—791no—
188#393—791no—
189#388—1581no—
190#387—1581no—
191#387—1581no—
192#385—1581no—
193#383—791no—
194#342—21no—
195#342C2×D7912yes—
196#324—1581no—
197#323—792no—
198C22×Q8—791no—
199C23×C4C22⋊D7911yescharacteristic
200C4×Q8—3161no—
201C4×Q8—3161no—
202#117—24no—
203#117—41no—
204#117—41no—
205#117—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—791no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4—1581no—
211C2×C4⋊C4C4×D7911yescharacteristic
212C2×C4⋊C4D31611yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—1581no—
215C2×C22⋊C4—1581no—
216C2×C22⋊C4C22⋊D7911yescharacteristic
217C2×C22⋊C4C22×D7911yescharacteristic
218C2×C42—41no—
219C2×C42—1581no—
220C2×C42C4×D7911yescharacteristic
221C2×C42D31611yescharacteristic
222C4⋊C8—3161no—
223C22⋊C8—1582no—
224#97—3161no—
225#95—791no—
226#95—791no—
227#95—1581no—
228#95—1581no—
229#95—1581no—
230#95—1581no—
231#95—1581no—
232#95—1581no—
233C24C2×C22⋊D7911yescharacteristic
234C2×Q8—3161no—
235C2×Q8—3162no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—1581no—
243C22×C4—1581no—
244C22×C4—1581no—
245C22×C4—1581no—
246C22×C4(C2×D79)⋊C411yescharacteristic
247C22×C4(C2×D79)⋊C411yescharacteristic
248C22×C4(C2×D79)⋊C411yescharacteristic
249C22×C4(C2×D79)⋊C411yescharacteristic
250C22×C4C2×C4×D7911yescharacteristic
251C22×C4C2×D31611yescharacteristic
252C2×C8—3162no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—44no—
256C4⋊C4—3161no—
257C4⋊C4—3162no—
258C4⋊C4—3161no—
259C4⋊C4—3161no—
260C4⋊C4—3162no—
261C22⋊C4—24no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—42no—
265C22⋊C4—3161no—
266C22⋊C4—3161no—
267C22⋊C4—3162no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—44no—
273C42—3162no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23C2×(C2×D79)⋊C411yescharacteristic
279C23#1457918711yescharacteristic
280C23#1457919211yescharacteristic
281Q8—3162no—
282Q8—3162no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—3161no—
303C2×C4—3161no—
304C2×C4—3161no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—3161no—
310C2×C4—3161no—
311C2×C4—3161no—
312C2×C4—3161no—
313C2×C4—1582no—
314C2×C4#1457907511yescharacteristic
315C2×C4#1457907511yescharacteristic
316C2×C4#1457908912yes—
317C8—6322no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22C2×#1457907511yescharacteristic
328C22#1909856211yescharacteristic
329C22C2×#1457908911yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—84no—
338C4—3162no—
339C4—6321no—
340C4—6321no—
341C4—6321no—
342C2—21no—
343C2—21no—
344C2—41no—
345C2—41no—
346C2#2376190211yescharacteristic
347C2#2376503911yescharacteristic
348C2#2376507411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1740C2
socleC22×C79C2×#99
prime radicalC24×C79C2×D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1740D79
hypocenterC79#11756
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic158C4semidirect
8. #19098493C4semidirect
10. #19098483C4semidirect
16. #19098431C4semidirect
180. C79#11756semidirect

Maximal subgroups

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

subgroupindex
#237689772
#237617342
#237614202
#237613992
C79×#17402
#1175679