Finite Group Databasebeta

← #34175299

Subgroups — #34175299

Order 20224.

422 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 75 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#23761730C211yesmaximal, characteristic
2#23761632C211yesmaximal, characteristic
3C79×#1821C211yesmaximal, characteristic
4C22×C4×Dic79C411yescharacteristic
5#19098527—21no—
6C2×C79⋊#95—21no—
7C2×C79⋊#95—21no—
8C2×C79⋊#95C411yescharacteristic
9#19098289—21no—
10#19098289C412yescomplemented
11C22×C4⋊C4×C79C2211yescharacteristic
12#407×C79—21no—
13#343×C79—21no—
14#343×C79C412yescomplemented
15C23×Dic79—21no—
16C23×Dic79—21no—
17C2×#11929608—41no—
18C2×C4×Dic79—41no—
19C2×C4×Dic79—42no—
20C22⋊(C79⋊C8)—22no—
21C22⋊(C79⋊C8)—41no—
22C79⋊#95—24no—
23C79⋊#95—42no—
24C79⋊#95—41no—
25C23×C4×C79C2×C411yescharacteristic
26C23×C4×C79D411yescharacteristic
27C23×C4×C79D411yescharacteristic
28C2×M16×C79—22no—
29C2×C4⋊C4×C79—41no—
30C2×C4⋊C4×C79—41no—
31C2×C4⋊C4×C79C2×C411yescharacteristic
32C2×C4⋊C4×C79C2×C411yescharacteristic
33C2×C4⋊C4×C79D411yescharacteristic
34C2×C4⋊C4×C79Q811yescharacteristic
35C22⋊C8×C79—41no—
36C22⋊C8×C79—41no—
37C22×Dic79—22no—
38C22×Dic79—22no—
39C22×Dic79—22no—
40C22×Dic79—22no—
41C22×Dic79—41no—
42C22×Dic79—41no—
43C22×Dic79—41no—
44C22×Dic79—41no—
45C22×Dic79—41no—
46C22×Dic79—41no—
47C22×Dic79—41no—
48C22×Dic79—41no—
49C22×Dic79—41no—
50C22×Dic79—41no—
51#11929608—81no—
52#11929608—81no—
53C2×C79⋊C8—42no—
54C4×Dic79—42no—
55C4×Dic79—42no—
56C24×C79C22⋊C411yescharacteristic
57C22×C4×C79—21no—
58C22×C4×C79—21no—
59C22×C4×C79—21no—
60C22×C4×C79—21no—
61C22×C4×C79—21no—
62C22×C4×C79—21no—
63C22×C4×C79—22no—
64C22×C4×C79—41no—
65C22×C4×C79—41no—
66C22×C4×C79—41no—
67C22×C4×C79—41no—
68C22×C4×C79—41no—
69C22×C4×C79C4211yescharacteristic
70C22×C4×C79C22⋊C411yescharacteristic
71C22×C4×C79C22⋊C411yescharacteristic
72C22×C4×C79C4⋊C411yescharacteristic
73C22×C4×C79C4⋊C411yescharacteristic
74C22×C4×C79C4⋊C411yescharacteristic
75M16×C79—42no—
76M16×C79—42no—
77C2×C8×C79—42no—
78C4⋊C4×C79—41no—
79C4⋊C4×C79—41no—
80C4⋊C4×C79—41no—
81C4⋊C4×C79—41no—
82C23×C79—21no—
83C23×C79—21no—
84C23×C79—21no—
85C23×C79—21no—
86C23×C79—21no—
87C23×C79—21no—
88C23×C79#9511yescharacteristic
89C23×C79#9911yescharacteristic
90C23×C79#9911yescharacteristic
91C2×C4×C79—21no—
92C2×C4×C79—21no—
93C2×C4×C79—21no—
94C2×C4×C79—21no—
95C2×C4×C79—21no—
96C2×C4×C79—21no—
97C2×C4×C79—21no—
98C2×C4×C79—21no—
99C2×C4×C79—21no—
100C2×C4×C79—42no—
101C2×C4×C79—42no—
102C2×C4×C79—42no—
103C2×C4×C79—41no—
104C2×C4×C79—41no—
105C2×C4×C79—41no—
106C2×C4×C79—41no—
107C2×C4×C79—41no—
108C2×C4×C79—41no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—81no—
112C2×C4×C79#10412yes—
113C2×Dic79—24no—
114C2×Dic79—42no—
115C2×Dic79—42no—
116C2×Dic79—42no—
117C2×Dic79—42no—
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—81no—
124C2×Dic79—81no—
125C2×Dic79—81no—
126C2×Dic79—81no—
127C8×C79—82no—
128C79⋊C8—82no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—21no—
139C22×C79—21no—
140C22×C79—21no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79H(3,ℤ/4)11yescharacteristic
144C22×C79#33911yescharacteristic
145C22×C79#34211yescharacteristic
146C22×C79AΣL(1,𝔽16)11yescharacteristic
147C22×C79#35211yescharacteristic
148C22×C79#35511yescharacteristic
149C22×C79#35611yescharacteristic
150C4×C79—22no—
151C4×C79—41no—
152C4×C79—41no—
153C4×C79—41no—
154C4×C79—41no—
155C4×C79—41no—
156Dic79—44no—
157Dic79—82no—
158#7215—791nomaximal, Sylow
159C2×C79—21no—
160C2×C79—21no—
161C2×C79—21no—
162C2×C79—21no—
163C2×C79—21no—
164C2×C79—21no—
165C2×C79#128411yescharacteristic
166C2×C79#139011yescharacteristic
167C2×C79#139411yescharacteristic
168#1821D7911yescharacteristic
169#1463—791no—
170#1436—791no—
171C79#721511yescharacteristic, complemented, Sylow
172C22×C4⋊C4C2×D7911yescharacteristic
173C22×C42—791no—
174#407—21no—
175#407—1581no—
176C2×#95—791no—
177C2×#95—1581no—
178C2×#95—1581no—
179#343—21no—
180#343Dic7912yescomplemented
181#324—792no—
182#324—1581no—
183C23×C4—1581no—
184C23×C4—1581no—
185C23×C4C4×D7911yescharacteristic
186C23×C4D31611yescharacteristic
187C23×C4C22⋊D7911yescharacteristic
188C2×M16—22no—
189C2×M16—3161no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4Dic15811yescharacteristic
193C2×C4⋊C4C4×D7911yescharacteristic
194C2×C4⋊C4C2×Dic7911yescharacteristic
195C2×C4⋊C4C22⋊D7911yescharacteristic
196C2×C42—3161no—
197C2×C42—3162no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—3161no—
201C22⋊C8—1582no—
202#95—3161no—
203#95—3162no—
204#95—1584no—
205C24(C2×D79)⋊C411yescharacteristic
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—3161no—
219C22×C4—3161no—
220C22×C4—3161no—
221C22×C4—3161no—
222C22×C4—3161no—
223C22×C4—3161no—
224C22×C4—3161no—
225C22×C4—3161no—
226C22×C4—3161no—
227C22×C4—3161no—
228C22×C4—1582no—
229C22×C4—1582no—
230C22×C4—1582no—
231C22×C4—1582no—
232C22×C4C4×Dic7911yescharacteristic
233C22×C4(C2×D79)⋊C411yescharacteristic
234C22×C4C22⋊Dic7911yescharacteristic
235C22×C4Dic79⋊C411yescharacteristic
236C22×C4Dic79⋊C411yescharacteristic
237C22×C4C4⋊Dic7911yescharacteristic
238M16—42no—
239M16—42no—
240M16—6321no—
241M16—6321no—
242C2×C8—42no—
243C2×C8—3162no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C42—3162no—
249C42—3162no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23C79⋊#9511yescharacteristic
257C23#1457907511yescharacteristic
258C23#1457907511yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—3162no—
281C2×C4—3162no—
282C2×C4—3162no—
283C2×C4—3162no—
284C2×C4—3162no—
285C2×C4—3162no—
286C2×C4—3162no—
287C2×C4—3162no—
288C2×C4—3162no—
289C2×C4—6321no—
290C2×C4—6321no—
291C2×C4—6321no—
292C2×C4—6321no—
293C2×C4—1584no—
294C2×C4#1457908812yes—
295C8—82no—
296C8—6322no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22#1909831911yescharacteristic
312C22#1909832211yescharacteristic
313C22#1909832911yescharacteristic
314C22#1909835111yescharacteristic
315C22#1909835411yescharacteristic
316C22#1909835911yescharacteristic
317C22#1909836111yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—3164no—
325C4—6322no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2376125211yescharacteristic
333C2#2376151911yescharacteristic
334C2#2376152611yescharacteristic
subgroupG/H
centerC22#19098329
commutatorC22×C4×C79C42
FrattiniC22×C4⋊C4C2×D79
FittingC79×#1821C2
socleC22×C79#342
prime radicalC24×C79C22⋊C4
cosocleC22×C4⋊C4×C79C22
hypercenter#1821D79
hypocenterC79#7215
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
10. #19098289C4semidirect
14. #343×C79C4semidirect
171. C79#7215semidirect
180. #343Dic79semidirect

Maximal subgroups

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

subgroupindex
#237617302
#237616322
C79×#18212
#721579