Finite Group Databasebeta

← #34199088

Subgroups — #34199088

Order 20224.

450 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 79 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#23769284C211yesmaximal, characteristic
2#23761738C211yesmaximal, characteristic, complemented
3#23761426C212yesmaximal, complemented
4#23761402C212yesmaximal, complemented
5C79×#1800C211yesmaximal, characteristic, complemented
6C2×#14579135C411yescharacteristic
7#19098527—21no—
8#19098502C2211yescharacteristic
9#19098499C411yescharacteristic
10#19098480C2211yescharacteristic
11#19098478—21no—
12#19098472C411yescharacteristic
13#19098471—21no—
14#19098454—21no—
15#19098442—21no—
16#19098426C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C2×C4×D4×C79C2211yescharacteristic
20#423×C79—21no—
21C2×#100×C79—21no—
22#343×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C22⋊D79—21no—
25C22×C22⋊D79—21no—
26C2×#11929608—41no—
27#14579135—41no—
28#14579135—41no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×Dic79⋊C4—21no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—21no—
33C2×(C2×D79)⋊C4—21no—
34C2×(C2×D79)⋊C4—21no—
35C2×(C2×D79)⋊C4—21no—
36C2×(C2×D79)⋊C4—21no—
37C22⋊(C79⋊C8)—22no—
38C22⋊(C79⋊C8)—41no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C22×D4×C79D411yescharacteristic
45C23×C4×C79C2×C411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—22no—
48C4×D4×C79—41no—
49C4×D4×C79—41no—
50C2×C4⋊C4×C79C2311yescharacteristic
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C4⋊C8×C79—41no—
55C4⋊C8×C79—41no—
56#100×C79—42no—
57C23×D79—21no—
58C2×D316—41no—
59C2×D316—81no—
60C2×C22⋊D79—42no—
61C2×C22⋊D79—42no—
62C2×C22⋊D79—81no—
63C2×C22⋊D79—81no—
64C2×C22⋊D79—81no—
65C2×C22⋊D79—81no—
66C22×Dic79—21no—
67C22×Dic79—21no—
68C22×Dic79—21no—
69#11929608—82no—
70C2×C79⋊C8—42no—
71C4⋊Dic79—41no—
72Dic79⋊C4—81no—
73C22⋊Dic79—81no—
74C22⋊Dic79—81no—
75(C2×D79)⋊C4—41no—
76(C2×D79)⋊C4—41no—
77(C2×D79)⋊C4—41no—
78(C2×D79)⋊C4—41no—
79(C2×D79)⋊C4—81no—
80(C2×D79)⋊C4—81no—
81C24×C79C22⋊C411yescharacteristic
82C24×C79C22⋊C411yescharacteristic
83C2×D4×C79—21no—
84C2×D4×C79—21no—
85C2×D4×C79—41no—
86C2×D4×C79—41no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—21no—
91C22×C4×C79—81no—
92C22×C4×C79—81no—
93C22×C4×C79C22⋊C411yescharacteristic
94C22×C4×C79C22⋊C411yescharacteristic
95C22×C4×C79C22×C411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79C2×D411yescharacteristic
98M16×C79—42no—
99M16×C79—42no—
100C2×C8×C79—42no—
101C4⋊C4×C79—41no—
102C22⋊C4×C79—41no—
103C22⋊C4×C79—41no—
104C42×C79—21no—
105C42×C79—21no—
106C23×C79—21no—
107C23×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—21no—
111C23×C79—21no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79—41no—
116C23×C79C2×C22⋊C411yescharacteristic
117C22×D79—41no—
118C22×D79—41no—
119C22×D79—41no—
120C22×D79—41no—
121C22×D79—41no—
122C22×D79—41no—
123C22×D79—41no—
124D4×C79—42no—
125D4×C79—42no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—41no—
132C2×C4×C79—42no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—81no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79#9911yescharacteristic
140C2×C4×C79#9911yescharacteristic
141C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C79#13711yescharacteristic
143C22⋊D79—84no—
144C22⋊D79—84no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154D316—82no—
155C8×C79—82no—
156C79⋊C8—82no—
157C22×C79—21no—
158C22×C79—21no—
159C22×C79—42no—
160C22×C79—41no—
161C22×C79—41no—
162C22×C79—41no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—41no—
166C22×C79—81no—
167C22×C79—81no—
168C22×C79—81no—
169C22×C79C2×#9911yescharacteristic
170C22×C79#41711yescharacteristic
171C22×C79#42111yescharacteristic
172C2×D79—81no—
173C2×D79—81no—
174C2×D79—81no—
175C2×D79—81no—
176C2×D79—81no—
177C2×D79—81no—
178C2×D79—81no—
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—41no—
184C4×C79—41no—
185C4×C79—41no—
186Dic79—81no—
187Dic79—81no—
188Dic79—81no—
189#11832—791nomaximal, Sylow
190C2×C79—21no—
191C2×C79—21no—
192C2×C79—81no—
193C2×C79—81no—
194C2×C79#151311yescharacteristic
195C2×C79#212111yescharacteristic
196C2×C79#213411yescharacteristic
197D79—161no—
198#2635—791no—
199#1800D7911yescharacteristic, complemented
200#1465—791no—
201#1353—792no—
202#1345—792no—
203C79#1183211yescharacteristic, complemented, Sylow
204C2×#121—791no—
205C2×C4×D4C2×D7911yescharacteristic
206#423—21no—
207C2×#100—21no—
208#407—1581no—
209#400—791no—
210#399—791no—
211#394—791no—
212#394—1581no—
213#392—791no—
214#392—1581no—
215#388—1581no—
216#386—1581no—
217#381—791no—
218#343C2×D7912yes—
219#324—792no—
220#323—1581no—
221C22×D4—791no—
222C22×D4—1581no—
223C22×D4—1581no—
224C22×D4C22⋊D7911yescharacteristic
225C23×C4C4×D7911yescharacteristic
226C23×C4D31611yescharacteristic
227C2×M16—22no—
228C2×M16—3161no—
229#121—3161no—
230#121—3161no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—791no—
234C2×C4⋊C4—1581no—
235C2×C4⋊C4C22×D7911yescharacteristic
236C2×C22⋊C4—1581no—
237C2×C22⋊C4—1581no—
238C2×C22⋊C4—1581no—
239C2×C22⋊C4—1581no—
240C2×C22⋊C4—1581no—
241C2×C22⋊C4—1581no—
242C2×C22⋊C4C4×D7911yescharacteristic
243C2×C22⋊C4D31611yescharacteristic
244C2×C42C22⋊D7911yescharacteristic
245C4⋊C8—41no—
246C4⋊C8—41no—
247#100—42no—
248C22⋊C8—3161no—
249C22⋊C8—1582no—
250#95—791no—
251#95—791no—
252#95—1581no—
253#95—1581no—
254#95—1581no—
255C24—1581no—
256C24(C2×D79)⋊C411yescharacteristic
257C24(C2×D79)⋊C411yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—3161no—
263C2×D4—3162no—
264C2×D4—3162no—
265C2×D4—6321no—
266C2×D4—6321no—
267C2×D4—6321no—
268C2×D4—6321no—
269C2×D4—6321no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—1581no—
277C22×C4—1581no—
278C22×C4—1581no—
279C22×C4(C2×D79)⋊C411yescharacteristic
280C22×C4(C2×D79)⋊C411yescharacteristic
281C22×C4C2×C4×D7911yescharacteristic
282C22×C4C2×C22⋊D7911yescharacteristic
283C22×C4C2×D31611yescharacteristic
284M16—42no—
285M16—42no—
286M16—6322no—
287C2×C8—42no—
288C2×C8—3162no—
289C4⋊C4—41no—
290C4⋊C4—3161no—
291C4⋊C4—6321no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—3161no—
295C22⋊C4—3161no—
296C22⋊C4—3161no—
297C22⋊C4—3161no—
298C22⋊C4—6321no—
299C22⋊C4—6321no—
300C22⋊C4—6321no—
301C22⋊C4—6321no—
302C42—21no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—3161no—
315C23—3161no—
316C23—3161no—
317C23—3161no—
318C23—3161no—
319C23—3161no—
320C23—3161no—
321C23C2×(C2×D79)⋊C411yescharacteristic
322D4—42no—
323D4—42no—
324D4—6322no—
325D4—6324no—
326D4—6324no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—3161no—
341C2×C4—3161no—
342C2×C4—3161no—
343C2×C4—3161no—
344C2×C4—3161no—
345C2×C4—3161no—
346C2×C4—3161no—
347C2×C4—3161no—
348C2×C4—3161no—
349C2×C4#1457907511yescharacteristic
350C2×C4#1457907511yescharacteristic
351C2×C4#1457918711yescharacteristic
352C2×C4#1457919211yescharacteristic
353C8—82no—
354C8—6322no—
355C22—21no—
356C22—21no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—6321no—
368C22—6321no—
369C22—6321no—
370C22—6321no—
371C22—6321no—
372C22—6321no—
373C22—6321no—
374C22C2×#1457907511yescharacteristic
375C22#1909856211yescharacteristic
376C22#1909858111yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—6321no—
385C4—6321no—
386C4—6321no—
387C2—21no—
388C2—21no—
389C2—81no—
390C2—81no—
391C2—12641no—
392C2#2376194111yescharacteristic
393C2#2376504711yescharacteristic
394C2#2376509111yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1800C2
socleC22×C79C2×#99
prime radical#23769284C2
cosocleC2×C4⋊C4×C79C23
hypercenter#1800D79
hypocenterC79#11832
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #23761738C2semidirect
3. #23761426C2semidirect
4. #23761402C2semidirect
5. C79×#1800C2semidirect
199. #1800D79semidirect
203. C79#11832semidirect

Maximal subgroups

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

subgroupindex
#237692842
#237617382
#237614262
#237614022
C79×#18002
#1183279