Finite Group Databasebeta

← #34176468

Subgroups — #34176468

Order 20224.

590 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C4×#14579135C211yesmaximal, characteristic
2#23763162C211yesmaximal, characteristic, complemented
3#23761224C212yesmaximal, complemented
4#23761220C212yesmaximal, complemented
5C79×#1817C211yesmaximal, characteristic, complemented
6C2×#14579135C411yescharacteristic
7C2×C4×D316C411yescharacteristic
8C2×C4×C22⋊D79—21no—
9C2×C22⋊(C79⋊C8)—21no—
10C2×#14579072—21no—
11#19098465C2211yescharacteristic
12#19098461C2211yescharacteristic
13#19098454—21no—
14#19098442—21no—
15#19098423C411yescharacteristic
16C4×C4⋊Dic79C411yescharacteristic
17C4×(C2×D79)⋊C4—21no—
18#19098318C2212yes—
19C22×C42×C79C2211yescharacteristic
20C2×C22⋊C8×C79—21no—
21#336×C79—21no—
22#336×C79C2212yes—
23C22×D316C2×C411yescharacteristic
24C22×C22⋊D79—21no—
25C22×C4×D79—21no—
26C22×C79⋊C8—22no—
27#14579135—22no—
28#14579135—22no—
29C4×D316—42no—
30C4×C22⋊D79—44no—
31C2×C4⋊Dic79C2×C411yescharacteristic
32C2×C4⋊Dic79C2×C411yescharacteristic
33C2×C4⋊Dic79C2×C411yescharacteristic
34C2×Dic79⋊C4—21no—
35C2×C22⋊Dic79—21no—
36C2×(C2×D79)⋊C4—21no—
37C2×(C2×D79)⋊C4—21no—
38C2×(C2×D79)⋊C4—21no—
39C2×(C2×D79)⋊C4—21no—
40C2×C4×Dic79—21no—
41C22⋊(C79⋊C8)—24no—
42#14579072—42no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C23×C4×C79C2×C411yescharacteristic
47C23×C4×C79D411yescharacteristic
48C23×C4×C79D411yescharacteristic
49C22×C8×C79—22no—
50C2×C42×C79—42no—
51C2×C42×C79C2×C411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C2×C42×C79C2311yescharacteristic
55C22⋊C8×C79—42no—
56C23×D79—21no—
57C2×D316—21no—
58C2×D316—41no—
59C2×D316—41no—
60C2×D316M1612yes—
61C2×C22⋊D79—24no—
62C2×C22⋊D79—42no—
63C2×C22⋊D79—42no—
64C22×Dic79—21no—
65C22×Dic79—21no—
66C22×Dic79—21no—
67C2×C4×D79—42no—
68C2×C4×D79—42no—
69C2×C4×D79—42no—
70C2×C79⋊C8—24no—
71C2×C79⋊C8—42no—
72C2×C79⋊C8—42no—
73C4⋊Dic79—21no—
74C4⋊Dic79—41no—
75C4⋊Dic79—41no—
76C4⋊Dic79M1612yes—
77Dic79⋊C4—42no—
78C22⋊Dic79—42no—
79(C2×D79)⋊C4—24no—
80(C2×D79)⋊C4—42no—
81(C2×D79)⋊C4—42no—
82(C2×D79)⋊C4—42no—
83C4×Dic79—42no—
84C24×C79C22⋊C411yescharacteristic
85C22×C4×C79—22no—
86C22×C4×C79—22no—
87C22×C4×C79—21no—
88C22×C4×C79—21no—
89C22×C4×C79—22no—
90C22×C4×C79—22no—
91C22×C4×C79—41no—
92C22×C4×C79—41no—
93C22×C4×C79—41no—
94C22×C4×C79—41no—
95C22×C4×C79C22⋊C411yescharacteristic
96C22×C4×C79C22⋊C411yescharacteristic
97C22×C4×C79C22⋊C411yescharacteristic
98C22×C4×C79C22×C411yescharacteristic
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79C2×D411yescharacteristic
101C2×C8×C79—42no—
102C2×C8×C79—42no—
103C2×C8×C79—42no—
104C42×C79—22no—
105C42×C79—22no—
106C42×C79—41no—
107C42×C79—41no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—21no—
111C23×C79—21no—
112C23×C79—21no—
113C23×C79—21no—
114C23×C79#10412yes—
115C23×C79C2×C22⋊C411yescharacteristic
116C22×D79—22no—
117C22×D79—41no—
118C22×D79—41no—
119C22×D79—41no—
120C22×D79—41no—
121C22×D79—41no—
122C22×D79—41no—
123C2×C4×C79—21no—
124C2×C4×C79—21no—
125C2×C4×C79—21no—
126C2×C4×C79—21no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—44no—
138C2×C4×C79—44no—
139C2×C4×C79—42no—
140C2×C4×C79—42no—
141C2×C4×C79#9912yes—
142C2×C4×C79C2×M1611yescharacteristic
143C2×C4×C79C2×M1611yescharacteristic
144C2×C4×C79#13512yes—
145C2×C4×C79AGL(1,ℤ/8)12yes—
146C22⋊D79—48no—
147C2×Dic79—22no—
148C2×Dic79—41no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156D316—42no—
157D316—42no—
158C4×D79—84no—
159C8×C79—82no—
160C79⋊C8—44no—
161C22×C79—22no—
162C22×C79—22no—
163C22×C79—22no—
164C22×C79—41no—
165C22×C79—41no—
166C22×C79—41no—
167C22×C79—41no—
168C22×C79#40711yescharacteristic
169C22×C79C2×#9911yescharacteristic
170C22×C79#41311yescharacteristic
171C22×C79#41611yescharacteristic
172C22×C79#41811yescharacteristic
173C22×C79C2×#10411yescharacteristic
174C22×C79#42111yescharacteristic
175C2×D79—42no—
176C2×D79—42no—
177C2×D79—42no—
178C2×D79—81no—
179C2×D79—81no—
180C2×D79—81no—
181C2×D79—81no—
182C4×C79—21no—
183C4×C79—21no—
184C4×C79—22no—
185C4×C79—22no—
186C4×C79—22no—
187C4×C79—41no—
188C4×C79—41no—
189Dic79—42no—
190Dic79—81no—
191Dic79—81no—
192#7620—791nomaximal, Sylow
193C2×C79—42no—
194C2×C79#146511yescharacteristic
195C2×C79#148811yescharacteristic
196C2×C79#149011yescharacteristic
197C2×C79#150111yescharacteristic
198C2×C79#151011yescharacteristic
199C2×C79#153211yescharacteristic
200C2×C79#155311yescharacteristic
201D79—82no—
202C4×#121—791no—
203#1817D7911yescharacteristic, complemented
204#1749—791no—
205#1271—792no—
206#1270—792no—
207C79#762011yescharacteristic, complemented, Sylow
208C2×#121—791no—
209C2×C4×D4—791no—
210C2×C4×D4—1581no—
211C22×C42C2×D7911yescharacteristic
212C2×C22⋊C8—21no—
213C2×C22⋊C8—1581no—
214C2×#97—1581no—
215#390—791no—
216#389—791no—
217#388—1581no—
218#386—1581no—
219#380—791no—
220C4×C4⋊C4—791no—
221C4×C22⋊C4—1581no—
222#336—21no—
223#336—792no—
224#336C2×D7912yes—
225C22×D4—791no—
226C22×D4—1581no—
227C23×C4—1581no—
228C23×C4C4×D7911yescharacteristic
229C23×C4D31611yescharacteristic
230C23×C4C22⋊D7911yescharacteristic
231C22×C8—22no—
232C22×C8—1582no—
233#121—1582no—
234#121—1582no—
235C4×D4—3162no—
236C4×D4—3164no—
237C2×C4⋊C4—791no—
238C2×C4⋊C4—791no—
239C2×C4⋊C4—791no—
240C2×C4⋊C4—1581no—
241C2×C22⋊C4—1581no—
242C2×C22⋊C4—1581no—
243C2×C22⋊C4—1581no—
244C2×C22⋊C4—1581no—
245C2×C22⋊C4—1581no—
246C2×C42—42no—
247C2×C42—1581no—
248C2×C42C4×D7911yescharacteristic
249C2×C42D31611yescharacteristic
250C2×C42C22⋊D7911yescharacteristic
251C2×C42C22×D7911yescharacteristic
252C22⋊C8—42no—
253C22⋊C8—1584no—
254#97—3162no—
255#95—1581no—
256#95—1581no—
257#95—1581no—
258C24—1581no—
259C24(C2×D79)⋊C411yescharacteristic
260C2×D4—3161no—
261C2×D4—3161no—
262C2×D4—3162no—
263C2×D4—3162no—
264C2×D4—792no—
265C2×D4—1584no—
266C2×D4—1581no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—3162no—
278C22×C4—3162no—
279C22×C4—3162no—
280C22×C4—1581no—
281C22×C4—1581no—
282C22×C4—1581no—
283C22×C4(C2×D79)⋊C411yescharacteristic
284C22×C4(C2×D79)⋊C411yescharacteristic
285C22×C4(C2×D79)⋊C411yescharacteristic
286C22×C4C2×C4×D7911yescharacteristic
287C22×C4C2×C22⋊D7911yescharacteristic
288C22×C4C2×D31611yescharacteristic
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—3162no—
293C2×C8—3162no—
294C2×C8—1584no—
295C4⋊C4—3161no—
296C4⋊C4—3161no—
297C4⋊C4—3162no—
298C4⋊C4—792no—
299C4⋊C4—1581no—
300C22⋊C4—3162no—
301C22⋊C4—3162no—
302C22⋊C4—3162no—
303C22⋊C4—3162no—
304C22⋊C4—1584no—
305C42—22no—
306C42—22no—
307C42—41no—
308C42—41no—
309C42—3162no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—3161no—
317C23—3161no—
318C23—3161no—
319C23—3161no—
320C23—3161no—
321C23—3161no—
322C23—1582no—
323C23#1457908712yes—
324C23C2×(C2×D79)⋊C411yescharacteristic
325D4—3162no—
326D4—3162no—
327D4—3168no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—3161no—
347C2×C4—3161no—
348C2×C4—3161no—
349C2×C4—3161no—
350C2×C4—3161no—
351C2×C4—3161no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—6324no—
355C2×C4—1582no—
356C2×C4#1457907512yes—
357C2×C4M16×D7911yescharacteristic
358C2×C4M16×D7911yescharacteristic
359C2×C4#1457918312yes—
360C2×C4#1457918612yes—
361C8—82no—
362C8—3164no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—3162no—
371C22—3162no—
372C22—3162no—
373C22—6321no—
374C22—6321no—
375C22—6321no—
376C22—6321no—
377C22#1909852611yescharacteristic
378C22C2×#1457907511yescharacteristic
379C22#1909854511yescharacteristic
380C22#1909856011yescharacteristic
381C22#1909856711yescharacteristic
382C22C2×#1457908711yescharacteristic
383C22#1909858111yescharacteristic
384C4—22no—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—3162no—
392C4—6321no—
393C4—6321no—
394C2—42no—
395C2—6322no—
396C2#2376173611yescharacteristic
397C2#2376183511yescharacteristic
398C2#2376184211yescharacteristic
399C2#2376189211yescharacteristic
400C2#2376192811yescharacteristic
401C2#2376202111yescharacteristic
402C2#2376210311yescharacteristic
subgroupG/H
centerC23C2×(C2×D79)⋊C4
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1817C2
socleC23×C79C2×C22⋊C4
prime radicalC2×#14579135C4
cosocleC2×C42×C79C23
hypercenter#1817D79
hypocenterC79#7620
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #23763162C2semidirect
3. #23761224C2semidirect
4. #23761220C2semidirect
5. C79×#1817C2semidirect
203. #1817D79semidirect
207. C79#7620semidirect

Maximal subgroups

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

subgroupindex
C4×#145791352
#237631622
#237612242
#237612202
C79×#18172
#762079