Finite Group Databasebeta

← #34193028

Subgroups — #34193028

Order 20224.

402 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764193C211yesmaximal, characteristic
2#23763527C211yesmaximal, characteristic, complemented
3#23763401C211yesmaximal, characteristic
4#23763397C211yesmaximal, characteristic
5#23761284C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C79×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D316C2211yescharacteristic
9C2×C4⋊(C79⋊C8)C2211yescharacteristic
10C2×#14579083—21no—
11C2×#14579081—21no—
12C2×(C2×D79)⋊C8—21no—
13C2×(C2×D79)⋊C8—21no—
14C2×C4×C79⋊C8C2211yescharacteristic
15#19098433C2211yescharacteristic
16#19098326—21no—
17#19098325—21no—
18#19098325—21no—
19#19098325—21no—
20C2×C4⋊C8×C79C2211yescharacteristic
21C2×C4⋊C8×C79C2211yescharacteristic
22#384×C79C2211yescharacteristic
23C22×D316D411yescharacteristic
24C22×C4×D79—21no—
25C22×C79⋊C8—21no—
26C22×C79⋊C8D411yescharacteristic
27C22×C79⋊C8D411yescharacteristic
28C4×D316—21no—
29C4×D316—21no—
30C4×D316—21no—
31C4×D316—21no—
32C2×C4⋊Dic79—21no—
33C2×C4⋊Dic79—21no—
34C2×C4⋊Dic79—21no—
35C2×C4⋊Dic79D411yescharacteristic
36C2×(C2×D79)⋊C4—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39#14579083—42no—
40#14579081—42no—
41(C2×D79)⋊C8—42no—
42(C2×D79)⋊C8—42no—
43C4×C79⋊C8—21no—
44C4×C79⋊C8—21no—
45C22×C8×C79—21no—
46C22×C8×C79—21no—
47C2×C4⋊C4×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C42×C79C2311yescharacteristic
51C4⋊C8×C79—21no—
52C4⋊C8×C79—21no—
53C4⋊C8×C79—21no—
54C4⋊C8×C79—21no—
55C23×D79—21no—
56C2×D316—21no—
57C2×D316—21no—
58C2×D316—21no—
59C2×D316—21no—
60C2×D316—21no—
61C2×D316SD1612yes—
62C22×Dic79—21no—
63C22×Dic79—41no—
64C2×C4×D79—41no—
65C2×C4×D79—41no—
66C2×C4×D79—41no—
67C2×C4×D79—41no—
68C2×C4×D79—41no—
69C2×C4×D79—41no—
70C2×C79⋊C8—21no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—21no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—21no—
76C2×C79⋊C8—41no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8—41no—
79C4⋊Dic79—21no—
80C4⋊Dic79—21no—
81C4⋊Dic79—41no—
82C4⋊Dic79—41no—
83C4⋊Dic79—41no—
84C4⋊Dic79—41no—
85C4⋊Dic79—81no—
86(C2×D79)⋊C4—41no—
87(C2×D79)⋊C4—41no—
88C22×C4×C79—41no—
89C22×C4×C79C2×D411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79#4111yescharacteristic
92C2×C8×C79—41no—
93C2×C8×C79—41no—
94C2×C8×C79—41no—
95C2×C8×C79—41no—
96C2×C8×C79—41no—
97C2×C8×C79—41no—
98C4⋊C4×C79—41no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—81no—
103C42×C79#4111yescharacteristic
104C42×C79#4111yescharacteristic
105C42×C79#4111yescharacteristic
106C42×C79#4111yescharacteristic
107C23×C79#12111yescharacteristic
108C22×D79—22no—
109C22×D79—41no—
110C22×D79—41no—
111C22×D79—41no—
112C22×D79—41no—
113C22×D79—41no—
114C22×D79—41no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—81no—
118C2×C4×C79—81no—
119C2×C4×C79—81no—
120C2×C4×C79#12311yescharacteristic
121C2×C4×C79#12311yescharacteristic
122C2×C4×C79#12311yescharacteristic
123C2×C4×C79#12411yescharacteristic
124C2×C4×C79#12611yescharacteristic
125C2×C4×C79#12611yescharacteristic
126C2×C4×C79C2×SD1611yescharacteristic
127C2×C4×C79#13511yescharacteristic
128C2×C4×C79#13511yescharacteristic
129C2×C4×C79#13511yescharacteristic
130C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C79#13711yescharacteristic
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—81no—
138C2×Dic79—81no—
139C2×Dic79—81no—
140D316—22no—
141D316—22no—
142D316—41no—
143D316—41no—
144C4×D79—81no—
145C4×D79—81no—
146C4×D79—81no—
147C4×D79—81no—
148C8×C79—81no—
149C8×C79—81no—
150C79⋊C8—41no—
151C79⋊C8—41no—
152C79⋊C8—81no—
153C22×C79#39711yescharacteristic
154C22×C79#46311yescharacteristic
155C22×C79#46311yescharacteristic
156C22×C79#46511yescharacteristic
157C22×C79#46811yescharacteristic
158C22×C79#47211yescharacteristic
159C22×C79#47311yescharacteristic
160C2×D79—42no—
161C2×D79—42no—
162C2×D79—42no—
163C2×D79—81no—
164C2×D79—81no—
165C2×D79—81no—
166C2×D79—81no—
167C4×C79—21no—
168C4×C79—21no—
169C4×C79—161no—
170C4×C79#48111yescharacteristic
171C4×C79#48211yescharacteristic
172C4×C79#48211yescharacteristic
173C4×C79#48211yescharacteristic
174C4×C79#48311yescharacteristic
175C4×C79#48511yescharacteristic
176C4×C79#48811yescharacteristic
177C4×C79#49011yescharacteristic
178Dic79—81no—
179Dic79—161no—
180#10625—791nomaximal, Sylow
181C2×C79#167711yescharacteristic
182C2×C79#167911yescharacteristic
183C2×C79#169711yescharacteristic
184C2×C79#169911yescharacteristic
185C2×C79#207311yescharacteristic
186C2×C79#207711yescharacteristic
187C2×C79#207811yescharacteristic
188D79—82no—
189#1954—791no—
190#1834—791no—
191#1802—791no—
192#1802—791no—
193#1296—791no—
194#1296D7911yescharacteristic, complemented
195#1277—791no—
196C79#1062511yescharacteristic, complemented, Sylow
197C2×C4×D4—791no—
198C2×C4⋊C8—791no—
199C2×C4⋊C8C2×D7911yescharacteristic
200C2×C4⋊C8C2×D7911yescharacteristic
201C2×#102—1581no—
202C2×#102—1581no—
203C2×C22⋊C8—1581no—
204C2×C22⋊C8—1581no—
205C2×C4×C8—791no—
206#384—791no—
207#384C2×D7911yescharacteristic
208#340—1581no—
209#340—1581no—
210#340—1581no—
211#340—1581no—
212C22×D4—791no—
213C23×C4—1581no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—791no—
217C22×C8—791no—
218C22×C8—1581no—
219C4×D4—1581no—
220C4×D4—1581no—
221C4×D4—1581no—
222C4×D4—1581no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—791no—
227C2×C4⋊C4—1581no—
228C2×C4⋊C4—1581no—
229C2×C4⋊C4—1581no—
230C2×C22⋊C4—1581no—
231C2×C42C22×D7911yescharacteristic
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—1581no—
237C4⋊C8—1581no—
238#102—3162no—
239#102—3162no—
240C22⋊C8—3162no—
241C22⋊C8—3162no—
242C4×C8—1581no—
243C4×C8—1581no—
244C24—1581no—
245C2×D4—792no—
246C2×D4—1581no—
247C2×D4—1581no—
248C2×D4—1581no—
249C2×D4—1581no—
250C2×D4—1581no—
251C22×C4—41no—
252C22×C4—3161no—
253C22×C4—3161no—
254C22×C4—3161no—
255C22×C4—3161no—
256C22×C4—3161no—
257C22×C4—3161no—
258C22×C4—3161no—
259C22×C4—1581no—
260C22×C4D4×D7911yescharacteristic
261C22×C4D4×D7911yescharacteristic
262C22×C4#1192962411yescharacteristic
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—3161no—
270C2×C8—3161no—
271C2×C8—3161no—
272C2×C8—1581no—
273C2×C8—1581no—
274C2×C8—1581no—
275C2×C8—1581no—
276C2×C8—1581no—
277C2×C8—1581no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—3161no—
284C4⋊C4—3161no—
285C4⋊C4—3161no—
286C4⋊C4—3161no—
287C4⋊C4—6321no—
288C4⋊C4—1581no—
289C4⋊C4—1581no—
290C22⋊C4—3161no—
291C22⋊C4—3161no—
292C42#1192962311yescharacteristic
293C42#1192962411yescharacteristic
294C42#1192962411yescharacteristic
295C42#1192962511yescharacteristic
296C23—3161no—
297C23—3161no—
298C23—3161no—
299C23—3161no—
300C23—3161no—
301C23—3161no—
302C23—1582no—
303C23#1457913411yescharacteristic
304D4—3161no—
305D4—3161no—
306D4—1582no—
307D4—1582no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—3161no—
314C2×C4—3161no—
315C2×C4—3161no—
316C2×C4—6321no—
317C2×C4—6321no—
318C2×C4—6321no—
319C2×C4—6321no—
320C2×C4—6321no—
321C2×C4—6321no—
322C2×C4—6321no—
323C2×C4#1457914311yescharacteristic
324C2×C4#1457914611yescharacteristic
325C2×C4#1457914611yescharacteristic
326C2×C4#1457914711yescharacteristic
327C2×C4#1457915511yescharacteristic
328C2×C4#1457915611yescharacteristic
329C2×C4SD16×D7911yescharacteristic
330C2×C4#1457918011yescharacteristic
331C2×C4#1457918211yescharacteristic
332C2×C4#1457918411yescharacteristic
333C2×C4#1457918511yescharacteristic
334C2×C4#1457918811yescharacteristic
335C2×C4#1457918811yescharacteristic
336C2×C4#1457919311yescharacteristic
337C8—81no—
338C8—81no—
339C8—3161no—
340C8—3161no—
341C8—6321no—
342C22—3162no—
343C22—3162no—
344C22—3162no—
345C22—6321no—
346C22—6321no—
347C22—6321no—
348C22—6321no—
349C22#1909849211yescharacteristic
350C22#1909877711yescharacteristic
351C22#1909878111yescharacteristic
352C22#1909879111yescharacteristic
353C22#1909881211yescharacteristic
354C22#1909883211yescharacteristic
355C22#1909883711yescharacteristic
356C4—21no—
357C4—21no—
358C4—161no—
359C4—6321no—
360C4—12641no—
361C4#1909888311yescharacteristic
362C4#1909888511yescharacteristic
363C4#1909888911yescharacteristic
364C4#1909889011yescharacteristic
365C4#1909889211yescharacteristic
366C4#1909890611yescharacteristic
367C4#1909891911yescharacteristic
368C4#1909892611yescharacteristic
369C2—6322no—
370C2#2376280811yescharacteristic
371C2#2376282611yescharacteristic
372C2#2376293611yescharacteristic
373C2#2376294611yescharacteristic
374C2#2376481711yescharacteristic
375C2#2376483811yescharacteristic
376C2#2376484411yescharacteristic
subgroupG/H
centerC23#14579134
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1296C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenter#1296D79
hypocenterC79#10625
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #23763527C2semidirect
5. #23761284C2semidirect
6. #23761234C2semidirect
7. C79×#1296C2semidirect
194. #1296D79semidirect
196. C79#10625semidirect

Maximal subgroups

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

subgroupindex
#237641932
#237635272
#237634012
#237633972
#237612842
#237612342
C79×#12962
#1062579