Finite Group Databasebeta

← #34192994

Subgroups — #34192994

Order 20224.

418 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764193C211yesmaximal, characteristic
2#23763556C211yesmaximal, characteristic, complemented
3#23763401C211yesmaximal, characteristic
4#23763397C211yesmaximal, characteristic
5#23761284C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C79×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D316C2211yescharacteristic
9C2×#14579092—21no—
10C2×#14579092—21no—
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×#14579083—21no—
13C2×#14579081—21no—
14C2×(C2×D79)⋊C8—21no—
15C2×(C2×D79)⋊C8—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17#19098433C2211yescharacteristic
18#19098326—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—
37#14579092—42no—
38#14579092—42no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41#14579083—42no—
42#14579081—42no—
43(C2×D79)⋊C8—42no—
44(C2×D79)⋊C8—42no—
45C4×C79⋊C8—21no—
46C4×C79⋊C8—21no—
47C22×C8×C79—21no—
48C22×C8×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79—21no—
52C2×C42×C79C2311yescharacteristic
53C4⋊C8×C79—21no—
54C4⋊C8×C79—21no—
55C4⋊C8×C79—21no—
56C4⋊C8×C79—21no—
57C23×D79—21no—
58C2×D316—21no—
59C2×D316—21no—
60C2×D316—21no—
61C2×D316—21no—
62C2×D316—21no—
63C2×D316SD1612yes—
64C22×Dic79—21no—
65C22×Dic79—41no—
66C2×C4×D79—41no—
67C2×C4×D79—41no—
68C2×C4×D79—41no—
69C2×C4×D79—41no—
70C2×C4×D79—41no—
71C2×C4×D79—41no—
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—
79C2×C79⋊C8D812yes—
80C2×C79⋊C8SD1612yes—
81C4⋊Dic79—21no—
82C4⋊Dic79—21no—
83C4⋊Dic79—41no—
84C4⋊Dic79—41no—
85C4⋊Dic79—41no—
86C4⋊Dic79—41no—
87C4⋊Dic79—81no—
88(C2×D79)⋊C4—41no—
89(C2×D79)⋊C4—41no—
90C22×C4×C79—41no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79#4111yescharacteristic
94C2×C8×C79—41no—
95C2×C8×C79—41no—
96C2×C8×C79—41no—
97C2×C8×C79—41no—
98C2×C8×C79—41no—
99C2×C8×C79—41no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79—81no—
105C42×C79#4111yescharacteristic
106C42×C79#4111yescharacteristic
107C42×C79#4111yescharacteristic
108C42×C79#4111yescharacteristic
109C23×C79#12111yescharacteristic
110C22×D79—22no—
111C22×D79—41no—
112C22×D79—41no—
113C22×D79—41no—
114C22×D79—41no—
115C22×D79—41no—
116C22×D79—41no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—81no—
120C2×C4×C79—81no—
121C2×C4×C79—81no—
122C2×C4×C79#12311yescharacteristic
123C2×C4×C79#12311yescharacteristic
124C2×C4×C79#12311yescharacteristic
125C2×C4×C79#12411yescharacteristic
126C2×C4×C79#12611yescharacteristic
127C2×C4×C79#12611yescharacteristic
128C2×C4×C79C2×D811yescharacteristic
129C2×C4×C79C2×SD1611yescharacteristic
130C2×C4×C79C2×SD1611yescharacteristic
131C2×C4×C79#13511yescharacteristic
132C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C79#13711yescharacteristic
135C2×C4×C79#13711yescharacteristic
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—81no—
140C2×Dic79—81no—
141C2×Dic79—81no—
142D316—22no—
143D316—22no—
144D316—41no—
145D316—41no—
146C4×D79—81no—
147C4×D79—81no—
148C4×D79—81no—
149C4×D79—81no—
150C8×C79—81no—
151C8×C79—81no—
152C79⋊C8—22no—
153C79⋊C8—22no—
154C79⋊C8—81no—
155C22×C79#39711yescharacteristic
156C22×C79#45911yescharacteristic
157C22×C79#46111yescharacteristic
158C22×C79#46511yescharacteristic
159C22×C79#46811yescharacteristic
160C22×C79#47111yescharacteristic
161C22×C79#47111yescharacteristic
162C2×D79—42no—
163C2×D79—42no—
164C2×D79—42no—
165C2×D79—81no—
166C2×D79—81no—
167C2×D79—81no—
168C2×D79—81no—
169C4×C79—21no—
170C4×C79—21no—
171C4×C79—161no—
172C4×C79#48011yescharacteristic
173C4×C79#48111yescharacteristic
174C4×C79#48111yescharacteristic
175C4×C79#48211yescharacteristic
176C4×C79#48311yescharacteristic
177C4×C79#48511yescharacteristic
178C4×C79#48611yescharacteristic
179C4×C79#48911yescharacteristic
180Dic79—81no—
181Dic79—161no—
182#10619—791nomaximal, Sylow
183C2×C79#166511yescharacteristic
184C2×C79#167111yescharacteristic
185C2×C79#168511yescharacteristic
186C2×C79#169111yescharacteristic
187C2×C79#207111yescharacteristic
188C2×C79#207711yescharacteristic
189C2×C79#207911yescharacteristic
190D79—82no—
191#1954—791no—
192#1844—791no—
193#1802—791no—
194#1802—791no—
195#1296—791no—
196#1296D7911yescharacteristic, complemented
197#1277—791no—
198C79#1061911yescharacteristic, complemented, Sylow
199C2×C4×D4—791no—
200C2×#106—1581no—
201C2×#106—1581no—
202C2×C4⋊C8—791no—
203C2×C4⋊C8C2×D7911yescharacteristic
204C2×C4⋊C8C2×D7911yescharacteristic
205C2×#102—1581no—
206C2×#102—1581no—
207C2×C22⋊C8—1581no—
208C2×C22⋊C8—1581no—
209C2×C4×C8—791no—
210#384—791no—
211#384C2×D7911yescharacteristic
212#340—1581no—
213#340—1581no—
214C22×D4—791no—
215C23×C4—1581no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—791no—
219C22×C8—791no—
220C22×C8—1581no—
221C4×D4—1581no—
222C4×D4—1581no—
223C4×D4—1581no—
224C4×D4—1581no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—791no—
229C2×C4⋊C4—1581no—
230C2×C4⋊C4—1581no—
231C2×C4⋊C4—1581no—
232C2×C22⋊C4—1581no—
233C2×C42C22×D7911yescharacteristic
234#106—3162no—
235#106—3162no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—1581no—
241C4⋊C8—1581no—
242#102—3162no—
243#102—3162no—
244C22⋊C8—3162no—
245C22⋊C8—3162no—
246C4×C8—1581no—
247C4×C8—1581no—
248C24—1581no—
249C2×D4—792no—
250C2×D4—1581no—
251C2×D4—1581no—
252C2×D4—1581no—
253C2×D4—1581no—
254C2×D4—1581no—
255C22×C4—41no—
256C22×C4—3161no—
257C22×C4—3161no—
258C22×C4—3161no—
259C22×C4—3161no—
260C22×C4—3161no—
261C22×C4—3161no—
262C22×C4—3161no—
263C22×C4—1581no—
264C22×C4D4×D7911yescharacteristic
265C22×C4D4×D7911yescharacteristic
266C22×C4#1192962411yescharacteristic
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—3161no—
274C2×C8—3161no—
275C2×C8—3161no—
276C2×C8—792no—
277C2×C8—792no—
278C2×C8—1581no—
279C2×C8—1581no—
280C2×C8—1581no—
281C2×C8—1581no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—3161no—
288C4⋊C4—3161no—
289C4⋊C4—3161no—
290C4⋊C4—3161no—
291C4⋊C4—6321no—
292C4⋊C4—1581no—
293C4⋊C4—1581no—
294C22⋊C4—3161no—
295C22⋊C4—3161no—
296C42#1192962311yescharacteristic
297C42#1192962411yescharacteristic
298C42#1192962411yescharacteristic
299C42#1192962511yescharacteristic
300C23—3161no—
301C23—3161no—
302C23—3161no—
303C23—3161no—
304C23—3161no—
305C23—3161no—
306C23—1582no—
307C23#1457913411yescharacteristic
308D4—3161no—
309D4—3161no—
310D4—1582no—
311D4—1582no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—3161no—
318C2×C4—3161no—
319C2×C4—3161no—
320C2×C4—6321no—
321C2×C4—6321no—
322C2×C4—6321no—
323C2×C4—6321no—
324C2×C4—6321no—
325C2×C4—6321no—
326C2×C4—6321no—
327C2×C4#1457914311yescharacteristic
328C2×C4#1457914611yescharacteristic
329C2×C4#1457914611yescharacteristic
330C2×C4#1457914711yescharacteristic
331C2×C4#1457915511yescharacteristic
332C2×C4#1457915611yescharacteristic
333C2×C4D8×D7911yescharacteristic
334C2×C4SD16×D7911yescharacteristic
335C2×C4SD16×D7911yescharacteristic
336C2×C4#1457918011yescharacteristic
337C2×C4#1457918811yescharacteristic
338C2×C4#1457918811yescharacteristic
339C2×C4#1457919311yescharacteristic
340C2×C4#1457919311yescharacteristic
341C8—81no—
342C8—81no—
343C8—6321no—
344C8—1582no—
345C8—1582no—
346C22—3162no—
347C22—3162no—
348C22—3162no—
349C22—6321no—
350C22—6321no—
351C22—6321no—
352C22—6321no—
353C22#1909849211yescharacteristic
354C22#1909875711yescharacteristic
355C22#1909876711yescharacteristic
356C22#1909879111yescharacteristic
357C22#1909881211yescharacteristic
358C22#1909882511yescharacteristic
359C22#1909882911yescharacteristic
360C4—21no—
361C4—21no—
362C4—161no—
363C4—6321no—
364C4—12641no—
365C4#1909887811yescharacteristic
366C4#1909888311yescharacteristic
367C4#1909888311yescharacteristic
368C4#1909888511yescharacteristic
369C4#1909889211yescharacteristic
370C4#1909890611yescharacteristic
371C4#1909890811yescharacteristic
372C4#1909892411yescharacteristic
373C2—6322no—
374C2#2376274011yescharacteristic
375C2#2376277411yescharacteristic
376C2#2376286811yescharacteristic
377C2#2376290211yescharacteristic
378C2#2376480711yescharacteristic
379C2#2376483811yescharacteristic
380C2#2376484911yescharacteristic
subgroupG/H
centerC23#14579134
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1296C2
socleC23×C79#121
prime radicalC22×D316D4
hypercenter#1296D79
hypocenterC79#10619
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. #23763556C2semidirect
5. #23761284C2semidirect
6. #23761234C2semidirect
7. C79×#1296C2semidirect
196. #1296D79semidirect
198. C79#10619semidirect

Maximal subgroups

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

subgroupindex
#237641932
#237635562
#237634012
#237633972
#237612842
#237612342
C79×#12962
#1061979