Finite Group Databasebeta

← #34192867

Subgroups — #34192867

Order 20224.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23764327C211yesmaximal, characteristic
2#23764205C211yesmaximal, characteristic
3#23764119C211yesmaximal, characteristic, complemented
4C2×#19098315C211yesmaximal, characteristic, complemented
5C2×#19098306C211yesmaximal, characteristic
6#23761283C211yesmaximal, characteristic, complemented
7C79×#1948C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9C2×C4⋊(C79⋊C8)—21no—
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×#14579081—21no—
13C2×#14579081—21no—
14C2×#14579081—21no—
15C2×#14579081—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17#19098318—21no—
18#19098315—22no—
19#19098306—22no—
20C2×#106×C79—21no—
21C2×C4⋊C8×C79C2211yescharacteristic
22#384×C79C2211yescharacteristic
23C4×C4⋊C4×C79C2211yescharacteristic
24#340×C79—21no—
25C22×D316—21no—
26C22×D316—21no—
27C22×D316—21no—
28C22×C79⋊C8—21no—
29C22×C79⋊C8—21no—
30C22×C79⋊C8D411yescharacteristic
31C22×C79⋊C8D411yescharacteristic
32#14579158—21no—
33#14579158—21no—
34#14579158—21no—
35#14579158Q812yes—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—42no—
41#14579081—42no—
42#14579081—42no—
43#14579081—42no—
44#14579081—42no—
45C4×C79⋊C8—21no—
46C4×C79⋊C8—21no—
47C22×C8×C79—21no—
48C2×C4⋊C4×C79—21no—
49C2×C4⋊C4×C79—21no—
50C2×C4⋊C4×C79—21no—
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C4⋊C4×C79D411yescharacteristic
53C2×C42×C79—21no—
54C2×C42×C79C2311yescharacteristic
55#106×C79—42no—
56C4⋊C8×C79—21no—
57C4⋊C8×C79D412yes—
58#95×C79—21no—
59C23×D79—41no—
60C2×D316—22no—
61C2×D316—22no—
62C2×D316—22no—
63C2×D316—41no—
64C2×D316—41no—
65C2×D316—41no—
66C2×D316—41no—
67C2×D316—41no—
68C2×D316—41no—
69C2×D316—41no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—81no—
75C2×D316—81no—
76C2×C79⋊C8—21no—
77C2×C79⋊C8—21no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8—21no—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C2×C79⋊C8SD1612yes—
88C22×C4×C79—21no—
89C22×C4×C79—21no—
90C22×C4×C79—41no—
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C22×C4×C79#4111yescharacteristic
94C2×C8×C79—22no—
95C2×C8×C79—41no—
96C2×C8×C79—41no—
97C4⋊C4×C79—21no—
98C4⋊C4×C79—21no—
99C4⋊C4×C79—21no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—41no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79—81no—
105C4⋊C4×C79D812yes—
106C42×C79—41no—
107C42×C79—41no—
108C42×C79C2×D411yescharacteristic
109C42×C79C2×Q811yescharacteristic
110C42×C79#4111yescharacteristic
111C42×C79#4111yescharacteristic
112C23×C79#12111yescharacteristic
113C22×D79—42no—
114C22×D79—81no—
115C22×D79—81no—
116C22×D79—81no—
117C22×D79—81no—
118C22×D79—81no—
119C22×D79—81no—
120C2×C4×C79—21no—
121C2×C4×C79—21no—
122C2×C4×C79—22no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—81no—
129C2×C4×C79—81no—
130C2×C4×C79—81no—
131C2×C4×C79#12111yescharacteristic
132C2×C4×C79#12111yescharacteristic
133C2×C4×C79C22⋊Q811yescharacteristic
134C2×C4×C79C22⋊Q811yescharacteristic
135C2×C4×C79C22⋊Q811yescharacteristic
136C2×C4×C79#12611yescharacteristic
137C2×C4×C79C2×D811yescharacteristic
138C2×C4×C79C2×SD1611yescharacteristic
139C2×C4×C79#13511yescharacteristic
140C2×C4×C79#13511yescharacteristic
141C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C79#13711yescharacteristic
145D316—42no—
146D316—42no—
147D316—42no—
148D316—42no—
149D316—82no—
150D316—82no—
151D316—81no—
152D316—81no—
153D316—81no—
154D316—81no—
155C8×C79—42no—
156C79⋊C8—22no—
157C79⋊C8—41no—
158C79⋊C8—81no—
159C79⋊C8—81no—
160C22×C79#39611yescharacteristic
161C22×C79#46011yescharacteristic
162C22×C79#46311yescharacteristic
163C22×C79#46611yescharacteristic
164C22×C79#46811yescharacteristic
165C22×C79#47111yescharacteristic
166C22×C79#47211yescharacteristic
167C2×D79—82no—
168C2×D79—82no—
169C2×D79—82no—
170C2×D79—161no—
171C2×D79—161no—
172C2×D79—161no—
173C2×D79—161no—
174C4×C79—21no—
175C4×C79—21no—
176C4×C79—42no—
177C4×C79—81no—
178C4×C79—161no—
179C4×C79#45911yescharacteristic
180C4×C79#46411yescharacteristic
181C4×C79#46511yescharacteristic
182C4×C79#46911yescharacteristic
183C4×C79#47411yescharacteristic
184C4×C79#47611yescharacteristic
185C4×C79#47811yescharacteristic
186C4×C79#47811yescharacteristic
187#10599—791nomaximal, Sylow
188C2×C79#166811yescharacteristic
189C2×C79#168111yescharacteristic
190C2×C79#168611yescharacteristic
191C2×C79#169711yescharacteristic
192C2×C79#206311yescharacteristic
193C2×C79#206511yescharacteristic
194C2×C79#206711yescharacteristic
195D79—162no—
196#1982—791no—
197#1957—791no—
198#1948D7911yescharacteristic, complemented
199#1939—791no—
200C2×#335—791no—
201C2×#331—791no—
202#1296—791no—
203C79#1059911yescharacteristic, complemented, Sylow
204C2×#127—791no—
205C2×#106—21no—
206C2×C4⋊C8—791no—
207C2×C4⋊C8—791no—
208C2×C4⋊C8—1581no—
209C2×C4⋊C8C2×D7911yescharacteristic
210C2×#102—1581no—
211C2×#102—1581no—
212C2×#102—1581no—
213C2×#102—1581no—
214C2×C4×C8—791no—
215#384C2×D7911yescharacteristic
216C4×C4⋊C4C2×D7911yescharacteristic
217#340—21no—
218#336—1581no—
219#335—1582no—
220#331—1582no—
221C22×D4—1581no—
222C22×D4—1581no—
223C22×D4—1581no—
224C22×C8—21no—
225C22×C8—791no—
226C22×C8—791no—
227C22×C8—1581no—
228C22×C8—1581no—
229#127—792no—
230#127—1581no—
231#127—1581no—
232#127—1581no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C22⋊D7911yescharacteristic
237C2×C4⋊C4C22⋊D7911yescharacteristic
238C2×C42—21no—
239C2×C42C22×D7911yescharacteristic
240#106—42no—
241C4⋊C8—21no—
242C4⋊C8—3162no—
243C4⋊C8—1581no—
244C4⋊C8—1581no—
245C4⋊C8—1581no—
246C4⋊C8—1581no—
247C4⋊C8D31612yes—
248#102—3162no—
249#102—3162no—
250#102—3162no—
251#102—3162no—
252C4×C8—1581no—
253C4×C8—1581no—
254#95—21no—
255C24—3161no—
256C2×D4—3161no—
257C2×D4—3161no—
258C2×D4—3161no—
259C2×D4—3161no—
260C2×D4—3161no—
261C2×D4—3161no—
262C2×D4—3161no—
263C2×D4—3161no—
264C2×D4—3161no—
265C2×D4—3161no—
266C2×D4—3161no—
267C2×D4—6321no—
268C2×D4—6321no—
269C2×D4—1582no—
270C2×D4—1582no—
271C2×D4—1582no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4D4×D7911yescharacteristic
276C22×C4C2×C22⋊D7911yescharacteristic
277C22×C4#1192962311yescharacteristic
278C2×C8—22no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—3161no—
282C2×C8—3161no—
283C2×C8—3161no—
284C2×C8—3161no—
285C2×C8—3161no—
286C2×C8—3161no—
287C2×C8—792no—
288C2×C8—1581no—
289C2×C8—1581no—
290C2×C8—1581no—
291C2×C8—1581no—
292C2×C8—1581no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C4⋊C4#1192960912yes—
302C42—41no—
303C42—41no—
304C42C2×D31611yescharacteristic
305C42Q8×D7911yescharacteristic
306C42#1192962411yescharacteristic
307C42#1192962511yescharacteristic
308C23—3162no—
309C23—6321no—
310C23—6321no—
311C23—6321no—
312C23—6321no—
313C23—6321no—
314C23—6321no—
315C23#1457913311yescharacteristic
316D4—3162no—
317D4—3162no—
318D4—3162no—
319D4—3162no—
320D4—6322no—
321D4—6322no—
322D4—6321no—
323D4—6321no—
324D4—6321no—
325D4—6321no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4#1457913511yescharacteristic
338C2×C4#1457913611yescharacteristic
339C2×C4#1457913711yescharacteristic
340C2×C4#1457913911yescharacteristic
341C2×C4#1457914111yescharacteristic
342C2×C4#1457915611yescharacteristic
343C2×C4C2×#1192960911yescharacteristic
344C2×C4SD16×D7911yescharacteristic
345C2×C4#1457918111yescharacteristic
346C2×C4#1457918211yescharacteristic
347C2×C4#1457918611yescharacteristic
348C2×C4#1457918811yescharacteristic
349C2×C4#1457918811yescharacteristic
350C2×C4#1457919211yescharacteristic
351C8—42no—
352C8—3161no—
353C8—6321no—
354C8—6321no—
355C8—1582no—
356C22—6322no—
357C22—6322no—
358C22—6322no—
359C22—12641no—
360C22—12641no—
361C22—12641no—
362C22—12641no—
363C22#1909848511yescharacteristic
364C22#1909876111yescharacteristic
365C22#1909878011yescharacteristic
366C22#1909879711yescharacteristic
367C22#1909881111yescharacteristic
368C22#1909882811yescharacteristic
369C22#1909883111yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—161no—
375C4#1909875811yescharacteristic
376C4#1909878511yescharacteristic
377C4#1909879311yescharacteristic
378C4#1909881711yescharacteristic
379C4#1909884011yescharacteristic
380C4#1909885211yescharacteristic
381C4#1909886011yescharacteristic
382C4#1909886211yescharacteristic
383C2—12642no—
384C2#2376275811yescharacteristic
385C2#2376283911yescharacteristic
386C2#2376287411yescharacteristic
387C2#2376293511yescharacteristic
388C2#2376475711yescharacteristic
389C2#2376476911yescharacteristic
390C2#2376478511yescharacteristic
subgroupG/H
centerC23#14579133
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1948C2
socleC23×C79#121
prime radicalC2×#14579158C22
hypercenter#1948D79
hypocenterC79#10599
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
3. #23764119C2semidirect
4. C2×#19098315C2semidirect
6. #23761283C2semidirect
7. C79×#1948C2semidirect
198. #1948D79semidirect
203. C79#10599semidirect

Maximal subgroups

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

subgroupindex
#237643272
#237642052
#237641192
C2×#190983152
C2×#190983062
#237612832
C79×#19482
#1059979