Finite Group Databasebeta

← #34191314

Subgroups — #34191314

Order 20224.

536 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23768431C211yesmaximal, characteristic, complemented
2#23764327C211yesmaximal, characteristic
3#23764326C211yesmaximal, characteristic
4#23764206C211yesmaximal, characteristic
5#23761284C211yesmaximal, characteristic, complemented
6#23761240C211yesmaximal, characteristic, complemented
7C79×#1279C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9C2×#14579152—21no—
10C2×C4⋊(C79⋊C8)C2211yescharacteristic
11C2×#14579083—21no—
12C2×#14579083—21no—
13C2×#14579083—22no—
14C2×#14579081—21no—
15C2×#14579081—21no—
16#19098495—22no—
17#19098462—21no—
18#19098461—21no—
19#19098433—21no—
20#19098433C2211yescharacteristic
21C4×C4⋊Dic79C2211yescharacteristic
22C2×C4⋊C8×C79C2211yescharacteristic
23C2×C4×C8×C79C2211yescharacteristic
24C4×C4⋊C4×C79C2211yescharacteristic
25C22×D316—21no—
26C22×D316—21no—
27C22×D316—21no—
28C22×C79⋊C8—21no—
29#14579158—21no—
30#14579158—21no—
31#14579158—21no—
32#14579158Q812yes—
33#14579152—42no—
34#14579152—42no—
35C2×C4⋊Dic79—22no—
36C2×C4⋊Dic79—21no—
37C2×C4⋊Dic79—21no—
38C2×C4⋊Dic79—22no—
39C2×C4⋊Dic79D411yescharacteristic
40C2×C4⋊Dic79D411yescharacteristic
41C2×C4⋊Dic79D412yes—
42C2×Dic79⋊C4—22no—
43C2×C4×Dic79—21no—
44C4⋊(C79⋊C8)—21no—
45C4⋊(C79⋊C8)—21no—
46#14579083—42no—
47#14579083—42no—
48#14579083—44no—
49#14579081—42no—
50#14579081—42no—
51C79⋊#95—22no—
52C79⋊#95—21no—
53C22×C8×C79—21no—
54C22×C8×C79—21no—
55C2×C4⋊C4×C79D411yescharacteristic
56C2×C4⋊C4×C79D411yescharacteristic
57C2×C42×C79—21no—
58C2×C42×C79C2311yescharacteristic
59C4⋊C8×C79—21no—
60C4⋊C8×C79—21no—
61C4×C8×C79—21no—
62C4×C8×C79—21no—
63#95×C79—21no—
64C23×D79—41no—
65C2×D316—22no—
66C2×D316—22no—
67C2×D316—22no—
68C2×D316—42no—
69C2×D316—42no—
70C2×D316—41no—
71C2×D316—41no—
72C2×D316—41no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—41no—
80C2×D316—41no—
81C22×Dic79—21no—
82C22×Dic79—21no—
83C22×Dic79—22no—
84C2×C79⋊C8—41no—
85C2×C79⋊C8—41no—
86C2×C79⋊C8—41no—
87C4⋊Dic79—21no—
88C4⋊Dic79—21no—
89C4⋊Dic79—21no—
90C4⋊Dic79—21no—
91C4⋊Dic79—42no—
92C4⋊Dic79—44no—
93C4⋊Dic79—41no—
94C4⋊Dic79—41no—
95C4⋊Dic79—41no—
96C4⋊Dic79—41no—
97C4⋊Dic79—82no—
98Dic79⋊C4—44no—
99C4×Dic79—41no—
100C4×Dic79—41no—
101C22×C4×C79—21no—
102C22×C4×C79—21no—
103C22×C4×C79C2×D411yescharacteristic
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C2×C8×C79—22no—
107C2×C8×C79—41no—
108C2×C8×C79—41no—
109C2×C8×C79—41no—
110C2×C8×C79—41no—
111C2×C8×C79—41no—
112C4⋊C4×C79—21no—
113C4⋊C4×C79—21no—
114C4⋊C4×C79D812yes—
115C4⋊C4×C79SD1612yes—
116C42×C79—41no—
117C42×C79—41no—
118C42×C79C2×Q811yescharacteristic
119C42×C79#4111yescharacteristic
120C42×C79#4111yescharacteristic
121C42×C79#4111yescharacteristic
122C23×C79#12011yescharacteristic
123C22×D79—42no—
124C22×D79—81no—
125C22×D79—81no—
126C22×D79—81no—
127C22×D79—81no—
128C22×D79—81no—
129C22×D79—81no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—22no—
133C2×C4×C79—22no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79C22⋊Q811yescharacteristic
139C2×C4×C79C22⋊Q811yescharacteristic
140C2×C4×C79C22⋊Q811yescharacteristic
141C2×C4×C79#12411yescharacteristic
142C2×C4×C79#12411yescharacteristic
143C2×C4×C79#12411yescharacteristic
144C2×C4×C79C2×D811yescharacteristic
145C2×C4×C79C2×SD1611yescharacteristic
146C2×C4×C79#13511yescharacteristic
147C2×C4×C79#13511yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C79#13711yescharacteristic
152C2×Dic79—41no—
153C2×Dic79—41no—
154C2×Dic79—42no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—42no—
158C2×Dic79—42no—
159C2×Dic79—41no—
160C2×Dic79—41no—
161D316—44no—
162D316—42no—
163D316—42no—
164D316—42no—
165D316—42no—
166D316—82no—
167D316—81no—
168D316—81no—
169D316—81no—
170D316—81no—
171C8×C79—42no—
172C8×C79—81no—
173C79⋊C8—81no—
174C22×C79#39311yescharacteristic
175C22×C79#44711yescharacteristic
176C22×C79#44811yescharacteristic
177C22×C79#44911yescharacteristic
178C22×C79#44911yescharacteristic
179C22×C79#45311yescharacteristic
180C22×C79#45411yescharacteristic
181C2×D79—82no—
182C2×D79—82no—
183C2×D79—82no—
184C2×D79—161no—
185C2×D79—161no—
186C2×D79—161no—
187C2×D79—161no—
188C4×C79—21no—
189C4×C79—21no—
190C4×C79—42no—
191C4×C79—42no—
192C4×C79#47411yescharacteristic
193C4×C79#47611yescharacteristic
194C4×C79#47811yescharacteristic
195C4×C79#47811yescharacteristic
196C4×C79#48611yescharacteristic
197C4×C79#48811yescharacteristic
198C4×C79#48911yescharacteristic
199C4×C79#49011yescharacteristic
200Dic79—81no—
201Dic79—82no—
202Dic79—81no—
203#10354—791nomaximal, Sylow
204C2×C79#162111yescharacteristic
205C2×C79#162211yescharacteristic
206C2×C79#162511yescharacteristic
207C2×C79#162611yescharacteristic
208C2×C79#202311yescharacteristic
209C2×C79#202511yescharacteristic
210C2×C79#202711yescharacteristic
211D79—162no—
212#2552—791no—
213#1982—791no—
214#1982—791no—
215#1957—791no—
216#1296—791no—
217#1279—791no—
218#1279D7911yescharacteristic, complemented
219C79#1035411yescharacteristic, complemented, Sylow
220C2×#127—791no—
221C2×#125—1581no—
222C2×C4⋊C8—791no—
223C2×C4⋊C8C2×D7911yescharacteristic
224C2×#102—1582no—
225C2×#102—1581no—
226C2×#102—1581no—
227C2×#102—1581no—
228C2×#102—1581no—
229C2×C4×C8C2×D7911yescharacteristic
230#398—1582no—
231#389—1581no—
232#389—1581no—
233#384—791no—
234#384—1581no—
235C4×C4⋊C4—791no—
236C4×C4⋊C4C2×D7911yescharacteristic
237C22×D4—1581no—
238C22×D4—1581no—
239C22×D4—1581no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—1581no—
243#127—792no—
244#127—1581no—
245#127—1581no—
246#127—1581no—
247#125—3162no—
248#125—3162no—
249C2×C4⋊C4—792no—
250C2×C4⋊C4—791no—
251C2×C4⋊C4—791no—
252C2×C4⋊C4—1582no—
253C2×C4⋊C4—1582no—
254C2×C4⋊C4—1581no—
255C2×C4⋊C4—1581no—
256C2×C4⋊C4—1582no—
257C2×C4⋊C4D31611yescharacteristic
258C2×C4⋊C4D31611yescharacteristic
259C2×C42—21no—
260C2×C42—1581no—
261C2×C42C22×D7911yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—1581no—
265C4⋊C8—1581no—
266#102—3164no—
267#102—3162no—
268#102—3162no—
269#102—3162no—
270#102—3162no—
271C4×C8—21no—
272C4×C8—21no—
273#95—21no—
274#95—1582no—
275#95—1581no—
276C24—3161no—
277C2×D4—3162no—
278C2×D4—3162no—
279C2×D4—3161no—
280C2×D4—3161no—
281C2×D4—3161no—
282C2×D4—3161no—
283C2×D4—3161no—
284C2×D4—3161no—
285C2×D4—3161no—
286C2×D4—3161no—
287C2×D4—3161no—
288C2×D4—3161no—
289C2×D4—3161no—
290C2×D4—1582no—
291C2×D4—1582no—
292C2×D4—1582no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—1581no—
296C22×C4—1581no—
297C22×C4—1582no—
298C22×C4D4×D7911yescharacteristic
299C22×C4D4×D7911yescharacteristic
300C22×C4C2×D31611yescharacteristic
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—3161no—
308C2×C8—3161no—
309C2×C8—3161no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—3162no—
313C4⋊C4—3164no—
314C4⋊C4—3164no—
315C4⋊C4—3161no—
316C4⋊C4—3161no—
317C4⋊C4—3161no—
318C4⋊C4—3161no—
319C4⋊C4—6322no—
320C4⋊C4—1581no—
321C4⋊C4—1581no—
322C4⋊C4—1581no—
323C4⋊C4—1581no—
324C4⋊C4D63212yes—
325C4⋊C4#1192961312yes—
326C42—41no—
327C42—41no—
328C42—3161no—
329C42—3161no—
330C42Q8×D7911yescharacteristic
331C42#1192962311yescharacteristic
332C42#1192962411yescharacteristic
333C42#1192962411yescharacteristic
334C23—3162no—
335C23—6321no—
336C23—6321no—
337C23—6321no—
338C23—6321no—
339C23—6321no—
340C23—6321no—
341C23#1457913111yescharacteristic
342D4—3164no—
343D4—3162no—
344D4—3162no—
345D4—3162no—
346D4—3162no—
347D4—6322no—
348D4—6321no—
349D4—6321no—
350D4—6321no—
351D4—6321no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—3161no—
361C2×C4—3161no—
362C2×C4—3162no—
363C2×C4—3161no—
364C2×C4—3162no—
365C2×C4—3161no—
366C2×C4—3162no—
367C2×C4—3161no—
368C2×C4—3161no—
369C2×C4#1457913911yescharacteristic
370C2×C4#1457913911yescharacteristic
371C2×C4#1457914111yescharacteristic
372C2×C4#1457914711yescharacteristic
373C2×C4#1457914711yescharacteristic
374C2×C4#1457914911yescharacteristic
375C2×C4C2×D63211yescharacteristic
376C2×C4C2×#1192961311yescharacteristic
377C2×C4#1457918211yescharacteristic
378C2×C4#1457918411yescharacteristic
379C2×C4#1457918511yescharacteristic
380C2×C4#1457918811yescharacteristic
381C2×C4#1457918911yescharacteristic
382C2×C4#1457919411yescharacteristic
383C8—42no—
384C8—81no—
385C8—6321no—
386C22—6322no—
387C22—6322no—
388C22—6322no—
389C22—12641no—
390C22—12641no—
391C22—12641no—
392C22—12641no—
393C22#1909847611yescharacteristic
394C22#1909869911yescharacteristic
395C22#1909870411yescharacteristic
396C22#1909870911yescharacteristic
397C22#1909871111yescharacteristic
398C22#1909873211yescharacteristic
399C22#1909873711yescharacteristic
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—6321no—
405C4—6322no—
406C4—6321no—
407C4#1909884411yescharacteristic
408C4#1909885411yescharacteristic
409C4#1909886211yescharacteristic
410C4#1909886611yescharacteristic
411C4#1909891011yescharacteristic
412C4#1909891911yescharacteristic
413C4#1909892311yescharacteristic
414C4#1909892611yescharacteristic
415C2—12642no—
416C2#2376246711yescharacteristic
417C2#2376247411yescharacteristic
418C2#2376249111yescharacteristic
419C2#2376249911yescharacteristic
420C2#2376453711yescharacteristic
421C2#2376454511yescharacteristic
422C2#2376455511yescharacteristic
subgroupG/H
centerC23#14579131
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1279C2
socleC23×C79#120
prime radicalC2×#14579158C22
hypercenter#1279D79
hypocenterC79#10354
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. #23768431C2semidirect
5. #23761284C2semidirect
6. #23761240C2semidirect
7. C79×#1279C2semidirect
218. #1279D79semidirect
219. C79#10354semidirect

Maximal subgroups

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

subgroupindex
#237684312
#237643272
#237643262
#237642062
#237612842
#237612402
C79×#12792
#1035479