Finite Group Databasebeta

← #34183551

Subgroups — #34183551

Order 20224.

618 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098911C211yesmaximal, characteristic
2#23764327C211yesmaximal, characteristic
3#23764327C211yesmaximal, characteristic
4C2×#19098309C211yesmaximal, characteristic, complemented
5#23761241C211yesmaximal, characteristic, complemented
6#23761241C211yesmaximal, characteristic, complemented
7C79×#2365C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9#19098911—22no—
10#19098911C414yescomplemented
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×C4⋊(C79⋊C8)C2211yescharacteristic
13C2×#14579081—22no—
14C2×#14579081—21no—
15C2×#14579081—21no—
16C2×#14579081—21no—
17C2×#14579081—21no—
18C2×C4×C79⋊C8C2211yescharacteristic
19#19098309—22no—
20C2×C4⋊Q8×C79C2211yescharacteristic
21C2×C4×Q8×C79—21no—
22#391×C79—21no—
23#389×C79—21no—
24#387×C79—22no—
25C4×C4⋊C4×C79—21no—
26C4×C4⋊C4×C79C2211yescharacteristic
27C4×C4⋊C4×C79C2211yescharacteristic
28C22×D316—21no—
29C22×D316—21no—
30C22×D316—21no—
31C22×C79⋊C8—21no—
32C22×C79⋊C8—21no—
33C22×C79⋊C8—21no—
34#14579158—21no—
35#14579158—21no—
36#14579158—21no—
37#14579158C2×C412yes—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C4⋊(C79⋊C8)—21no—
42#14579081—28no—
43#14579081—42no—
44#14579081—42no—
45#14579081—42no—
46#14579081—42no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8C2×C412yes—
49C22×Q8×C79—21no—
50C4⋊Q8×C79—21no—
51C4⋊Q8×C79—21no—
52C4⋊Q8×C79C2×C412yes—
53C4⋊Q8×C79D412yes—
54C4×Q8×C79—42no—
55C4×Q8×C79—42no—
56C2×C4⋊C4×C79—21no—
57C2×C4⋊C4×C79—22no—
58C2×C4⋊C4×C79D411yescharacteristic
59C2×C4⋊C4×C79D411yescharacteristic
60C2×C4⋊C4×C79D412yes—
61C2×C4⋊C4×C79D411yescharacteristic
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C42×C79—22no—
64C2×C42×C79—21no—
65C2×C42×C79—21no—
66C2×C42×C79C2311yescharacteristic
67#95×C79—22no—
68#95×C79—22no—
69#95×C79—21no—
70#95×C79—21no—
71C23×D79—41no—
72C2×D316—22no—
73C2×D316—22no—
74C2×D316—22no—
75C2×D316—42no—
76C2×D316—42no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—41no—
82C2×D316—41no—
83C2×D316—41no—
84C2×D316—41no—
85C2×D316—41no—
86C2×D316—41no—
87C2×D316—41no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C2×C79⋊C8—41no—
94C2×C79⋊C8—41no—
95C2×C79⋊C8—41no—
96C2×C79⋊C8—41no—
97C2×Q8×C79—22no—
98C2×Q8×C79—22no—
99C2×Q8×C79—41no—
100C2×Q8×C79—41no—
101C2×Q8×C79—41no—
102C2×Q8×C79—41no—
103C22×C4×C79—21no—
104C22×C4×C79—22no—
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—21no—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×D411yescharacteristic
110C22×C4×C79C2×D411yescharacteristic
111C4⋊C4×C79—22no—
112C4⋊C4×C79—22no—
113C4⋊C4×C79—22no—
114C4⋊C4×C79—21no—
115C4⋊C4×C79—21no—
116C4⋊C4×C79—21no—
117C4⋊C4×C79—21no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—44no—
121C4⋊C4×C79C22⋊C414yes—
122C4⋊C4×C79D812yes—
123C4⋊C4×C79SD1612yes—
124C42×C79—44no—
125C42×C79—41no—
126C42×C79—41no—
127C42×C79—41no—
128C42×C79—41no—
129C42×C79C22×C411yescharacteristic
130C42×C79C2×D411yescharacteristic
131C42×C79#4111yescharacteristic
132C42×C79#4111yescharacteristic
133C23×C79#12011yescharacteristic
134C22×D79—42no—
135C22×D79—81no—
136C22×D79—81no—
137C22×D79—81no—
138C22×D79—81no—
139C22×D79—81no—
140C22×D79—81no—
141Q8×C79—42no—
142Q8×C79—42no—
143Q8×C79—42no—
144Q8×C79—42no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—24no—
148C2×C4×C79—22no—
149C2×C4×C79—22no—
150C2×C4×C79—41no—
151C2×C4×C79—42no—
152C2×C4×C79—41no—
153C2×C4×C79—42no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—41no—
159C2×C4×C79—41no—
160C2×C4×C79—41no—
161C2×C4×C79—41no—
162C2×C4×C79C2×C22⋊C411yescharacteristic
163C2×C4×C79C4×D411yescharacteristic
164C2×C4×C79C4×D411yescharacteristic
165C2×C4×C79#12111yescharacteristic
166C2×C4×C79#12111yescharacteristic
167C2×C4×C79#12311yescharacteristic
168C2×C4×C79C2×D811yescharacteristic
169C2×C4×C79C2×SD1611yescharacteristic
170C2×C4×C79#13511yescharacteristic
171C2×C4×C79#13511yescharacteristic
172C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
173C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C79#13711yescharacteristic
176D316—44no—
177D316—42no—
178D316—42no—
179D316—42no—
180D316—42no—
181D316—82no—
182D316—81no—
183D316—81no—
184D316—81no—
185D316—81no—
186C79⋊C8—42no—
187C79⋊C8—81no—
188C79⋊C8—81no—
189C22×C79#38611yescharacteristic
190C22×C79#44711yescharacteristic
191C22×C79#44811yescharacteristic
192C22×C79#44911yescharacteristic
193C22×C79#44911yescharacteristic
194C22×C79#45411yescharacteristic
195C22×C79#45411yescharacteristic
196C2×D79—82no—
197C2×D79—82no—
198C2×D79—82no—
199C2×D79—161no—
200C2×D79—161no—
201C2×D79—161no—
202C2×D79—161no—
203C4×C79—21no—
204C4×C79—21no—
205C4×C79—44no—
206C4×C79—42no—
207C4×C79—42no—
208C4×C79—81no—
209C4×C79—81no—
210C4×C79C4×D811yescharacteristic
211C4×C79C4×SD1611yescharacteristic
212C4×C79#44011yescharacteristic
213C4×C79#44211yescharacteristic
214C4×C79#45911yescharacteristic
215C4×C79#46011yescharacteristic
216C4×C79#46311yescharacteristic
217C4×C79#46411yescharacteristic
218#9062—791nomaximal, Sylow
219C2×C79#162511yescharacteristic
220C2×C79#162611yescharacteristic
221C2×C79#162711yescharacteristic
222C2×C79#162811yescharacteristic
223C2×C79#187911yescharacteristic
224C2×C79#188011yescharacteristic
225C2×C79#188811yescharacteristic
226D79—162no—
227C2×#486—791no—
228#2365D7911yescharacteristic, complemented
229#1982—791no—
230#1982—791no—
231C2×#332—791no—
232#1279—791no—
233#1279—791no—
234C79#906211yescharacteristic, complemented, Sylow
235C2×C4⋊Q8C2×D7911yescharacteristic
236C2×#127—791no—
237C2×C4×Q8—21no—
238#486—794no—
239#486—1582no—
240C2×C4⋊C8—791no—
241C2×C4⋊C8—791no—
242C2×#102—1581no—
243C2×#102—1581no—
244C2×#102—1581no—
245C2×#102—1581no—
246C2×#102—1582no—
247C2×C4×C8—791no—
248#391—21no—
249#389—21no—
250#387—22no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4C2×D7911yescharacteristic
253C4×C4⋊C4C2×D7911yescharacteristic
254#332—1582no—
255C22×Q8—21no—
256C22×D4—1581no—
257C22×D4—1581no—
258C22×D4—1581no—
259C22×C8—1581no—
260C22×C8—1581no—
261C22×C8—1581no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8C4×D7912yes—
265C4⋊Q8D31612yes—
266#127—792no—
267#127—1581no—
268#127—1581no—
269#127—1581no—
270C4×Q8—42no—
271C4×Q8—42no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4C22⋊D7911yescharacteristic
275C2×C4⋊C4C22⋊D7911yescharacteristic
276C2×C4⋊C4C22⋊D7912yes—
277C2×C4⋊C4C22⋊D7911yescharacteristic
278C2×C4⋊C4C22⋊D7911yescharacteristic
279C2×C42—22no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42C22×D7911yescharacteristic
283C4⋊C8—1581no—
284C4⋊C8—1581no—
285C4⋊C8—1581no—
286C4⋊C8—1581no—
287#102—3162no—
288#102—3162no—
289#102—3162no—
290#102—3162no—
291#102—1588no—
292C4×C8—792no—
293C4×C8—1581no—
294#95—22no—
295#95—22no—
296#95—21no—
297#95—21no—
298C24—3161no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—3162no—
306C2×D4—3162no—
307C2×D4—3161no—
308C2×D4—3161no—
309C2×D4—3161no—
310C2×D4—3161no—
311C2×D4—3161no—
312C2×D4—3161no—
313C2×D4—3161no—
314C2×D4—3161no—
315C2×D4—3161no—
316C2×D4—3161no—
317C2×D4—3161no—
318C2×D4—1582no—
319C2×D4—1582no—
320C2×D4—1582no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4C2×C22⋊D7911yescharacteristic
327C22×C4C2×C22⋊D7911yescharacteristic
328C22×C4C2×C22⋊D7911yescharacteristic
329C2×C8—3161no—
330C2×C8—3161no—
331C2×C8—3161no—
332C2×C8—3161no—
333C2×C8—3161no—
334C2×C8—3161no—
335C2×C8—3161no—
336C2×C8—3161no—
337C2×C8—1582no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—44no—
348C4⋊C4(C2×D79)⋊C414yes—
349C4⋊C4#1192960912yes—
350C4⋊C4#1192961112yes—
351C42—44no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42C2×C4×D7911yescharacteristic
357C42C2×D31611yescharacteristic
358C42#1192962411yescharacteristic
359C42#1192962411yescharacteristic
360C23—3162no—
361C23—6321no—
362C23—6321no—
363C23—6321no—
364C23—6321no—
365C23—6321no—
366C23—6321no—
367C23#1457913011yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372D4—3164no—
373D4—3162no—
374D4—3162no—
375D4—3162no—
376D4—3162no—
377D4—6322no—
378D4—6321no—
379D4—6321no—
380D4—6321no—
381D4—6321no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4C2×(C2×D79)⋊C411yescharacteristic
400C2×C4C4×C22⋊D7911yescharacteristic
401C2×C4C4×C22⋊D7911yescharacteristic
402C2×C4#1457913511yescharacteristic
403C2×C4#1457913511yescharacteristic
404C2×C4#1457914411yescharacteristic
405C2×C4C2×#1192960911yescharacteristic
406C2×C4C2×#1192961111yescharacteristic
407C2×C4#1457918111yescharacteristic
408C2×C4#1457918111yescharacteristic
409C2×C4#1457918611yescharacteristic
410C2×C4#1457918611yescharacteristic
411C2×C4#1457918711yescharacteristic
412C2×C4#1457919211yescharacteristic
413C8—3162no—
414C8—6321no—
415C8—6321no—
416C22—6322no—
417C22—6322no—
418C22—6322no—
419C22—12641no—
420C22—12641no—
421C22—12641no—
422C22—12641no—
423C22#1909844211yescharacteristic
424C22#1909869711yescharacteristic
425C22#1909870211yescharacteristic
426C22#1909870811yescharacteristic
427C22#1909870811yescharacteristic
428C22#1909873611yescharacteristic
429C22#1909873611yescharacteristic
430C4—21no—
431C4—21no—
432C4—44no—
433C4—42no—
434C4—42no—
435C4—81no—
436C4—81no—
437C4C4×#1192960911yescharacteristic
438C4C4×#1192961111yescharacteristic
439C4#1909866111yescharacteristic
440C4#1909867311yescharacteristic
441C4#1909875811yescharacteristic
442C4#1909876311yescharacteristic
443C4#1909877811yescharacteristic
444C4#1909878511yescharacteristic
445C2—12642no—
446C2#2376248911yescharacteristic
447C2#2376249611yescharacteristic
448C2#2376250311yescharacteristic
449C2#2376251011yescharacteristic
450C2#2376374811yescharacteristic
451C2#2376375511yescharacteristic
452C2#2376380311yescharacteristic
subgroupG/H
centerC23#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2365C2
socleC23×C79#120
prime radicalC2×#14579158C22
cosocleC2×C42×C79C23
hypercenter#2365D79
hypocenterC79#9062
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2365

Complements

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

NH ≅ G/Ntype
4. C2×#19098309C2semidirect
5. #23761241C2semidirect
6. #23761241C2semidirect
7. C79×#2365C2semidirect
10. #19098911C4semidirect
228. #2365D79semidirect
234. C79#9062semidirect

Maximal subgroups

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

subgroupindex
C2×#190989112
#237643272
C2×#190983092
#237612412
C79×#23652
#906279