Finite Group Databasebeta

← #34191648

Subgroups — #34191648

Order 20224.

572 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
1#23768399C211yesmaximal, characteristic
2#23764205C211yesmaximal, characteristic
3#23764156C211yesmaximal, characteristic, complemented
4C2×#19098306C211yesmaximal, characteristic
5C2×Q8⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
6#23761239C211yesmaximal, characteristic, complemented
7C79×#1803C211yesmaximal, characteristic, complemented
8C2×#14579158C2211yescharacteristic
9C2×#14579150—21no—
10C2×#14579094—21no—
11C2×C4⋊(C79⋊C8)C2211yescharacteristic
12C2×#14579081—22no—
13C2×C4×C79⋊C8C2211yescharacteristic
14#19098480—22no—
15#19098465—21no—
16#19098465—21no—
17#19098464—21no—
18C4×C4⋊Dic79C2211yescharacteristic
19#19098325—21no—
20#19098306—22no—
21Q8⋊(C79⋊C8)—22no—
22C2×C4×Q8×C79C2211yescharacteristic
23C2×C4⋊C8×C79C2211yescharacteristic
24C2×#103×C79—21no—
25#384×C79C2211yescharacteristic
26#340×C79—21no—
27C22×D316—21no—
28C22×D316—21no—
29C22×D316D412yes—
30C22×C79⋊C8—21no—
31C22×C79⋊C8—21no—
32#14579158—21no—
33#14579158—21no—
34#14579158—21no—
35#14579158—21no—
36#14579150—42no—
37#14579150—42no—
38C2×C4⋊Dic79D411yescharacteristic
39C2×C4⋊Dic79D411yescharacteristic
40C2×(C2×D79)⋊C4—22no—
41C2×(C2×D79)⋊C4—22no—
42C2×(C2×D79)⋊C4—22no—
43C2×(C2×D79)⋊C4—22no—
44C2×C4×Dic79—21no—
45#14579094—42no—
46C4⋊(C79⋊C8)—21no—
47C4⋊(C79⋊C8)D412yes—
48#14579081—44no—
49C4×C79⋊C8—21no—
50C4×C79⋊C8—21no—
51C79⋊#95—21no—
52C22×Q8×C79D411yescharacteristic
53C22×C8×C79—21no—
54C4×Q8×C79—21no—
55C4×Q8×C79—21no—
56C4×Q8×C79—21no—
57C4×Q8×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C4⋊C4×C79—21no—
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C42×C79—21no—
64C2×C42×C79C2311yescharacteristic
65C4⋊C8×C79—21no—
66C4⋊C8×C79—21no—
67#103×C79—42no—
68C23×D79—22no—
69C2×D316—22no—
70C2×D316—22no—
71C2×D316—42no—
72C2×D316—42no—
73C2×D316—41no—
74C2×D316—41no—
75C2×D316—41no—
76C2×D316—41no—
77C2×D316—41no—
78C2×D316—41no—
79C2×D316—41no—
80C2×D316—41no—
81C2×D316—41no—
82C2×D316—41no—
83C2×D316—41no—
84C2×D316—41no—
85C22×Dic79—21no—
86C22×Dic79—21no—
87C2×C79⋊C8—22no—
88C2×C79⋊C8—22no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—41no—
91C2×C79⋊C8—41no—
92C2×C79⋊C8—41no—
93C4⋊Dic79—21no—
94C4⋊Dic79—21no—
95C4⋊Dic79—21no—
96C4⋊Dic79—21no—
97(C2×D79)⋊C4—44no—
98(C2×D79)⋊C4—44no—
99(C2×D79)⋊C4—82no—
100(C2×D79)⋊C4—82no—
101C4×Dic79—41no—
102C4×Dic79—41no—
103C2×Q8×C79—21no—
104C2×Q8×C79—21no—
105C2×Q8×C79—21no—
106C2×Q8×C79—21no—
107C2×Q8×C79—21no—
108C2×Q8×C79D812yes—
109C22×C4×C79—21no—
110C22×C4×C79—21no—
111C22×C4×C79—41no—
112C22×C4×C79C2×D411yescharacteristic
113C22×C4×C79C2×D411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C2×C8×C79—41no—
116C2×C8×C79—41no—
117C2×C8×C79—41no—
118C4⋊C4×C79—21no—
119C4⋊C4×C79—21no—
120C4⋊C4×C79—42no—
121C4⋊C4×C79—41no—
122C4⋊C4×C79—41no—
123C4⋊C4×C79—41no—
124C4⋊C4×C79—41no—
125C4⋊C4×C79—41no—
126C4⋊C4×C79—41no—
127C42×C79—41no—
128C42×C79—41no—
129C42×C79C2×D411yescharacteristic
130C42×C79#4111yescharacteristic
131C42×C79#4111yescharacteristic
132C42×C79#4111yescharacteristic
133C23×C79#12011yescharacteristic
134C22×D79—42no—
135C22×D79—42no—
136C22×D79—42no—
137C22×D79—42no—
138C22×D79—42no—
139C22×D79—42no—
140C22×D79—42no—
141Q8×C79—22no—
142Q8×C79—22no—
143Q8×C79—41no—
144Q8×C79—41no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—22no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—81no—
154C2×C4×C79—81no—
155C2×C4×C79—81no—
156C2×C4×C79#12111yescharacteristic
157C2×C4×C79#12111yescharacteristic
158C2×C4×C79#12111yescharacteristic
159C2×C4×C79#12311yescharacteristic
160C2×C4×C79#12311yescharacteristic
161C2×C4×C79#12411yescharacteristic
162C2×C4×C79C2×D811yescharacteristic
163C2×C4×C79#13511yescharacteristic
164C2×C4×C79#13511yescharacteristic
165C2×C4×C79#13511yescharacteristic
166C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C79#13711yescharacteristic
169C2×C4×C79#13711yescharacteristic
170C2×Dic79—41no—
171C2×Dic79—41no—
172C2×Dic79—41no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79—41no—
176D316—44no—
177D316—82no—
178D316—81no—
179D316—81no—
180D316—81no—
181D316—81no—
182D316—81no—
183D316—81no—
184D316—81no—
185D316—81no—
186C8×C79—81no—
187C79⋊C8—42no—
188C79⋊C8—42no—
189C22×C79#39411yescharacteristic
190C22×C79#44711yescharacteristic
191C22×C79#44911yescharacteristic
192C22×C79#45211yescharacteristic
193C22×C79#45211yescharacteristic
194C22×C79#45311yescharacteristic
195C22×C79#45511yescharacteristic
196C2×D79—82no—
197C2×D79—82no—
198C2×D79—82no—
199C2×D79—82no—
200C2×D79—82no—
201C2×D79—82no—
202C2×D79—82no—
203C4×C79—21no—
204C4×C79—21no—
205C4×C79—42no—
206C4×C79—81no—
207C4×C79—161no—
208C4×C79#46311yescharacteristic
209C4×C79#46411yescharacteristic
210C4×C79#46611yescharacteristic
211C4×C79#46811yescharacteristic
212C4×C79#48011yescharacteristic
213C4×C79#48211yescharacteristic
214C4×C79#48811yescharacteristic
215C4×C79#48911yescharacteristic
216Dic79—81no—
217Dic79—81no—
218#10411—791nomaximal, Sylow
219C2×C79#162111yescharacteristic
220C2×C79#163011yescharacteristic
221C2×C79#164011yescharacteristic
222C2×C79#164111yescharacteristic
223C2×C79#203311yescharacteristic
224C2×C79#203711yescharacteristic
225C2×C79#204211yescharacteristic
226D79—162no—
227#2548—791no—
228#1957—791no—
229#1947—791no—
230#1803D7911yescharacteristic, complemented
231C2×#331—791no—
232C2×Q8⋊C8—791no—
233#1279—791no—
234C79#1041111yescharacteristic, complemented, Sylow
235C2×#127—791no—
236C2×#124—1581no—
237C2×C4×Q8C2×D7911yescharacteristic
238C2×#107—1581no—
239C2×C4⋊C8—791no—
240C2×C4⋊C8C2×D7911yescharacteristic
241C2×#103—21no—
242C2×#102—1582no—
243C2×C4×C8—791no—
244#394—1582no—
245#390—1581no—
246#390—1581no—
247#390—1581no—
248#384C2×D7911yescharacteristic
249C4×C4⋊C4—791no—
250#340—21no—
251#340—1581no—
252#331—1582no—
253Q8⋊C8—1582no—
254C22×Q8C22⋊D7911yescharacteristic
255C22×D4—792no—
256C22×D4—1581no—
257C22×D4—1581no—
258C22×C8—21no—
259C22×C8—1581no—
260C22×C8—1581no—
261#127—1581no—
262#127—1581no—
263#127—1581no—
264#127—1581no—
265#124—3162no—
266#124—3162no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—791no—
276C2×C4⋊C4—791no—
277C2×C4⋊C4C22⋊D7911yescharacteristic
278C2×C22⋊C4—1582no—
279C2×C22⋊C4—1582no—
280C2×C22⋊C4—1582no—
281C2×C22⋊C4—1582no—
282C2×C42—21no—
283C2×C42—1581no—
284C2×C42C22×D7911yescharacteristic
285#107—3162no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—792no—
289C4⋊C8—1581no—
290#103—42no—
291#102—3164no—
292C4×C8—1581no—
293C4×C8—1581no—
294#95—1581no—
295C24—1582no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8#1192960912yes—
302C2×D4—3162no—
303C2×D4—3162no—
304C2×D4—3161no—
305C2×D4—3161no—
306C2×D4—3161no—
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—1582no—
317C2×D4—1582no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—1581no—
322C22×C4—1581no—
323C22×C4D4×D7911yescharacteristic
324C22×C4D4×D7911yescharacteristic
325C22×C4C2×C22⋊D7911yescharacteristic
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—3161no—
330C2×C8—3161no—
331C2×C8—3161no—
332C2×C8—3161no—
333C2×C8—1582no—
334C2×C8—1582no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—1581no—
345C4⋊C4—1581no—
346C4⋊C4—1581no—
347C4⋊C4—1581no—
348C22⋊C4—3164no—
349C22⋊C4—3164no—
350C22⋊C4—6322no—
351C22⋊C4—6322no—
352C42—41no—
353C42—41no—
354C42—3161no—
355C42—3161no—
356C42D4×D7911yescharacteristic
357C42#1192962311yescharacteristic
358C42#1192962411yescharacteristic
359C42#1192962411yescharacteristic
360C23—3162no—
361C23—3162no—
362C23—3162no—
363C23—3162no—
364C23—3162no—
365C23—3162no—
366C23—3162no—
367C23#1457912911yescharacteristic
368Q8—22no—
369Q8—22no—
370Q8—41no—
371Q8—41no—
372D4—3164no—
373D4—6322no—
374D4—6321no—
375D4—6321no—
376D4—6321no—
377D4—6321no—
378D4—6321no—
379D4—6321no—
380D4—6321no—
381D4—6321no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—3161no—
394C2×C4—3161no—
395C2×C4—3161no—
396C2×C4—3161no—
397C2×C4—3161no—
398C2×C4—3161no—
399C2×C4#1457913311yescharacteristic
400C2×C4#1457913411yescharacteristic
401C2×C4#1457913411yescharacteristic
402C2×C4#1457914311yescharacteristic
403C2×C4#1457914411yescharacteristic
404C2×C4#1457914711yescharacteristic
405C2×C4C2×#1192960911yescharacteristic
406C2×C4#1457918111yescharacteristic
407C2×C4#1457918211yescharacteristic
408C2×C4#1457918411yescharacteristic
409C2×C4#1457918611yescharacteristic
410C2×C4#1457918811yescharacteristic
411C2×C4#1457919211yescharacteristic
412C2×C4#1457919311yescharacteristic
413C8—81no—
414C8—3162no—
415C8—3162no—
416C22—6322no—
417C22—6322no—
418C22—6322no—
419C22—6322no—
420C22—6322no—
421C22—6322no—
422C22—6322no—
423C22#1909847811yescharacteristic
424C22#1909869611yescharacteristic
425C22#1909870711yescharacteristic
426C22#1909872311yescharacteristic
427C22#1909872411yescharacteristic
428C22#1909872911yescharacteristic
429C22#1909873911yescharacteristic
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—161no—
435C4—6321no—
436C4—6321no—
437C4#1909878111yescharacteristic
438C4#1909878811yescharacteristic
439C4#1909879711yescharacteristic
440C4#1909881111yescharacteristic
441C4#1909887611yescharacteristic
442C4#1909888711yescharacteristic
443C4#1909891911yescharacteristic
444C4#1909892411yescharacteristic
445C2—12642no—
446C2#2376246311yescharacteristic
447C2#2376252011yescharacteristic
448C2#2376258511yescharacteristic
449C2#2376259111yescharacteristic
450C2#2376458211yescharacteristic
451C2#2376460811yescharacteristic
452C2#2376463711yescharacteristic
subgroupG/H
centerC23#14579129
commutator, cosocleC2×C42×C79C23
FrattiniC2×C42C22×D79
FittingC79×#1803C2
socleC23×C79#120
prime radicalC2×#14579158C22
hypercenter#1803D79
hypocenterC79#10411
LCSG ▷ C2×C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
3. #23764156C2semidirect
5. C2×Q8⋊(C79⋊C8)C2semidirect
6. #23761239C2semidirect
7. C79×#1803C2semidirect
230. #1803D79semidirect
234. C79#10411semidirect

Maximal subgroups

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

subgroupindex
#237683992
#237642052
#237641562
C2×#190983062
C2×Q8⋊(C79⋊C8)2
#237612392
C79×#18032
#1041179