Finite Group Databasebeta

← #34340384

Subgroups — #34340384

Order 20224.

664 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773193C211yesmaximal, characteristic
2C2×#19098840C211yesmaximal, characteristic
3#23772349C211yesmaximal, characteristic, complemented
4#23764908C212yesmaximal, complemented
5#23764757C212yesmaximal
6#23764390C211yesmaximal, characteristic, complemented
7#23764182C212yesmaximal, complemented
8#23764175C211yesmaximal, characteristic, complemented
9#23764061C212yesmaximal
10C4×#14579077C211yesmaximal, characteristic
11C79×#3165C211yesmaximal, characteristic, complemented
12C2×C4×D316C2211yescharacteristic
13#19098860—22no—
14#19098850—21no—
15#19098840—21no—
16#19098840—21no—
17#19098840—22no—
18#19098840—21no—
19#19098694—21no—
20#19098694—22no—
21Q8×C79⋊C8—21no—
22C2×#14579096C2212yes—
23#19098606C2212yes—
24C2×#14579094C2212yes—
25#19098590C2212yes—
26#19098587C2211yescharacteristic
27C2×C4⋊(C79⋊C8)C2211yescharacteristic
28#19098567C2212yes—
29C2×#14579081C2212yes—
30C2×#14579077C2211yescharacteristic
31C2×#14579077C2211yescharacteristic
32#19098519C2212yes—
33C4×#11929608C2211yescharacteristic
34#19098332C2212yes—
35#19098332C2212yes—
36#19098331C2211yescharacteristic
37#19098331C2211yescharacteristic
38#19098327C2212yes—
39#533×C79C2211yescharacteristic
40C2×C4⋊Q8×C79C2211yescharacteristic
41#501×C79—22no—
42#500×C79—21no—
43#498×C79—21no—
44#428×C79C2212yes—
45#428×C79C2212yes—
46C4×M16×C79C2211yescharacteristic
47C22×D316C2311yescharacteristic
48C22×C4×D79—21no—
49C2×#11929608C2311yescharacteristic
50C2×#11929608C2312yes—
51C2×#11929608C2311yescharacteristic
52C22×C79⋊C8C2312yes—
53C4×D316—21no—
54C4×D316—21no—
55C4×D316—21no—
56C4×D316—21no—
57C2×C4⋊Dic79C2311yescharacteristic
58C2×(C2×D79)⋊C4—21no—
59#14579096—24no—
60#14579094—22no—
61#14579094—22no—
62#14579094—22no—
63#14579092—22no—
64C4⋊(C79⋊C8)—21no—
65C4⋊(C79⋊C8)—21no—
66C4⋊(C79⋊C8)—22no—
67C4⋊(C79⋊C8)—22no—
68C4⋊(C79⋊C8)D411yescharacteristic
69C4⋊(C79⋊C8)D412yes—
70C4⋊(C79⋊C8)D411yescharacteristic
71#14579081—22no—
72#14579081—22no—
73#14579081—22no—
74#14579081—22no—
75#14579077—22no—
76#14579077Q814yescomplemented
77#14579072—21no—
78#14579072—22no—
79C4×C79⋊C8—21no—
80C4×C79⋊C8—22no—
81C22×Q8×C79—21no—
82C2×M16×C79C2311yescharacteristic
83C2×M16×C79C2311yescharacteristic
84C4⋊Q8×C79—21no—
85C4⋊Q8×C79—21no—
86C4⋊Q8×C79—21no—
87C4⋊Q8×C79—21no—
88C4⋊Q8×C79D411yescharacteristic
89C4⋊Q8×C79D411yescharacteristic
90#125×C79D412yes—
91C22⋊Q8×C79—21no—
92C22⋊Q8×C79—21no—
93C4×Q8×C79—21no—
94C4×Q8×C79—21no—
95#117×C79C2312yes—
96C2×C4⋊C4×C79—21no—
97C2×C4⋊C4×C79C2312yes—
98C2×C42×C79C2311yescharacteristic
99#107×C79—22no—
100#107×C79—22no—
101#106×C79—22no—
102#106×C79—22no—
103#97×C79—21no—
104C4×C8×C79—21no—
105C23×D79—21no—
106C2×D316—21no—
107C2×D316—21no—
108C2×D316—21no—
109C2×D316—21no—
110C2×D316—21no—
111C2×D316C2×Q812yes—
112C22×Dic79—21no—
113C2×C4×D79—41no—
114C2×C4×D79—41no—
115C2×C4×D79—82no—
116#11929608—21no—
117#11929608—22no—
118#11929608—21no—
119#11929608—21no—
120#11929608—22no—
121#11929608C2×Q812yes—
122C2×C79⋊C8—21no—
123C2×C79⋊C8—21no—
124C2×C79⋊C8—22no—
125C2×C79⋊C8—22no—
126C2×C79⋊C8—22no—
127C2×C79⋊C8C2×D412yes—
128C2×C79⋊C8C2×D412yes—
129C4⋊Dic79—21no—
130C4⋊Dic79—21no—
131(C2×D79)⋊C4—41no—
132(C2×D79)⋊C4—41no—
133C2×Q8×C79—21no—
134C2×Q8×C79—21no—
135C2×Q8×C79—41no—
136C2×Q8×C79—41no—
137C2×Q8×C79—82no—
138C22×C4×C79—22no—
139C22×C4×C79#4111yescharacteristic
140C22×C4×C79#4111yescharacteristic
141C22×C4×C79C2411yescharacteristic
142M16×C79—21no—
143M16×C79—21no—
144M16×C79C2×Q812yes—
145M16×C79#4112yes—
146C2×C8×C79—21no—
147C2×C8×C79—21no—
148C4⋊C4×C79—22no—
149C4⋊C4×C79—22no—
150C4⋊C4×C79—21no—
151C4⋊C4×C79—21no—
152C4⋊C4×C79—22no—
153C4⋊C4×C79—22no—
154C4⋊C4×C79—41no—
155C4⋊C4×C79—41no—
156C4⋊C4×C79C2×D412yes—
157C4⋊C4×C79C2×D412yes—
158C22⋊C4×C79—22no—
159C42×C79—22no—
160C42×C79C2×D411yescharacteristic
161C42×C79C2×D411yescharacteristic
162C42×C79C2×D411yescharacteristic
163C42×C79C2×D411yescharacteristic
164C23×C79C2×#4111yescharacteristic
165C22×D79—22no—
166C22×D79—41no—
167C22×D79—41no—
168C22×D79—81no—
169C22×D79—81no—
170Q8×C79—42no—
171Q8×C79—81no—
172Q8×C79—82no—
173Q8×C79—82no—
174C2×C4×C79—22no—
175C2×C4×C79—22no—
176C2×C4×C79—22no—
177C2×C4×C79—41no—
178C2×C4×C79—42no—
179C2×C4×C79—82no—
180C2×C4×C79#12111yescharacteristic
181C2×C4×C79#12111yescharacteristic
182C2×C4×C79#12111yescharacteristic
183C2×C4×C79#12111yescharacteristic
184C2×C4×C79C22×D411yescharacteristic
185C2×C4×C79C22×D411yescharacteristic
186C2×C4×C79C22×Q811yescharacteristic
187C2×C4×C79C2×#4111yescharacteristic
188C2×C4×C792−1+411yescharacteristic
189C2×C4×C792−1+411yescharacteristic
190C2×Dic79—41no—
191C2×Dic79—81no—
192D316—22no—
193D316—22no—
194D316—41no—
195D316—41no—
196C4×D79—82no—
197C4×D79—82no—
198C8×C79—22no—
199C8×C79—22no—
200C79⋊C8—22no—
201C79⋊C8—22no—
202C79⋊C8—22no—
203C79⋊C8—41no—
204C79⋊C8—42no—
205C22×C79—21no—
206C22×C79—21no—
207C22×C79C2×#12111yescharacteristic
208C22×C79#53311yescharacteristic
209C22×C79#54111yescharacteristic
210C2×D79—42no—
211C2×D79—82no—
212C2×D79—81no—
213C2×D79—81no—
214C2×D79—82no—
215C4×C79—21no—
216C4×C79—21no—
217C4×C79—21no—
218C4×C79—42no—
219C4×C79—42no—
220C4×C79—82no—
221C4×C79#47212yes—
222C4×C79#54711yescharacteristic
223C4×C79#54811yescharacteristic
224C4×C79D4×Q811yescharacteristic
225C4×C79D4×Q811yescharacteristic
226Dic79—81no—
227#25489—791nomaximal, Sylow
228C2×C79—21no—
229C2×C79—21no—
230C2×C79#242911yescharacteristic
231C2×C79C2×#47211yescharacteristic
232C2×C79#306911yescharacteristic
233D79—82no—
234#3165D7911yescharacteristic, complemented
235#3077—791no—
236C2×#474—791no—
237#2962—791no—
238#2089—792no—
239#2063—792no—
240#1997—791no—
241#1952—792no—
242#1950—791no—
243#1931—792no—
244C4×#100—791no—
245C79#2548911yescharacteristic, complemented, Sylow
246#533C2×D7911yescharacteristic
247C2×C4⋊Q8C2×D7911yescharacteristic
248C2×C4×D4—791no—
249#501—22no—
250#500—21no—
251#498—21no—
252#478—1582no—
253#476—1581no—
254#474—1581no—
255#474—1582no—
256#474—1581no—
257#474—1581no—
258#446—1582no—
259#446—1581no—
260C8×Q8—1581no—
261C2×#108—792no—
262#428—792no—
263#428C2×D7912yes—
264#428C2×D7912yes—
265C2×#107—792no—
266#424—792no—
267#423—791no—
268C2×C4⋊C8—791no—
269#418—792no—
270C2×#102—792no—
271C2×#100—791no—
272C2×#100—791no—
273#405—792no—
274C4×M16—791no—
275C4×M16C2×D7911yescharacteristic
276#343—791no—
277#343—791no—
278#343—792no—
279#343—792no—
280#341—792no—
281C22×Q8—21no—
282C22×D4—791no—
283C23×C4—1581no—
284C2×M16—791no—
285C2×M16—792no—
286C2×M16—791no—
287C2×M16C22×D7911yescharacteristic
288C2×M16C22×D7911yescharacteristic
289C22×C8—792no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8C22⋊D7911yescharacteristic
295C4⋊Q8C22⋊D7911yescharacteristic
296#125C22⋊D7912yes—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×D4—1581no—
302C4×D4—1581no—
303C4×D4—1581no—
304C4×D4—1581no—
305#117C22×D7912yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—791no—
308C2×C4⋊C4C22×D7912yes—
309C2×C22⋊C4—1581no—
310C2×C42C22×D7911yescharacteristic
311#108—1584no—
312#107—22no—
313#107—22no—
314#107—1582no—
315#107—1582no—
316#107—1582no—
317#106—22no—
318#106—22no—
319#106—1582no—
320C4⋊C8—791no—
321C4⋊C8—791no—
322C4⋊C8—792no—
323C4⋊C8—1582no—
324C4⋊C8—1582no—
325C4⋊C8—1581no—
326C4⋊C8—1581no—
327#102—1582no—
328#102—1582no—
329#102—1582no—
330#102—1582no—
331#100—794no—
332#100—1582no—
333#97—21no—
334#97—1582no—
335#97—1581no—
336C4×C8—21no—
337C4×C8—1582no—
338C4×C8—1581no—
339C24—1581no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×D4—792no—
346C2×D4—1581no—
347C2×D4—1581no—
348C2×D4—1581no—
349C2×D4—1581no—
350C2×D4—1581no—
351C22×C4—22no—
352C22×C4—3161no—
353C22×C4—3161no—
354C22×C4—6322no—
355C22×C4—1581no—
356C22×C4#1192962311yescharacteristic
357C22×C4#1192962311yescharacteristic
358C22×C4C23×D7911yescharacteristic
359M16—21no—
360M16—21no—
361M16—792no—
362M16—1581no—
363M16—1582no—
364M16—1581no—
365M16—1582no—
366M16—1581no—
367M16Q8×D7912yes—
368M16#1192962512yes—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—792no—
372C2×C8—792no—
373C2×C8—1581no—
374C2×C8—1582no—
375C2×C8—1582no—
376C2×C8—1582no—
377C2×C8—1581no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—1581no—
387C4⋊C4—1581no—
388C4⋊C4C2×C22⋊D7912yes—
389C4⋊C4C2×C22⋊D7912yes—
390C22⋊C4—22no—
391C22⋊C4—3161no—
392C22⋊C4—3161no—
393C42—22no—
394C42D4×D7911yescharacteristic
395C42D4×D7911yescharacteristic
396C42C2×C22⋊D7911yescharacteristic
397C42C2×C22⋊D7911yescharacteristic
398C23—3161no—
399C23—3161no—
400C23—6321no—
401C23—6321no—
402C23—1582no—
403C23C2×#1192962311yescharacteristic
404Q8—42no—
405Q8—81no—
406Q8—82no—
407Q8—82no—
408D4—3161no—
409D4—3161no—
410D4—1582no—
411D4—1582no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—82no—
418C2×C4—3161no—
419C2×C4—6322no—
420C2×C4—6322no—
421C2×C4—6321no—
422C2×C4#1457913311yescharacteristic
423C2×C4#1457913311yescharacteristic
424C2×C4#1457913311yescharacteristic
425C2×C4#1457913311yescharacteristic
426C2×C4C2×D4×D7911yescharacteristic
427C2×C4C22×C22⋊D7911yescharacteristic
428C2×C4C2×Q8×D7911yescharacteristic
429C2×C4C2×#1192962511yescharacteristic
430C2×C4#1457920911yescharacteristic
431C2×C4#1457920911yescharacteristic
432C8—22no—
433C8—22no—
434C8—3161no—
435C8—3162no—
436C8—1582no—
437C8—1582no—
438C8—1582no—
439C22—21no—
440C22—21no—
441C22—3162no—
442C22—6322no—
443C22—6322no—
444C22—6321no—
445C22—6321no—
446C22C2×#1457913311yescharacteristic
447C22#1909911811yescharacteristic
448C22#1909917411yescharacteristic
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—6321no—
456C4#1909883112yes—
457C4#1909921611yescharacteristic
458C4#1909922411yescharacteristic
459C4Q8×C22⋊D7911yescharacteristic
460C4#1909922811yescharacteristic
461C2—21no—
462C2—21no—
463C2—6322no—
464C2#2376700211yescharacteristic
465C2C2×#1909883111yescharacteristic
466C2#2377313411yescharacteristic
subgroupG/H
centerC22C2×#14579133
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#3165C2
socleC22×C79C2×#121
prime radicalC22×D316C23
hypercenter#3165D79
hypocenterC79#25489
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
3. #23772349C2semidirect
4. #23764908C2semidirect
6. #23764390C2semidirect
7. #23764182C2semidirect
8. #23764175C2semidirect
11. C79×#3165C2semidirect
76. #14579077Q8semidirect
234. #3165D79semidirect
245. C79#25489semidirect

Maximal subgroups

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

subgroupindex
#237731932
C2×#190988402
#237723492
#237649082
#237647572
#237643902
#237641822
#237641752
#237640612
C4×#145790772
C79×#31652
#2548979