Finite Group Databasebeta

← #34183752

Subgroups — #34183752

Order 20224.

520 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772367C211yesmaximal, characteristic, complemented
2#23761787C211yesmaximal, characteristic, complemented
3C2×D4⋊(C79⋊C8)C211yesmaximal, characteristic, complemented
4#23761721C211yesmaximal, characteristic, complemented
5#23761251C211yesmaximal, characteristic, complemented
6#23761238C211yesmaximal, characteristic, complemented
7C4×AGL(1,ℤ/8)×C79C211yesmaximal, characteristic
8#19098694—21no—
9#19098641—21no—
10#19098635—22no—
11#19098616—21no—
12#19098616C2211yescharacteristic
13#19098616C2211yescharacteristic
14C2×C4⋊(C79⋊C8)C2211yescharacteristic
15C4×#11929608—21no—
16C2×C4×C79⋊C8C2211yescharacteristic
17Q8⋊(C79⋊C8)—21no—
18D4⋊(C79⋊C8)—21no—
19D4⋊(C79⋊C8)—21no—
20D4⋊(C79⋊C8)—21no—
21#19098287—22no—
22C2×AGL(1,ℤ/8)×C79C411yescharacteristic
23C4×#41×C79C2211yescharacteristic
24C2×C4×D4×C79C2211yescharacteristic
25#442×C79—21no—
26#440×C79—21no—
27C4×SD16×C79—21no—
28C4×D8×C79—21no—
29#428×C79C411yescharacteristic
30#418×C79C411yescharacteristic
31#417×C79C411yescharacteristic
32C4×M16×C79C2211yescharacteristic
33C2×#11929608—22no—
34C22×C79⋊C8—21no—
35C22×C79⋊C8—21no—
36C4⋊(C79⋊C8)—22no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41C22⋊(C79⋊C8)—22no—
42C22⋊(C79⋊C8)—21no—
43C22⋊(C79⋊C8)D412yes—
44#14579072—22no—
45C4×C79⋊C8—21no—
46C4×C79⋊C8—21no—
47C4×C79⋊C8—21no—
48C4×C79⋊C8D411yescharacteristic
49C4×C79⋊C8D411yescharacteristic
50C2×#41×C79C2×C411yescharacteristic
51C22×D4×C79C2×C411yescharacteristic
52C23×C4×C79—21no—
53AGL(1,ℤ/8)×C79—22no—
54AGL(1,ℤ/8)×C79—22no—
55C2×SD16×C79—21no—
56C2×D8×C79—21no—
57C2×M16×C79C2×C411yescharacteristic
58C2×M16×C79C2×C411yescharacteristic
59C4×Q8×C79D411yescharacteristic
60C4×Q8×C79D411yescharacteristic
61C4×D4×C79—21no—
62C4×D4×C79—21no—
63C4×D4×C79—21no—
64C4×D4×C79—21no—
65C4×D4×C79—21no—
66C4×D4×C79—21no—
67C4×D4×C79D411yescharacteristic
68C4×D4×C79D411yescharacteristic
69#117×C79—21no—
70#117×C79C2×C411yescharacteristic
71C2×C4⋊C4×C79C2×C411yescharacteristic
72C2×C22⋊C4×C79—21no—
73C2×C42×C79—21no—
74C2×C42×C79C2311yescharacteristic
75#107×C79—21no—
76#106×C79—21no—
77#103×C79—21no—
78#102×C79—21no—
79#102×C79—21no—
80#102×C79—21no—
81#97×C79—21no—
82C4×C8×C79—21no—
83#11929608—44no—
84C2×C79⋊C8—21no—
85C2×C79⋊C8—21no—
86C2×C79⋊C8—22no—
87C2×C79⋊C8—42no—
88C2×C79⋊C8—41no—
89C2×C79⋊C8—41no—
90C2×C79⋊C8—81no—
91C24×C79—21no—
92#41×C79—21no—
93#41×C79—21no—
94#41×C79—21no—
95#41×C79—21no—
96C2×Q8×C79C22⋊C411yescharacteristic
97C2×D4×C79—21no—
98C2×D4×C79—21no—
99C2×D4×C79—21no—
100C2×D4×C79—21no—
101C2×D4×C79—21no—
102C2×D4×C79—21no—
103C2×D4×C79C22⋊C411yescharacteristic
104C2×D4×C79M1612yescomplemented
105C22×C4×C79—21no—
106C22×C4×C79—21no—
107C22×C4×C79—21no—
108C22×C4×C79—21no—
109C22×C4×C79—41no—
110C22×C4×C79—41no—
111C22×C4×C79—81no—
112C22×C4×C79—81no—
113C22×C4×C79C22×C411yescharacteristic
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79#4111yescharacteristic
116SD16×C79—24no—
117D8×C79—24no—
118M16×C79—21no—
119M16×C79—21no—
120M16×C79—21no—
121M16×C79M1612yescomplemented
122C2×C8×C79—21no—
123C2×C8×C79—21no—
124C4⋊C4×C79—21no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79—21no—
127C4⋊C4×C79—21no—
128C4⋊C4×C79C22⋊C411yescharacteristic
129C4⋊C4×C79C22⋊C411yescharacteristic
130C22⋊C4×C79—21no—
131C22⋊C4×C79—21no—
132C22⋊C4×C79—21no—
133C22⋊C4×C79—41no—
134C22⋊C4×C79—41no—
135C42×C79—21no—
136C42×C79—21no—
137C42×C79—21no—
138C42×C79C2×D411yescharacteristic
139C42×C79C2×D411yescharacteristic
140C42×C79C2×D411yescharacteristic
141C42×C79#4111yescharacteristic
142C23×C79—21no—
143C23×C79—22no—
144C23×C79—41no—
145C23×C79—41no—
146C23×C79—81no—
147C23×C79—81no—
148C23×C79C4×D411yescharacteristic
149Q8×C79—21no—
150Q8×C79—21no—
151D4×C79—21no—
152D4×C79—22no—
153D4×C79—21no—
154D4×C79—22no—
155D4×C79—41no—
156D4×C79—41no—
157D4×C79—41no—
158D4×C79—41no—
159C2×C4×C79—21no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—21no—
164C2×C4×C79—41no—
165C2×C4×C79—42no—
166C2×C4×C79—41no—
167C2×C4×C79—41no—
168C2×C4×C79—41no—
169C2×C4×C79—41no—
170C2×C4×C79—41no—
171C2×C4×C79—82no—
172C2×C4×C79—81no—
173C2×C4×C79—81no—
174C2×C4×C79—81no—
175C2×C4×C79C2×C22⋊C411yescharacteristic
176C2×C4×C79C4×D411yescharacteristic
177C2×C4×C79#12011yescharacteristic
178C2×C4×C79#12111yescharacteristic
179C2×C4×C79#12111yescharacteristic
180C2×C4×C79#12311yescharacteristic
181C2×C4×C79C2×M1611yescharacteristic
182C2×C4×C79C2×M1611yescharacteristic
183C2×C4×C79#13111yescharacteristic
184C2×C4×C79#13111yescharacteristic
185C8×C79—22no—
186C8×C79—41no—
187C79⋊C8—22no—
188C79⋊C8—41no—
189C79⋊C8—42no—
190C79⋊C8—81no—
191C79⋊C8—81no—
192C22×C79—21no—
193C22×C79—21no—
194C22×C79—41no—
195C22×C79—42no—
196C22×C79—41no—
197C22×C79—41no—
198C22×C79—82no—
199C22×C79—82no—
200C22×C79—81no—
201C22×C79—81no—
202C22×C79#38611yescharacteristic
203C22×C79#43511yescharacteristic
204C22×C79#43611yescharacteristic
205C4×C79—21no—
206C4×C79—21no—
207C4×C79—21no—
208C4×C79—41no—
209C4×C79—41no—
210C4×C79—41no—
211C4×C79—81no—
212C4×C79#40711yescharacteristic
213C4×C79#40811yescharacteristic
214C4×C79#43511yescharacteristic
215C4×C79#43611yescharacteristic
216C4×C79#47111yescharacteristic
217C4×C79#47211yescharacteristic
218#9097—791nomaximal, Sylow
219C2×C79—21no—
220C2×C79—21no—
221C2×C79—81no—
222C2×C79—82no—
223C2×C79#187011yescharacteristic
224C2×C79#190611yescharacteristic
225C2×C79#190711yescharacteristic
226#2965—791no—
227C4×AGL(1,ℤ/8)D7911yescharacteristic
228#1477—791no—
229C2×D4⋊C8—791no—
230#1460—791no—
231#1283—791no—
232#1278—791no—
233C79#909711yescharacteristic, complemented, Sylow
234C2×AGL(1,ℤ/8)Dic7911yescharacteristic
235C4×#41C2×D7911yescharacteristic
236C2×C4×D4C2×D7911yescharacteristic
237#446—1581no—
238#442—21no—
239#440—21no—
240C4×SD16—21no—
241C4×D8—21no—
242#436—1581no—
243#435—1582no—
244#431—791no—
245#431—791no—
246#431—1581no—
247#428Dic7911yescharacteristic
248C2×C4⋊C8—791no—
249#418Dic7911yescharacteristic
250#417Dic7911yescharacteristic
251C4×M16—1581no—
252C4×M16C2×D7911yescharacteristic
253C2×C4×C8—791no—
254Q8⋊C8—1581no—
255D4⋊C8—1581no—
256D4⋊C8—1581no—
257D4⋊C8—1581no—
258#323—1582no—
259C2×#41C2×Dic7911yescharacteristic
260C22×D4C2×Dic7911yescharacteristic
261C23×C4—21no—
262AGL(1,ℤ/8)—22no—
263AGL(1,ℤ/8)—22no—
264C2×SD16—21no—
265C2×D8—21no—
266C2×M16—1582no—
267C2×M16C2×Dic7911yescharacteristic
268C2×M16C2×Dic7911yescharacteristic
269C22×C8—1581no—
270C22×C8—1581no—
271C4×Q8C22⋊D7911yescharacteristic
272C4×Q8C22⋊D7911yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4C22⋊D7911yescharacteristic
280C4×D4C22⋊D7911yescharacteristic
281#117—21no—
282#117C2×Dic7911yescharacteristic
283C2×C4⋊C4C2×Dic7911yescharacteristic
284C2×C22⋊C4—21no—
285C2×C42—21no—
286C2×C42C22×D7911yescharacteristic
287#107—21no—
288#106—21no—
289C4⋊C8—1582no—
290C4⋊C8—1581no—
291C4⋊C8—1581no—
292C4⋊C8—1581no—
293C4⋊C8—1581no—
294#103—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298C22⋊C8—792no—
299C22⋊C8—1581no—
300C22⋊C8—1582no—
301#97—21no—
302#97—1582no—
303C4×C8—21no—
304C4×C8—791no—
305C4×C8—791no—
306C4×C8—1581no—
307C4×C8—1581no—
308C4×C8—1581no—
309C24—21no—
310#41—21no—
311#41—21no—
312#41—21no—
313#41—21no—
314C2×Q8C22⋊Dic7911yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4C22⋊Dic7911yescharacteristic
322C2×D4#1192960812yescomplemented
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4C22×Dic7911yescharacteristic
332C22×C4D4×D7911yescharacteristic
333C22×C4#1192962311yescharacteristic
334SD16—24no—
335D8—24no—
336M16—21no—
337M16—21no—
338M16—21no—
339M16—3164no—
340M16#1192960812yescomplemented
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—3162no—
344C2×C8—3161no—
345C2×C8—3161no—
346C2×C8—6321no—
347C2×C8—1581no—
348C2×C8—1582no—
349C2×C8—1581no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4C22⋊Dic7911yescharacteristic
355C4⋊C4C22⋊Dic7911yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42D4×D7911yescharacteristic
365C42C2×C22⋊D7911yescharacteristic
366C42C2×C22⋊D7911yescharacteristic
367C42#1192962311yescharacteristic
368C23—21no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—81no—
373C23—81no—
374C23D4×Dic7911yescharacteristic
375Q8—21no—
376Q8—21no—
377D4—22no—
378D4—21no—
379D4—22no—
380D4—21no—
381D4—41no—
382D4—41no—
383D4—41no—
384D4—41no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4C2×C22⋊Dic7911yescharacteristic
402C2×C4D4×Dic7911yescharacteristic
403C2×C4#1457912911yescharacteristic
404C2×C4#1457913311yescharacteristic
405C2×C4#1457913311yescharacteristic
406C2×C4#1457914211yescharacteristic
407C2×C4C2×#1192960811yescharacteristic
408C2×C4C2×#1192960811yescharacteristic
409C2×C4#1457916811yescharacteristic
410C2×C4#1457916811yescharacteristic
411C8—22no—
412C8—41no—
413C8—3161no—
414C8—3162no—
415C8—6321no—
416C8—6321no—
417C8—1582no—
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22#1909844311yescharacteristic
429C22#1909863511yescharacteristic
430C22#1909864111yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4#1909852711yescharacteristic
439C4#1909853011yescharacteristic
440C4#1909863511yescharacteristic
441C4#1909864111yescharacteristic
442C4#1909882411yescharacteristic
443C4#1909883111yescharacteristic
444C2—21no—
445C2—21no—
446C2—81no—
447C2—82no—
448C2#2376369811yescharacteristic
449C2#2376390311yescharacteristic
450C2#2376391011yescharacteristic
subgroupG/H
centerC2×C4#14579133
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC4×AGL(1,ℤ/8)×C79C2
socleC22×C79#386
prime radicalC2×AGL(1,ℤ/8)×C79C4
cosocleC2×C42×C79C23
hypercenterC4×AGL(1,ℤ/8)D79
hypocenterC79#9097
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23772367C2semidirect
2. #23761787C2semidirect
3. C2×D4⋊(C79⋊C8)C2semidirect
4. #23761721C2semidirect
5. #23761251C2semidirect
6. #23761238C2semidirect
104. C2×D4×C79M16semidirect
121. M16×C79M16semidirect
233. C79#9097semidirect
322. C2×D4#11929608semidirect
340. M16#11929608semidirect

Maximal subgroups

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

subgroupindex
#237723672
#237617872
C2×D4⋊(C79⋊C8)2
#237617212
#237612512
#237612382
C4×AGL(1,ℤ/8)×C792
#909779