Finite Group Databasebeta

← #34183156

Subgroups — #34183156

Order 20224.

530 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772855C211yesmaximal, characteristic
2#23761850C211yesmaximal, characteristic, complemented
3#23761798C211yesmaximal, characteristic, complemented
4#23761782C211yesmaximal, characteristic, complemented
5#23761295C211yesmaximal, characteristic, complemented
6#23761234C211yesmaximal, characteristic, complemented
7C79×#2906C211yesmaximal, characteristic, complemented
8C2×#14579181—21no—
9#19099113C2211yescharacteristic
10#19098785—21no—
11#19098778—21no—
12#19098773C411yescharacteristic
13#19098768C411yescharacteristic
14#19098763C411yescharacteristic
15#19098758C411yescharacteristic
16#19098616C2211yescharacteristic
17C2×C4⋊(C79⋊C8)C2211yescharacteristic
18#19098557—21no—
19C2×C4×C79⋊C8C2211yescharacteristic
20Q8⋊(C79⋊C8)—21no—
21D4⋊(C79⋊C8)—21no—
22#19098290—21no—
23#19098290—21no—
24#19098290—22no—
25C2×#131×C79—21no—
26C4×#41×C79C2211yescharacteristic
27#424×C79—21no—
28#424×C79—22no—
29#423×C79—21no—
30#423×C79C2211yescharacteristic
31C2×C4⋊C8×C79—21no—
32C2×C4⋊C8×C79C2211yescharacteristic
33C2×#11929624—21no—
34#14579181—42no—
35#14579181—42no—
36C2×#11929614—21no—
37C2×#11929612—21no—
38C2×#11929611—21no—
39C2×#11929609—21no—
40C22×C79⋊C8—21no—
41C22×C79⋊C8—21no—
42#14579161C2×C411yescharacteristic
43#14579158C2×C411yescharacteristic
44#14579149—21no—
45#14579135—21no—
46#14579135—21no—
47C4×D316—21no—
48C4×D316—21no—
49C4⋊(C79⋊C8)—21no—
50C4⋊(C79⋊C8)—21no—
51C4⋊(C79⋊C8)C2×C411yescharacteristic
52C4⋊(C79⋊C8)C2×C411yescharacteristic
53#14579084—21no—
54#14579084—21no—
55#14579081—21no—
56#14579081—21no—
57C22⋊(C79⋊C8)—21no—
58C4×C79⋊C8—21no—
59C4×C79⋊C8—21no—
60C4×C79⋊C8—21no—
61C2×#41×C79D411yescharacteristic
62#131×C79—42no—
63#131×C79—42no—
64C2×M16×C79—22no—
65C2×M16×C79—21no—
66C22×C8×C79—22no—
67C22×C8×C79—21no—
68C4×Q8×C79C2×C411yescharacteristic
69C4×Q8×C79D411yescharacteristic
70C4×D4×C79—21no—
71C4×D4×C79—21no—
72C4×D4×C79C2×C411yescharacteristic
73C4×D4×C79D411yescharacteristic
74#117×C79—21no—
75#117×C79D411yescharacteristic
76C2×C42×C79—21no—
77C2×C42×C79C2311yescharacteristic
78C4⋊C8×C79—22no—
79C4⋊C8×C79—22no—
80C4⋊C8×C79—21no—
81C4⋊C8×C79—21no—
82C4⋊C8×C79D411yescharacteristic
83C4⋊C8×C79D411yescharacteristic
84C4⋊C8×C79D412yes—
85#11929624—41no—
86#11929624—41no—
87#11929624—41no—
88#11929624—41no—
89C2×Dic158—21no—
90C2×D316—21no—
91C2×D316—21no—
92C2×D316—21no—
93C2×C22⋊D79—41no—
94C2×C4×D79—41no—
95#11929614—42no—
96#11929612—42no—
97#11929611—42no—
98#11929609—42no—
99C2×C79⋊C8—22no—
100C2×C79⋊C8—21no—
101C2×C79⋊C8—21no—
102C2×C79⋊C8—41no—
103C2×C79⋊C8—41no—
104C2×C79⋊C8—41no—
105C2×C79⋊C8—41no—
106C2×C79⋊C8—41no—
107C4⋊Dic79—21no—
108C4⋊Dic79—21no—
109(C2×D79)⋊C4—41no—
110(C2×D79)⋊C4—41no—
111#41×C79—21no—
112#41×C79—21no—
113#41×C79D812yes—
114#41×C79SD1612yes—
115C2×Q8×C79C22⋊C411yescharacteristic
116C2×D4×C79—21no—
117C2×D4×C79C22⋊C411yescharacteristic
118C22×C4×C79—21no—
119C22×C4×C79—21no—
120C22×C4×C79—21no—
121C22×C4×C79C2×D411yescharacteristic
122C22×C4×C79#4111yescharacteristic
123C22×C4×C79#4111yescharacteristic
124M16×C79—44no—
125M16×C79—41no—
126M16×C79—41no—
127C2×C8×C79—21no—
128C2×C8×C79—22no—
129C2×C8×C79—21no—
130C2×C8×C79—44no—
131C2×C8×C79—41no—
132C2×C8×C79—41no—
133C4⋊C4×C79—21no—
134C4⋊C4×C79—21no—
135C4⋊C4×C79C22⋊C411yescharacteristic
136C4⋊C4×C79C22⋊C411yescharacteristic
137C22⋊C4×C79—21no—
138C22⋊C4×C79—21no—
139C22⋊C4×C79—21no—
140C42×C79—21no—
141C42×C79—21no—
142C42×C79—21no—
143C42×C79C22×C411yescharacteristic
144C42×C79C2×D411yescharacteristic
145C42×C79C2×D411yescharacteristic
146C42×C79C2×D411yescharacteristic
147C23×C79—21no—
148C23×C79#12311yescharacteristic
149C22×D79—41no—
150Q8×C79—21no—
151Q8×C79#10212yes—
152D4×C79—22no—
153D4×C79—22no—
154D4×C79—21no—
155D4×C79#10212yes—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—22no—
160C2×C4×C79—22no—
161C2×C4×C79—21no—
162C2×C4×C79—21no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—81no—
167C2×C4×C79C2×C22⋊C411yescharacteristic
168C2×C4×C79C4×D411yescharacteristic
169C2×C4×C79C4×D411yescharacteristic
170C2×C4×C79#12011yescharacteristic
171C2×C4×C79#12111yescharacteristic
172C2×C4×C79#12111yescharacteristic
173C2×C4×C79C2×D811yescharacteristic
174C2×C4×C79C2×SD1611yescharacteristic
175C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
176C2×C4×C79#13711yescharacteristic
177C22⋊D79—81no—
178C22⋊D79—81no—
179C2×Dic79—41no—
180D316—42no—
181D316—41no—
182D316—41no—
183D316—41no—
184D316—41no—
185C4×D79—81no—
186C4×D79—81no—
187Dic158—41no—
188Dic158—41no—
189C8×C79—41no—
190C8×C79—42no—
191C8×C79—41no—
192C79⋊C8—42no—
193C79⋊C8—41no—
194C79⋊C8—41no—
195C79⋊C8—81no—
196C22×C79—21no—
197C22×C79—21no—
198C22×C79—22no—
199C22×C79—41no—
200C22×C79—41no—
201C22×C79#38611yescharacteristic
202C22×C79#48011yescharacteristic
203C22×C79#48111yescharacteristic
204C2×D79—81no—
205C2×D79—81no—
206C2×D79—81no—
207C4×C79—22no—
208C4×C79—21no—
209C4×C79—21no—
210C4×C79—21no—
211C4×C79—41no—
212C4×C79—41no—
213C4×C79C2×#10211yescharacteristic
214C4×C79#41711yescharacteristic
215C4×C79#44311yescharacteristic
216C4×C79#44411yescharacteristic
217C4×C79#44711yescharacteristic
218C4×C79#44811yescharacteristic
219Dic79—81no—
220#8995—791nomaximal, Sylow
221C2×C79—21no—
222C2×C79—21no—
223C2×C79—42no—
224C2×C79#187411yescharacteristic
225C2×C79#187611yescharacteristic
226C2×C79#188111yescharacteristic
227D79—161no—
228#3035—791no—
229#2906D7911yescharacteristic, complemented
230#1492—791no—
231#1480—791no—
232#1476—791no—
233#1301—791no—
234#1277—791no—
235C79#899511yescharacteristic, complemented, Sylow
236C2×#135—1581no—
237C2×#131—21no—
238#532—791no—
239C4×#41C2×D7911yescharacteristic
240#464—1581no—
241#463—1581no—
242#462—791no—
243#461—791no—
244#460—791no—
245#459—791no—
246#431—791no—
247#424—21no—
248#424—22no—
249#423—21no—
250#423C2×D7911yescharacteristic
251C2×C4⋊C8—21no—
252C2×C4⋊C8—791no—
253C2×C4⋊C8C2×D7911yescharacteristic
254#416—1581no—
255C2×C4×C8—791no—
256Q8⋊C8—1581no—
257D4⋊C8—1581no—
258D4⋊C8—1581no—
259D4⋊C8—1581no—
260D4⋊C8—1582no—
261C2×#41—1581no—
262C2×#41C22⋊D7911yescharacteristic
263#135—3162no—
264#135—3162no—
265C2×Q16—1581no—
266C2×SD16—1581no—
267C2×SD16—1581no—
268C2×D8—1581no—
269#131—42no—
270#131—42no—
271C2×M16—22no—
272C2×M16—21no—
273C22×C8—22no—
274C22×C8—21no—
275C22×C8—1581no—
276C22×C8—1581no—
277C4⋊Q8—791no—
278#127—791no—
279#124—1581no—
280#121—1581no—
281#121—1581no—
282C4×Q8C4×D7911yescharacteristic
283C4×Q8D31611yescharacteristic
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—1581no—
287C4×D4—1581no—
288C4×D4C4×D7911yescharacteristic
289C4×D4D31611yescharacteristic
290#117—21no—
291#117C22⋊D7911yescharacteristic
292C2×C42—21no—
293C2×C42C22×D7911yescharacteristic
294C4⋊C8—22no—
295C4⋊C8—22no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—791no—
299C4⋊C8—791no—
300C4⋊C8—1581no—
301C4⋊C8—1581no—
302C4⋊C8C22⋊D7911yescharacteristic
303C4⋊C8C22⋊D7911yescharacteristic
304C4⋊C8C22⋊D7912yes—
305#103—1581no—
306#103—1581no—
307#102—1581no—
308#102—1581no—
309C22⋊C8—1581no—
310C4×C8—1581no—
311C4×C8—1581no—
312C4×C8—1581no—
313#41—21no—
314#41—21no—
315#41—3161no—
316#41—3161no—
317#41—3161no—
318#41—3161no—
319#41#1192960912yes—
320#41#1192961112yes—
321C2×Q8—1581no—
322C2×Q8(C2×D79)⋊C411yescharacteristic
323C2×D4—21no—
324C2×D4—3161no—
325C2×D4—1581no—
326C2×D4—1581no—
327C2×D4—1581no—
328C2×D4(C2×D79)⋊C411yescharacteristic
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—3161no—
333C22×C4C2×C22⋊D7911yescharacteristic
334C22×C4#1192962411yescharacteristic
335C22×C4#1192962411yescharacteristic
336Q16—3162no—
337SD16—3162no—
338SD16—3162no—
339D8—3162no—
340M16—44no—
341M16—41no—
342M16—41no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—44no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—3161no—
350C2×C8—3161no—
351C2×C8—3161no—
352C2×C8—3161no—
353C2×C8—3161no—
354C2×C8—1582no—
355C2×C8—1581no—
356C2×C8—1581no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—1581no—
360C4⋊C4—1581no—
361C4⋊C4(C2×D79)⋊C411yescharacteristic
362C4⋊C4(C2×D79)⋊C411yescharacteristic
363C22⋊C4—21no—
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—3161no—
367C22⋊C4—3161no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42C2×C4×D7911yescharacteristic
372C42C2×C22⋊D7911yescharacteristic
373C42C2×C22⋊D7911yescharacteristic
374C42C2×D31611yescharacteristic
375C23—21no—
376C23—3161no—
377C23#1457914411yescharacteristic
378Q8—21no—
379Q8—3161no—
380Q8—3161no—
381Q8#1457908112yes—
382D4—22no—
383D4—22no—
384D4—21no—
385D4—3162no—
386D4—3161no—
387D4—3161no—
388D4—3161no—
389D4—3161no—
390D4—6321no—
391D4—6321no—
392D4#1457908112yes—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—3161no—
405C2×C4—6321no—
406C2×C4—6321no—
407C2×C4C2×(C2×D79)⋊C411yescharacteristic
408C2×C4C4×C22⋊D7911yescharacteristic
409C2×C4C4×C22⋊D7911yescharacteristic
410C2×C4#1457913011yescharacteristic
411C2×C4#1457913511yescharacteristic
412C2×C4#1457913511yescharacteristic
413C2×C4C2×#1192960911yescharacteristic
414C2×C4C2×#1192961111yescharacteristic
415C2×C4#1457918711yescharacteristic
416C2×C4#1457919211yescharacteristic
417C8—41no—
418C8—42no—
419C8—41no—
420C8—3162no—
421C8—3161no—
422C8—3161no—
423C8—6321no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—6321no—
430C22—6321no—
431C22—6321no—
432C22#1909844211yescharacteristic
433C22#1909887611yescharacteristic
434C22#1909888111yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—6321no—
442C4C2×#1457908111yescharacteristic
443C4#1909856211yescharacteristic
444C4#1909867811yescharacteristic
445C4#1909868311yescharacteristic
446C4#1909869711yescharacteristic
447C4#1909870211yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—12641no—
452C2#2376371711yescharacteristic
453C2#2376372711yescharacteristic
454C2#2376376211yescharacteristic
subgroupG/H
centerC2×C4#14579130
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2906C2
socleC22×C79#386
prime radical#23772855C2
cosocleC2×C42×C79C23
hypercenter#2906D79
hypocenterC79#8995
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #23761850C2semidirect
3. #23761798C2semidirect
4. #23761782C2semidirect
5. #23761295C2semidirect
6. #23761234C2semidirect
7. C79×#2906C2semidirect
229. #2906D79semidirect
235. C79#8995semidirect

Maximal subgroups

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

subgroupindex
#237728552
#237618502
#237617982
#237617822
#237612952
#237612342
C79×#29062
#899579