Finite Group Databasebeta

← #35152675

Subgroups — #35152675

Order 20224.

1810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776803C211yesmaximal, characteristic
2#23772348C218yesmaximal, complemented
3#23772345C218yesmaximal, complemented
4#23772340C211yesmaximal, characteristic, complemented
5#23772339C212yesmaximal, complemented
6#23772335C211yesmaximal, characteristic, complemented
7#23772332C212yesmaximal, complemented
8#23772330C211yesmaximal, characteristic, complemented
9#23771962C211yesmaximal, characteristic
10#424×D79C212yesmaximal
11#23771816C212yesmaximal
12C2×#19098520C211yesmaximal, characteristic
13C79×#2961C211yesmaximal, characteristic, complemented
14C22×#11929607C2211yescharacteristic
15C2×M16×D79C2212yes—
16C22×C8×D79C2211yescharacteristic
17#19099193C414yes—
18#19099192C414yes—
19#19099190C414yes—
20#19099188C414yes—
21#19099120C2211yescharacteristic
22#19099117C2218yes—
23#19099116C2212yes—
24C2×#14579118C2211yescharacteristic
25#117×D79C2212yes—
26#19098694C2214yescomplemented
27#19098692C2218yescomplemented
28#19098691C2218yescomplemented
29#19098691C2218yescomplemented
30#19098690C2214yescomplemented
31#19098690C2218yescomplemented
32Q8×C79⋊C8C2214yescomplemented
33C8×Dic158C2218yescomplemented
34#19098687C2214yescomplemented
35#19098591C2212yes—
36#19098590C2212yes—
37#19098589C2212yes—
38#19098588C2218yes—
39#19098588C2218yes—
40#19098587C2211yescharacteristic
41#19098586C2212yes—
42C4⋊C8×D79C2218yes—
43#19098521C2212yes—
44#19098520C2214yes—
45#19098520C2214yes—
46#19098520C2214yes—
47#19098519C2212yes—
48#19098518C2212yes—
49#19098517C2214yes—
50#19098515C2214yes—
51C4×#11929607C2214yes—
52C2×#14579071C2211yescharacteristic
53C2×C4×C79⋊C8C2211yescharacteristic
54C2×C8×Dic79C2211yescharacteristic
55#533×C79C2211yescharacteristic
56#446×C79C2218yescomplemented
57#424×C79C2212yes—
58#423×C79C2211yescharacteristic
59#405×C79C2212yes—
60C2×#97×C79C2211yescharacteristic
61C22×C4×D79C2311yescharacteristic
62C2×#11929608C2312yes—
63C2×#11929607Q818yescomplemented
64C2×#11929607C2314yes—
65C2×#11929607C2312yes—
66C2×#11929607C2312yes—
67M16×D79—28no—
68C22×C79⋊C8C2311yescharacteristic
69C22×C79⋊C8C2311yescharacteristic
70C2×C8×D79—24no—
71C2×C8×D79C2314yes—
72C2×C8×D79C2314yes—
73#14579161C2×C412yes—
74#14579160C2×C414yes—
75#14579160C2×C418yes—
76#14579159C2×C418yes—
77#14579153C2×C412yes—
78#14579152C2×C414yes—
79#14579151C2×C418yes—
80#14579151C2×C418yes—
81#14579141C2×C418yes—
82#14579140C2×C414yes—
83#14579139C2×C418yes—
84#14579139C2×C418yes—
85#14579138C2×C418yes—
86#14579137C2×C418yes—
87C4×Dic158C2314yes—
88C4×Dic158C2318yes—
89#14579120C2312yes—
90#14579119C2×C414yes—
91#14579118C2×C412yes—
92#14579118C2×C412yes—
93#14579118C2314yes—
94#14579118C2314yes—
95#14579117C2×C414yes—
96C2×Dic79⋊C4C2×C412yes—
97C4⋊C4×D79C2×C414yes—
98C2×(C2×D79)⋊C4C2×C412yes—
99C22⋊C4×D79C2×C414yes—
100C2×C4×Dic79C2311yescharacteristic
101C42×D79C2314yes—
102C4⋊(C79⋊C8)C2318yes—
103C4⋊(C79⋊C8)C2314yes—
104Dic79⋊C8C2318yes—
105Dic79⋊C8C2318yes—
106Dic79⋊C8C2318yes—
107#14579072C2314yes—
108#14579071C2312yes—
109#14579071C2314yes—
110#14579071C2314yes—
111#14579071C2312yes—
112C4×C79⋊C8C2312yes—
113C4×C79⋊C8C2312yes—
114C4×C79⋊C8C2314yes—
115C8×Dic79C2314yes—
116C8×Dic79C2314yes—
117C8×Dic79C2314yes—
118C2×M16×C79C2312yes—
119C22×C8×C79C2311yescharacteristic
120C22×C8×C79C2311yescharacteristic
121C4⋊Q8×C79C2×C412yes—
122#125×C79C2×C412yes—
123C22⋊Q8×C79C2×C414yes—
124C4×Q8×C79C2314yes—
125#117×C79C2312yes—
126C2×C42×C79C2311yescharacteristic
127C4⋊C8×C79C2314yes—
128C4⋊C8×C79C2318yes—
129#97×C79C2314yes—
130#97×C79C2314yes—
131C4×C8×C79C2314yes—
132C23×D79C22×C411yescharacteristic
133C2×Dic158C22×C418yes—
134C2×Dic158C22×C414yes—
135C22×Dic79C22×C412yes—
136C22×Dic79C22×C411yescharacteristic
137C2×C4×D79C22×C418yes—
138C2×C4×D79C22×C412yes—
139C2×C4×D79C2×Q814yes—
140C2×C4×D79#4114yes—
141C2×C4×D79C2412yes—
142#11929608—24no—
143#11929607—28no—
144#11929607C2×Q818yes—
145#11929607C2×Q818yes—
146C2×C79⋊C8—22no—
147C2×C79⋊C8C2×Q814yes—
148C2×C79⋊C8C2412yes—
149C2×C79⋊C8C2414yes—
150C2×C79⋊C8C2411yescharacteristic
151C2×C79⋊C8C2411yescharacteristic
152C8×D79—28no—
153C8×D79—28no—
154C4⋊Dic79C22×C414yes—
155C4⋊Dic79C22×C418yes—
156C4⋊Dic79C22×C414yes—
157Dic79⋊C4C22×C418yes—
158Dic79⋊C4C22×C414yes—
159Dic79⋊C4C22×C414yes—
160Dic79⋊C4C22×C418yes—
161Dic79⋊C4C22×C418yes—
162Dic79⋊C4C22×C418yes—
163C22⋊Dic79C22×C414yes—
164(C2×D79)⋊C4C22×C418yes—
165(C2×D79)⋊C4C22×C414yes—
166(C2×D79)⋊C4C22×C414yes—
167C4×Dic79C22×C418yes—
168C4×Dic79C22×C412yes—
169C4×Dic79C2414yes—
170C4×Dic79C2412yes—
171C2×Q8×C79C22×C414yes—
172C22×C4×C79C22×C412yes—
173C22×C4×C79C2411yescharacteristic
174M16×C79—24no—
175C2×C8×C79—22no—
176C2×C8×C79C2×Q814yes—
177C2×C8×C79C2414yes—
178C2×C8×C79C2412yes—
179C2×C8×C79C2412yes—
180C4⋊C4×C79C22×C414yes—
181C4⋊C4×C79C22×C418yes—
182C4⋊C4×C79C22×C414yes—
183C22⋊C4×C79C22×C414yes—
184C42×C79C22×C411yescharacteristic
185C42×C79C22×C411yescharacteristic
186C42×C79C2414yes—
187C42×C79C2412yes—
188C23×C79C23×C411yescharacteristic
189C22×D79—42no—
190C22×D79C4×Q814yes—
191C22×D79C23×C412yes—
192Q8×C79—44no—
193C2×C4×C79—42no—
194C2×C4×C79C23×C418yes—
195C2×C4×C79C23×C412yes—
196C2×C4×C79C23×C412yes—
197C2×C4×C79C23×C411yescharacteristic
198C2×C4×C79C22×Q812yes—
199C2×C4×C79C2×#4112yes—
200C2×C4×C79C2511yescharacteristic
201C2×Dic79—42no—
202C2×Dic79C4×Q814yes—
203C2×Dic79C23×C414yes—
204C2×Dic79C23×C418yes—
205C2×Dic79C23×C412yes—
206C4×D79—48no—
207C4×D79C4×Q818yes—
208C4×D79C22×Q814yes—
209C4×D79C2×#4114yes—
210Dic158—44no—
211Dic158—48no—
212C8×C79—22no—
213C8×C79—24no—
214C8×C79C22×Q814yes—
215C79⋊C8—22no—
216C79⋊C8—24no—
217C79⋊C8C22×Q812yes—
218C79⋊C8C22×Q812yes—
219C22×C79—22no—
220C22×C79C2×C4×Q812yes—
221C22×C79C24×C411yescharacteristic
222C2×D79—24no—
223C2×D79—44no—
224C2×D79C2×C4×Q814yes—
225C4×C79—22no—
226C4×C79—22no—
227C4×C79—28no—
228C4×C79C2×C4×Q812yes—
229C4×C79C23×Q811yescharacteristic
230C4×C79C22×#4111yescharacteristic
231Dic79—28no—
232Dic79—24no—
233Dic79C2×C4×Q814yes—
234#60193—791nomaximal, Sylow
235C2×C79—22no—
236C2×C79C22×C4×Q811yescharacteristic
237C2×C79#357312yes—
238D79—24no—
239#3458—791no—
240#2962—798no—
241#2962—798no—
242#2961—792no—
243#2961—791no—
244#2961D7911yescharacteristic, complemented
245#2960—791no—
246#2960—792no—
247#2960—791no—
248#2905—791no—
249C2×#424—792no—
250#2873—792no—
251C2×#405—791no—
252C79#6019311yescharacteristic, complemented, Sylow
253C22×M16—792no—
254C22×M16—791no—
255C23×C8—791no—
256#544—794no—
257#544—794no—
258#544—794no—
259#543—794no—
260#533—798no—
261#533—792no—
262#533—791no—
263#533C2×D7911yescharacteristic
264C2×#117—792no—
265C2×#117—791no—
266#446—794no—
267#446—794no—
268#446—798no—
269#446—798no—
270#446—798no—
271#446—798no—
272#446C2×D7918yescomplemented
273C8×Q8—794no—
274C8×Q8—794no—
275C8×Q8—798no—
276#424—792no—
277#424—792no—
278#424—792no—
279#424—798no—
280#424—798no—
281#424C2×D7912yes—
282#423—792no—
283#423—791no—
284#423C2×D7911yescharacteristic
285C2×C4⋊C8—798no—
286#405—792no—
287#405—792no—
288#405—792no—
289#405—794no—
290#405—794no—
291#405—794no—
292#405—794no—
293#405C2×D7912yes—
294C4×M16—794no—
295C4×M16—794no—
296C2×#97—791no—
297C2×#97C2×D7911yescharacteristic
298C2×C4×C8—791no—
299C2×C4×C8—791no—
300C23×C4—791no—
301C2×M16—792no—
302C2×M16—798no—
303C2×M16—792no—
304C2×M16—792no—
305C2×M16—794no—
306C2×M16—1588no—
307C2×M16C22×D7912yes—
308C22×C8—791no—
309C22×C8—791no—
310C22×C8—794no—
311C22×C8—794no—
312C22×C8—1584no—
313C22×C8C22×D7911yescharacteristic
314C22×C8C22×D7911yescharacteristic
315C4⋊Q8—798no—
316C4⋊Q8—798no—
317C4⋊Q8—794no—
318C4⋊Q8—792no—
319C4⋊Q8C4×D7912yes—
320#125—798no—
321#125—798no—
322#125—794no—
323#125—792no—
324#125C4×D7912yes—
325C22⋊Q8—798no—
326C22⋊Q8—794no—
327C22⋊Q8—798no—
328C22⋊Q8—798no—
329C22⋊Q8—798no—
330C22⋊Q8—798no—
331C22⋊Q8C4×D7914yes—
332C4×Q8—794no—
333C4×Q8—798no—
334C4×Q8C22×D7914yes—
335#117—792no—
336#117—794no—
337#117—794no—
338#117—792no—
339#117—792no—
340#117—794no—
341#117—794no—
342#117C22×D7912yes—
343C2×C4⋊C4—792no—
344C2×C4⋊C4—794no—
345C2×C22⋊C4—792no—
346C2×C22⋊C4—794no—
347C2×C42—791no—
348C2×C42—794no—
349C2×C42C22×D7911yescharacteristic
350C4⋊C8—798no—
351C4⋊C8—798no—
352C4⋊C8—798no—
353C4⋊C8—798no—
354C4⋊C8—794no—
355C4⋊C8C22×D7918yes—
356C4⋊C8C22×D7914yes—
357#97—792no—
358#97—792no—
359#97—794no—
360#97—794no—
361#97—794no—
362#97C22×D7914yes—
363#97C22×D7914yes—
364C4×C8—794no—
365C4×C8—794no—
366C4×C8—794no—
367C4×C8—792no—
368C4×C8—792no—
369C4×C8—794no—
370C4×C8C22×D7914yes—
371C24—791no—
372C2×Q8—798no—
373C2×Q8—794no—
374C2×Q8C2×C4×D7914yes—
375C22×C4—792no—
376C22×C4—798no—
377C22×C4—791no—
378C22×C4—794no—
379C22×C4—794no—
380C22×C4—792no—
381C22×C4—792no—
382C22×C4C2×C4×D7912yes—
383C22×C4C23×D7911yescharacteristic
384M16—24no—
385M16—798no—
386M16—798no—
387M16—1588no—
388M16—1584no—
389C2×C8—22no—
390C2×C8—794no—
391C2×C8—791no—
392C2×C8—791no—
393C2×C8—792no—
394C2×C8—794no—
395C2×C8—1588no—
396C2×C8—1588no—
397C2×C8—1582no—
398C2×C8Q8×D7914yes—
399C2×C8C23×D7914yes—
400C2×C8C23×D7912yes—
401C2×C8C23×D7912yes—
402C4⋊C4—798no—
403C4⋊C4—794no—
404C4⋊C4—794no—
405C4⋊C4—798no—
406C4⋊C4—798no—
407C4⋊C4—798no—
408C4⋊C4—794no—
409C4⋊C4—798no—
410C4⋊C4—794no—
411C4⋊C4C2×C4×D7914yes—
412C4⋊C4C2×C4×D7914yes—
413C4⋊C4C2×C4×D7918yes—
414C22⋊C4—794no—
415C22⋊C4—798no—
416C22⋊C4—794no—
417C22⋊C4—794no—
418C22⋊C4C2×C4×D7914yes—
419C42—794no—
420C42—792no—
421C42—798no—
422C42—792no—
423C42C2×C4×D7911yescharacteristic
424C42C2×C4×D7911yescharacteristic
425C42C23×D7912yes—
426C42C23×D7914yes—
427C23—3162no—
428C23—794no—
429C23—792no—
430C23C22×C4×D7911yescharacteristic
431Q8—44no—
432Q8—3164no—
433Q8—3168no—
434C2×C4—42no—
435C2×C4—3168no—
436C2×C4—3162no—
437C2×C4—794no—
438C2×C4—792no—
439C2×C4—798no—
440C2×C4—794no—
441C2×C4—794no—
442C2×C4—798no—
443C2×C4—794no—
444C2×C4C22×C4×D7918yes—
445C2×C4C22×C4×D7912yes—
446C2×C4C22×C4×D7912yes—
447C2×C4C22×C4×D7911yescharacteristic
448C2×C4C2×Q8×D7912yes—
449C2×C4#41×D7912yes—
450C2×C4C24×D7911yescharacteristic
451C8—22no—
452C8—24no—
453C8—792no—
454C8—792no—
455C8—1582no—
456C8—1584no—
457C8C2×Q8×D7914yes—
458C22—22no—
459C22—3164no—
460C22—794no—
461C22—1584no—
462C22C4×Q8×D7912yes—
463C22C23×C4×D7911yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—794no—
468C4—1588no—
469C4—1584no—
470C4C4×Q8×D7912yes—
471C4C22×Q8×D7911yescharacteristic
472C4C2×#41×D7911yescharacteristic
473C2—22no—
474C2—1584no—
475C2C2×C4×Q8×D7911yescharacteristic
476C2#2377811312yes—
subgroupG/H
center, FrattiniC2×C4C24×D79
commutator, socleC22×C79C24×C4
FittingC79×#2961C2
prime radicalC23×D79C22×C4
cosocleC2×C4×C79C25
hypercenter#2961D79
hypocenterC79#60193
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ #2961

Complements

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

NH ≅ G/Ntype
2. #23772348C2semidirect
3. #23772345C2semidirect
4. #23772340C2semidirect
5. #23772339C2semidirect
6. #23772335C2semidirect
7. #23772332C2semidirect
8. #23772330C2semidirect
13. C79×#2961C2semidirect
26. #19098694C22semidirect
27. #19098692C22semidirect
28. #19098691C22semidirect
29. #19098691C22semidirect
30. #19098690C22semidirect
31. #19098690C22semidirect
32. Q8×C79⋊C8C22semidirect
33. C8×Dic158C22semidirect
34. #19098687C22semidirect
56. #446×C79C22semidirect
63. C2×#11929607Q8semidirect
244. #2961D79semidirect
252. C79#60193semidirect
272. #446C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237768032
#237723482
#237723452
#237723402
#237723392
#237723352
#237723322
#237723302
#237719622
#424×D792
#237718162
C2×#190985202
C79×#29612
#6019379