Finite Group Databasebeta

← #34205283

Subgroups — #34205283

Order 20224.

492 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773120C211yesmaximal, characteristic
2#23765135C211yesmaximal, characteristic, complemented
3C2×#19098372C211yesmaximal, characteristic, complemented
4#23765028C211yesmaximal, characteristic, complemented
5#23761516C211yesmaximal, characteristic, complemented
6#23761498C211yesmaximal, characteristic, complemented
7C79×#2912C211yesmaximal, characteristic, complemented
8#19099361—21no—
9C2×#14579183C2211yescharacteristic
10C2×#14579101C2211yescharacteristic
11C2×#14579101C2211yescharacteristic
12#19098838C412yescomplemented
13#19098833C412yescomplemented
14#19098826—22no—
15C2×#14579097C2211yescharacteristic
16#19098545—21no—
17#19098372—22no—
18#19098372—21no—
19#19098372—21no—
20#19098347—22no—
21C2×#131×C79C2211yescharacteristic
22#520×C79—21no—
23#428×C79—21no—
24#428×C79—22no—
25#427×C79—21no—
26#427×C79C2211yescharacteristic
27C2×#106×C79—21no—
28C2×#106×C79C2211yescharacteristic
29C2×#11929624—21no—
30#14579192—41no—
31#14579191—22no—
32#14579187—41no—
33#14579186—22no—
34#14579183—21no—
35#14579183—21no—
36#14579183—21no—
37#14579183—21no—
38#14579181—42no—
39C2×Dic316C2×C411yescharacteristic, complemented
40C2×#11929613—21no—
41C2×D632C2×C411yescharacteristic, complemented
42C2×#11929608—21no—
43#14579101—21no—
44#14579101—21no—
45#14579101—21no—
46#14579101—21no—
47C2×C79⋊C16—21no—
48C2×C79⋊C16—21no—
49#14579097—21no—
50#14579097C2×C412yescomplemented
51#14579079—22no—
52#14579077—22no—
53C2×#41×C79D411yescharacteristic
54#131×C79—21no—
55#131×C79—21no—
56#131×C79C2×C412yes—
57#131×C79D412yes—
58C2×M16×C79—21no—
59C2×M16×C79D411yescharacteristic
60C22×C8×C79—21no—
61C22×C8×C79C2311yescharacteristic
62C4×Q8×C79—41no—
63C4×D4×C79—42no—
64C4×D4×C79—41no—
65#117×C79—22no—
66#117×C79—21no—
67C2×C4⋊C4×C79—22no—
68C2×C4⋊C4×C79—21no—
69#107×C79—22no—
70#107×C79—21no—
71#106×C79—22no—
72#106×C79—21no—
73#106×C79D411yescharacteristic
74#106×C79D412yes—
75#106×C79D411yescharacteristic
76#11929624—41no—
77#11929624—41no—
78#11929624—41no—
79#11929624—41no—
80C2×Dic158—21no—
81C2×D316—21no—
82C2×C22⋊D79—41no—
83C2×C4×D79—41no—
84Dic316—21no—
85Dic316—21no—
86#11929614—42no—
87#11929613—41no—
88#11929613—41no—
89#11929612—42no—
90#11929611—42no—
91D632—21no—
92D632—21no—
93#11929609—42no—
94#11929608—22no—
95#11929608—41no—
96C2×C79⋊C8—41no—
97C79⋊C16—41no—
98C79⋊C16—41no—
99#41×C79—21no—
100#41×C79—21no—
101#41×C79C22⋊C412yes—
102#41×C79SD1612yes—
103C2×Q8×C79D811yescharacteristic
104C2×D4×C79—21no—
105C2×D4×C79D811yescharacteristic
106C22×C4×C79—21no—
107C22×C4×C79—42no—
108C22×C4×C79—41no—
109C22×C4×C79C2×D411yescharacteristic
110M16×C79—22no—
111M16×C79—22no—
112M16×C79—21no—
113M16×C79C22⋊C412yes—
114C2×C8×C79—22no—
115C2×C8×C79—22no—
116C2×C8×C79—21no—
117C2×C8×C79C22×C411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79C2×D411yescharacteristic
120C2×C8×C79C2×D411yescharacteristic
121C2×C8×C79#4111yescharacteristic
122C2×C8×C79#4111yescharacteristic
123C4⋊C4×C79—21no—
124C4⋊C4×C79—22no—
125C4⋊C4×C79—21no—
126C4⋊C4×C79—42no—
127C4⋊C4×C79—41no—
128C22⋊C4×C79—42no—
129C22⋊C4×C79—41no—
130C42×C79—42no—
131C42×C79—41no—
132C23×C79—21no—
133C23×C79AGL(1,ℤ/8)11yescharacteristic
134C22×D79—41no—
135Q8×C79—21no—
136Q8×C79#10212yes—
137D4×C79—22no—
138D4×C79—22no—
139D4×C79—21no—
140D4×C79#10212yes—
141C2×C4×C79—21no—
142C2×C4×C79—22no—
143C2×C4×C79—22no—
144C2×C4×C79—41no—
145C2×C4×C79—42no—
146C2×C4×C79—41no—
147C2×C4×C79—161no—
148C2×C4×C79—162no—
149C2×C4×C79C2×C22⋊C411yescharacteristic
150C2×C4×C79#12011yescharacteristic
151C2×C4×C79#12311yescharacteristic
152C2×C4×C79C2×D811yescharacteristic
153C2×C4×C79C2×SD1611yescharacteristic
154C2×C4×C79#13711yescharacteristic
155C22⋊D79—81no—
156C22⋊D79—81no—
157C2×Dic79—41no—
158D316—41no—
159D316—41no—
160C4×D79—81no—
161C4×D79—81no—
162Dic158—41no—
163Dic158—41no—
164C8×C79—22no—
165C8×C79C4×D411yescharacteristic
166C8×C79C4×D411yescharacteristic
167C8×C79#12111yescharacteristic
168C8×C79#12111yescharacteristic
169C79⋊C8—42no—
170C22×C79—22no—
171C22×C79—41no—
172C22×C79—42no—
173C22×C79#41711yescharacteristic
174C22×C79#44711yescharacteristic
175C22×C79#48111yescharacteristic
176C2×D79—81no—
177C2×D79—161no—
178C4×C79—22no—
179C4×C79—81no—
180C4×C79—82no—
181C4×C79—81no—
182C4×C79#38611yescharacteristic
183C4×C79C2×#10211yescharacteristic
184C4×C79#44811yescharacteristic
185C4×C79#48011yescharacteristic
186Dic79—81no—
187#13005—791nomaximal, Sylow
188C2×C79—21no—
189C2×C79—21no—
190C2×C79—21no—
191C2×C79—42no—
192C2×C79#187411yescharacteristic
193D79—161no—
194#3067—791no—
195#2912D7911yescharacteristic, complemented
196#2145—791no—
197C2×#360—791no—
198#2117—791no—
199#1389—791no—
200#1383—791no—
201C79#1300511yescharacteristic, complemented, Sylow
202#575—1581no—
203C2×#135—791no—
204C2×#131C2×D7911yescharacteristic
205#520—21no—
206C2×M32—791no—
207C2×M32—791no—
208#473—792no—
209#472—792no—
210#471—1582no—
211C2×#108—791no—
212#428—21no—
213#428—22no—
214#427—21no—
215#427C2×D7911yescharacteristic
216C2×#106—21no—
217C2×#106C2×D7911yescharacteristic
218#413—1581no—
219#360—1582no—
220#360—1581no—
221#360—1581no—
222#350—1582no—
223C2×#41—1581no—
224C2×#41C22⋊D7911yescharacteristic
225#137—3161no—
226#137—1582no—
227AGL(1,ℤ/8)—3161no—
228AGL(1,ℤ/8)—1582no—
229#135—3162no—
230#135—1581no—
231#135—1581no—
232#135—1581no—
233#135—1581no—
234C2×Q16—791no—
235C2×SD16—1581no—
236C2×D8—791no—
237#131—21no—
238#131—21no—
239#131C4×D7912yes—
240#131D31612yes—
241C2×M16—21no—
242C2×M16—1581no—
243C2×M16C22⋊D7911yescharacteristic
244C22×C8—21no—
245C22×C8C22×D7911yescharacteristic
246C4×Q8—41no—
247C4×D4—42no—
248C4×D4—41no—
249#117—22no—
250#117—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253M32—1581no—
254M32—1581no—
255M32—1581no—
256M32—1581no—
257C2×C16—1581no—
258C2×C16—1581no—
259#108—792no—
260#108—1581no—
261#107—22no—
262#107—21no—
263#106—22no—
264#106—21no—
265#106C22⋊D7911yescharacteristic
266#106C22⋊D7912yes—
267#106C22⋊D7911yescharacteristic
268#101—1582no—
269#100—1582no—
270#41—21no—
271#41—21no—
272#41—3161no—
273#41—3161no—
274#41—3161no—
275#41—3161no—
276#41(C2×D79)⋊C412yes—
277#41#1192961112yes—
278C2×Q8—1581no—
279C2×Q8#1192960911yescharacteristic
280C2×D4—21no—
281C2×D4—3161no—
282C2×D4—1581no—
283C2×D4#1192960911yescharacteristic
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—3161no—
288C22×C4C2×C22⋊D7911yescharacteristic
289Q16—3162no—
290Q16—1581no—
291Q16—1581no—
292SD16—3162no—
293SD16—3162no—
294SD16—3161no—
295SD16—3161no—
296D8—3162no—
297D8—1581no—
298D8—1581no—
299M16—21no—
300M16—22no—
301M16—22no—
302M16—3161no—
303M16—1582no—
304M16(C2×D79)⋊C412yes—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—3161no—
309C2×C8C2×C4×D7911yescharacteristic
310C2×C8C2×C22⋊D7911yescharacteristic
311C2×C8C2×C22⋊D7911yescharacteristic
312C2×C8C2×D31611yescharacteristic
313C2×C8#1192962411yescharacteristic
314C2×C8#1192962411yescharacteristic
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—41no—
322C42—42no—
323C42—41no—
324C16—3161no—
325C16—3161no—
326C23—21no—
327C23—3161no—
328C23#1457918711yescharacteristic
329Q8—21no—
330Q8—3161no—
331Q8—3161no—
332Q8#1457908112yes—
333D4—22no—
334D4—22no—
335D4—21no—
336D4—3161no—
337D4—3161no—
338D4—6321no—
339D4—6321no—
340D4#1457908112yes—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—161no—
348C2×C4—162no—
349C2×C4—3161no—
350C2×C4—6321no—
351C2×C4—6321no—
352C2×C4C2×(C2×D79)⋊C411yescharacteristic
353C2×C4#1457913011yescharacteristic
354C2×C4#1457914411yescharacteristic
355C2×C4C2×#1192960911yescharacteristic
356C2×C4C2×#1192961111yescharacteristic
357C2×C4#1457919211yescharacteristic
358C8—22no—
359C8—3162no—
360C8C4×C22⋊D7911yescharacteristic
361C8C4×C22⋊D7911yescharacteristic
362C8#1457913511yescharacteristic
363C8#1457913511yescharacteristic
364C22—22no—
365C22—41no—
366C22—42no—
367C22—6321no—
368C22—12641no—
369C22#1909856211yescharacteristic
370C22#1909869711yescharacteristic
371C22#1909888111yescharacteristic
372C4—22no—
373C4—82no—
374C4—81no—
375C4—81no—
376C4—6321no—
377C4#1909844211yescharacteristic
378C4C2×#1457908111yescharacteristic
379C4#1909870211yescharacteristic
380C4#1909887611yescharacteristic
381C2—21no—
382C2—21no—
383C2—21no—
384C2—42no—
385C2—12641no—
386C2#2376371711yescharacteristic
subgroupG/H
centerC2#23763717
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#2912C2
socleC2×C79#1874
prime radical#23773120C2
cosocleC22×C8×C79C23
hypercenter#2912D79
hypocenterC79#13005
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #2912

Complements

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

NH ≅ G/Ntype
2. #23765135C2semidirect
3. C2×#19098372C2semidirect
4. #23765028C2semidirect
5. #23761516C2semidirect
6. #23761498C2semidirect
7. C79×#2912C2semidirect
12. #19098838C4semidirect
13. #19098833C4semidirect
39. C2×Dic316C2×C4semidirect
41. C2×D632C2×C4semidirect
50. #14579097C2×C4semidirect
195. #2912D79semidirect
201. C79#13005semidirect

Maximal subgroups

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

subgroupindex
#237731202
#237651352
C2×#190983722
#237650282
#237615162
#237614982
C79×#29122
#1300579