Finite Group Databasebeta

← #34205452

Subgroups — #34205452

Order 20224.

476 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776534C211yesmaximal, characteristic
2#23765136C211yesmaximal, characteristic, complemented
3C2×#19098372C211yesmaximal, characteristic, complemented
4#23764298C211yesmaximal, characteristic, complemented
5#23761507C211yesmaximal, characteristic, complemented
6#23761486C211yesmaximal, characteristic, complemented
7C79×#2952C211yesmaximal, characteristic, complemented
8C2×#14579183C2211yescharacteristic
9#19099004—21no—
10#19099003C412yescomplemented
11#19098999—21no—
12#19098998C412yescomplemented
13#19098993—22no—
14C2×#14579101C2211yescharacteristic
15C2×#14579101C2211yescharacteristic
16#19098601C2211yescharacteristic
17#19098579—21no—
18#19098536—21no—
19#19098373—22no—
20#19098372—22no—
21C2×#135×C79C2211yescharacteristic
22#520×C79—21no—
23#442×C79—21no—
24#441×C79—21no—
25C4×SD16×C79—22no—
26C2×#106×C79C2211yescharacteristic
27#417×C79—21no—
28#405×C79C2211yescharacteristic
29C2×#11929624—21no—
30#14579183—21no—
31#14579183—21no—
32#14579183—21no—
33#14579183—21no—
34C2×Dic316C2×C411yescharacteristic, complemented
35C2×#11929613—21no—
36C2×D632C2×C411yescharacteristic, complemented
37#14579120—21no—
38#14579107—22no—
39#14579105—22no—
40#14579104—22no—
41#14579102—22no—
42#14579101—21no—
43#14579101—21no—
44#14579101—21no—
45#14579101C2×C412yescomplemented
46C2×C79⋊C16—21no—
47C2×C79⋊C16—21no—
48#14579095—21no—
49#14579093—21no—
50#14579087—42no—
51#14579075—42no—
52C2×#41×C79—21no—
53#135×C79—21no—
54#135×C79—21no—
55#135×C79C2×C412yes—
56#135×C79D412yes—
57C2×Q16×C79D411yescharacteristic
58C2×SD16×C79—21no—
59C2×D8×C79D411yescharacteristic
60C2×M16×C79D411yescharacteristic
61C22×C8×C79C2311yescharacteristic
62C4×Q8×C79—22no—
63C4×D4×C79—22no—
64C4×D4×C79—42no—
65#117×C79—41no—
66#117×C79D411yescharacteristic
67C2×C4⋊C4×C79—21no—
68C2×C4⋊C4×C79—41no—
69#106×C79—21no—
70#106×C79D412yes—
71#103×C79—22no—
72#102×C79—22no—
73#97×C79—21no—
74C4×C8×C79—21no—
75#11929624—41no—
76#11929624—41no—
77#11929624—41no—
78#11929624—41no—
79C2×Dic158—21no—
80C2×D316—21no—
81C2×C22⋊D79—41no—
82C2×C4×D79—41no—
83Dic316—21no—
84Dic316—21no—
85#11929613—41no—
86#11929613—41no—
87D632—21no—
88D632—21no—
89C4⋊Dic79—41no—
90C22⋊Dic79—41no—
91C4×Dic79—41no—
92C79⋊C16—22no—
93C79⋊C16—41no—
94#41×C79—22no—
95#41×C79—22no—
96#41×C79—41no—
97#41×C79—41no—
98C2×Q8×C79—21no—
99C2×D4×C79—21no—
100C2×D4×C79—41no—
101C22×C4×C79—42no—
102C22×C4×C79—41no—
103C22×C4×C79—41no—
104C22×C4×C79C2×D411yescharacteristic
105Q16×C79—21no—
106Q16×C79C22⋊C412yes—
107SD16×C79—22no—
108SD16×C79—22no—
109D8×C79—21no—
110D8×C79C22⋊C412yes—
111M16×C79—21no—
112M16×C79—21no—
113C2×C8×C79—21no—
114C2×C8×C79C22×C411yescharacteristic
115C2×C8×C79C2×D411yescharacteristic
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79C2×D411yescharacteristic
118C2×C8×C79#4111yescharacteristic
119C2×C8×C79#4111yescharacteristic
120C4⋊C4×C79—21no—
121C4⋊C4×C79—22no—
122C4⋊C4×C79—42no—
123C4⋊C4×C79—41no—
124C22⋊C4×C79—42no—
125C22⋊C4×C79D812yes—
126C42×C79—21no—
127C42×C79—42no—
128C23×C79—41no—
129C23×C79C2×D811yescharacteristic
130C22×D79—41no—
131Q8×C79—22no—
132Q8×C79—41no—
133D4×C79—22no—
134D4×C79—42no—
135D4×C79—42no—
136D4×C79—41no—
137C2×C4×C79—22no—
138C2×C4×C79—41no—
139C2×C4×C79—42no—
140C2×C4×C79—42no—
141C2×C4×C79—42no—
142C2×C4×C79—162no—
143C2×C4×C79—161no—
144C2×C4×C79C4×D411yescharacteristic
145C2×C4×C79#12011yescharacteristic
146C2×C4×C79#12111yescharacteristic
147C2×C4×C79#13511yescharacteristic
148C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
150C22⋊D79—81no—
151C22⋊D79—81no—
152C2×Dic79—41no—
153C2×Dic79—81no—
154D316—41no—
155D316—41no—
156C4×D79—81no—
157C4×D79—81no—
158Dic158—41no—
159Dic158—41no—
160C8×C79—41no—
161C8×C79C2×C22⋊C411yescharacteristic
162C8×C79C4×D411yescharacteristic
163C8×C79#12111yescharacteristic
164C8×C79#12311yescharacteristic
165C22×C79—41no—
166C22×C79—42no—
167C22×C79—81no—
168C22×C79—81no—
169C22×C79C4×D811yescharacteristic
170C22×C79#44711yescharacteristic
171C22×C79#45911yescharacteristic
172C2×D79—81no—
173C2×D79—161no—
174C4×C79—42no—
175C4×C79—42no—
176C4×C79—82no—
177C4×C79#38611yescharacteristic
178C4×C79#44211yescharacteristic
179C4×C79#44911yescharacteristic
180C4×C79#46311yescharacteristic
181Dic79—81no—
182Dic79—161no—
183#13030—791nomaximal, Sylow
184C2×C79—21no—
185C2×C79—21no—
186C2×C79—21no—
187C2×C79—82no—
188C2×C79#187911yescharacteristic
189D79—161no—
190#3414—791no—
191#2952D7911yescharacteristic, complemented
192#2145—791no—
193C2×#360—791no—
194#1976—791no—
195#1386—791no—
196#1378—791no—
197C79#1303011yescharacteristic, complemented, Sylow
198C2×#135—791no—
199C2×#135C2×D7911yescharacteristic
200#520—21no—
201#510—792no—
202#510—1581no—
203#509—792no—
204#509—1581no—
205#508—1582no—
206C2×M32—791no—
207C2×M32—791no—
208#442—21no—
209#441—21no—
210C4×SD16—22no—
211#427—791no—
212C2×#106C2×D7911yescharacteristic
213#421—1581no—
214#417—21no—
215#410—1581no—
216#405C2×D7911yescharacteristic
217#360—1582no—
218#360—1582no—
219C2×#41—21no—
220C2×#41—1581no—
221#135—21no—
222#135—21no—
223#135—1581no—
224#135—1581no—
225#135—1581no—
226#135—1581no—
227#135C4×D7912yes—
228#135D31612yes—
229C2×Q16—791no—
230C2×Q16C22⋊D7911yescharacteristic
231C2×SD16—21no—
232C2×SD16—1581no—
233C2×D8—791no—
234C2×D8C22⋊D7911yescharacteristic
235C2×M16C22⋊D7911yescharacteristic
236C22×C8C22×D7911yescharacteristic
237C4×Q8—22no—
238C4×D4—22no—
239C4×D4—42no—
240#117—41no—
241#117—1581no—
242#117C22⋊D7911yescharacteristic
243C2×C4⋊C4—21no—
244C2×C4⋊C4—41no—
245Q32—1582no—
246SD32—1582no—
247SD32—1582no—
248D16—1582no—
249M32—792no—
250M32—1581no—
251M32—1581no—
252M32—1581no—
253C2×C16—1581no—
254C2×C16—1581no—
255#107—1581no—
256#106—21no—
257#106—1581no—
258#106C22⋊D7912yes—
259#104—3162no—
260#103—22no—
261#102—22no—
262#99—3162no—
263#97—21no—
264C4×C8—21no—
265#41—22no—
266#41—22no—
267#41—41no—
268#41—41no—
269#41—3161no—
270#41—3161no—
271#41—3161no—
272#41—3161no—
273C2×Q8—21no—
274C2×Q8—1581no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—3161no—
278C2×D4—1581no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—3161no—
283C22×C4C2×C22⋊D7911yescharacteristic
284Q16—21no—
285Q16—1581no—
286Q16—1581no—
287Q16(C2×D79)⋊C412yes—
288SD16—22no—
289SD16—22no—
290SD16—3161no—
291SD16—3161no—
292D8—21no—
293D8—1581no—
294D8—1581no—
295D8(C2×D79)⋊C412yes—
296M16—21no—
297M16—21no—
298C2×C8—21no—
299C2×C8C2×C4×D7911yescharacteristic
300C2×C8C2×C22⋊D7911yescharacteristic
301C2×C8C2×C22⋊D7911yescharacteristic
302C2×C8C2×D31611yescharacteristic
303C2×C8#1192962411yescharacteristic
304C2×C8#1192962411yescharacteristic
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—3161no—
310C22⋊C4—42no—
311C22⋊C4—3161no—
312C22⋊C4#1192960912yes—
313C42—21no—
314C42—42no—
315C42—3161no—
316C16—3161no—
317C16—1582no—
318C23—41no—
319C23—3161no—
320C23C2×#1192960911yescharacteristic
321Q8—22no—
322Q8—41no—
323Q8—3161no—
324Q8—3161no—
325D4—22no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—3161no—
330D4—3161no—
331D4—6321no—
332D4—6321no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—162no—
339C2×C4—161no—
340C2×C4—3161no—
341C2×C4—6321no—
342C2×C4—6321no—
343C2×C4—6321no—
344C2×C4C4×C22⋊D7911yescharacteristic
345C2×C4#1457913011yescharacteristic
346C2×C4#1457913511yescharacteristic
347C2×C4#1457918111yescharacteristic
348C2×C4#1457918611yescharacteristic
349C2×C4#1457918711yescharacteristic
350C8—41no—
351C8C2×(C2×D79)⋊C411yescharacteristic
352C8C4×C22⋊D7911yescharacteristic
353C8#1457913511yescharacteristic
354C8#1457914411yescharacteristic
355C22—41no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—6321no—
360C22—12641no—
361C22C4×#1192960911yescharacteristic
362C22#1909869711yescharacteristic
363C22#1909875811yescharacteristic
364C4—42no—
365C4—42no—
366C4—82no—
367C4—6321no—
368C4—12641no—
369C4#1909844211yescharacteristic
370C4#1909867311yescharacteristic
371C4#1909870811yescharacteristic
372C4#1909877811yescharacteristic
373C2—21no—
374C2—21no—
375C2—21no—
376C2—82no—
377C2—12641no—
378C2#2376374811yescharacteristic
subgroupG/H
centerC2#23763748
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC79×#2952C2
socleC2×C79#1879
prime radical#23776534C2
cosocleC22×C8×C79C23
hypercenter#2952D79
hypocenterC79#13030
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2952

Complements

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

NH ≅ G/Ntype
2. #23765136C2semidirect
3. C2×#19098372C2semidirect
4. #23764298C2semidirect
5. #23761507C2semidirect
6. #23761486C2semidirect
7. C79×#2952C2semidirect
10. #19099003C4semidirect
12. #19098998C4semidirect
34. C2×Dic316C2×C4semidirect
36. C2×D632C2×C4semidirect
45. #14579101C2×C4semidirect
191. #2952D79semidirect
197. C79#13030semidirect

Maximal subgroups

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

subgroupindex
#237765342
#237651362
C2×#190983722
#237642982
#237615072
#237614862
C79×#29522
#1303079