Finite Group Databasebeta

← #34205431

Subgroups — #34205431

Order 20224.

484 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23776534C211yesmaximal, characteristic
2#23765135C211yesmaximal, characteristic, complemented
3#23765128C211yesmaximal, characteristic, complemented
4#23764288C211yesmaximal, characteristic, complemented
5#23761516C211yesmaximal, characteristic, complemented
6#23761482C211yesmaximal, characteristic, complemented
7C4×#135×C79C211yesmaximal, characteristic, complemented
8C2×#14579183C2211yescharacteristic
9#19099004—21no—
10#19099003C412yescomplemented
11#19098999—21no—
12#19098998C412yescomplemented
13#19098993—22no—
14C2×#14579101C2211yescharacteristic
15C2×#14579101C2211yescharacteristic
16C2×#14579097C2211yescharacteristic
17C2×#14579089—21no—
18#19098529—21no—
19#19098372—22no—
20#19098370—22no—
21C2×#135×C79C2211yescharacteristic
22C4×#41×C79—21no—
23C4×Q16×C79—21no—
24C4×SD16×C79—22no—
25C4×D8×C79—21no—
26#427×C79C2211yescharacteristic
27#416×C79—21no—
28C2×C4×C8×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
37C2×#11929608—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#14579097—21no—
49#14579097—21no—
50#14579089—41no—
51#14579089—41no—
52(C2×D79)⋊C8—42no—
53C2×#41×C79—21no—
54#135×C79—21no—
55#135×C79—21no—
56#135×C79C2×C412yes—
57#135×C79D412yes—
58C2×Q16×C79D411yescharacteristic
59C2×SD16×C79—21no—
60C2×D8×C79D411yescharacteristic
61C22×C8×C79D411yescharacteristic
62C22×C8×C79C2311yescharacteristic
63C4×Q8×C79—22no—
64C4×D4×C79—22no—
65C4×D4×C79—42no—
66#117×C79—21no—
67#117×C79—41no—
68C2×C42×C79—41no—
69C2×C42×C79D411yescharacteristic
70#107×C79—21no—
71#106×C79D412yes—
72#103×C79—22no—
73#102×C79—22no—
74C4×C8×C79—21no—
75C4×C8×C79—21no—
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#11929613—41no—
87#11929613—41no—
88D632—21no—
89D632—21no—
90#11929608—41no—
91#11929608—41no—
92C2×C79⋊C8—41no—
93C79⋊C16—22no—
94C79⋊C16—41no—
95#41×C79—22no—
96#41×C79—22no—
97#41×C79—41no—
98#41×C79—41no—
99C2×Q8×C79—21no—
100C2×D4×C79—21no—
101C2×D4×C79—41no—
102C22×C4×C79—21no—
103C22×C4×C79—42no—
104C22×C4×C79—41no—
105C22×C4×C79C2×D411yescharacteristic
106Q16×C79—21no—
107Q16×C79C22⋊C412yes—
108SD16×C79—22no—
109SD16×C79—22no—
110D8×C79—21no—
111D8×C79C22⋊C412yes—
112C2×C8×C79—21no—
113C2×C8×C79—21no—
114C2×C8×C79—21no—
115C2×C8×C79C22×C411yescharacteristic
116C2×C8×C79C2×D411yescharacteristic
117C2×C8×C79C2×D411yescharacteristic
118C2×C8×C79C2×D411yescharacteristic
119C2×C8×C79#4111yescharacteristic
120C2×C8×C79#4111yescharacteristic
121C4⋊C4×C79—22no—
122C4⋊C4×C79—42no—
123C22⋊C4×C79—42no—
124C22⋊C4×C79—41no—
125C42×C79—21no—
126C42×C79—42no—
127C42×C79—41no—
128C42×C79D811yescharacteristic
129C42×C79D811yescharacteristic
130C23×C79—41no—
131C23×C79AGL(1,ℤ/8)11yescharacteristic
132C22×D79—41no—
133Q8×C79—22no—
134Q8×C79—41no—
135D4×C79—22no—
136D4×C79—42no—
137D4×C79—42no—
138D4×C79—41no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—42no—
142C2×C4×C79—41no—
143C2×C4×C79—42no—
144C2×C4×C79—42no—
145C2×C4×C79—42no—
146C2×C4×C79—84no—
147C2×C4×C79C4×D411yescharacteristic
148C2×C4×C79#12011yescharacteristic
149C2×C4×C79#12111yescharacteristic
150C2×C4×C79C2×D811yescharacteristic
151C2×C4×C79#13511yescharacteristic
152C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
153C22⋊D79—81no—
154C22⋊D79—81no—
155C2×Dic79—41no—
156D316—41no—
157D316—41no—
158C4×D79—81no—
159C4×D79—81no—
160Dic158—41no—
161Dic158—41no—
162C8×C79—41no—
163C8×C79C2×C22⋊C411yescharacteristic
164C8×C79C4×D411yescharacteristic
165C8×C79#12111yescharacteristic
166C8×C79#12311yescharacteristic
167C79⋊C8—81no—
168C22×C79—41no—
169C22×C79—42no—
170C22×C79—82no—
171C22×C79#44211yescharacteristic
172C22×C79#44711yescharacteristic
173C22×C79#46311yescharacteristic
174C2×D79—82no—
175C2×D79—81no—
176C4×C79—22no—
177C4×C79—41no—
178C4×C79—42no—
179C4×C79—82no—
180C4×C79#38611yescharacteristic
181C4×C79C4×D811yescharacteristic
182C4×C79#44911yescharacteristic
183C4×C79#45911yescharacteristic
184Dic79—81no—
185#13027—791nomaximal, Sylow
186C2×C79—21no—
187C2×C79—21no—
188C2×C79—21no—
189C2×C79—82no—
190C2×C79#187911yescharacteristic
191D79—161no—
192#3414—791no—
193C4×#135D7911yescharacteristic, complemented
194#2145—791no—
195#2143—791no—
196#1974—791no—
197#1389—791no—
198#1376—791no—
199C79#1302711yescharacteristic, complemented, Sylow
200C2×#135—791no—
201C2×#135C2×D7911yescharacteristic
202C4×#41—21no—
203#510—792no—
204#510—1581no—
205#509—792no—
206#509—1581no—
207#508—1582no—
208C2×M32—791no—
209C2×M32—791no—
210C4×Q16—21no—
211C4×SD16—22no—
212C4×D8—21no—
213C2×#108—791no—
214#427C2×D7911yescharacteristic
215C2×#104—1581no—
216#416—21no—
217#408—1581no—
218C2×C4×C8C2×D7911yescharacteristic
219#360—1582no—
220#359—1582no—
221C2×#41—21no—
222C2×#41—1581no—
223#135—21no—
224#135—21no—
225#135—1581no—
226#135—1581no—
227#135—1581no—
228#135—1581no—
229#135C4×D7912yes—
230#135D31612yes—
231C2×Q16—791no—
232C2×Q16C22⋊D7911yescharacteristic
233C2×SD16—21no—
234C2×SD16—1581no—
235C2×D8—791no—
236C2×D8C22⋊D7911yescharacteristic
237C2×M16—1581no—
238C22×C8C22⋊D7911yescharacteristic
239C22×C8C22×D7911yescharacteristic
240C4×Q8—22no—
241C4×D4—22no—
242C4×D4—42no—
243#117—21no—
244#117—41no—
245C2×C42—41no—
246C2×C42C22⋊D7911yescharacteristic
247Q32—1582no—
248SD32—1582no—
249SD32—1582no—
250D16—1582no—
251M32—792no—
252M32—1581no—
253M32—1581no—
254M32—1581no—
255C2×C16—1581no—
256C2×C16—1581no—
257#108—1581no—
258#108—1581no—
259#107—21no—
260#106C22⋊D7912yes—
261#104—3161no—
262#104—3161no—
263#103—22no—
264#102—22no—
265C22⋊C8—3162no—
266C4×C8—21no—
267C4×C8—21no—
268#41—22no—
269#41—22no—
270#41—41no—
271#41—41no—
272#41—3161no—
273#41—3161no—
274#41—3161no—
275#41—3161no—
276C2×Q8—21no—
277C2×Q8—1581no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—3161no—
281C2×D4—1581no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—3161no—
286C22×C4C2×C22⋊D7911yescharacteristic
287Q16—21no—
288Q16—1581no—
289Q16—1581no—
290Q16(C2×D79)⋊C412yes—
291SD16—22no—
292SD16—22no—
293SD16—3161no—
294SD16—3161no—
295D8—21no—
296D8—1581no—
297D8—1581no—
298D8(C2×D79)⋊C412yes—
299M16—3161no—
300M16—3161no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—3161no—
305C2×C8C2×C4×D7911yescharacteristic
306C2×C8C2×C22⋊D7911yescharacteristic
307C2×C8C2×C22⋊D7911yescharacteristic
308C2×C8C2×D31611yescharacteristic
309C2×C8#1192962411yescharacteristic
310C2×C8#1192962411yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C42—21no—
316C42—42no—
317C42—41no—
318C42#1192960911yescharacteristic
319C42#1192960911yescharacteristic
320C16—3161no—
321C16—1582no—
322C23—41no—
323C23—3161no—
324C23#1457918711yescharacteristic
325Q8—22no—
326Q8—41no—
327Q8—3161no—
328Q8—3161no—
329D4—22no—
330D4—42no—
331D4—42no—
332D4—41no—
333D4—3161no—
334D4—3161no—
335D4—6321no—
336D4—6321no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—84no—
345C2×C4—3161no—
346C2×C4—6321no—
347C2×C4—6321no—
348C2×C4C4×C22⋊D7911yescharacteristic
349C2×C4#1457913011yescharacteristic
350C2×C4#1457913511yescharacteristic
351C2×C4C2×#1192960911yescharacteristic
352C2×C4#1457918111yescharacteristic
353C2×C4#1457918611yescharacteristic
354C8—41no—
355C8—6321no—
356C8C2×(C2×D79)⋊C411yescharacteristic
357C8C4×C22⋊D7911yescharacteristic
358C8#1457913511yescharacteristic
359C8#1457914411yescharacteristic
360C22—41no—
361C22—42no—
362C22—82no—
363C22—6322no—
364C22—6321no—
365C22#1909867311yescharacteristic
366C22#1909869711yescharacteristic
367C22#1909877811yescharacteristic
368C4—22no—
369C4—41no—
370C4—42no—
371C4—82no—
372C4—6321no—
373C4#1909844211yescharacteristic
374C4C4×#1192960911yescharacteristic
375C4#1909870811yescharacteristic
376C4#1909875811yescharacteristic
377C2—21no—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—12641no—
382C2#2376374811yescharacteristic
subgroupG/H
centerC4#19098758
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC4×#135×C79C2
socleC2×C79#1879
prime radical#23776534C2
cosocleC22×C8×C79C23
hypercenterC4×#135D79
hypocenterC79#13027
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
2. #23765135C2semidirect
3. #23765128C2semidirect
4. #23764288C2semidirect
5. #23761516C2semidirect
6. #23761482C2semidirect
7. C4×#135×C79C2semidirect
10. #19099003C4semidirect
12. #19098998C4semidirect
34. C2×Dic316C2×C4semidirect
36. C2×D632C2×C4semidirect
45. #14579101C2×C4semidirect
193. C4×#135D79semidirect
199. C79#13027semidirect

Maximal subgroups

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

subgroupindex
#237765342
#237651352
#237651282
#237642882
#237615162
#237614822
C4×#135×C792
#1302779