Finite Group Databasebeta

← #34185017

Subgroups — #34185017

Order 20224.

480 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774528C211yesmaximal, characteristic
2#23764232C211yesmaximal, characteristic
3#23762085C211yesmaximal, characteristic, complemented
4#23761959C211yesmaximal, characteristic
5#23761258C211yesmaximal, characteristic, complemented
6#23761229C211yesmaximal, characteristic, complemented
7C79×#2331C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098959C412yescomplemented
10#19098911C412yescomplemented
11#19098881—22no—
12#19098850—22no—
13#19098616C2211yescharacteristic
14#19098616C2211yescharacteristic
15#19098587C2211yescharacteristic
16C2×#14579089—21no—
17C2×#14579089—21no—
18#19098309—22no—
19#19098296—22no—
20C2×C4⋊Q8×C79C2211yescharacteristic
21#391×C79—22no—
22#389×C79—21no—
23#387×C79—22no—
24#382×C79C2211yescharacteristic
25C4×C4⋊C4×C79—21no—
26#376×C79C2211yescharacteristic
27C2×#11929624—21no—
28C2×#11929608—21no—
29#14579161C2×C411yescharacteristic
30#14579158C2×C411yescharacteristic
31#14579149—21no—
32#14579135—21no—
33#14579135—21no—
34C4×D316—21no—
35C4×D316—21no—
36#14579092—24no—
37C4⋊(C79⋊C8)—21no—
38C4⋊(C79⋊C8)—21no—
39C4⋊(C79⋊C8)—21no—
40C4⋊(C79⋊C8)—21no—
41#14579089—42no—
42#14579089—42no—
43#14579081—24no—
44C22⋊(C79⋊C8)—21no—
45C22⋊(C79⋊C8)—21no—
46C4×C79⋊C8—21no—
47C4×C79⋊C8C2×C412yes—
48C22×Q8×C79—21no—
49C4⋊Q8×C79—21no—
50C4⋊Q8×C79—21no—
51C4⋊Q8×C79C2×C412yes—
52C4⋊Q8×C79D412yes—
53C2×C4⋊C4×C79—21no—
54C2×C4⋊C4×C79—21no—
55C2×C4⋊C4×C79—22no—
56C2×C4⋊C4×C79D412yes—
57C2×C42×C79—22no—
58C2×C42×C79D411yescharacteristic
59C2×C42×C79D411yescharacteristic
60C2×C42×C79C2311yescharacteristic
61#95×C79—22no—
62#95×C79—22no—
63#95×C79—21no—
64#95×C79—21no—
65#95×C79—21no—
66#95×C79D412yes—
67#11929624—41no—
68#11929624—41no—
69#11929624—41no—
70#11929624—41no—
71C2×Dic158—21no—
72C2×D316—21no—
73C2×D316—21no—
74C2×D316—21no—
75C2×C22⋊D79—41no—
76C2×C4×D79—41no—
77#11929608—41no—
78#11929608—41no—
79C2×C79⋊C8—22no—
80C2×C79⋊C8—41no—
81C2×C79⋊C8—41no—
82C4⋊Dic79—21no—
83C4⋊Dic79—21no—
84(C2×D79)⋊C4—41no—
85(C2×D79)⋊C4—41no—
86C2×Q8×C79—22no—
87C2×Q8×C79—22no—
88C2×Q8×C79—82no—
89C22×C4×C79—21no—
90C22×C4×C79—22no—
91C22×C4×C79—21no—
92C22×C4×C79—22no—
93C22×C4×C79C2×D411yescharacteristic
94C22×C4×C79C2×D411yescharacteristic
95C22×C4×C79C2×D411yescharacteristic
96C4⋊C4×C79—22no—
97C4⋊C4×C79—22no—
98C4⋊C4×C79—22no—
99C4⋊C4×C79—41no—
100C4⋊C4×C79—41no—
101C4⋊C4×C79—82no—
102C4⋊C4×C79C22⋊C414yes—
103C42×C79—21no—
104C42×C79—21no—
105C42×C79—21no—
106C42×C79—82no—
107C42×C79SD1612yes—
108C42×C79C22×C411yescharacteristic
109C42×C79C2×D411yescharacteristic
110C42×C79#4111yescharacteristic
111C42×C79#4111yescharacteristic
112C23×C79#12011yescharacteristic
113C22×D79—41no—
114Q8×C79—82no—
115Q8×C79—82no—
116C2×C4×C79—24no—
117C2×C4×C79—22no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—42no—
122C2×C4×C79—42no—
123C2×C4×C79—41no—
124C2×C4×C79—42no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—42no—
128C2×C4×C79—41no—
129C2×C4×C79—42no—
130C2×C4×C79C2×C22⋊C411yescharacteristic
131C2×C4×C79C4×D411yescharacteristic
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79#12111yescharacteristic
134C2×C4×C79#12111yescharacteristic
135C2×C4×C79#12311yescharacteristic
136C2×C4×C79C2×SD1611yescharacteristic
137C2×C4×C79#13511yescharacteristic
138C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C79#13711yescharacteristic
140C22⋊D79—81no—
141C22⋊D79—81no—
142C2×Dic79—41no—
143D316—42no—
144D316—41no—
145D316—41no—
146D316—81no—
147C4×D79—81no—
148C4×D79—81no—
149Dic158—41no—
150Dic158—41no—
151C79⋊C8—42no—
152C79⋊C8—81no—
153C79⋊C8—81no—
154C22×C79—21no—
155C22×C79—21no—
156C22×C79#38611yescharacteristic
157C22×C79#44811yescharacteristic
158C22×C79#44911yescharacteristic
159C2×D79—81no—
160C2×D79—81no—
161C2×D79—81no—
162C4×C79—21no—
163C4×C79—21no—
164C4×C79—21no—
165C4×C79—21no—
166C4×C79—42no—
167C4×C79—44no—
168C4×C79—82no—
169C4×C79—81no—
170C4×C79C4×SD1611yescharacteristic
171C4×C79#44011yescharacteristic
172C4×C79#46011yescharacteristic
173C4×C79#46411yescharacteristic
174Dic79—81no—
175#9302—791nomaximal, Sylow
176C2×C79—21no—
177C2×C79—21no—
178C2×C79#188011yescharacteristic
179C2×C79#190411yescharacteristic
180C2×C79#191211yescharacteristic
181D79—161no—
182#3227—791no—
183#2331D7911yescharacteristic, complemented
184#1963—791no—
185#1549—791no—
186#1517—791no—
187#1286—791no—
188#1274—791no—
189C79#930211yescharacteristic, complemented, Sylow
190#532—791no—
191C2×C4⋊Q8C2×D7911yescharacteristic
192#498—792no—
193#486—792no—
194#481—1582no—
195#476—1582no—
196#431—791no—
197#431—791no—
198#423—791no—
199C2×#104—1581no—
200C2×#104—1581no—
201#391—22no—
202#389—21no—
203#387—22no—
204#382C2×D7911yescharacteristic
205C4×C4⋊C4—21no—
206#376C2×D7911yescharacteristic
207#332—1582no—
208#327—1582no—
209C2×#41—1581no—
210C22×Q8—21no—
211C2×M16—1581no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—791no—
215C4⋊Q8C4×D7912yes—
216C4⋊Q8D31612yes—
217#127—791no—
218#124—1581no—
219#121—1581no—
220#121—1581no—
221C4×D4—1581no—
222C4×D4—1581no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C22⋊D7912yes—
227C2×C42—22no—
228C2×C42C22⋊D7911yescharacteristic
229C2×C42C22⋊D7911yescharacteristic
230C2×C42C22×D7911yescharacteristic
231#106—1584no—
232C4⋊C8—1581no—
233C4⋊C8—1581no—
234C4⋊C8—1581no—
235C4⋊C8—1581no—
236#104—3162no—
237#104—3162no—
238#102—1584no—
239C22⋊C8—1581no—
240C22⋊C8—1581no—
241C4×C8—792no—
242C4×C8—1581no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95C22⋊D7912yes—
249#41—3161no—
250#41—3161no—
251#41—3161no—
252#41—3161no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—82no—
256C2×Q8—1581no—
257C2×D4—3161no—
258C2×D4—1581no—
259C2×D4—1581no—
260C2×D4—1581no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—3161no—
266C22×C4C2×C22⋊D7911yescharacteristic
267C22×C4C2×C22⋊D7911yescharacteristic
268C22×C4C2×C22⋊D7911yescharacteristic
269M16—3161no—
270M16—3161no—
271C2×C8—3161no—
272C2×C8—3161no—
273C2×C8—1582no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—82no—
280C4⋊C4—1581no—
281C4⋊C4—1581no—
282C4⋊C4(C2×D79)⋊C414yes—
283C22⋊C4—3161no—
284C22⋊C4—3161no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—82no—
289C42#1192961112yes—
290C42C2×C4×D7911yescharacteristic
291C42C2×D31611yescharacteristic
292C42#1192962411yescharacteristic
293C42#1192962411yescharacteristic
294C23—3161no—
295C23#1457913011yescharacteristic
296Q8—82no—
297Q8—82no—
298Q8—3161no—
299Q8—3161no—
300D4—3162no—
301D4—3161no—
302D4—3161no—
303D4—6321no—
304D4—6321no—
305D4—6321no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—3161no—
321C2×C4—6321no—
322C2×C4—6321no—
323C2×C4C2×(C2×D79)⋊C411yescharacteristic
324C2×C4C4×C22⋊D7911yescharacteristic
325C2×C4C4×C22⋊D7911yescharacteristic
326C2×C4#1457913511yescharacteristic
327C2×C4#1457913511yescharacteristic
328C2×C4#1457914411yescharacteristic
329C2×C4C2×#1192961111yescharacteristic
330C2×C4#1457918111yescharacteristic
331C2×C4#1457918611yescharacteristic
332C2×C4#1457919211yescharacteristic
333C8—3162no—
334C8—6321no—
335C8—6321no—
336C22—21no—
337C22—21no—
338C22—6321no—
339C22—6321no—
340C22—6321no—
341C22#1909844211yescharacteristic
342C22#1909870211yescharacteristic
343C22#1909870811yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—44no—
350C4—82no—
351C4—81no—
352C4—6321no—
353C4C4×#1192961111yescharacteristic
354C4#1909866111yescharacteristic
355C4#1909876311yescharacteristic
356C4#1909878511yescharacteristic
357C2—21no—
358C2—21no—
359C2—12641no—
360C2#2376375511yescharacteristic
361C2#2376388711yescharacteristic
362C2#2376393711yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2331C2
socleC22×C79#386
prime radical#19099113C22
cosocleC2×C42×C79C23
hypercenter#2331D79
hypocenterC79#9302
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #23762085C2semidirect
5. #23761258C2semidirect
6. #23761229C2semidirect
7. C79×#2331C2semidirect
9. #19098959C4semidirect
10. #19098911C4semidirect
183. #2331D79semidirect
189. C79#9302semidirect

Maximal subgroups

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

subgroupindex
#237745282
#237642322
#237620852
#237619592
#237612582
#237612292
C79×#23312
#930279