Finite Group Databasebeta

← #34184928

Subgroups — #34184928

Order 20224.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772118C211yesmaximal, characteristic
2#23764374C211yesmaximal, characteristic, complemented
3#23762085C211yesmaximal, characteristic, complemented
4#23761960C211yesmaximal, characteristic
5#23761259C211yesmaximal, characteristic, complemented
6#23761230C211yesmaximal, characteristic, complemented
7C79×#3206C211yesmaximal, characteristic
8C2×#14579168—21no—
9#19098635—22no—
10#19098635—22no—
11#19098616C2211yescharacteristic
12#19098587C2211yescharacteristic
13#19098530—21no—
14C2×#14579072C2211yescharacteristic
15#19098429C2211yescharacteristic
16#19098325—21no—
17#19098325—21no—
18#19098309—22no—
19#19098297—22no—
20#532×C79C2211yescharacteristic
21C2×C4⋊Q8×C79C2211yescharacteristic
22#492×C79C412yescomplemented
23#487×C79C412yescomplemented
24#461×C79—22no—
25#448×C79—22no—
26#431×C79C2211yescharacteristic
27#14579168—42no—
28#14579168—42no—
29C2×#11929608—22no—
30C2×#11929608—21no—
31C22×C79⋊C8—22no—
32C22×C79⋊C8D411yescharacteristic
33C22×C79⋊C8D411yescharacteristic
34C2×C4⋊Dic79—21no—
35C4⋊(C79⋊C8)—21no—
36C4⋊(C79⋊C8)—21no—
37C4⋊(C79⋊C8)D412yes—
38C22⋊(C79⋊C8)—24no—
39C22⋊(C79⋊C8)—21no—
40#14579072—21no—
41#14579072—21no—
42C4×C79⋊C8—21no—
43C79⋊#95—21no—
44C79⋊#95—21no—
45C2×#41×C79—21no—
46C22×Q8×C79—21no—
47C2×SD16×C79—24no—
48C4⋊Q8×C79—21no—
49C4⋊Q8×C79—21no—
50C4⋊Q8×C79—21no—
51C4⋊Q8×C79C2×C412yes—
52C4⋊Q8×C79C2×C411yescharacteristic
53#127×C79C2×C411yescharacteristic
54#124×C79—21no—
55#121×C79—21no—
56#121×C79D412yes—
57C4×D4×C79—21no—
58C4×D4×C79D412yes—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79—21no—
61C2×C42×C79C2311yescharacteristic
62C4⋊C8×C79—21no—
63#103×C79—24no—
64C22⋊C8×C79—21no—
65C4×C8×C79C2×C412yes—
66C22×Dic79—41no—
67#11929608—44no—
68#11929608—41no—
69#11929608—41no—
70C2×C79⋊C8—21no—
71C2×C79⋊C8—21no—
72C2×C79⋊C8—22no—
73C2×C79⋊C8—21no—
74C2×C79⋊C8—21no—
75C2×C79⋊C8—21no—
76C2×C79⋊C8—44no—
77C2×C79⋊C8—41no—
78C2×C79⋊C8SD1612yescomplemented
79C4⋊Dic79—41no—
80C4⋊Dic79—41no—
81#41×C79—41no—
82#41×C79—41no—
83#41×C79—41no—
84#41×C79—41no—
85C2×Q8×C79—21no—
86C2×Q8×C79—22no—
87C2×Q8×C79—41no—
88C2×Q8×C79—82no—
89C2×D4×C79—22no—
90C2×D4×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79C22⋊C412yes—
93C22×C4×C79—22no—
94C22×C4×C79—41no—
95C22×C4×C79C2×D411yescharacteristic
96C22×C4×C79C2×D411yescharacteristic
97C22×C4×C79#4111yescharacteristic
98SD16×C79—84no—
99C2×C8×C79—22no—
100C4⋊C4×C79—21no—
101C4⋊C4×C79—22no—
102C4⋊C4×C79—22no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79—41no—
105C4⋊C4×C79C22⋊C412yes—
106C22⋊C4×C79—22no—
107C22⋊C4×C79—22no—
108C42×C79C22×C411yescharacteristic
109C42×C79C2×D411yescharacteristic
110C42×C79C2×D411yescharacteristic
111C42×C79#4111yescharacteristic
112C23×C79—22no—
113C23×C79#12111yescharacteristic
114Q8×C79—44no—
115Q8×C79—41no—
116Q8×C79—41no—
117Q8×C79—82no—
118D4×C79—42no—
119D4×C79—42no—
120D4×C79—42no—
121D4×C79—41no—
122D4×C79—41no—
123D4×C79—81no—
124C2×C4×C79—22no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—42no—
130C2×C4×C79—82no—
131C2×C4×C79C2×C22⋊C411yescharacteristic
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79C4×D411yescharacteristic
134C2×C4×C79#12011yescharacteristic
135C2×C4×C79#12111yescharacteristic
136C2×C4×C79#12311yescharacteristic
137C2×C4×C79C2×SD1611yescharacteristic
138C2×C4×C79#13511yescharacteristic
139C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C79#13711yescharacteristic
141C2×Dic79—81no—
142C2×Dic79—81no—
143C2×Dic79—81no—
144C8×C79—42no—
145C79⋊C8—22no—
146C79⋊C8—42no—
147C79⋊C8—41no—
148C79⋊C8—81no—
149C22×C79—21no—
150C22×C79—21no—
151C22×C79—42no—
152C22×C79—42no—
153C22×C79—42no—
154C22×C79#38611yescharacteristic
155C22×C79#46111yescharacteristic
156C22×C79#46311yescharacteristic
157C4×C79—21no—
158C4×C79—21no—
159C4×C79—21no—
160C4×C79—21no—
161C4×C79—42no—
162C4×C79—82no—
163C4×C79C4×SD1611yescharacteristic
164C4×C79#44011yescharacteristic
165C4×C79#45011yescharacteristic
166C4×C79#45211yescharacteristic
167Dic79—161no—
168#9289—791nomaximal, Sylow
169C2×C79—21no—
170C2×C79—21no—
171C2×C79—82no—
172C2×C79#187711yescharacteristic
173C2×C79#190211yescharacteristic
174C2×C79#191011yescharacteristic
175#3206D7911yescharacteristic
176#2932—791no—
177#1993—791no—
178#1549—791no—
179#1517—791no—
180#1286—791no—
181#1275—791no—
182C79#928911yescharacteristic, complemented, Sylow
183C2×#131—1581no—
184#532C2×D7911yescharacteristic
185C2×C4⋊Q8C2×D7911yescharacteristic
186#492Dic7912yescomplemented
187#487Dic7912yescomplemented
188#461—22no—
189#448—22no—
190#435—1582no—
191#435—1582no—
192#431—791no—
193#431C2×D7911yescharacteristic
194#423—791no—
195#408—1581no—
196C2×#97—791no—
197#382—791no—
198#340—1581no—
199#340—1581no—
200#332—1582no—
201#327—1582no—
202C2×#41—21no—
203C22×Q8—21no—
204C2×SD16—24no—
205#131—3162no—
206#131—3162no—
207C2×M16—1582no—
208C2×M16—1581no—
209C22×C8—791no—
210C22×C8—791no—
211C22×C8—1582no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C2×Dic7912yes—
216C4⋊Q8C2×Dic7911yescharacteristic
217#127C2×Dic7911yescharacteristic
218#124—21no—
219#121—21no—
220#121C22⋊D7912yes—
221C4×D4—21no—
222C4×D4C22⋊D7912yes—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1581no—
226C2×C42C22×D7911yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—792no—
229C4⋊C8—1581no—
230C4⋊C8—1581no—
231#103—24no—
232C22⋊C8—21no—
233C22⋊C8—1581no—
234C22⋊C8—1584no—
235#97—1581no—
236#97—1581no—
237C4×C8—1581no—
238C4×C8C2×Dic7912yes—
239#95—1581no—
240#95—1581no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—82no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4C22⋊Dic7912yes—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—3161no—
256C22×C4D4×D7911yescharacteristic
257C22×C4C2×C22⋊D7911yescharacteristic
258C22×C4#1192962311yescharacteristic
259SD16—84no—
260M16—3161no—
261M16—3161no—
262M16—3164no—
263C2×C8—22no—
264C2×C8—3164no—
265C2×C8—3161no—
266C2×C8—792no—
267C2×C8—1581no—
268C2×C8—1582no—
269C2×C8—1581no—
270C2×C8—1581no—
271C2×C8—1581no—
272C2×C8—1581no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—3161no—
279C4⋊C4—3161no—
280C4⋊C4C22⋊Dic7912yes—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C42C22×Dic7911yescharacteristic
284C42D4×D7911yescharacteristic
285C42C2×C22⋊D7911yescharacteristic
286C42#1192962311yescharacteristic
287C23—22no—
288C23#1457913311yescharacteristic
289Q8—44no—
290Q8—41no—
291Q8—41no—
292Q8—82no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—41no—
297D4—41no—
298D4—81no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—6321no—
307C2×C4—6321no—
308C2×C4—6321no—
309C2×C4C2×C22⋊Dic7911yescharacteristic
310C2×C4D4×Dic7911yescharacteristic
311C2×C4D4×Dic7911yescharacteristic
312C2×C4#1457912911yescharacteristic
313C2×C4#1457913311yescharacteristic
314C2×C4#1457914211yescharacteristic
315C2×C4SD16×D7911yescharacteristic
316C2×C4#1457918211yescharacteristic
317C2×C4#1457918811yescharacteristic
318C2×C4#1457919011yescharacteristic
319C8—42no—
320C8—3162no—
321C8—3161no—
322C8—6321no—
323C8—1582no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#1909844311yescharacteristic
330C22#1909876611yescharacteristic
331C22#1909878011yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—82no—
338C4—12641no—
339C4SD16×Dic7911yescharacteristic
340C4#1909866411yescharacteristic
341C4#1909871211yescharacteristic
342C4#1909872211yescharacteristic
343C2—21no—
344C2—21no—
345C2—82no—
346C2#2376373611yescharacteristic
347C2#2376387511yescharacteristic
348C2#2376392811yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3206C2
socleC22×C79#386
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#3206D79
hypocenterC79#9289
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
2. #23764374C2semidirect
3. #23762085C2semidirect
5. #23761259C2semidirect
6. #23761230C2semidirect
22. #492×C79C4semidirect
23. #487×C79C4semidirect
78. C2×C79⋊C8SD16semidirect
182. C79#9289semidirect
186. #492Dic79semidirect
187. #487Dic79semidirect

Maximal subgroups

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

subgroupindex
#237721182
#237643742
#237620852
#237619602
#237612592
#237612302
C79×#32062
#928979