Finite Group Databasebeta

← #34184914

Subgroups — #34184914

Order 20224.

478 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772118C211yesmaximal, characteristic
2#23764374C211yesmaximal, characteristic, complemented
3#23762081C211yesmaximal, characteristic, complemented
4#23761960C211yesmaximal, characteristic
5#23761259C211yesmaximal, characteristic, complemented
6#23761230C211yesmaximal, characteristic, complemented
7C79×#3227C211yesmaximal, 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#19098312—22no—
19#19098297—22no—
20#532×C79C2211yescharacteristic
21C2×C4⋊Q8×C79C2211yescharacteristic
22#498×C79C412yescomplemented
23#486×C79C412yescomplemented
24#481×C79—22no—
25#476×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—
47C4⋊Q8×C79—21no—
48C4⋊Q8×C79—21no—
49C4⋊Q8×C79—21no—
50C4⋊Q8×C79C2×C412yes—
51C4⋊Q8×C79C2×C411yescharacteristic
52#127×C79C2×C411yescharacteristic
53#124×C79—21no—
54#121×C79—21no—
55#121×C79D412yes—
56C4×D4×C79—21no—
57C4×D4×C79D412yes—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C42×C79C2311yescharacteristic
61#106×C79—24no—
62C4⋊C8×C79—21no—
63#102×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⋊C8SD1612yes—
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
98C2×C8×C79—22no—
99C4⋊C4×C79—21no—
100C4⋊C4×C79—22no—
101C4⋊C4×C79—22no—
102C4⋊C4×C79—41no—
103C4⋊C4×C79—41no—
104C4⋊C4×C79C22⋊C412yes—
105C22⋊C4×C79—22no—
106C22⋊C4×C79—22no—
107C42×C79C22×C411yescharacteristic
108C42×C79C2×D411yescharacteristic
109C42×C79C2×D411yescharacteristic
110C42×C79#4111yescharacteristic
111C23×C79—22no—
112C23×C79#12111yescharacteristic
113Q8×C79—41no—
114Q8×C79—41no—
115Q8×C79—82no—
116Q8×C79—82no—
117D4×C79—42no—
118D4×C79—42no—
119D4×C79—42no—
120D4×C79—41no—
121D4×C79—41no—
122D4×C79—81no—
123C2×C4×C79—22no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—42no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—82no—
130C2×C4×C79C2×C22⋊C411yescharacteristic
131C2×C4×C79C4×D411yescharacteristic
132C2×C4×C79C4×D411yescharacteristic
133C2×C4×C79#12011yescharacteristic
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
140C2×Dic79—81no—
141C2×Dic79—81no—
142C2×Dic79—81no—
143C8×C79—42no—
144C79⋊C8—22no—
145C79⋊C8—42no—
146C79⋊C8—41no—
147C79⋊C8—81no—
148C22×C79—21no—
149C22×C79—21no—
150C22×C79—42no—
151C22×C79—42no—
152C22×C79—42no—
153C22×C79#38611yescharacteristic
154C22×C79#46111yescharacteristic
155C22×C79#46311yescharacteristic
156C4×C79—21no—
157C4×C79—21no—
158C4×C79—21no—
159C4×C79—21no—
160C4×C79—42no—
161C4×C79—82no—
162C4×C79C4×SD1611yescharacteristic
163C4×C79#44011yescharacteristic
164C4×C79#45011yescharacteristic
165C4×C79#45211yescharacteristic
166Dic79—161no—
167#9287—791nomaximal, Sylow
168C2×C79—21no—
169C2×C79—21no—
170C2×C79—82no—
171C2×C79#187711yescharacteristic
172C2×C79#190311yescharacteristic
173C2×C79#190911yescharacteristic
174#3227D7911yescharacteristic
175#2932—791no—
176#1993—791no—
177#1548—791no—
178#1517—791no—
179#1286—791no—
180#1275—791no—
181C79#928711yescharacteristic, complemented, Sylow
182C2×#131—1581no—
183#532C2×D7911yescharacteristic
184C2×C4⋊Q8C2×D7911yescharacteristic
185#498Dic7912yescomplemented
186#486Dic7912yescomplemented
187#481—22no—
188#476—22no—
189#435—1582no—
190#435—1582no—
191#431—791no—
192#431C2×D7911yescharacteristic
193#423—791no—
194#408—1581no—
195C2×#97—791no—
196#382—791no—
197#340—1581no—
198#340—1581no—
199#333—1582no—
200#327—1582no—
201C2×#41—21no—
202C22×Q8—21no—
203#131—3162no—
204#131—3162no—
205C2×M16—1582no—
206C2×M16—1581no—
207C22×C8—791no—
208C22×C8—791no—
209C22×C8—1582no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8C2×Dic7912yes—
214C4⋊Q8C2×Dic7911yescharacteristic
215#127C2×Dic7911yescharacteristic
216#124—21no—
217#121—21no—
218#121C22⋊D7912yes—
219C4×D4—21no—
220C4×D4C22⋊D7912yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—1581no—
224C2×C42C22×D7911yescharacteristic
225#106—24no—
226C4⋊C8—21no—
227C4⋊C8—792no—
228C4⋊C8—1581no—
229C4⋊C8—1581no—
230#102—24no—
231C22⋊C8—21no—
232C22⋊C8—1581no—
233C22⋊C8—1584no—
234#97—1581no—
235#97—1581no—
236C4×C8—1581no—
237C4×C8C2×Dic7912yes—
238#95—1581no—
239#95—1581no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—82no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊Dic7912yes—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—3161no—
255C22×C4D4×D7911yescharacteristic
256C22×C4C2×C22⋊D7911yescharacteristic
257C22×C4#1192962311yescharacteristic
258M16—3161no—
259M16—3161no—
260M16—3164no—
261C2×C8—22no—
262C2×C8—3164no—
263C2×C8—3161no—
264C2×C8—792no—
265C2×C8—1581no—
266C2×C8—1582no—
267C2×C8—1581no—
268C2×C8—1581no—
269C2×C8—1581no—
270C2×C8—1581no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—3161no—
277C4⋊C4—3161no—
278C4⋊C4C22⋊Dic7912yes—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C42C22×Dic7911yescharacteristic
282C42D4×D7911yescharacteristic
283C42C2×C22⋊D7911yescharacteristic
284C42#1192962311yescharacteristic
285C23—22no—
286C23#1457913311yescharacteristic
287Q8—41no—
288Q8—41no—
289Q8—82no—
290Q8—82no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—41no—
295D4—41no—
296D4—81no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—6321no—
305C2×C4—6321no—
306C2×C4—6321no—
307C2×C4C2×C22⋊Dic7911yescharacteristic
308C2×C4D4×Dic7911yescharacteristic
309C2×C4D4×Dic7911yescharacteristic
310C2×C4#1457912911yescharacteristic
311C2×C4#1457913311yescharacteristic
312C2×C4#1457914211yescharacteristic
313C2×C4SD16×D7911yescharacteristic
314C2×C4#1457918211yescharacteristic
315C2×C4#1457918811yescharacteristic
316C2×C4#1457919011yescharacteristic
317C8—42no—
318C8—3162no—
319C8—3161no—
320C8—6321no—
321C8—1582no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22#1909844311yescharacteristic
328C22#1909876611yescharacteristic
329C22#1909878011yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—42no—
335C4—82no—
336C4—12641no—
337C4SD16×Dic7911yescharacteristic
338C4#1909866411yescharacteristic
339C4#1909871211yescharacteristic
340C4#1909872211yescharacteristic
341C2—21no—
342C2—21no—
343C2—82no—
344C2#2376373611yescharacteristic
345C2#2376388211yescharacteristic
346C2#2376392311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#3227C2
socleC22×C79#386
prime radical#532×C79C22
cosocleC2×C42×C79C23
hypercenter#3227D79
hypocenterC79#9287
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
2. #23764374C2semidirect
3. #23762081C2semidirect
5. #23761259C2semidirect
6. #23761230C2semidirect
22. #498×C79C4semidirect
23. #486×C79C4semidirect
181. C79#9287semidirect
185. #498Dic79semidirect
186. #486Dic79semidirect

Maximal subgroups

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

subgroupindex
#237721182
#237643742
#237620812
#237619602
#237612592
#237612302
C79×#32272
#928779