Finite Group Databasebeta

← #34448288

Subgroups — #34448288

Order 20224.

5430 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11929625C211yesmaximal, characteristic
2C2×#19099037C212yesmaximal, complemented
3#23766229C2116yesmaximal, complemented
4#23765986C211yesmaximal, characteristic, complemented
5#23765977C214yesmaximal, complemented
6#23765779C212yesmaximal, complemented
7#381×D79C214yesmaximal, complemented
8C79×#2310C211yesmaximal, characteristic, complemented
9C22×#11929625—216no—
10C22×#11929625C4116yescomplemented
11C22×#11929625C2218yes—
12C23×D316C2212yes—
13C23×D316C2211yescharacteristic
14C23×C4×D79C2212yes—
15C23×C4×D79C2211yescharacteristic
16#19099037—216no—
17C2×C4×D316C2218yescomplemented
18C2×#14579123C2218yescomplemented
19C2×#14579120C2212yes—
20C2×C22⋊C4×D79C2218yes—
21#19098470C2218yescomplemented
22#19098441C22132yescomplemented
23#19098426C2214yes—
24#19098425C22116yescomplemented
25#19098424C2214yes—
26#19098420C2218yescomplemented
27#19098418C2214yes—
28C4×(C2×D79)⋊C4C2218yescomplemented
29#95×D79C22116yescomplemented
30C23×Q8×C79C2211yescharacteristic
31C2×#117×C79C2212yes—
32#391×C79C2218yescomplemented
33#381×C79C2214yes—
34C24×D79C2311yescharacteristic
35C24×D79C2312yes—
36C2×#11929625—2128no—
37C2×#11929625—232no—
38C2×#11929625—232no—
39C2×#11929625C2×C4164yescomplemented
40C22×D316—216no—
41C22×D316C2×C418yes—
42C22×D316C2×C4116yescomplemented
43C22×D316D4116yes—
44C22×D316C2318yes—
45C22×D316C2314yes—
46C23×Dic79C2311yescharacteristic
47C22×C4×D79—28no—
48C22×C4×D79—216no—
49C22×C4×D79C2×C4116yescomplemented
50C22×C4×D79C2×C418yescomplemented
51C22×C4×D79C2318yes—
52C22×C4×D79C2314yes—
53C22×C4×D79C23116yes—
54C4×D316—232no—
55#14579123—232no—
56#14579120—28no—
57C2×C4⋊Dic79C2314yes—
58C2×C22⋊Dic79C2314yes—
59C2×(C2×D79)⋊C4C23116yes—
60C2×(C2×D79)⋊C4C23116yes—
61C22⋊C4×D79—232no—
62C2×C4×Dic79C2314yes—
63C79⋊#95C23116yes—
64C79⋊#95C23116yes—
65C22×Q8×C79—28no—
66C22×Q8×C79C2×C418yescomplemented
67C22×Q8×C79C2314yes—
68C23×C4×C79C2312yes—
69C23×C4×C79C2311yescharacteristic
70#117×C79—28no—
71C2×C4⋊C4×C79C2314yes—
72C2×C22⋊C4×C79C2314yes—
73C2×C42×C79C2314yes—
74#95×C79C23116yes—
75C23×D79—24no—
76C23×D79—48no—
77C23×D79—44no—
78C23×D79C22×C414yes—
79C23×D79C22×C418yes—
80C23×D79C2×D4116yes—
81C23×D79C2412yes—
82C23×D79C2414yes—
83#11929625—2256no—
84C2×D316—264no—
85C2×D316—232no—
86C2×D316C22⋊C4164yescomplemented
87C2×D316C22×C4132yes—
88C2×D316C22×C4116yes—
89C2×D316C2×D4116yes—
90C2×D316C2×D4116yes—
91C22×Dic79—24no—
92C22×Dic79C22×C414yes—
93C22×Dic79C2418yes—
94C22×Dic79C2412yes—
95C2×C4×D79—264no—
96C2×C4×D79—232no—
97C2×C4×D79—232no—
98C2×C4×D79—232no—
99C2×C4×D79—28no—
100C2×C4×D79—28no—
101C2×C4×D79—432no—
102C2×C4×D79C22×C4132yes—
103C2×C4×D79C22×C4116yes—
104C4⋊Dic79—28no—
105C22⋊Dic79—28no—
106(C2×D79)⋊C4—232no—
107(C2×D79)⋊C4—416no—
108C4×Dic79—28no—
109C24×C79C2411yescharacteristic
110C2×Q8×C79—232no—
111C2×Q8×C79—28no—
112C2×Q8×C79—28no—
113C2×Q8×C79C22×C4116yes—
114C22×C4×C79—28no—
115C22×C4×C79C22×C418yes—
116C22×C4×C79C22×C414yes—
117C22×C4×C79C2×D418yes—
118C22×C4×C79C2418yes—
119C22×C4×C79C2414yes—
120C22×C4×C79C2412yes—
121C4⋊C4×C79—28no—
122C22⋊C4×C79—28no—
123C42×C79—28no—
124C23×C79—24no—
125C23×C79C23×C412yes—
126C23×C79C22×D414yes—
127C23×C79C2511yescharacteristic
128C22×D79—216no—
129C22×D79—28no—
130C22×D79—24no—
131C22×D79—28no—
132C22×D79—416no—
133C22×D79—48no—
134C22×D79—432no—
135C22×D79—416no—
136C22×D79C2×C22⋊C4132yes—
137C22×D79C23×C414yes—
138C22×D79C23×C418yes—
139C22×D79C22×D4116yes—
140Q8×C79—232no—
141C2×C4×C79—216no—
142C2×C4×C79—28no—
143C2×C4×C79—28no—
144C2×C4×C79—216no—
145C2×C4×C79C2×C22⋊C4116yes—
146C2×C4×C79C23×C418yes—
147C2×C4×C79C23×C414yes—
148C2×C4×C79C22×D414yes—
149C2×C4×C79C22×D414yes—
150C2×Dic79—28no—
151C2×Dic79—24no—
152C2×Dic79—28no—
153C2×Dic79—216no—
154C2×Dic79C23×C414yes—
155D316—264no—
156D316C2×C22⋊C4164yes—
157C4×D79—264no—
158C4×D79—232no—
159C4×D79—832no—
160C22×C79—24no—
161C22×C79—28no—
162C22×C79C22×C22⋊C414yes—
163C22×C79C24×C411yescharacteristic
164C22×C79C23×D412yes—
165C22×C79#58514yes—
166C2×D79—216no—
167C2×D79—232no—
168C2×D79—24no—
169C2×D79—28no—
170C2×D79—432no—
171C2×D79—416no—
172C2×D79—48no—
173C2×D79C22×C22⋊C4116yes—
174C4×C79—28no—
175C4×C79—48no—
176C4×C79C22×C22⋊C418yes—
177Dic79—24no—
178Dic79—48no—
179#34172—791nomaximal, Sylow
180C2×C79—24no—
181C2×C79C23×C22⋊C411yescharacteristic
182C2×C79#342812yes—
183C2×C79#348514yes—
184D79—216no—
185D79—44no—
186C23×#41—791no—
187C2×#518—792no—
188#2339—7916no—
189#2310—791no—
190#2310D7911yescharacteristic, complemented
191#2308—794no—
192#2280—792no—
193C2×#381—794no—
194C79#3417211yescharacteristic, complemented, Sylow
195C22×#41—798no—
196C22×#41—7916no—
197C22×#41—15816no—
198C23×Q8C2×D7911yescharacteristic
199C23×D4—791no—
200C23×D4—792no—
201C24×C4—792no—
202C24×C4—791no—
203#518—15816no—
204C2×C4×D4—798no—
205C2×C4×D4—798no—
206C2×#117—792no—
207C2×#117C2×D7912yes—
208C22×C22⋊C4—798no—
209#391—798no—
210#391C2×D7918yescomplemented
211#386—7932no—
212#381—794no—
213#381—794no—
214#381—7916no—
215#381C2×D7914yes—
216#380—798no—
217#379—794no—
218C4×C22⋊C4—798no—
219C2×#95—7916no—
220C25—792no—
221C25—791no—
222C2×#41—7964no—
223C2×#41—158128no—
224C2×#41—15832no—
225C2×#41—15832no—
226C22×Q8—28no—
227C22×Q8C4×D7918yescomplemented
228C22×Q8C22×D7914yes—
229C22×D4—798no—
230C22×D4—7916no—
231C22×D4—7916no—
232C22×D4—794no—
233C22×D4—798no—
234C22×D4—15816no—
235C23×C4—7916no—
236C23×C4—791no—
237C23×C4—7916no—
238C23×C4—798no—
239C23×C4—798no—
240C23×C4—794no—
241C23×C4—15816no—
242C23×C4—1588no—
243C23×C4C22×D7912yes—
244C23×C4C22×D7911yescharacteristic
245C4×D4—15832no—
246C4×D4—15832no—
247#117—28no—
248#117—1588no—
249C2×C4⋊C4—794no—
250C2×C4⋊C4C22×D7914yes—
251C2×C22⋊C4—7916no—
252C2×C22⋊C4—794no—
253C2×C22⋊C4—7916no—
254C2×C22⋊C4—15832no—
255C2×C22⋊C4C22×D7914yes—
256C2×C42—794no—
257C2×C42C22×D7914yes—
258#95—7916no—
259#95—7916no—
260#95C22×D79116yes—
261C24—3164no—
262C24—3168no—
263C24—798no—
264C24—7916no—
265C24—794no—
266C24—794no—
267C24—792no—
268C24—1584no—
269C24C23×D7911yescharacteristic
270#41—158256no—
271C2×Q8—232no—
272C2×Q8—28no—
273C2×Q8—28no—
274C2×Q8C2×C4×D79116yes—
275C2×D4—7964no—
276C2×D4—7916no—
277C2×D4—7932no—
278C2×D4—7916no—
279C2×D4—7916no—
280C2×D4—15864no—
281C2×D4—15832no—
282C22×C4—28no—
283C22×C4—31632no—
284C22×C4—792no—
285C22×C4—794no—
286C22×C4—7932no—
287C22×C4—7916no—
288C22×C4—798no—
289C22×C4—15832no—
290C22×C4—15864no—
291C22×C4—15832no—
292C22×C4—1584no—
293C22×C4—15832no—
294C22×C4—1588no—
295C22×C4—1588no—
296C22×C4C2×C4×D7918yes—
297C22×C4C2×C4×D7914yes—
298C22×C4D4×D7918yes—
299C22×C4C23×D7918yes—
300C22×C4C23×D7914yes—
301C22×C4C23×D7912yes—
302C4⋊C4—28no—
303C4⋊C4—1588no—
304C22⋊C4—28no—
305C22⋊C4—31616no—
306C22⋊C4—15832no—
307C22⋊C4—1588no—
308C42—28no—
309C42—1588no—
310C23—24no—
311C23—31616no—
312C23—31632no—
313C23—3168no—
314C23—31616no—
315C23—7932no—
316C23—798no—
317C23—7916no—
318C23—794no—
319C23—1588no—
320C23—1588no—
321C23—1584no—
322C23—15816no—
323C23C22×C4×D7912yes—
324C23C2×D4×D7914yes—
325C23C24×D7911yescharacteristic
326Q8—232no—
327D4—7964no—
328D4—15864no—
329C2×C4—216no—
330C2×C4—28no—
331C2×C4—216no—
332C2×C4—28no—
333C2×C4—794no—
334C2×C4—63232no—
335C2×C4—15864no—
336C2×C4—15816no—
337C2×C4—1588no—
338C2×C4—1588no—
339C2×C4—1584no—
340C2×C4—15832no—
341C2×C4C22⋊C4×D79116yes—
342C2×C4C22×C4×D7918yes—
343C2×C4C22×C4×D7914yes—
344C2×C4C2×D4×D7914yes—
345C2×C4C2×D4×D7914yes—
346C22—28no—
347C22—24no—
348C22—31616no—
349C22—31632no—
350C22—3168no—
351C22—7916no—
352C22—1584no—
353C22—1588no—
354C22—15832no—
355C22—15816no—
356C22C2×C22⋊C4×D7914yes—
357C22C23×C4×D7911yescharacteristic
358C22C22×D4×D7912yes—
359C22#1909940014yes—
360C4—28no—
361C4—48no—
362C4—3168no—
363C4—1584no—
364C4C2×C22⋊C4×D7918yes—
365C2—24no—
366C2—3164no—
367C2—15816no—
368C2C22×C22⋊C4×D7911yescharacteristic
369C2#2377660212yes—
370C2#2377704314yes—
subgroupG/H
center, FrattiniC23C24×D79
commutatorC22×C79C24×C4
FittingC79×#2310C2
socle, cosocleC23×C79C25
prime radicalC23×#11929625C2
hypercenter#2310D79
hypocenterC79#34172
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C23 ◁ #2310

Complements

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

NH ≅ G/Ntype
2. C2×#19099037C2semidirect
3. #23766229C2semidirect
4. #23765986C2semidirect
5. #23765977C2semidirect
6. #23765779C2semidirect
7. #381×D79C2semidirect
8. C79×#2310C2semidirect
10. C22×#11929625C4semidirect
17. C2×C4×D316C22semidirect
18. C2×#14579123C22semidirect
21. #19098470C22semidirect
22. #19098441C22semidirect
24. #19098425C22semidirect
26. #19098420C22semidirect
28. C4×(C2×D79)⋊C4C22semidirect
29. #95×D79C22semidirect
32. #391×C79C22semidirect
39. C2×#11929625C2×C4semidirect
42. C22×D316C2×C4semidirect
49. C22×C4×D79C2×C4semidirect
50. C22×C4×D79C2×C4semidirect
66. C22×Q8×C79C2×C4semidirect
86. C2×D316C22⋊C4semidirect
190. #2310D79semidirect
194. C79#34172semidirect
210. #391C2×D79semidirect
227. C22×Q8C4×D79semidirect

Maximal subgroups

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

subgroupindex
C23×#119296252
C2×#190990372
#237662292
#237659862
#237659772
#237657792
#381×D792
C79×#23102
#3417279