Finite Group Databasebeta

← #34215284

Subgroups — #34215284

Order 20224.

1590 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098462C213yesmaximal, complemented
2C2×#19098456C213yesmaximal
3#23761668C213yesmaximal, complemented
4#23761665C211yesmaximal, characteristic, complemented
5#23761645C213yesmaximal
6#95×Dic79C211yesmaximal, characteristic, complemented
7C2×#399×C79C211yesmaximal, characteristic
8C22×C4⋊Dic79C2213yes—
9C22×C4⋊Dic79C2213yes—
10C22×C22⋊Dic79C2211yescharacteristic
11C22×C22⋊Dic79C2213yes—
12C22×C4×Dic79C2213yes—
13#19098462—212no—
14#19098456—212no—
15C2×C79⋊#95C2213yes—
16C2×C79⋊#95C2213yes—
17C2×C79⋊#95C2216yes—
18C2×C79⋊#95C2213yes—
19C22×C4⋊C4×C79C2213yes—
20C22×C22⋊C4×C79C2213yes—
21#399×C79C418yescomplemented
22C2×#95×C79C2211yescharacteristic
23C23×Dic79C2311yescharacteristic
24C23×Dic79C2313yes—
25C23×Dic79C2313yes—
26C23×Dic79C2311yescharacteristic
27C2×C4⋊Dic79—26no—
28C2×C4⋊Dic79—26no—
29C2×C4⋊Dic79—212no—
30C2×C4⋊Dic79—46no—
31C2×C22⋊Dic79—26no—
32C2×C22⋊Dic79—26no—
33C2×C22⋊Dic79—46no—
34C2×C4×Dic79—46no—
35C2×C4×Dic79D4112yes—
36C79⋊#95—26no—
37C79⋊#95—26no—
38C79⋊#95—26no—
39C79⋊#95—212no—
40C25×C79C2311yescharacteristic
41C23×C4×C79C2313yes—
42C23×C4×C79C2313yes—
43C2×C4⋊C4×C79—26no—
44C2×C4⋊C4×C79—26no—
45C2×C4⋊C4×C79C2×C4112yes—
46C2×C22⋊C4×C79—26no—
47C2×C22⋊C4×C79—26no—
48C2×C22⋊C4×C79C2×C4112yes—
49#95×C79C2×C414yes—
50C22×Dic79—23no—
51C22×Dic79—23no—
52C22×Dic79—26no—
53C22×Dic79—26no—
54C22×Dic79—23no—
55C22×Dic79—26no—
56C22×Dic79—26no—
57C22×Dic79—23no—
58C22×Dic79—21no—
59C22×Dic79—23no—
60C22×Dic79—23no—
61C22×Dic79—23no—
62C22×Dic79—21no—
63C22×Dic79—23no—
64C22×Dic79C2×D416yes—
65C22×Dic79C2×D416yes—
66C4⋊Dic79—412no—
67C4⋊Dic79—412no—
68C22⋊Dic79—412no—
69C22⋊Dic79—44no—
70C4×Dic79—412no—
71C24×C79—23no—
72C24×C79—23no—
73C24×C79—23no—
74C24×C79—21no—
75C24×C79—21no—
76C24×C79—23no—
77C24×C79C22×C412yes—
78C24×C79C2411yescharacteristic
79C22×C4×C79—26no—
80C22×C4×C79—26no—
81C22×C4×C79—23no—
82C22×C4×C79—26no—
83C22×C4×C79—23no—
84C22×C4×C79—26no—
85C22×C4×C79C22×C416yes—
86C22×C4×C79C22×C416yes—
87C22×C4×C79C2×D416yes—
88C22×C4×C79#4116yes—
89C4⋊C4×C79—224no—
90C22⋊C4×C79—224no—
91C23×C79—26no—
92C23×C79—22no—
93C23×C79—26no—
94C23×C79—41no—
95C23×C79—43no—
96C23×C79—43no—
97C23×C79—43no—
98C23×C79—43no—
99C23×C79—43no—
100C23×C79—46no—
101C23×C79—43no—
102C23×C79—43no—
103C23×C79C23×C411yescharacteristic
104C23×C79C22×D413yes—
105C23×C79C2×#4113yes—
106C23×C79H(5,𝔽2)13yes—
107C23×C792−1+413yes—
108C23×C792−1+411yescharacteristic
109C23×C792−1+411yescharacteristic
110C2×C4×C79—212no—
111C2×C4×C79—26no—
112C2×C4×C79—26no—
113C2×C4×C79—212no—
114C2×C4×C79—26no—
115C2×C4×C79—212no—
116C2×C4×C79—43no—
117C2×C4×C79—43no—
118C2×C4×C79—46no—
119C2×C4×C79C4×D4112yes—
120C2×Dic79—26no—
121C2×Dic79—26no—
122C2×Dic79—26no—
123C2×Dic79—26no—
124C2×Dic79—26no—
125C2×Dic79—46no—
126C2×Dic79—46no—
127C2×Dic79—41no—
128C2×Dic79—43no—
129C2×Dic79—43no—
130C2×Dic79—46no—
131C2×Dic79—43no—
132C2×Dic79—43no—
133C2×Dic79—43no—
134C2×Dic79—46no—
135C2×Dic79#12714yes—
136C22×C79—42no—
137C22×C79—46no—
138C22×C79—46no—
139C22×C79—81no—
140C22×C79—81no—
141C22×C79—83no—
142C22×C79—83no—
143C22×C79C2×C4×D413yes—
144C22×C79#51911yescharacteristic
145C22×C79#52013yes—
146C22×C79C2×#12711yescharacteristic
147C22×C79#53213yes—
148C22×C79#53413yes—
149C22×C79#53613yes—
150C22×C79#53613yes—
151C22×C79#53713yes—
152C22×C79#53916yes—
153C22×C79#54113yes—
154C22×C79#54213yes—
155C4×C79—212no—
156C4×C79—46no—
157Dic79—24no—
158Dic79—46no—
159Dic79—83no—
160Dic79—81no—
161#14322—791nomaximal, Sylow
162C2×C79—82no—
163C2×C79C4×#12711yescharacteristic
164C2×C79#232213yes—
165C2×C79#232313yes—
166C2×C79#265611yescharacteristic
167C2×C79#265811yescharacteristic
168C2×C79#266313yes—
169C2×C79#266413yes—
170C2×#399D7911yescharacteristic
171C2×#389—793no—
172C2×#388—793no—
173#1445—793no—
174#1444—791no—
175#1439—793no—
176C4×#95—791no—
177C79#1432211yescharacteristic, complemented, Sylow
178C22×C4⋊C4—793no—
179C22×C4⋊C4—793no—
180C22×C4⋊C4C2×D7913yes—
181C22×C22⋊C4—793no—
182C22×C22⋊C4—791no—
183C22×C22⋊C4C2×D7913yes—
184C22×C42—793no—
185#399Dic7918yescomplemented
186#389—15812no—
187#388—15812no—
188C2×#95—793no—
189C2×#95—796no—
190C2×#95—793no—
191C2×#95—793no—
192C2×#95C2×D7911yescharacteristic
193C25C22×D7911yescharacteristic
194C23×C4—791no—
195C23×C4—793no—
196C23×C4—793no—
197C23×C4—791no—
198C23×C4C22×D7913yes—
199C23×C4C22×D7913yes—
200C2×C4⋊C4—26no—
201C2×C4⋊C4—26no—
202C2×C4⋊C4—3166no—
203C2×C4⋊C4—1586no—
204C2×C4⋊C4—15812no—
205C2×C4⋊C4—1586no—
206C2×C4⋊C4C2×Dic79112yes—
207C2×C22⋊C4—26no—
208C2×C22⋊C4—26no—
209C2×C22⋊C4—3166no—
210C2×C22⋊C4—1586no—
211C2×C22⋊C4—1586no—
212C2×C22⋊C4C2×Dic79112yes—
213C2×C42—3166no—
214C2×C42—7912no—
215#95—1586no—
216#95—15812no—
217#95—1586no—
218#95—1586no—
219#95C2×Dic7914yes—
220C24—23no—
221C24—23no—
222C24—23no—
223C24—21no—
224C24—21no—
225C24—23no—
226C24C22×Dic7912yes—
227C24C23×D7911yescharacteristic
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—23no—
231C22×C4—26no—
232C22×C4—23no—
233C22×C4—26no—
234C22×C4—796no—
235C22×C4—796no—
236C22×C4—1586no—
237C22×C4—1583no—
238C22×C4—1583no—
239C22×C4—1581no—
240C22×C4—1583no—
241C22×C4—1583no—
242C22×C4—1583no—
243C22×C4—1586no—
244C22×C4—1586no—
245C22×C4—1586no—
246C22×C4—1583no—
247C22×C4—1583no—
248C22×C4—1583no—
249C22×C4—1581no—
250C22×C4C22×Dic7916yes—
251C22×C4C22×Dic7916yes—
252C22×C4D4×D7916yes—
253C22×C4#1192962316yes—
254C4⋊C4—224no—
255C4⋊C4—31612no—
256C4⋊C4—31612no—
257C22⋊C4—224no—
258C22⋊C4—3164no—
259C22⋊C4—31612no—
260C42—31612no—
261C23—26no—
262C23—22no—
263C23—26no—
264C23—43no—
265C23—43no—
266C23—46no—
267C23—43no—
268C23—43no—
269C23—41no—
270C23—43no—
271C23—43no—
272C23—43no—
273C23C23×Dic7911yescharacteristic
274C23C2×D4×D7913yes—
275C23C2×#1192962313yes—
276C23#1457920713yes—
277C23#1457920813yes—
278C23#1457920911yescharacteristic
279C23#1457920911yescharacteristic
280C2×C4—212no—
281C2×C4—212no—
282C2×C4—26no—
283C2×C4—26no—
284C2×C4—26no—
285C2×C4—212no—
286C2×C4—43no—
287C2×C4—46no—
288C2×C4—43no—
289C2×C4—3166no—
290C2×C4—3166no—
291C2×C4—3163no—
292C2×C4—3166no—
293C2×C4—3163no—
294C2×C4—3161no—
295C2×C4—3163no—
296C2×C4—3163no—
297C2×C4—3166no—
298C2×C4—3163no—
299C2×C4—794no—
300C2×C4—1586no—
301C2×C4—1586no—
302C2×C4—1586no—
303C2×C4—1586no—
304C2×C4—1586no—
305C2×C4D4×Dic79112yes—
306C22—46no—
307C22—46no—
308C22—42no—
309C22—81no—
310C22—81no—
311C22—83no—
312C22—83no—
313C22C2×D4×Dic7913yes—
314C22#1909904111yescharacteristic
315C22#1909904613yes—
316C22#127×D7911yescharacteristic
317C22#1909911113yes—
318C22#1909912413yes—
319C22#1909913513yes—
320C22#1909913613yes—
321C22#1909914313yes—
322C22#1909915716yes—
323C22#1909917313yes—
324C22#1909918613yes—
325C4—212no—
326C4—46no—
327C4—3166no—
328C4—6323no—
329C4—6321no—
330C4—1584no—
331C2—82no—
332C2#127×Dic7911yescharacteristic
333C2#2376607213yes—
334C2#2376608413yes—
335C2#2376950311yescharacteristic
336C2#2376950911yescharacteristic
337C2#2376955313yes—
338C2#2376956413yes—
subgroupG/H
center, FrattiniC24C23×D79
commutatorC23×C79C23×C4
FittingC2×#399×C79C2
socle, cosocleC24×C79C24
prime radicalC25×C79C23
hypercenterC2×#399D79
hypocenterC79#14322
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C24 ◁ C2×#399

Complements

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

NH ≅ G/Ntype
1. C2×#19098462C2semidirect
3. #23761668C2semidirect
4. #23761665C2semidirect
6. #95×Dic79C2semidirect
21. #399×C79C4semidirect
177. C79#14322semidirect
185. #399Dic79semidirect

Maximal subgroups

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

subgroupindex
C2×#190984622
C2×#190984562
#237616682
#237616652
#237616452
#95×Dic792
C2×#399×C792
#1432279