Finite Group Databasebeta

← #34205219

Subgroups — #34205219

Order 20224.

530 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098801C211yesmaximal, characteristic
2C2×#19098600C211yesmaximal, characteristic
3C2×#19098367C211yesmaximal, characteristic, complemented
4C2×(C2×D79)⋊C16C211yesmaximal, characteristic
5#23761492C211yesmaximal, characteristic, complemented
6#23761491C211yesmaximal, characteristic, complemented
7C2×#358×C79C211yesmaximal, characteristic, complemented
8C22×C8×D79C2211yescharacteristic
9C2×#14579137—21no—
10C2×#14579119—21no—
11C22×C79⋊C16C2211yescharacteristic
12#19098801—22no—
13#19098801C414yescomplemented
14#19098600—22no—
15#19098600—22no—
16C2×#14579095C2211yescharacteristic
17C2×#14579095C2211yescharacteristic
18C2×#14579092—21no—
19C2×#14579085—21no—
20#19098367—22no—
21(C2×D79)⋊C16—22no—
22C22×Q16×C79C2211yescharacteristic
23C22×C16×C79C2211yescharacteristic
24C2×#107×C79C2211yescharacteristic
25#358×C79—22no—
26C22×C4×D79D411yescharacteristic
27C22×C79⋊C8D411yescharacteristic
28C2×C8×D79—21no—
29C2×C8×D79—21no—
30C2×C8×D79—21no—
31C2×C8×D79—21no—
32C2×C8×D79C2×C412yes—
33C2×C8×D79D412yes—
34#14579137—24no—
35#14579137—42no—
36#14579119—24no—
37#14579119—42no—
38C2×C4⋊Dic79—21no—
39C2×C4⋊Dic79—21no—
40C2×Dic79⋊C4—41no—
41C2×(C2×D79)⋊C4—41no—
42C2×(C2×D79)⋊C4—41no—
43C2×C4×Dic79—41no—
44C2×C79⋊C16—21no—
45C2×C79⋊C16—21no—
46C2×C79⋊C16—21no—
47#14579095—21no—
48#14579095—21no—
49#14579095C2×C412yes—
50#14579095D412yes—
51#14579092—24no—
52#14579085—24no—
53C22×Q8×C79—21no—
54C2×Q16×C79—21no—
55C2×Q16×C79—21no—
56C2×Q16×C79—21no—
57C2×Q16×C79—21no—
58C2×Q16×C79—21no—
59C2×Q16×C79C2×C412yes—
60C22×C8×C79C2311yescharacteristic
61C2×C4⋊C4×C79—21no—
62C2×C16×C79—21no—
63C2×C16×C79—21no—
64C2×C16×C79—21no—
65#107×C79—21no—
66#107×C79D412yes—
67C23×D79D811yescharacteristic
68C22×Dic79—41no—
69C22×Dic79—41no—
70C22×Dic79D811yescharacteristic
71C2×C4×D79—21no—
72C2×C4×D79—21no—
73C2×C4×D79—21no—
74C2×C4×D79—21no—
75C2×C4×D79C22⋊C412yes—
76C2×C4×D79SD1612yes—
77C2×C79⋊C8—21no—
78C2×C79⋊C8—21no—
79C2×C79⋊C8C22⋊C412yes—
80C8×D79—22no—
81C8×D79—22no—
82C8×D79—22no—
83C8×D79—22no—
84C4⋊Dic79—22no—
85C4⋊Dic79—22no—
86C4⋊Dic79—41no—
87C4⋊Dic79—41no—
88Dic79⋊C4—44no—
89(C2×D79)⋊C4—44no—
90(C2×D79)⋊C4—44no—
91C4×Dic79—44no—
92C79⋊C16—41no—
93C2×Q8×C79—22no—
94C2×Q8×C79—41no—
95C2×Q8×C79—41no—
96C2×Q8×C79—41no—
97C2×Q8×C79—41no—
98C2×Q8×C79—41no—
99C22×C4×C79—41no—
100C22×C4×C79—41no—
101C22×C4×C79C2×D411yescharacteristic
102Q16×C79—22no—
103Q16×C79—22no—
104Q16×C79—41no—
105Q16×C79—41no—
106C2×C8×C79C22×C411yescharacteristic
107C2×C8×C79C2×D411yescharacteristic
108C2×C8×C79C2×D411yescharacteristic
109C2×C8×C79C2×D411yescharacteristic
110C2×C8×C79#4111yescharacteristic
111C2×C8×C79#4111yescharacteristic
112C4⋊C4×C79—22no—
113C4⋊C4×C79—41no—
114C16×C79—41no—
115C23×C79C2×D811yescharacteristic
116C22×D79—21no—
117C22×D79—21no—
118C22×D79—21no—
119C22×D79—21no—
120C22×D79—21no—
121C22×D79—21no—
122C22×D79#10212yes—
123Q8×C79—42no—
124Q8×C79—42no—
125Q8×C79—81no—
126Q8×C79—81no—
127C2×C4×C79—42no—
128C2×C4×C79—42no—
129C2×C4×C79—81no—
130C2×C4×C79—81no—
131C2×C4×C79—81no—
132C2×C4×C79—81no—
133C2×C4×C79C2×C22⋊C411yescharacteristic
134C2×C4×C79#12011yescharacteristic
135C2×C4×C79#12311yescharacteristic
136C2×C4×C79C2×SD1611yescharacteristic
137C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C79#13711yescharacteristic
139C2×Dic79—21no—
140C2×Dic79—21no—
141C2×Dic79—42no—
142C2×Dic79—42no—
143C2×Dic79—81no—
144C2×Dic79—81no—
145C2×Dic79—81no—
146C2×Dic79—81no—
147C2×Dic79#10212yes—
148C4×D79—22no—
149C4×D79—22no—
150C4×D79—22no—
151C4×D79—22no—
152C8×C79C4×D411yescharacteristic
153C8×C79C4×D411yescharacteristic
154C8×C79#12111yescharacteristic
155C8×C79#12111yescharacteristic
156C79⋊C8—22no—
157C22×C79C2×#10211yescharacteristic
158C22×C79#44711yescharacteristic
159C22×C79#48011yescharacteristic
160C22×C79#50811yescharacteristic
161C22×C79#50811yescharacteristic
162C22×C79#51011yescharacteristic
163C22×C79#51011yescharacteristic
164C2×D79—22no—
165C2×D79—22no—
166C2×D79—22no—
167C2×D79—41no—
168C2×D79—41no—
169C2×D79—41no—
170C2×D79—41no—
171C4×C79—82no—
172C4×C79—82no—
173C4×C79#38611yescharacteristic
174C4×C79#41711yescharacteristic
175C4×C79#44811yescharacteristic
176C4×C79#48111yescharacteristic
177Dic79—22no—
178Dic79—82no—
179Dic79—82no—
180#12994—791nomaximal, Sylow
181C2×C79#187411yescharacteristic
182C2×C79#213811yescharacteristic
183C2×C79#214111yescharacteristic
184C2×C79#218911yescharacteristic
185C2×C79#218911yescharacteristic
186C2×C79#223711yescharacteristic
187C2×C79#223711yescharacteristic
188D79—42no—
189C2×#467—791no—
190C2×#427—791no—
191C2×#358—791no—
192C2×#358D7911yescharacteristic, complemented
193C2×C22⋊C16—791no—
194#1380—791no—
195#1380—791no—
196C79#1299411yescharacteristic, complemented, Sylow
197C22×Q16C2×D7911yescharacteristic
198C23×C8—791no—
199C2×C22⋊Q8—1581no—
200C2×#117—1581no—
201C22×C16—791no—
202C22×C16C2×D7911yescharacteristic
203#467—794no—
204#467—1582no—
205#427—1582no—
206#427—1582no—
207C2×#107—791no—
208C2×#107—791no—
209C2×#107C2×D7911yescharacteristic
210C2×#106—1581no—
211C2×#103—1581no—
212#358—22no—
213#358—1582no—
214C22⋊C16—1582no—
215C22×Q8—21no—
216C23×C4—791no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16C4×D7912yes—
223C22×C8—791no—
224C22×C8—792no—
225C22×C8—792no—
226C22×C8—1581no—
227C22×C8—1581no—
228C22×C8—1581no—
229C22×C8—1581no—
230C22×C8C22×D7911yescharacteristic
231C22⋊Q8—3162no—
232C22⋊Q8—1584no—
233#117—3162no—
234#117—1584no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—3161no—
237C2×C4⋊C4—1581no—
238C2×C4⋊C4—1581no—
239C2×C22⋊C4—3161no—
240C2×C22⋊C4—3161no—
241C2×C42—3161no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1581no—
246C2×C16—1581no—
247C2×C16—1581no—
248#107—21no—
249#107—792no—
250#107—792no—
251#107—1581no—
252#107—1581no—
253#107C22⋊D7912yes—
254#106—1584no—
255#103—1584no—
256C24—791no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—3161no—
266C22×C4—3161no—
267C22×C4—791no—
268C22×C4—792no—
269C22×C4—792no—
270C22×C4—1581no—
271C22×C4—1581no—
272C22×C4—1581no—
273C22×C4—1581no—
274C22×C4D4×D7911yescharacteristic
275Q16—22no—
276Q16—22no—
277Q16—41no—
278Q16—41no—
279C2×C8—792no—
280C2×C8—1581no—
281C2×C8—1581no—
282C2×C8—1582no—
283C2×C8—1582no—
284C2×C8—1582no—
285C2×C8—1582no—
286C2×C8C2×C4×D7911yescharacteristic
287C2×C8D4×D7911yescharacteristic
288C2×C8D4×D7911yescharacteristic
289C2×C8C2×C22⋊D7911yescharacteristic
290C2×C8#1192962311yescharacteristic
291C2×C8#1192962411yescharacteristic
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—3164no—
295C4⋊C4—3161no—
296C4⋊C4—3161no—
297C4⋊C4—1582no—
298C4⋊C4—1582no—
299C22⋊C4—3164no—
300C22⋊C4—3164no—
301C42—3164no—
302C16—41no—
303C16—3161no—
304C23—792no—
305C23—1581no—
306C23—1581no—
307C23—1581no—
308C23—1581no—
309C23—1581no—
310C23—1581no—
311C23D8×D7911yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—81no—
315Q8—81no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—3162no—
323C2×C4—3162no—
324C2×C4—792no—
325C2×C4—6321no—
326C2×C4—6321no—
327C2×C4—6321no—
328C2×C4—6321no—
329C2×C4—1581no—
330C2×C4—1581no—
331C2×C4—1582no—
332C2×C4—1582no—
333C2×C4—1582no—
334C2×C4—1582no—
335C2×C4C22⋊C4×D7911yescharacteristic
336C2×C4#1457912911yescharacteristic
337C2×C4#1457914311yescharacteristic
338C2×C4SD16×D7911yescharacteristic
339C2×C4#1457918511yescharacteristic
340C2×C4#1457919011yescharacteristic
341C8—1582no—
342C8(C22⋊D79)⋊C411yescharacteristic
343C8C4×C22⋊D7911yescharacteristic
344C8#1457913211yescharacteristic
345C8#1457913411yescharacteristic
346C22—3161no—
347C22—3161no—
348C22—3161no—
349C22—3161no—
350C22—1582no—
351C22—1582no—
352C22—1582no—
353C22#102×D7911yescharacteristic
354C22#1909869511yescharacteristic
355C22#1909887511yescharacteristic
356C22#1909899411yescharacteristic
357C22#1909899611yescharacteristic
358C22#1909900211yescharacteristic
359C22#1909900511yescharacteristic
360C4—82no—
361C4—82no—
362C4—6322no—
363C4—6322no—
364C4—1582no—
365C4#1909843911yescharacteristic
366C4#1909856111yescharacteristic
367C4#1909870011yescharacteristic
368C4#1909888011yescharacteristic
369C2—3162no—
370C2#2376371411yescharacteristic
371C2#2376510611yescharacteristic
372C2#2376511811yescharacteristic
373C2#2376533711yescharacteristic
374C2#2376533911yescharacteristic
375C2#2376560611yescharacteristic
376C2#2376560711yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#358×C79C2
socleC23×C79C2×D8
prime radicalC23×D79D8
cosocleC22×C8×C79C23
hypercenterC2×#358D79
hypocenterC79#12994
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
3. C2×#19098367C2semidirect
5. #23761492C2semidirect
6. #23761491C2semidirect
7. C2×#358×C79C2semidirect
13. #19098801C4semidirect
192. C2×#358D79semidirect
196. C79#12994semidirect

Maximal subgroups

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

subgroupindex
C2×#190988012
C2×#190986002
C2×#190983672
C2×(C2×D79)⋊C162
#237614922
#237614912
C2×#358×C792
#1299479