Finite Group Databasebeta

← #34205209

Subgroups — #34205209

Order 20224.

582 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098796C211yesmaximal, characteristic
2C2×#19098600C211yesmaximal, characteristic, complemented
3C2×#19098364C211yesmaximal, characteristic, complemented
4C2×(C2×D79)⋊C16C211yesmaximal, characteristic, complemented
5#23761492C211yesmaximal, characteristic, complemented
6#23761491C211yesmaximal, characteristic, complemented
7C2×#357×C79C211yesmaximal, characteristic, complemented
8C22×C8×D79C2211yescharacteristic
9C2×#14579132—21no—
10C2×#14579119—21no—
11C22×C79⋊C16C2211yescharacteristic
12#19098796—22no—
13#19098796C414yescomplemented
14#19098600—22no—
15#19098600—22no—
16C2×#14579095C2211yescharacteristic
17C2×#14579095C2211yescharacteristic
18C2×#14579092—21no—
19C2×#14579082—21no—
20#19098364—22no—
21(C2×D79)⋊C16—22no—
22C22×D8×C79C2211yescharacteristic
23C22×C16×C79C2211yescharacteristic
24C2×#107×C79C2211yescharacteristic
25#357×C79—22no—
26C22×C22⋊D79—41no—
27C22×C4×D79D411yescharacteristic
28C22×C79⋊C8D411yescharacteristic
29C2×C8×D79—21no—
30C2×C8×D79—21no—
31C2×C8×D79—21no—
32C2×C8×D79—21no—
33C2×C8×D79C2×C412yes—
34C2×C8×D79D412yes—
35#14579132—24no—
36#14579132—42no—
37#14579119—24no—
38#14579119—42no—
39C2×C4⋊Dic79—21no—
40C2×C4⋊Dic79—21no—
41C2×C22⋊Dic79—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#14579095D412yescomplemented
51#14579092—24no—
52#14579082—24no—
53C22×D4×C79—21no—
54C2×D8×C79—21no—
55C2×D8×C79—21no—
56C2×D8×C79—21no—
57C2×D8×C79—21no—
58C2×D8×C79—21no—
59C2×D8×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×C79D412yescomplemented
67C23×D79D811yescharacteristic
68C2×C22⋊D79—44no—
69C2×C22⋊D79—82no—
70C2×C22⋊D79—82no—
71C22×Dic79—41no—
72C22×Dic79—41no—
73C22×Dic79D811yescharacteristic
74C2×C4×D79—21no—
75C2×C4×D79—21no—
76C2×C4×D79—21no—
77C2×C4×D79—21no—
78C2×C4×D79C22⋊C412yes—
79C2×C4×D79SD1612yes—
80C2×C79⋊C8—21no—
81C2×C79⋊C8—21no—
82C2×C79⋊C8C22⋊C412yes—
83C8×D79—22no—
84C8×D79—22no—
85C8×D79—22no—
86C8×D79—22no—
87C4⋊Dic79—22no—
88C4⋊Dic79—22no—
89C4⋊Dic79—41no—
90C4⋊Dic79—41no—
91C22⋊Dic79—44no—
92(C2×D79)⋊C4—44no—
93C4×Dic79—44no—
94C79⋊C16—41no—
95C24×C79—41no—
96C2×D4×C79—22no—
97C2×D4×C79—41no—
98C2×D4×C79—41no—
99C2×D4×C79—41no—
100C2×D4×C79—41no—
101C2×D4×C79—41no—
102C22×C4×C79—41no—
103C22×C4×C79C2×D411yescharacteristic
104D8×C79—22no—
105D8×C79—22no—
106D8×C79—41no—
107D8×C79—41no—
108C2×C8×C79C22×C411yescharacteristic
109C2×C8×C79C2×D411yescharacteristic
110C2×C8×C79C2×D411yescharacteristic
111C2×C8×C79C2×D411yescharacteristic
112C2×C8×C79#4111yescharacteristic
113C2×C8×C79#4111yescharacteristic
114C4⋊C4×C79—22no—
115C4⋊C4×C79—41no—
116C16×C79—41no—
117C23×C79—42no—
118C23×C79—81no—
119C23×C79—81no—
120C23×C79—81no—
121C23×C79—81no—
122C23×C79—81no—
123C23×C79—81no—
124C23×C79C2×D811yescharacteristic
125C22×D79—21no—
126C22×D79—21no—
127C22×D79—21no—
128C22×D79—21no—
129C22×D79—21no—
130C22×D79—21no—
131C22×D79#10212yes—
132D4×C79—42no—
133D4×C79—42no—
134D4×C79—81no—
135D4×C79—81no—
136C2×C4×C79—42no—
137C2×C4×C79—81no—
138C2×C4×C79—81no—
139C2×C4×C79C2×C22⋊C411yescharacteristic
140C2×C4×C79#12011yescharacteristic
141C2×C4×C79#12311yescharacteristic
142C2×C4×C79C2×SD1611yescharacteristic
143C2×C4×C79AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C79#13711yescharacteristic
145C22⋊D79—88no—
146C2×Dic79—21no—
147C2×Dic79—21no—
148C2×Dic79—42no—
149C2×Dic79—42no—
150C2×Dic79—81no—
151C2×Dic79—81no—
152C2×Dic79—81no—
153C2×Dic79—81no—
154C2×Dic79#10212yes—
155C4×D79—22no—
156C4×D79—22no—
157C4×D79—22no—
158C4×D79—22no—
159C8×C79C4×D411yescharacteristic
160C8×C79C4×D411yescharacteristic
161C8×C79#12111yescharacteristic
162C8×C79#12111yescharacteristic
163C79⋊C8—22no—
164C22×C79—82no—
165C22×C79—82no—
166C22×C79—82no—
167C22×C79—161no—
168C22×C79—161no—
169C22×C79—161no—
170C22×C79—161no—
171C22×C79C2×#10211yescharacteristic
172C22×C79#44711yescharacteristic
173C22×C79#48011yescharacteristic
174C22×C79#50811yescharacteristic
175C22×C79#50811yescharacteristic
176C22×C79#50911yescharacteristic
177C22×C79#50911yescharacteristic
178C2×D79—22no—
179C2×D79—22no—
180C2×D79—22no—
181C2×D79—41no—
182C2×D79—41no—
183C2×D79—41no—
184C2×D79—41no—
185C4×C79—82no—
186C4×C79#38611yescharacteristic
187C4×C79#41711yescharacteristic
188C4×C79#44811yescharacteristic
189C4×C79#48111yescharacteristic
190Dic79—22no—
191Dic79—82no—
192Dic79—82no—
193#12992—791nomaximal, Sylow
194C2×C79—162no—
195C2×C79#187411yescharacteristic
196C2×C79#213811yescharacteristic
197C2×C79#214011yescharacteristic
198C2×C79#218611yescharacteristic
199C2×C79#218611yescharacteristic
200C2×C79#223411yescharacteristic
201C2×C79#223411yescharacteristic
202D79—42no—
203C2×#466—791no—
204C2×#427—791no—
205C2×#357—791no—
206C2×#357D7911yescharacteristic, complemented
207C2×C22⋊C16—791no—
208#1380—791no—
209#1380—791no—
210C79#1299211yescharacteristic, complemented, Sylow
211C22×D8C2×D7911yescharacteristic
212C23×C8—791no—
213C2×#121—1581no—
214C2×#117—1581no—
215C22×C16—791no—
216C22×C16C2×D7911yescharacteristic
217#466—794no—
218#466—1582no—
219#427—1582no—
220#427—1582no—
221C2×#107—791no—
222C2×#107—791no—
223C2×#107C2×D7911yescharacteristic
224C2×#106—1581no—
225C2×#102—1581no—
226#357—22no—
227#357—1582no—
228C22⋊C16—1582no—
229C22×D4—21no—
230C22×D4—3161no—
231C23×C4—791no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8C4×D7912yes—
238C22×C8—791no—
239C22×C8—792no—
240C22×C8—792no—
241C22×C8—1581no—
242C22×C8—1581no—
243C22×C8—1581no—
244C22×C8—1581no—
245C22×C8C22×D7911yescharacteristic
246#121—3162no—
247#121—1584no—
248#117—3162no—
249#117—1584no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—1581no—
252C2×C4⋊C4—1581no—
253C2×C22⋊C4—3161no—
254C2×C22⋊C4—3161no—
255C2×C42—3161no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—1581no—
260C2×C16—1581no—
261C2×C16—1581no—
262#107—21no—
263#107—792no—
264#107—792no—
265#107—1581no—
266#107—1581no—
267#107C22⋊D7912yescomplemented
268#106—1584no—
269#102—1584no—
270C24—41no—
271C24—791no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—3164no—
279C2×D4—6322no—
280C2×D4—6322no—
281C22×C4—41no—
282C22×C4—3161no—
283C22×C4—3161no—
284C22×C4—791no—
285C22×C4—792no—
286C22×C4—792no—
287C22×C4—1581no—
288C22×C4—1581no—
289C22×C4—1581no—
290C22×C4—1581no—
291C22×C4D4×D7911yescharacteristic
292D8—22no—
293D8—22no—
294D8—41no—
295D8—41no—
296C2×C8—792no—
297C2×C8—1581no—
298C2×C8—1581no—
299C2×C8—1582no—
300C2×C8—1582no—
301C2×C8—1582no—
302C2×C8—1582no—
303C2×C8C2×C4×D7911yescharacteristic
304C2×C8D4×D7911yescharacteristic
305C2×C8D4×D7911yescharacteristic
306C2×C8C2×C22⋊D7911yescharacteristic
307C2×C8#1192962311yescharacteristic
308C2×C8#1192962411yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—3161no—
312C4⋊C4—3161no—
313C4⋊C4—1582no—
314C4⋊C4—1582no—
315C22⋊C4—3164no—
316C22⋊C4—3164no—
317C42—3164no—
318C16—41no—
319C16—3161no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—792no—
328C23—1581no—
329C23—1581no—
330C23—1581no—
331C23—1581no—
332C23—1581no—
333C23—1581no—
334C23D8×D7911yescharacteristic
335D4—42no—
336D4—42no—
337D4—81no—
338D4—81no—
339D4—6328no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—3162no—
344C2×C4—3162no—
345C2×C4—792no—
346C2×C4—6321no—
347C2×C4—6321no—
348C2×C4—6321no—
349C2×C4—6321no—
350C2×C4—1581no—
351C2×C4—1581no—
352C2×C4—1582no—
353C2×C4—1582no—
354C2×C4—1582no—
355C2×C4—1582no—
356C2×C4C22⋊C4×D7911yescharacteristic
357C2×C4#1457912911yescharacteristic
358C2×C4#1457914311yescharacteristic
359C2×C4SD16×D7911yescharacteristic
360C2×C4#1457918511yescharacteristic
361C2×C4#1457919011yescharacteristic
362C8—1582no—
363C8(C22⋊D79)⋊C411yescharacteristic
364C8C4×C22⋊D7911yescharacteristic
365C8#1457913211yescharacteristic
366C8#1457913411yescharacteristic
367C22—82no—
368C22—82no—
369C22—82no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—3161no—
375C22—3161no—
376C22—3161no—
377C22—3161no—
378C22—1582no—
379C22—1582no—
380C22—1582no—
381C22#102×D7911yescharacteristic
382C22#1909869511yescharacteristic
383C22#1909887511yescharacteristic
384C22#1909899211yescharacteristic
385C22#1909899411yescharacteristic
386C22#1909899711yescharacteristic
387C22#1909900011yescharacteristic
388C4—82no—
389C4—6322no—
390C4—6322no—
391C4—1582no—
392C4#1909843911yescharacteristic
393C4#1909856111yescharacteristic
394C4#1909870011yescharacteristic
395C4#1909888011yescharacteristic
396C2—162no—
397C2—3162no—
398C2#2376371411yescharacteristic
399C2#2376510411yescharacteristic
400C2#2376511411yescharacteristic
401C2#2376532011yescharacteristic
402C2#2376532211yescharacteristic
403C2#2376558911yescharacteristic
404C2#2376559011yescharacteristic
subgroupG/H
centerC23D8×D79
commutatorC2×C8×C79C22×C4
FrattiniC22×C8C22×D79
FittingC2×#357×C79C2
socleC23×C79C2×D8
prime radicalC2×#19098796C2
cosocleC22×C8×C79C23
hypercenterC2×#357D79
hypocenterC79#12992
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#19098600C2semidirect
3. C2×#19098364C2semidirect
4. C2×(C2×D79)⋊C16C2semidirect
5. #23761492C2semidirect
6. #23761491C2semidirect
7. C2×#357×C79C2semidirect
13. #19098796C4semidirect
50. #14579095D4semidirect
66. #107×C79D4semidirect
206. C2×#357D79semidirect
210. C79#12992semidirect
267. #107C22⋊D79semidirect

Maximal subgroups

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

subgroupindex
C2×#190987962
C2×#190986002
C2×#190983642
C2×(C2×D79)⋊C162
#237614922
#237614912
C2×#357×C792
#1299279