Finite Group Databasebeta

← #34344255

Subgroups — #34344255

Order 20224.

1038 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×Dic158C211yesmaximal, characteristic
2C2×#19098961C211yesmaximal, characteristic
3C2×#19098849C211yesmaximal, characteristic
4#23768385C211yesmaximal, characteristic
5#23764734C214yesmaximal
6#23764347C211yesmaximal, characteristic
7#23763671C214yesmaximal
8C4×#14579086C211yesmaximal, characteristic
9C79×#1984C211yesmaximal, characteristic
10Q8×Dic158—24no—
11Q8×Dic158—24no—
12C2×#14579161C2211yescharacteristic
13C2×#14579161C2212yes—
14C2×#14579161C2211yescharacteristic
15C2×#14579160—21no—
16C2×#14579160—22no—
17C2×#14579159—21no—
18C2×Q8×Dic79—21no—
19C2×C4×Dic158C2212yes—
20C2×C4×Dic158C2211yescharacteristic
21#19098961—24no—
22#19098849—24no—
23C2×#14579093C2214yes—
24C2×#14579093C2212yes—
25C2×#14579086C2212yes—
26C2×#14579086C2212yes—
27#19098493—22no—
28#19098468—21no—
29#19098461C2214yes—
30C4×C4⋊Dic79C2211yescharacteristic
31#19098326C2212yes—
32#19098326C2214yes—
33C2×C4×Q8×C79C2211yescharacteristic
34C2×C4⋊C8×C79C2211yescharacteristic
35C2×C4×C8×C79C2211yescharacteristic
36#336×C79C2214yes—
37C22×Dic158—22no—
38C22×Dic158—21no—
39C22×Dic158—21no—
40C22×Dic158C2311yescharacteristic
41#14579161—22no—
42#14579161—24no—
43#14579161—24no—
44#14579161—22no—
45#14579161—22no—
46#14579161D414yes—
47#14579160—216no—
48#14579160—28no—
49#14579159—44no—
50Q8×Dic79—28no—
51C4×Dic158—22no—
52C4×Dic158—24no—
53C4×Dic158—24no—
54C4×Dic158D414yes—
55C2×C4⋊Dic79—22no—
56C2×C4⋊Dic79—21no—
57C2×C4⋊Dic79—21no—
58C2×C4⋊Dic79C2312yes—
59C2×C4⋊Dic79C2312yes—
60C2×C4⋊Dic79C2311yescharacteristic
61C2×C4⋊Dic79C2312yes—
62C2×Dic79⋊C4—22no—
63C2×Dic79⋊C4—21no—
64C2×Dic79⋊C4—24no—
65C2×C4×Dic79—22no—
66C2×C4×Dic79—21no—
67C2×C4×Dic79—21no—
68#14579093—28no—
69#14579093—24no—
70#14579086—24no—
71#14579086Q818yescomplemented
72C79⋊#95—21no—
73C79⋊#95—24no—
74C22×Q8×C79D411yescharacteristic
75C22×C8×C79C2312yes—
76C22×C8×C79C2312yes—
77C4×Q8×C79—22no—
78C4×Q8×C79—22no—
79C2×C4⋊C4×C79D411yescharacteristic
80C2×C4⋊C4×C79D412yes—
81C2×C42×C79C2312yes—
82C2×C42×C79C2311yescharacteristic
83C4⋊C8×C79—22no—
84C4×C8×C79—22no—
85C2×Dic158—28no—
86C2×Dic158—24no—
87C2×Dic158—21no—
88C2×Dic158—22no—
89C2×Dic158—42no—
90C2×Dic158—42no—
91C2×Dic158—42no—
92C2×Dic158—82no—
93C2×Dic158—84no—
94C2×Dic158C2×D412yes—
95C2×Dic158C2×Q814yes—
96C22×Dic79—21no—
97C22×Dic79—22no—
98C22×Dic79—21no—
99C22×Dic79—22no—
100C22×Dic79—22no—
101C4⋊Dic79—28no—
102C4⋊Dic79—24no—
103C4⋊Dic79—22no—
104C4⋊Dic79—21no—
105C4⋊Dic79—22no—
106C4⋊Dic79—22no—
107C4⋊Dic79—22no—
108C4⋊Dic79—42no—
109C4⋊Dic79C2×D414yes—
110C4⋊Dic79C2×D412yes—
111C4⋊Dic79C2×Q814yes—
112Dic79⋊C4—28no—
113Dic79⋊C4—24no—
114Dic79⋊C4—48no—
115C4×Dic79—28no—
116C4×Dic79—24no—
117C4×Dic79—42no—
118C2×Q8×C79—22no—
119C2×Q8×C79—22no—
120C2×Q8×C79SD1614yes—
121C22×C4×C79C2×D412yes—
122C22×C4×C79C2×D411yescharacteristic
123C22×C4×C79C2×D412yes—
124C22×C4×C79C2×D411yescharacteristic
125C22×C4×C79C2411yescharacteristic
126C2×C8×C79—22no—
127C2×C8×C79—22no—
128C2×C8×C79—22no—
129C2×C8×C79—24no—
130C2×C8×C79C2×Q814yes—
131C4⋊C4×C79—24no—
132C4⋊C4×C79—22no—
133C42×C79—22no—
134C42×C79—22no—
135C42×C79C2×D412yes—
136C42×C79C2×D412yes—
137C23×C79C22×D411yescharacteristic
138Q8×C79—28no—
139C2×C4×C79—22no—
140C2×C4×C79—22no—
141C2×C4×C79—22no—
142C2×C4×C79—22no—
143C2×C4×C79—24no—
144C2×C4×C79#12012yes—
145C2×C4×C79#12012yes—
146C2×C4×C79C2×SD1614yes—
147C2×C4×C79C2×SD1612yes—
148C2×C4×C79#13712yes—
149C2×C4×C79C22×D411yescharacteristic
150C2×C4×C79C22×D411yescharacteristic
151C2×C4×C79C22×Q812yes—
152C2×C4×C792−1+412yes—
153C2×Dic79—22no—
154C2×Dic79—24no—
155C2×Dic79—24no—
156C2×Dic79—22no—
157C2×Dic79—24no—
158C2×Dic79—42no—
159C2×Dic79—42no—
160C2×Dic79—44no—
161C2×Dic79—42no—
162C2×Dic79—42no—
163C2×Dic79—42no—
164C2×Dic79—42no—
165Dic158—22no—
166Dic158—24no—
167Dic158—84no—
168Dic158—88no—
169Dic158—84no—
170Dic158C4⋊Q814yescomplemented
171C8×C79—24no—
172C8×C79—42no—
173C22×C79C2×#12011yescharacteristic
174C22×C79D4×Q812yes—
175C22×C79C22×SD1611yescharacteristic
176C22×C79C2×#13711yescharacteristic
177C22×C79#57712yes—
178C4×C79—24no—
179C4×C79—22no—
180C4×C79—42no—
181C4×C79#44814yes—
182C4×C79C2×C4⋊Q811yescharacteristic
183C4×C79#53611yescharacteristic
184C4×C79D4×Q812yes—
185Dic79—24no—
186Dic79—44no—
187Dic79—44no—
188Dic79—42no—
189Dic79—82no—
190#25837—791nomaximal, Sylow
191C2×C79#243411yescharacteristic
192C2×C79C2×#44811yescharacteristic
193C2×C79#300911yescharacteristic
194C2×C79Q8×SD1612yes—
195C2×C79#338712yes—
196C2×Q82—791no—
197C2×#498—791no—
198C2×#475—791no—
199#2546—791no—
200#2058—794no—
201#1987—791no—
202#1984D7911yescharacteristic
203#1865—794no—
204C4×#103—791no—
205C79#2583711yescharacteristic, complemented, Sylow
206Q82—1584no—
207Q82—1584no—
208C2×C4⋊Q8—791no—
209C2×C4⋊Q8—791no—
210C2×C4⋊Q8—792no—
211C2×C4⋊Q8—1581no—
212C2×C4⋊Q8—1581no—
213C2×C4⋊Q8—1582no—
214C2×C4×Q8—791no—
215C2×C4×Q8—792no—
216C2×C4×Q8—1581no—
217C2×C4×Q8C2×D7911yescharacteristic
218#498—1584no—
219#475—1584no—
220C2×#106—792no—
221C2×#106—794no—
222C2×C4⋊C8C2×D7911yescharacteristic
223C2×#103—792no—
224C2×#103—792no—
225C2×C4×C8C2×D7911yescharacteristic
226#398—1582no—
227#391—1581no—
228#389—794no—
229C4×C4⋊C4—791no—
230#340—792no—
231#340—794no—
232#336C2×D7914yes—
233C22×Q8—791no—
234C22×Q8—1581no—
235C22×Q8—1582no—
236C22×Q8—1581no—
237C22×Q8D31611yescharacteristic
238C22×C8C22×D7912yes—
239C22×C8C22×D7912yes—
240C4⋊Q8—3164no—
241C4⋊Q8—794no—
242C4⋊Q8—1584no—
243C4⋊Q8—1584no—
244C4⋊Q8—15816no—
245C4⋊Q8—1588no—
246C4⋊Q8—1582no—
247C4⋊Q8—1582no—
248C4⋊Q8—1582no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—794no—
252C4×Q8—1584no—
253C4×Q8—1584no—
254C4×Q8—1588no—
255C4×Q8—1582no—
256C2×C4⋊C4—791no—
257C2×C4⋊C4—792no—
258C2×C4⋊C4—792no—
259C2×C4⋊C4—792no—
260C2×C4⋊C4—1581no—
261C2×C4⋊C4—1584no—
262C2×C4⋊C4—1582no—
263C2×C4⋊C4—1581no—
264C2×C4⋊C4—1582no—
265C2×C4⋊C4—1581no—
266C2×C4⋊C4D31611yescharacteristic
267C2×C4⋊C4D31612yes—
268C2×C42—1581no—
269C2×C42—1582no—
270C2×C42—1581no—
271C2×C42C22×D7912yes—
272C2×C42C22×D7911yescharacteristic
273#106—1584no—
274#106—1588no—
275C4⋊C8—22no—
276#103—798no—
277#103—1584no—
278C4×C8—22no—
279#95—1581no—
280#95—1584no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—3162no—
284C2×Q8—3162no—
285C2×Q8—3162no—
286C2×Q8—794no—
287C2×Q8—792no—
288C2×Q8—6322no—
289C2×Q8—6324no—
290C2×Q8—1588no—
291C2×Q8—1582no—
292C2×Q8—1584no—
293C2×Q8—1581no—
294C2×Q8#1192961314yes—
295C22×C4—1582no—
296C22×C4—1582no—
297C22×C4—1582no—
298C22×C4—1581no—
299C22×C4—1581no—
300C22×C4C2×D31612yes—
301C22×C4C2×D31611yescharacteristic
302C22×C4C2×D31612yes—
303C22×C4C2×D31611yescharacteristic
304C22×C4C23×D7911yescharacteristic
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—22no—
309C2×C8Q8×D7914yes—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—3162no—
313C4⋊C4—3168no—
314C4⋊C4—794no—
315C4⋊C4—792no—
316C4⋊C4—794no—
317C4⋊C4—1588no—
318C4⋊C4—1588no—
319C4⋊C4—1582no—
320C4⋊C4—1582no—
321C4⋊C4—1582no—
322C4⋊C4—1584no—
323C4⋊C4—1584no—
324C4⋊C4—1581no—
325C4⋊C4—1582no—
326C42—22no—
327C42—22no—
328C42—3162no—
329C42—1588no—
330C42—1584no—
331C42D4×D7912yes—
332C42D4×D7912yes—
333C23C22×D31611yescharacteristic
334Q8—28no—
335Q8—794no—
336Q8—6324no—
337Q8—6328no—
338Q8—6324no—
339Q8—1584no—
340Q8—1582no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—3162no—
347C2×C4—3164no—
348C2×C4—3162no—
349C2×C4—3162no—
350C2×C4—3162no—
351C2×C4—3162no—
352C2×C4—3162no—
353C2×C4—1584no—
354C2×C4—1584no—
355C2×C4—1582no—
356C2×C4—1584no—
357C2×C4—1582no—
358C2×C4#1457913112yes—
359C2×C4#1457913112yes—
360C2×C4C2×#1192961314yes—
361C2×C4C2×#1192961312yes—
362C2×C4#1457919412yes—
363C2×C4C2×D4×D7911yescharacteristic
364C2×C4C2×D4×D7911yescharacteristic
365C2×C4C2×Q8×D7912yes—
366C2×C4#1457920812yes—
367C8—24no—
368C8—42no—
369C22C2×#1457913111yescharacteristic
370C22Q8×D31612yes—
371C22C22×#1192961311yescharacteristic
372C22C2×#1457919411yescharacteristic
373C22#1909937512yes—
374C4—22no—
375C4—24no—
376C4—42no—
377C4—3164no—
378C4—3164no—
379C4—3162no—
380C4—6322no—
381C4—1584no—
382C4#1909870414yes—
383C4C4⋊Q8×D7911yescharacteristic
384C4#1909913311yescharacteristic
385C4#1909922612yes—
386C2#2376704311yescharacteristic
387C2C2×#1909870411yescharacteristic
388C2#2377267911yescharacteristic
389C2Q8×#1192961312yes—
390C2#2377633412yes—
subgroupG/H
centerC23C22×D316
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC79×#1984C2
socle, prime radicalC23×C79C22×D4
hypercenter#1984D79
hypocenterC79#25837
LCSG ▷ C22×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
71. #14579086Q8semidirect
170. Dic158C4⋊Q8semidirect
205. C79#25837semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1582
C2×#190989612
C2×#190988492
#237683852
#237647342
#237643472
#237636712
C4×#145790862
C79×#19842
#2583779