Finite Group Databasebeta

← #34290358

Subgroups — #34290358

Order 20224.

914 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19098518C211yesmaximal, characteristic
2#23765883C211yesmaximal, characteristic
3#23763961C214yesmaximal, complemented
4#23763954C214yesmaximal, complemented
5#23763096C211yesmaximal, characteristic
6#23763096C211yesmaximal, characteristic
7#23763096C211yesmaximal, characteristic
8#23763094C211yesmaximal, characteristic
9C22×C4⋊C8×C79C211yesmaximal, characteristic
10C22×#11929608C2211yescharacteristic
11C22×#11929608C2211yescharacteristic
12C2×#14579120C2211yescharacteristic
13C2×#14579120C2211yescharacteristic
14#19098518—24no—
15C2×#14579071C2212yes—
16C2×C8×Dic79C2212yes—
17#19098461C412yes—
18#19098461C412yes—
19#19098436C412yes—
20#19098436C412yes—
21C4×Dic79⋊C4C2214yes—
22#19098318C2214yes—
23#19098317C2212yes—
24#19098317C2214yes—
25#19098317C2214yes—
26#19098317C2212yes—
27C23×C8×C79C2211yescharacteristic
28C23×C8×C79C2211yescharacteristic
29C22×C42×C79C2211yescharacteristic
30C2×C4⋊C8×C79—24no—
31C2×C4⋊C8×C79C2212yes—
32C2×C4⋊C8×C79C2212yes—
33C2×#11929608—22no—
34C2×#11929608—22no—
35C2×#11929608—22no—
36C2×#11929608—22no—
37C2×#11929608—22no—
38C2×#11929608Q814yes—
39C22×C79⋊C8C2312yes—
40C22×C79⋊C8C2312yes—
41#14579120—22no—
42#14579120—22no—
43#14579120—22no—
44#14579120—22no—
45C2×C4⋊Dic79C2×C412yes—
46C2×C4⋊Dic79C2×C411yescharacteristic
47C2×C4⋊Dic79C2×C411yescharacteristic
48C2×Dic79⋊C4C2×C414yes—
49C2×Dic79⋊C4C2×C414yes—
50C2×C22⋊Dic79C2×C412yes—
51C2×C22⋊Dic79C2×C412yes—
52C2×C4×Dic79C2312yes—
53C2×C4×Dic79C2312yes—
54#14579071—24no—
55C8×Dic79—24no—
56C79⋊#95C2×C414yes—
57C79⋊#95C2×C414yes—
58C23×C4×C79C2×C411yescharacteristic
59C23×C4×C79C2×C411yescharacteristic
60C23×C4×C79C2311yescharacteristic
61C22×C8×C79—22no—
62C22×C8×C79—22no—
63C22×C8×C79—22no—
64C22×C8×C79—22no—
65C22×C8×C79—22no—
66C22×C8×C79D414yes—
67C22×C8×C79C2312yes—
68C22×C8×C79C2311yescharacteristic
69C22×C8×C79C2311yescharacteristic
70C2×C42×C79—22no—
71C2×C42×C79—22no—
72C2×C42×C79C2×C412yes—
73C2×C42×C79C2312yes—
74C4⋊C8×C79—24no—
75C4⋊C8×C79—24no—
76C22×Dic79C22×C414yes—
77C22×Dic79C22×C412yes—
78C22×Dic79C22×C412yes—
79#11929608—28no—
80#11929608—44no—
81C2×C79⋊C8—22no—
82C2×C79⋊C8—22no—
83C2×C79⋊C8—22no—
84C2×C79⋊C8—22no—
85C2×C79⋊C8—22no—
86C2×C79⋊C8C2×Q814yes—
87C4⋊Dic79—24no—
88C4⋊Dic79—22no—
89C4⋊Dic79—22no—
90Dic79⋊C4—28no—
91Dic79⋊C4—44no—
92C22⋊Dic79—24no—
93C22⋊Dic79—24no—
94C4×Dic79—22no—
95C4×Dic79—22no—
96C4×Dic79—22no—
97C4×Dic79—22no—
98C24×C79C22×C411yescharacteristic
99C22×C4×C79—22no—
100C22×C4×C79—24no—
101C22×C4×C79—22no—
102C22×C4×C79—42no—
103C22×C4×C79C22×C412yes—
104C22×C4×C79C22×C412yes—
105C22×C4×C79C22×C411yescharacteristic
106C22×C4×C79C2×D412yes—
107C22×C4×C79C2×Q812yes—
108C22×C4×C79#4112yes—
109C22×C4×C79#4112yes—
110C22×C4×C79#4112yes—
111C22×C4×C79#4112yes—
112C22×C4×C79C2411yescharacteristic
113C2×C8×C79—28no—
114C2×C8×C79—22no—
115C2×C8×C79—21no—
116C2×C8×C79—21no—
117C2×C8×C79—21no—
118C2×C8×C79—21no—
119C2×C8×C79—22no—
120C2×C8×C79—22no—
121C2×C8×C79—44no—
122C2×C8×C79C2×D412yes—
123C2×C8×C79C2×D412yes—
124C42×C79—24no—
125C42×C79—22no—
126C42×C79—22no—
127C23×C79—22no—
128C23×C79—22no—
129C23×C79#11712yes—
130C23×C79C4×D412yes—
131C23×C79C4×Q812yes—
132C23×C79C23×C411yescharacteristic
133C2×C4×C79—28no—
134C2×C4×C79—22no—
135C2×C4×C79—22no—
136C2×C4×C79—22no—
137C2×C4×C79—22no—
138C2×C4×C79—22no—
139C2×C4×C79—44no—
140C2×C4×C79#11712yes—
141C2×C4×C79C4×D412yes—
142C2×C4×C79C4×Q812yes—
143C2×C4×C79C22⋊Q814yes—
144C2×C4×C79#12314yes—
145C2×C4×C79#12514yes—
146C2×C4×C79#12614yes—
147C2×C4×C79#13114yes—
148C2×C4×C79C22×D411yescharacteristic
149C2×C4×C79C22×Q811yescharacteristic
150C2×C4×C79C2×#4111yescharacteristic
151C2×C4×C79C2×#4111yescharacteristic
152C2×C4×C79C2×#4111yescharacteristic
153C2×C4×C79C2×#4111yescharacteristic
154C2×Dic79—24no—
155C2×Dic79—22no—
156C2×Dic79—22no—
157C2×Dic79—24no—
158C2×Dic79—24no—
159C2×Dic79—24no—
160C2×Dic79—24no—
161C8×C79—22no—
162C8×C79—22no—
163C8×C79—42no—
164C79⋊C8—24no—
165C79⋊C8—42no—
166C22×C79—24no—
167C22×C79—22no—
168C22×C79—22no—
169C22×C79—24no—
170C22×C79#38714yes—
171C22×C79C2×#11711yescharacteristic
172C22×C79C2×C4×D411yescharacteristic
173C22×C79C2×C4×Q811yescharacteristic
174C22×C79C2×#13112yes—
175C22×C79#56812yes—
176C4×C79—24no—
177C4×C79—42no—
178C4×C79#38714yes—
179C4×C79C2×C22⋊Q811yescharacteristic
180C4×C79C2×#12311yescharacteristic
181C4×C79C2×#12511yescharacteristic
182C4×C79C2×#12611yescharacteristic
183Dic79—44no—
184Dic79—42no—
185Dic79—42no—
186#21024—791nomaximal, Sylow
187C2×C79—24no—
188C2×C79C2×#38711yescharacteristic
189C2×C79#292512yes—
190C2×C79#292911yescharacteristic
191C2×C79#293011yescharacteristic
192C2×C79#296212yes—
193C22×C4⋊C8D7911yescharacteristic
194C2×#405—791no—
195#2296—791no—
196#1917—794no—
197#1916—794no—
198#1729—791no—
199#1729—791no—
200#1729—791no—
201#1728—791no—
202C79#2102411yescharacteristic, complemented, Sylow
203C22×M16—791no—
204C22×M16—791no—
205C23×C8C2×D7911yescharacteristic
206C23×C8C2×D7911yescharacteristic
207C2×#117—791no—
208C2×#117—791no—
209C22×C42C2×D7911yescharacteristic
210C2×C4⋊C8—24no—
211C2×C4⋊C8C2×D7912yes—
212C2×C4⋊C8C2×D7912yes—
213#405—1584no—
214C2×#97—792no—
215C2×C4×C8—792no—
216#389—792no—
217#389—792no—
218#385—792no—
219#385—792no—
220C4×C4⋊C4—794no—
221#336—792no—
222#336—794no—
223#336—794no—
224#336—792no—
225#336—794no—
226C23×C4C4×D7911yescharacteristic
227C23×C4C4×D7911yescharacteristic
228C23×C4C22×D7911yescharacteristic
229C2×M16—794no—
230C2×M16—1582no—
231C2×M16—1582no—
232C2×M16—1582no—
233C2×M16—1582no—
234C2×M16—1582no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—792no—
241C22×C8—792no—
242C22×C8C22⋊D7914yes—
243C22×C8C22×D7912yes—
244C22×C8C22×D7911yescharacteristic
245C22×C8C22×D7911yescharacteristic
246#117—1582no—
247#117—1582no—
248#117—1582no—
249#117—1582no—
250C2×C4⋊C4—794no—
251C2×C4⋊C4—792no—
252C2×C4⋊C4—791no—
253C2×C4⋊C4—791no—
254C2×C4⋊C4—794no—
255C2×C22⋊C4—792no—
256C2×C22⋊C4—792no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—792no—
260C2×C42—792no—
261C2×C42C4×D7912yes—
262C2×C42C22×D7912yes—
263C4⋊C8—24no—
264C4⋊C8—24no—
265#97—1584no—
266C4×C8—1584no—
267#95—794no—
268#95—794no—
269C24C2×C4×D7911yescharacteristic
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—794no—
275C22×C4—792no—
276C22×C4—792no—
277C22×C4C2×C4×D7912yes—
278C22×C4C2×C4×D7912yes—
279C22×C4C2×C4×D7911yescharacteristic
280C22×C4C2×C22⋊D7912yes—
281C22×C4Q8×D7912yes—
282C22×C4#1192962312yes—
283C22×C4#1192962312yes—
284C22×C4#1192962412yes—
285C22×C4#1192962512yes—
286C22×C4C23×D7911yescharacteristic
287M16—3164no—
288M16—1588no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—28no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—44no—
298C2×C8—794no—
299C2×C8—1582no—
300C2×C8—1582no—
301C2×C8—1582no—
302C2×C8—1582no—
303C2×C8—1582no—
304C2×C8C2×C22⋊D7912yes—
305C2×C8C2×C22⋊D7912yes—
306C4⋊C4—3164no—
307C4⋊C4—1584no—
308C4⋊C4—1582no—
309C4⋊C4—1582no—
310C4⋊C4—1588no—
311C22⋊C4—1584no—
312C22⋊C4—1584no—
313C42—22no—
314C42—24no—
315C42—22no—
316C42—1582no—
317C42—1582no—
318C42—1582no—
319C42—1582no—
320C23—22no—
321C23—22no—
322C23#1457911912yes—
323C23C4×C22⋊D7912yes—
324C23#1457912612yes—
325C23C22×C4×D7911yescharacteristic
326C2×C4—28no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—44no—
333C2×C4—1584no—
334C2×C4—1582no—
335C2×C4—1582no—
336C2×C4—1584no—
337C2×C4—1584no—
338C2×C4—1584no—
339C2×C4—1584no—
340C2×C4#1457911912yes—
341C2×C4C4×C22⋊D7912yes—
342C2×C4#1457912612yes—
343C2×C4#1457913714yes—
344C2×C4#1457914214yes—
345C2×C4#1457915114yes—
346C2×C4#1457915614yes—
347C2×C4#1457916714yes—
348C2×C4C22×C22⋊D7911yescharacteristic
349C2×C4C2×Q8×D7911yescharacteristic
350C2×C4C2×#1192962311yescharacteristic
351C2×C4C2×#1192962311yescharacteristic
352C2×C4C2×#1192962411yescharacteristic
353C2×C4C2×#1192962511yescharacteristic
354C8—22no—
355C8—22no—
356C8—42no—
357C8—3162no—
358C8—1584no—
359C22—24no—
360C22—24no—
361C22—22no—
362C22—22no—
363C22#1909844414yes—
364C22C2×#1457911911yescharacteristic
365C22C2×C4×C22⋊D7911yescharacteristic
366C22C2×#1457912611yescharacteristic
367C22C2×#1457916712yes—
368C22#1909932912yes—
369C4—24no—
370C4—42no—
371C4—3164no—
372C4—3162no—
373C4—3162no—
374C4#1909844414yes—
375C4C2×#1457913711yescharacteristic
376C4C2×#1457914211yescharacteristic
377C4C2×#1457915111yescharacteristic
378C4C2×#1457915611yescharacteristic
379C2—24no—
380C2C2×#1909844411yescharacteristic
381C2#2377206512yes—
382C2#2377209411yescharacteristic
383C2#2377210311yescharacteristic
384C2#2377234312yes—
subgroupG/H
center, FrattiniC22×C4C23×D79
commutator, socleC23×C79C23×C4
FittingC22×C4⋊C8×C79C2
prime radicalC24×C79C22×C4
cosocleC22×C4×C79C24
hypercenterC22×C4⋊C8D79
hypocenterC79#21024
LCSG ▷ C23×C79 ▷ C79
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #23763961C2semidirect
4. #23763954C2semidirect
202. C79#21024semidirect

Maximal subgroups

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

subgroupindex
C2×#190985182
#237658832
#237639612
#237639542
#237630962
#237630942
C22×C4⋊C8×C792
#2102479