Finite Group Databasebeta

← #34298858

Subgroups — #34298858

Order 20224.

762 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
1#23766023C211yesmaximal, characteristic
2#23766009C211yesmaximal, characteristic
3#23764170C212yesmaximal, complemented
4#23764164C212yesmaximal, complemented
5#23763601C212yesmaximal
6#23763595C212yesmaximal
7#23763568C211yesmaximal, characteristic, complemented
8#23763557C211yesmaximal, characteristic, complemented
9#23763534C211yesmaximal, characteristic, complemented
10#23763531C211yesmaximal, characteristic, complemented
11C2×#433×C79C211yesmaximal, characteristic
12C2×#14579120C2211yescharacteristic
13C22×C4⋊Dic79C2211yescharacteristic
14C2×#14579095C2212yes—
15C2×#14579093C2212yes—
16C2×#14579093C2212yes—
17C2×#14579093C2212yes—
18#19098462—22no—
19#19098456—22no—
20#19098449—22no—
21#19098438—22no—
22#19098433C2211yescharacteristic
23#19098433C2211yescharacteristic
24#19098433C2211yescharacteristic
25#19098429C2211yescharacteristic
26#19098429C2211yescharacteristic
27#19098429C2211yescharacteristic
28#19098423C2212yes—
29C4×C4⋊Dic79C2212yes—
30#19098326C2212yes—
31#19098326C2212yes—
32#19098326C2212yes—
33#19098326C2212yes—
34C2×#117×C79C2211yescharacteristic
35#433×C79C418yescomplemented
36C2×C4⋊C8×C79C2212yes—
37C2×C22⋊C8×C79C2212yes—
38C2×#97×C79C2211yescharacteristic
39C2×C4×C8×C79C2211yescharacteristic
40C23×Dic79—21no—
41#14579120—42no—
42C2×C4⋊Dic79—21no—
43C2×C4⋊Dic79—22no—
44C2×C4⋊Dic79—22no—
45C2×C4⋊Dic79—22no—
46C2×C4⋊Dic79—42no—
47C2×C4⋊Dic79C2312yes—
48C2×C4⋊Dic79C2312yes—
49C2×C4⋊Dic79C2311yescharacteristic
50C2×C4⋊Dic79C2311yescharacteristic
51C2×C4⋊Dic79C2311yescharacteristic
52C2×C4⋊Dic79C2311yescharacteristic
53C2×C22⋊Dic79—21no—
54C2×C22⋊Dic79—22no—
55C2×C4×Dic79—21no—
56C2×C4×Dic79—22no—
57#14579095—42no—
58#14579093—42no—
59#14579093—42no—
60#14579093—42no—
61C79⋊#95—22no—
62C79⋊#95—22no—
63C79⋊#95—22no—
64C79⋊#95—22no—
65C79⋊#95—22no—
66C23×C4×C79C2311yescharacteristic
67C22×C8×C79C2312yes—
68C22×C8×C79C2312yes—
69#117×C79C2×C414yes—
70#117×C79Q814yes—
71C2×C4⋊C4×C79D412yes—
72C2×C22⋊C4×C79D412yes—
73C2×C42×C79C2311yescharacteristic
74C2×C42×C79C2311yescharacteristic
75C4⋊C8×C79C2×C418yescomplemented
76C22⋊C8×C79C2×C418yes—
77#97×C79C2×C414yescomplemented
78C4×C8×C79C2×C414yescomplemented
79C22×Dic79—21no—
80C22×Dic79—22no—
81C22×Dic79—22no—
82C22×Dic79—21no—
83C22×Dic79—21no—
84C22×Dic79—21no—
85C22×Dic79—42no—
86C22×Dic79—82no—
87C4⋊Dic79—22no—
88C4⋊Dic79—22no—
89C4⋊Dic79—22no—
90C4⋊Dic79—22no—
91C4⋊Dic79—22no—
92C4⋊Dic79—22no—
93C4⋊Dic79—24no—
94C4⋊Dic79—42no—
95C4⋊Dic79—82no—
96C4⋊Dic79—82no—
97C4⋊Dic79—82no—
98C22⋊Dic79—42no—
99C22⋊Dic79—82no—
100C4×Dic79—42no—
101C4×Dic79—82no—
102C24×C79C2×D411yescharacteristic
103C22×C4×C79—22no—
104C22×C4×C79—22no—
105C22×C4×C79C22×C412yes—
106C22×C4×C79C2×D412yes—
107C22×C4×C79C2×D412yes—
108C22×C4×C79C2×D411yescharacteristic
109C22×C4×C79C2×Q812yes—
110C22×C4×C79C2411yescharacteristic
111C2×C8×C79—22no—
112C2×C8×C79—22no—
113C2×C8×C79—22no—
114C2×C8×C79—22no—
115C2×C8×C79C22×C414yes—
116C2×C8×C79C22×C414yes—
117C4⋊C4×C79C4⋊C418yes—
118C22⋊C4×C79C4⋊C418yes—
119C42×C79C22×C412yes—
120C42×C79C22×C412yes—
121C42×C79C2×Q812yes—
122C42×C79C2×Q812yes—
123C23×C79—21no—
124C23×C79—21no—
125C23×C79—22no—
126C23×C79—22no—
127C23×C79C2×C4⋊C412yes—
128C23×C79C22×D411yescharacteristic
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—22no—
132C2×C4×C79—22no—
133C2×C4×C79—22no—
134C2×C4×C79—22no—
135C2×C4×C79—24no—
136C2×C4×C79—24no—
137C2×C4×C79C2×C4⋊C414yes—
138C2×C4×C79C2×C4⋊C414yes—
139C2×C4×C79C2×C4⋊C412yes—
140C2×C4×C79C23×C411yescharacteristic
141C2×C4×C79C22×Q811yescharacteristic
142C2×C4×C79H(5,𝔽2)12yes—
143C2×C4×C792−1+412yes—
144C2×Dic79—41no—
145C2×Dic79—41no—
146C2×Dic79—41no—
147C2×Dic79—41no—
148C2×Dic79—42no—
149C2×Dic79—42no—
150C2×Dic79—42no—
151C2×Dic79—42no—
152C2×Dic79—42no—
153C2×Dic79—42no—
154C2×Dic79—42no—
155C2×Dic79—42no—
156C2×Dic79—42no—
157C2×Dic79—42no—
158C2×Dic79—84no—
159C8×C79—24no—
160C8×C79—24no—
161C22×C79—22no—
162C22×C79—24no—
163C22×C79—41no—
164C22×C79—41no—
165C22×C79—42no—
166C22×C79C22×C4⋊C411yescharacteristic
167C22×C79#53511yescharacteristic
168C22×C79#53611yescharacteristic
169C22×C79#57712yes—
170C22×C79#57712yes—
171C4×C79—22no—
172C4×C79—24no—
173C4×C79—24no—
174C4×C79#52012yes—
175C4×C79#54412yes—
176Dic79—81no—
177Dic79—82no—
178Dic79—81no—
179Dic79—81no—
180Dic79—81no—
181Dic79—82no—
182#21990—791nomaximal, Sylow
183C2×C79—42no—
184C2×C79#231711yescharacteristic
185C2×C79#291212yes—
186C2×C79#322111yescharacteristic
187C2×C79#322211yescharacteristic
188C2×C79#324412yes—
189C2×#433D7911yescharacteristic
190#2317—791no—
191#2314—791no—
192#1949—792no—
193#1948—792no—
194#1854—792no—
195#1853—792no—
196#1847—791no—
197#1844—791no—
198#1836—791no—
199#1835—791no—
200C79#2199011yescharacteristic, complemented, Sylow
201C2×#117—791no—
202C2×#117C2×D7911yescharacteristic
203C22×C4⋊C4—791no—
204#433Dic7918yescomplemented
205C2×#107—792no—
206C2×#106—792no—
207C2×#106—792no—
208C2×#106—792no—
209C2×C4⋊C8C2×D7912yes—
210C2×C22⋊C8C2×D7912yes—
211C2×#97C2×D7911yescharacteristic
212C2×C4×C8C2×D7911yescharacteristic
213#389—1582no—
214#388—1582no—
215#387—1582no—
216#385—1582no—
217#384—791no—
218#384—791no—
219#384—791no—
220#382—791no—
221#382—791no—
222#382—791no—
223#380—792no—
224C4×C4⋊C4—792no—
225#340—792no—
226#340—792no—
227#340—792no—
228#340—792no—
229C23×C4—1581no—
230C23×C4C22×D7911yescharacteristic
231C22×C8C22×D7912yes—
232C22×C8C22×D7912yes—
233#117—3162no—
234#117Dic15814yes—
235#117C2×Dic7914yes—
236C2×C4⋊C4—3162no—
237C2×C4⋊C4—791no—
238C2×C4⋊C4—791no—
239C2×C4⋊C4—791no—
240C2×C4⋊C4—791no—
241C2×C4⋊C4—792no—
242C2×C4⋊C4—792no—
243C2×C4⋊C4—1581no—
244C2×C4⋊C4—1582no—
245C2×C4⋊C4—1582no—
246C2×C4⋊C4—1582no—
247C2×C4⋊C4D31612yes—
248C2×C22⋊C4—1581no—
249C2×C22⋊C4—1582no—
250C2×C22⋊C4D31612yes—
251C2×C42—1581no—
252C2×C42—1582no—
253C2×C42C22×D7911yescharacteristic
254C2×C42C22×D7911yescharacteristic
255#107—3162no—
256#106—3162no—
257#106—3162no—
258#106—3162no—
259C4⋊C8C2×Dic7918yescomplemented
260C22⋊C8C2×Dic7918yes—
261#97C2×Dic7914yescomplemented
262C4×C8C2×Dic7914yescomplemented
263#95—1582no—
264#95—1582no—
265#95—1582no—
266#95—1582no—
267#95—1582no—
268C24C2×D31611yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—3162no—
272C22×C4—6322no—
273C22×C4—1582no—
274C22×C4—1581no—
275C22×C4—1582no—
276C22×C4—1581no—
277C22×C4—1581no—
278C22×C4—1581no—
279C22×C4C22×Dic7912yes—
280C22×C4C2×D31612yes—
281C22×C4C2×D31611yescharacteristic
282C22×C4C2×D31612yes—
283C22×C4C2×Dic15812yes—
284C22×C4C23×D7911yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8C22×Dic7914yes—
290C2×C8C22×Dic7914yes—
291C4⋊C4—3162no—
292C4⋊C4—6322no—
293C4⋊C4—6322no—
294C4⋊C4—6322no—
295C4⋊C4—1582no—
296C4⋊C4—1582no—
297C4⋊C4—1582no—
298C4⋊C4—1582no—
299C4⋊C4—1582no—
300C4⋊C4—1582no—
301C4⋊C4—1584no—
302C4⋊C4C4⋊Dic7918yes—
303C22⋊C4—3162no—
304C22⋊C4—6322no—
305C22⋊C4C4⋊Dic7918yes—
306C42—3162no—
307C42—6322no—
308C42C22×Dic7912yes—
309C42C22×Dic7912yes—
310C42C2×Dic15812yes—
311C42C2×Dic15812yes—
312C23—21no—
313C23—22no—
314C23—22no—
315C23—21no—
316C23C2×C4⋊Dic7912yes—
317C23C22×D31611yescharacteristic
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—3162no—
327C2×C4—3162no—
328C2×C4—3162no—
329C2×C4—3162no—
330C2×C4—3162no—
331C2×C4—3162no—
332C2×C4—3162no—
333C2×C4—3162no—
334C2×C4—3162no—
335C2×C4—3161no—
336C2×C4—3161no—
337C2×C4—3161no—
338C2×C4—3161no—
339C2×C4—3162no—
340C2×C4—6324no—
341C2×C4C2×C4⋊Dic7914yes—
342C2×C4C2×C4⋊Dic7914yes—
343C2×C4C2×C4⋊Dic7912yes—
344C2×C4C23×Dic7911yescharacteristic
345C2×C4C22×Dic15811yescharacteristic
346C2×C4#1457920712yes—
347C2×C4#1457920812yes—
348C8—24no—
349C8—24no—
350C22—22no—
351C22—24no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22C22×C4⋊Dic7911yescharacteristic
356C22#1909913211yescharacteristic
357C22#1909913911yescharacteristic
358C22#1909937512yes—
359C22#1909937512yes—
360C4—22no—
361C4—24no—
362C4—24no—
363C4—6321no—
364C4—6321no—
365C4—6321no—
366C4—6322no—
367C4—6322no—
368C4—6321no—
369C4#1909904612yes—
370C4#1909919112yes—
371C2—42no—
372C2#2376602311yescharacteristic
373C2#2377199512yes—
374C2#2377446311yescharacteristic
375C2#2377447411yescharacteristic
376C2#2377470012yes—
subgroupG/H
centerC23C22×D316
commutatorC2×C4×C79C23×C4
FrattiniC22×C4C23×D79
FittingC2×#433×C79C2
socleC23×C79C22×D4
prime radicalC24×C79C2×D4
cosocleC22×C4×C79C24
hypercenterC2×#433D79
hypocenterC79#21990
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
3. #23764170C2semidirect
4. #23764164C2semidirect
7. #23763568C2semidirect
8. #23763557C2semidirect
9. #23763534C2semidirect
10. #23763531C2semidirect
35. #433×C79C4semidirect
75. C4⋊C8×C79C2×C4semidirect
77. #97×C79C2×C4semidirect
78. C4×C8×C79C2×C4semidirect
200. C79#21990semidirect
204. #433Dic79semidirect
259. C4⋊C8C2×Dic79semidirect
261. #97C2×Dic79semidirect
262. C4×C8C2×Dic79semidirect

Maximal subgroups

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

subgroupindex
#237660232
#237660092
#237641702
#237641642
#237636012
#237635952
#237635682
#237635572
#237635342
#237635312
C2×#433×C792
#2199079