Finite Group Databasebeta

← #34262192

Subgroups — #34262192

Order 20224.

940 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23774172C212yesmaximal
2#23769213C211yesmaximal, characteristic
3#23762442C214yesmaximal, complemented
4#23762374C214yesmaximal, complemented
5#23761972C212yesmaximal
6C2×#19098289C211yesmaximal, characteristic
7C2×#544×C79C211yesmaximal, characteristic
8C2×#14579140C2211yescharacteristic
9#19098898—22no—
10#19098896C2214yes—
11#19098802—24no—
12#19098719—22no—
13#19098718C2214yes—
14C2×C22⋊(C79⋊C8)C2212yes—
15#19098502C2212yes—
16#19098496—22no—
17#19098483C2212yes—
18#19098436—21no—
19#19098426—21no—
20#19098423C2211yescharacteristic
21#19098300C2218yes—
22#19098289C2214yes—
23#544×C79—24no—
24#544×C79—22no—
25#544×C79C2214yes—
26C2×C4⋊Q8×C79—22no—
27C2×#125×C79—22no—
28C2×C22⋊Q8×C79—21no—
29C2×C22⋊Q8×C79C2212yes—
30C2×#117×C79—21no—
31C22×C4⋊C4×C79C2211yescharacteristic
32C22×Dic158C2311yescharacteristic
33C2×#11929614—24no—
34C22×C79⋊C8—22no—
35#14579140—22no—
36#14579140—22no—
37C2×C4⋊Dic79—21no—
38C2×C4⋊Dic79C2311yescharacteristic
39C2×Dic79⋊C4—21no—
40C2×Dic79⋊C4—22no—
41C2×C22⋊Dic79—21no—
42C2×C22⋊Dic79—21no—
43#14579094—24no—
44#14579085—24no—
45#14579084—24no—
46C22⋊(C79⋊C8)—22no—
47C22⋊(C79⋊C8)C2314yes—
48C79⋊#95—22no—
49C79⋊#95—22no—
50C79⋊#95—22no—
51C79⋊#95C2312yes—
52C22×Q8×C79—21no—
53C22×Q8×C79D412yes—
54C23×C4×C79D411yescharacteristic
55C23×C4×C79D411yescharacteristic
56C23×C4×C79C2311yescharacteristic
57C4⋊Q8×C79—28no—
58C4⋊Q8×C79—28no—
59#125×C79—28no—
60#125×C79—28no—
61C22⋊Q8×C79—24no—
62C22⋊Q8×C79—24no—
63C22⋊Q8×C79—22no—
64C22⋊Q8×C79—22no—
65C22⋊Q8×C79D414yes—
66C22⋊Q8×C79C2314yes—
67#117×C79—24no—
68#117×C79—24no—
69C2×C4⋊C4×C79—21no—
70C2×C4⋊C4×C79—21no—
71C2×C4⋊C4×C79—21no—
72C2×C4⋊C4×C79—22no—
73C2×C4⋊C4×C79—22no—
74C2×C4⋊C4×C79—22no—
75C2×C4⋊C4×C79—22no—
76C2×C4⋊C4×C79D412yes—
77C2×C4⋊C4×C79D412yes—
78C2×C4⋊C4×C79C2312yes—
79C2×C22⋊C4×C79—22no—
80C2×C42×C79—22no—
81C2×Dic158—22no—
82C2×Dic158—81no—
83C22×Dic79—21no—
84C22×Dic79—21no—
85C22×Dic79—22no—
86#11929614—84no—
87C2×C79⋊C8—24no—
88C2×C79⋊C8—82no—
89C4⋊Dic79—22no—
90C4⋊Dic79—81no—
91Dic79⋊C4—42no—
92Dic79⋊C4—82no—
93C22⋊Dic79—41no—
94C22⋊Dic79—41no—
95C22⋊Dic79—42no—
96C24×C79C2×D411yescharacteristic
97C2×Q8×C79—24no—
98C2×Q8×C79—22no—
99C2×Q8×C79—82no—
100C2×Q8×C79—82no—
101C2×Q8×C79C2×D414yes—
102C22×C4×C79—21no—
103C22×C4×C79—21no—
104C22×C4×C79—22no—
105C22×C4×C79—22no—
106C22×C4×C79—22no—
107C22×C4×C79—21no—
108C22×C4×C79—21no—
109C22×C4×C79—81no—
110C22×C4×C79—81no—
111C22×C4×C79Q1614yescomplemented
112C22×C4×C79C2×D412yes—
113C22×C4×C79C2×D412yes—
114C22×C4×C79C2×D411yescharacteristic
115C22×C4×C79C2×D412yes—
116C22×C4×C79C2×D412yes—
117C22×C4×C79C2411yescharacteristic
118C4⋊C4×C79—24no—
119C4⋊C4×C79—24no—
120C4⋊C4×C79—24no—
121C4⋊C4×C79—28no—
122C4⋊C4×C79—24no—
123C4⋊C4×C79—24no—
124C4⋊C4×C79—22no—
125C4⋊C4×C79—24no—
126C4⋊C4×C79—24no—
127C4⋊C4×C79C2×D414yes—
128C22⋊C4×C79—24no—
129C22⋊C4×C79—24no—
130C42×C79—28no—
131C23×C79—41no—
132C23×C79—41no—
133C23×C79#12011yescharacteristic
134C23×C79#12011yescharacteristic
135C23×C79C2×Q1612yes—
136C23×C79#13712yes—
137C23×C79C22×D411yescharacteristic
138Q8×C79—44no—
139Q8×C79—82no—
140Q8×C79—84no—
141C2×C4×C79—22no—
142C2×C4×C79—22no—
143C2×C4×C79—24no—
144C2×C4×C79—24no—
145C2×C4×C79—24no—
146C2×C4×C79—24no—
147C2×C4×C79—22no—
148C2×C4×C79—41no—
149C2×C4×C79—42no—
150C2×C4×C79—41no—
151C2×C4×C79—42no—
152C2×C4×C79—44no—
153C2×C4×C79—42no—
154C2×C4×C79—44no—
155C2×C4×C79—42no—
156C2×C4×C79—81no—
157C2×C4×C79#12012yes—
158C2×C4×C79C2×Q1614yes—
159C2×C4×C79C22×D412yes—
160C2×Dic79—42no—
161C2×Dic79—42no—
162C2×Dic79—42no—
163C2×Dic79—42no—
164C2×Dic79—41no—
165C2×Dic79—41no—
166C2×Dic79—42no—
167Dic158—44no—
168C79⋊C8—44no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79—21no—
172C22×C79—21no—
173C22×C79—21no—
174C22×C79—21no—
175C22×C79—42no—
176C22×C79—42no—
177C22×C79#45112yes—
178C22×C79#45112yes—
179C22×C79C2×#12011yescharacteristic
180C22×C79C22×Q1611yescharacteristic
181C22×C79C2×#13711yescharacteristic
182C4×C79—24no—
183C4×C79—22no—
184C4×C79—22no—
185C4×C79—44no—
186C4×C79—42no—
187C4×C79—44no—
188Dic79—82no—
189Dic79—81no—
190Dic79—81no—
191#18295—791nomaximal, Sylow
192C2×C79—21no—
193C2×C79—21no—
194C2×C79—22no—
195C2×C79—22no—
196C2×C79C2×#45111yescharacteristic
197C2×C79#302011yescharacteristic
198C2×C79#302611yescharacteristic
199C2×#544D7911yescharacteristic
200#3189—792no—
201#2627—791no—
202#1617—794no—
203#1605—794no—
204#1520—792no—
205C2×#324—791no—
206C79#1829511yescharacteristic, complemented, Sylow
207#544—24no—
208#544—22no—
209#544C2×D7914yes—
210C2×C4⋊Q8—22no—
211C2×#125—22no—
212C2×C22⋊Q8—21no—
213C2×C22⋊Q8—791no—
214C2×C22⋊Q8C2×D7912yes—
215C2×#117—21no—
216C22×C4⋊C4C2×D7911yescharacteristic
217#484—794no—
218#484—1582no—
219#467—1584no—
220#451—794no—
221#451—1582no—
222C2×C22⋊C8—792no—
223#400—792no—
224#398—1582no—
225#395—792no—
226#385—1581no—
227#381—1581no—
228#380—791no—
229#328—798no—
230#324—794no—
231C22×Q8—21no—
232C22×Q8—791no—
233C22×Q8C22⋊D7912yes—
234C23×C4C22⋊D7911yescharacteristic
235C23×C4C22⋊D7911yescharacteristic
236C23×C4C22×D7911yescharacteristic
237C2×Q16—1584no—
238C22×C8—1582no—
239C4⋊Q8—28no—
240C4⋊Q8—28no—
241#125—28no—
242#125—28no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—1582no—
248C22⋊Q8—1582no—
249C22⋊Q8C22⋊D7914yes—
250C22⋊Q8C22×D7914yes—
251#117—24no—
252#117—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—791no—
261C2×C4⋊C4—1581no—
262C2×C4⋊C4—1581no—
263C2×C4⋊C4—1582no—
264C2×C4⋊C4C22⋊D7912yes—
265C2×C4⋊C4C22⋊D7912yes—
266C2×C4⋊C4C22×D7912yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—1581no—
269C2×C22⋊C4—1581no—
270C2×C42—22no—
271#107—1584no—
272#103—1584no—
273#103—1584no—
274C22⋊C8—794no—
275C22⋊C8—1582no—
276#95—792no—
277#95—1582no—
278#95—1582no—
279#95—1582no—
280C24C2×C22⋊D7911yescharacteristic
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—82no—
284C2×Q8—82no—
285C2×Q8—6321no—
286C2×Q8—1582no—
287C2×Q8C2×C22⋊D7914yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—1582no—
298C22×C4—1581no—
299C22×C4—1581no—
300C22×C4#1192961414yescomplemented
301C22×C4C2×C22⋊D7912yes—
302C22×C4C2×C22⋊D7912yes—
303C22×C4C2×C22⋊D7911yescharacteristic
304C22×C4C2×C22⋊D7912yes—
305C22×C4C2×C22⋊D7912yes—
306C22×C4C23×D7911yescharacteristic
307Q16—6324no—
308C2×C8—6322no—
309C2×C8—1584no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—3162no—
320C4⋊C4—6322no—
321C4⋊C4—6321no—
322C4⋊C4—1582no—
323C4⋊C4C2×C22⋊D7914yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—3161no—
327C22⋊C4—3161no—
328C22⋊C4—3162no—
329C42—28no—
330C23—41no—
331C23—41no—
332C23#1457913011yescharacteristic
333C23#1457913011yescharacteristic
334C23C2×#1192961412yes—
335C23#1457919212yes—
336C23C22×C22⋊D7911yescharacteristic
337Q8—44no—
338Q8—82no—
339Q8—84no—
340Q8—3164no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—3162no—
358C2×C4—3162no—
359C2×C4—3162no—
360C2×C4—3162no—
361C2×C4—3161no—
362C2×C4—3161no—
363C2×C4—3162no—
364C2×C4#1457913012yes—
365C2×C4C2×#1192961414yes—
366C2×C4C22×C22⋊D7912yes—
367C8—3164no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1909871912yes—
377C22#1909871912yes—
378C22C2×#1457913011yescharacteristic
379C22C22×#1192961411yescharacteristic
380C22C2×#1457919211yescharacteristic
381C4—22no—
382C4—22no—
383C4—24no—
384C4—44no—
385C4—42no—
386C4—44no—
387C4—6322no—
388C4—6321no—
389C4—6321no—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2C2×#1909871911yescharacteristic
395C2#2377276411yescharacteristic
396C2#2377279711yescharacteristic
subgroupG/H
centerC22C2×#14579130
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#544×C79C2
socleC22×C79C2×#120
prime radicalC24×C79C2×D4
hypercenterC2×#544D79
hypocenterC79#18295
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #23762442C2semidirect
4. #23762374C2semidirect
111. C22×C4×C79Q16semidirect
206. C79#18295semidirect
300. C22×C4#11929614semidirect

Maximal subgroups

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

subgroupindex
#237741722
#237692132
#237624422
#237623742
#237619722
C2×#190982892
C2×#544×C792
#1829579