Finite Group Databasebeta

← #34260685

Subgroups — #34260685

Order 20224.

1054 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1054 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23773344C212yesmaximal, complemented
2#23767674C211yesmaximal, characteristic
3#23762425C214yesmaximal, complemented
4#23762391C214yesmaximal, complemented
5#23761933C212yesmaximal, complemented
6#23761727C211yesmaximal, characteristic
7C2×#543×C79C211yesmaximal, characteristic
8C2×#14579120C2211yescharacteristic
9#19098901—24no—
10#19098896C2214yescomplemented
11#19098891C2214yescomplemented
12#19098603—22no—
13#19098572—22no—
14#19098527C2212yes—
15#19098499C2212yes—
16#19098488C2212yes—
17#19098451—22no—
18#19098436—21no—
19#19098426C2211yescharacteristic
20#19098419—21no—
21#19098300C2218yescomplemented
22#19098287C2214yes—
23#543×C79—24no—
24#543×C79—22no—
25#543×C79C2214yes—
26C2×C22⋊Q8×C79—22no—
27C2×C22⋊Q8×C79—21no—
28C2×C22⋊Q8×C79—22no—
29C2×C22⋊Q8×C79C2212yes—
30C22×C22⋊C4×C79—21no—
31C22×C22⋊C4×C79C2211yescharacteristic
32C2×#11929608—22no—
33#14579120—22no—
34#14579120—22no—
35C2×C4⋊Dic79C2311yescharacteristic
36C2×C4⋊Dic79C2311yescharacteristic
37C2×Dic79⋊C4—22no—
38C2×C22⋊Dic79—21no—
39C2×C22⋊Dic79—21no—
40C2×C22⋊Dic79—22no—
41C2×C22⋊Dic79—22no—
42C2×C4×Dic79—21no—
43#14579094—24no—
44#14579092—24no—
45#14579085—24no—
46#14579085—24no—
47C22⋊(C79⋊C8)—22no—
48C22⋊(C79⋊C8)C2314yes—
49C79⋊#95—21no—
50C79⋊#95—22no—
51C79⋊#95C2312yes—
52C25×C79D411yescharacteristic
53C22×Q8×C79—21no—
54C22×Q8×C79D412yes—
55C23×C4×C79—21no—
56C23×C4×C79—21no—
57C23×C4×C79D411yescharacteristic
58C23×C4×C79C2311yescharacteristic
59C22⋊Q8×C79—28no—
60C22⋊Q8×C79—28no—
61C22⋊Q8×C79—24no—
62C22⋊Q8×C79—28no—
63C22⋊Q8×C79—24no—
64C22⋊Q8×C79—22no—
65C22⋊Q8×C79—22no—
66C22⋊Q8×C79—28no—
67C22⋊Q8×C79D414yes—
68C22⋊Q8×C79C2314yes—
69C2×C4⋊C4×C79—21no—
70C2×C4⋊C4×C79—22no—
71C2×C4⋊C4×C79—22no—
72C2×C4⋊C4×C79D412yes—
73C2×C22⋊C4×C79—22no—
74C2×C22⋊C4×C79—24no—
75C2×C22⋊C4×C79—22no—
76C2×C22⋊C4×C79—24no—
77C2×C22⋊C4×C79—22no—
78C2×C22⋊C4×C79—22no—
79C2×C22⋊C4×C79D412yes—
80C2×C22⋊C4×C79C2312yes—
81C22×Dic79—22no—
82C22×Dic79—21no—
83C22×Dic79—21no—
84#11929608—44no—
85C2×C79⋊C8—24no—
86C4⋊Dic79—22no—
87C4⋊Dic79—22no—
88Dic79⋊C4—82no—
89C22⋊Dic79—42no—
90C22⋊Dic79—41no—
91C22⋊Dic79—41no—
92C22⋊Dic79—44no—
93C22⋊Dic79—82no—
94C4×Dic79—42no—
95C24×C79—21no—
96C24×C79—21no—
97C24×C79—24no—
98C24×C79—42no—
99C24×C79—81no—
100C24×C79C2×D411yescharacteristic
101C24×C79C2×D412yes—
102C2×Q8×C79—24no—
103C2×Q8×C79—22no—
104C2×Q8×C79—44no—
105C2×Q8×C79—82no—
106C2×Q8×C79C2×D414yes—
107C22×C4×C79—22no—
108C22×C4×C79—21no—
109C22×C4×C79—22no—
110C22×C4×C79—24no—
111C22×C4×C79—24no—
112C22×C4×C79—21no—
113C22×C4×C79—42no—
114C22×C4×C79—81no—
115C22×C4×C79—82no—
116C22×C4×C79—82no—
117C22×C4×C79C2×D412yes—
118C22×C4×C79C2×D412yes—
119C22×C4×C79C2×D411yescharacteristic
120C22×C4×C79C2×D412yes—
121C22×C4×C79C2411yescharacteristic
122C4⋊C4×C79—24no—
123C4⋊C4×C79—28no—
124C4⋊C4×C79—24no—
125C4⋊C4×C79—22no—
126C4⋊C4×C79—24no—
127C4⋊C4×C79C2×D414yes—
128C22⋊C4×C79—28no—
129C22⋊C4×C79—24no—
130C22⋊C4×C79—24no—
131C22⋊C4×C79—24no—
132C22⋊C4×C79—24no—
133C23×C79—24no—
134C23×C79—24no—
135C23×C79—22no—
136C23×C79—22no—
137C23×C79—22no—
138C23×C79—22no—
139C23×C79—41no—
140C23×C79—42no—
141C23×C79—42no—
142C23×C79—42no—
143C23×C79—42no—
144C23×C79—81no—
145C23×C79—81no—
146C23×C79—88no—
147C23×C79#12011yescharacteristic
148C23×C79#12011yescharacteristic
149C23×C79#13714yes—
150C23×C79C22×D411yescharacteristic
151Q8×C79—44no—
152Q8×C79—82no—
153Q8×C79—84no—
154C2×C4×C79—24no—
155C2×C4×C79—24no—
156C2×C4×C79—22no—
157C2×C4×C79—24no—
158C2×C4×C79—41no—
159C2×C4×C79—42no—
160C2×C4×C79—41no—
161C2×C4×C79—42no—
162C2×C4×C79—44no—
163C2×C4×C79—44no—
164C2×C4×C79—42no—
165C2×C4×C79—81no—
166C2×C4×C79—84no—
167C2×C4×C79—84no—
168C2×C4×C79#12012yes—
169C2×C4×C79C22×D412yes—
170C2×Dic79—42no—
171C2×Dic79—42no—
172C2×Dic79—42no—
173C2×Dic79—41no—
174C2×Dic79—41no—
175C2×Dic79—42no—
176C2×Dic79—42no—
177C79⋊C8—44no—
178C22×C79—21no—
179C22×C79—21no—
180C22×C79—21no—
181C22×C79—21no—
182C22×C79—21no—
183C22×C79—24no—
184C22×C79—21no—
185C22×C79—21no—
186C22×C79—44no—
187C22×C79—44no—
188C22×C79—48no—
189C22×C79—42no—
190C22×C79—42no—
191C22×C79—82no—
192C22×C79—84no—
193C22×C79#45812yes—
194C22×C79C2×#12011yescharacteristic
195C22×C79C2×#13712yes—
196C4×C79—22no—
197C4×C79—22no—
198C4×C79—42no—
199C4×C79—44no—
200C4×C79—44no—
201Dic79—82no—
202Dic79—81no—
203Dic79—81no—
204#18138—791nomaximal, Sylow
205C2×C79—22no—
206C2×C79—22no—
207C2×C79—41no—
208C2×C79—44no—
209C2×C79#300411yescharacteristic
210C2×C79#301911yescharacteristic
211C2×C79C2×#45811yescharacteristic
212C2×#543D7911yescharacteristic
213#3095—792no—
214#2482—791no—
215#1614—794no—
216#1608—794no—
217#1511—792no—
218#1462—791no—
219C79#1813811yescharacteristic, complemented, Sylow
220#543—24no—
221#543—22no—
222#543C2×D7914yes—
223C2×C22⋊Q8—22no—
224C2×C22⋊Q8—21no—
225C2×C22⋊Q8—22no—
226C2×C22⋊Q8C2×D7912yes—
227C2×#117—791no—
228C22×C22⋊C4—21no—
229C22×C22⋊C4C2×D7911yescharacteristic
230#485—1584no—
231#484—794no—
232#483—794no—
233#428—1582no—
234#419—1582no—
235#407—792no—
236#399—792no—
237#397—792no—
238#388—1582no—
239#385—1581no—
240#381—791no—
241#379—1581no—
242#328—798no—
243#323—794no—
244C25C22⋊D7911yescharacteristic
245C22×Q8—21no—
246C22×Q8C22⋊D7912yes—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C22⋊D7911yescharacteristic
250C23×C4C22×D7911yescharacteristic
251C2×M16—1582no—
252C22⋊Q8—28no—
253C22⋊Q8—28no—
254C22⋊Q8—24no—
255C22⋊Q8—28no—
256C22⋊Q8—24no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—28no—
260C22⋊Q8C22⋊D7914yes—
261C22⋊Q8C22×D7914yes—
262#117—1582no—
263#117—1582no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—791no—
268C2×C4⋊C4—791no—
269C2×C4⋊C4—1582no—
270C2×C4⋊C4C22⋊D7912yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—1581no—
278C2×C22⋊C4—1581no—
279C2×C22⋊C4—1582no—
280C2×C22⋊C4—1582no—
281C2×C22⋊C4C22⋊D7912yes—
282C2×C22⋊C4C22×D7912yes—
283C2×C42—1581no—
284#107—1584no—
285#106—1584no—
286#103—1584no—
287#103—1584no—
288C22⋊C8—794no—
289C22⋊C8—1582no—
290#95—792no—
291#95—1581no—
292#95—1582no—
293C24—21no—
294C24—21no—
295C24—24no—
296C24—42no—
297C24—81no—
298C24C2×C22⋊D7911yescharacteristic
299C24C2×C22⋊D7912yes—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—44no—
303C2×Q8—82no—
304C2×Q8C2×C22⋊D7914yes—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—1582no—
316C22×C4—1581no—
317C22×C4—1581no—
318C22×C4C2×C22⋊D7912yes—
319C22×C4C2×C22⋊D7912yes—
320C22×C4C2×C22⋊D7911yescharacteristic
321C22×C4C2×C22⋊D7912yes—
322C22×C4C23×D7911yescharacteristic
323M16—3164no—
324C2×C8—1584no—
325C4⋊C4—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—6322no—
331C4⋊C4—1582no—
332C4⋊C4—1582no—
333C4⋊C4C2×C22⋊D7914yes—
334C22⋊C4—28no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—3162no—
340C22⋊C4—3161no—
341C22⋊C4—3161no—
342C22⋊C4—3164no—
343C22⋊C4—6322no—
344C42—3162no—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23—88no—
359C23#1457913011yescharacteristic
360C23#1457913011yescharacteristic
361C23#1457919214yes—
362C23C22×C22⋊D7911yescharacteristic
363Q8—44no—
364Q8—82no—
365Q8—84no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—3162no—
381C2×C4—3162no—
382C2×C4—3162no—
383C2×C4—3161no—
384C2×C4—3161no—
385C2×C4—3162no—
386C2×C4—3162no—
387C2×C4#1457913012yes—
388C2×C4C22×C22⋊D7912yes—
389C8—3164no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—24no—
398C22—44no—
399C22—44no—
400C22—48no—
401C22—42no—
402C22—42no—
403C22—82no—
404C22—84no—
405C22#1909875312yes—
406C22C2×#1457913011yescharacteristic
407C22C2×#1457919212yes—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—44no—
412C4—44no—
413C4—6322no—
414C4—6321no—
415C4—6321no—
416C2—22no—
417C2—22no—
418C2—41no—
419C2—44no—
420C2#2377263411yescharacteristic
421C2#2377275711yescharacteristic
422C2C2×#1909875311yescharacteristic
subgroupG/H
centerC22C2×#14579130
commutator, cosocleC22×C4×C79C24
FrattiniC22×C4C23×D79
FittingC2×#543×C79C2
socleC22×C79C2×#120
prime radicalC25×C79D4
hypercenterC2×#543D79
hypocenterC79#18138
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #23773344C2semidirect
3. #23762425C2semidirect
4. #23762391C2semidirect
5. #23761933C2semidirect
10. #19098896C22semidirect
11. #19098891C22semidirect
21. #19098300C22semidirect
219. C79#18138semidirect

Maximal subgroups

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

subgroupindex
#237733442
#237676742
#237624252
#237623912
#237619332
#237617272
C2×#543×C792
#1813879