Finite Group Databasebeta

← #34186254

Subgroups — #34186254

Order 20224.

558 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 103 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#23772038C211yesmaximal, characteristic, complemented
2#23763483C211yesmaximal, characteristic, complemented
3#23761727C212yesmaximal
4#23761308C212yesmaximal, complemented
5C79×#2482C211yesmaximal, characteristic
6#19098624—24no—
7#19098587—21no—
8#19098527C2212yes—
9#19098527C2212yes—
10C4×#11929608—21no—
11#19098332—21no—
12#19098332—21no—
13#19098287—24no—
14C2×#117×C79C2211yescharacteristic
15C22×C22⋊C4×C79C2211yescharacteristic
16#399×C79—21no—
17#397×C79—21no—
18#388×C79—21no—
19#388×C79—21no—
20#385×C79C411yescharacteristic
21#385×C79C411yescharacteristic
22#381×C79C2211yescharacteristic
23#379×C79C411yescharacteristic
24#379×C79C411yescharacteristic
25C2×#11929608—22no—
26C2×#11929608—22no—
27C4⋊(C79⋊C8)—24no—
28C22⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)D414yes—
32#14579072—22no—
33C4×C79⋊C8—22no—
34C25×C79C2×C411yescharacteristic
35C23×C4×C79C2×C411yescharacteristic
36C23×C4×C79C2311yescharacteristic
37#117×C79—22no—
38#117×C79—22no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—21no—
41C2×C4⋊C4×C79D411yescharacteristic
42C2×C4⋊C4×C79D411yescharacteristic
43C2×C22⋊C4×C79—21no—
44C2×C22⋊C4×C79—21no—
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79—21no—
47C2×C22⋊C4×C79—21no—
48C2×C22⋊C4×C79—22no—
49C2×C22⋊C4×C79—21no—
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79C2×C411yescharacteristic
53C2×C22⋊C4×C79C2×C411yescharacteristic
54C2×C22⋊C4×C79C2×C411yescharacteristic
55C2×C42×C79D411yescharacteristic
56C2×C42×C79D411yescharacteristic
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60#95×C79—21no—
61#11929608—42no—
62#11929608—42no—
63#11929608—42no—
64#11929608—42no—
65C2×C79⋊C8—24no—
66C2×C79⋊C8—42no—
67C24×C79—21no—
68C24×C79—22no—
69C24×C79—42no—
70C24×C79—41no—
71C24×C79—41no—
72C24×C79—81no—
73C24×C79C22×C411yescharacteristic
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—22no—
78C22×C4×C79—41no—
79C22×C4×C79—41no—
80C22×C4×C79—81no—
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C22×C4×C79C2×D412yes—
88C22×C4×C79#4112yes—
89C4⋊C4×C79—22no—
90C4⋊C4×C79—22no—
91C4⋊C4×C79—81no—
92C4⋊C4×C79—81no—
93C22⋊C4×C79—22no—
94C22⋊C4×C79—22no—
95C22⋊C4×C79—42no—
96C22⋊C4×C79—42no—
97C22⋊C4×C79—41no—
98C22⋊C4×C79—42no—
99C22⋊C4×C79—41no—
100C22⋊C4×C79—41no—
101C22⋊C4×C79—41no—
102C22⋊C4×C79—81no—
103C22⋊C4×C79—81no—
104C42×C79—21no—
105C42×C79—21no—
106C42×C79—21no—
107C42×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—22no—
111C23×C79—24no—
112C23×C79—41no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79—42no—
118C23×C79—41no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79—82no—
122C23×C79—82no—
123C23×C79—84no—
124C23×C79—84no—
125C23×C79C2×C22⋊C411yescharacteristic
126C23×C79C4×D412yes—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—21no—
133C2×C4×C79—21no—
134C2×C4×C79—21no—
135C2×C4×C79—42no—
136C2×C4×C79—42no—
137C2×C4×C79—41no—
138C2×C4×C79—41no—
139C2×C4×C79—41no—
140C2×C4×C79—41no—
141C2×C4×C79—41no—
142C2×C4×C79—41no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—82no—
151C2×C4×C79#12011yescharacteristic
152C2×C4×C79#12112yes—
153C2×C4×C79#12311yescharacteristic
154C79⋊C8—44no—
155C79⋊C8—82no—
156C22×C79—21no—
157C22×C79—21no—
158C22×C79—22no—
159C22×C79—22no—
160C22×C79—22no—
161C22×C79—42no—
162C22×C79—42no—
163C22×C79—42no—
164C22×C79—42no—
165C22×C79—42no—
166C22×C79—81no—
167C22×C79—81no—
168C22×C79—84no—
169C22×C79—84no—
170C22×C79—81no—
171C22×C79—81no—
172C22×C79#38611yescharacteristic
173C22×C79#41011yescharacteristic
174C22×C79#41011yescharacteristic
175C4×C79—21no—
176C4×C79—22no—
177C4×C79—21no—
178C4×C79—41no—
179C4×C79—41no—
180C4×C79—41no—
181C4×C79—41no—
182C4×C79—81no—
183C4×C79—81no—
184C4×C79—81no—
185#9512—791nomaximal, Sylow
186C2×C79—21no—
187C2×C79—21no—
188C2×C79—42no—
189C2×C79—82no—
190C2×C79#189111yescharacteristic
191C2×C79#189911yescharacteristic
192C2×C79#191411yescharacteristic
193#2921—791no—
194#2482D7911yescharacteristic
195#1822—791no—
196#1462—792no—
197#1306—792no—
198C79#951211yescharacteristic, complemented, Sylow
199C2×#117C2×D7911yescharacteristic
200C22×C22⋊C4C2×D7911yescharacteristic
201#433—1584no—
202#423—1581no—
203#407—792no—
204#407—792no—
205C4×M16—1581no—
206#399—21no—
207#397—21no—
208#388—21no—
209#388—21no—
210#385Dic7911yescharacteristic
211#385Dic7911yescharacteristic
212#381C2×D7911yescharacteristic
213#379Dic7911yescharacteristic
214#379Dic7911yescharacteristic
215#343—1581no—
216#343—1581no—
217#323—1584no—
218C25C2×Dic7911yescharacteristic
219C23×C4C2×Dic7911yescharacteristic
220C23×C4C22×D7911yescharacteristic
221C2×M16—1582no—
222C2×M16—1582no—
223#117—22no—
224#117—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C22⋊D7911yescharacteristic
228C2×C4⋊C4C22⋊D7911yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4C2×Dic7911yescharacteristic
238C2×C22⋊C4C2×Dic7911yescharacteristic
239C2×C22⋊C4C2×Dic7911yescharacteristic
240C2×C22⋊C4C2×Dic7911yescharacteristic
241C2×C42C22⋊D7911yescharacteristic
242C2×C42C22⋊D7911yescharacteristic
243C4⋊C8—1584no—
244C22⋊C8—794no—
245C22⋊C8—1582no—
246C22⋊C8—1582no—
247C22⋊C8—1582no—
248#97—1582no—
249C4×C8—1582no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254C24—21no—
255C24—22no—
256C24—42no—
257C24—41no—
258C24—41no—
259C24—81no—
260C24C22×Dic7911yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—81no—
268C22×C4C22⋊Dic7911yescharacteristic
269C22×C4C22⋊Dic7911yescharacteristic
270C22×C4C22⋊Dic7911yescharacteristic
271C22×C4C22⋊Dic7911yescharacteristic
272C22×C4D4×D7912yes—
273C22×C4C2×C22⋊D7911yescharacteristic
274C22×C4C2×C22⋊D7911yescharacteristic
275C22×C4#1192962312yes—
276M16—3162no—
277M16—3162no—
278M16—3162no—
279M16—3162no—
280C2×C8—3162no—
281C2×C8—1584no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—24no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—82no—
315C23—82no—
316C23—84no—
317C23—84no—
318C23C2×C22⋊Dic7911yescharacteristic
319C23D4×Dic7912yes—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—82no—
344C2×C4#1457912911yescharacteristic
345C2×C4#1457913312yes—
346C2×C4#1457914211yescharacteristic
347C8—3164no—
348C8—6322no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—81no—
360C22—81no—
361C22—84no—
362C22—84no—
363C22—81no—
364C22—81no—
365C22#1909844311yescharacteristic
366C22#1909853711yescharacteristic
367C22#1909853711yescharacteristic
368C4—21no—
369C4—22no—
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—82no—
382C2#2376381911yescharacteristic
383C2#2376385911yescharacteristic
384C2#2376394811yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2482C2
socleC22×C79#386
prime radicalC25×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#2482D79
hypocenterC79#9512
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
1. #23772038C2semidirect
2. #23763483C2semidirect
4. #23761308C2semidirect
198. C79#9512semidirect

Maximal subgroups

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

subgroupindex
#237720382
#237634832
#237617272
#237613082
C79×#24822
#951279