Finite Group Databasebeta

← #34186264

Subgroups — #34186264

Order 20224.

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

normal subgroups only

This group has 478 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
2#23763483C211yesmaximal, characteristic
3#23761730C212yesmaximal
4#23761308C212yesmaximal
5C79×#2484C211yesmaximal, characteristic
6#19098624—24no—
7#19098587—21no—
8#19098527C2212yes—
9#19098527C2212yes—
10C4×#11929608—21no—
11#19098332—21no—
12#19098332—21no—
13#19098289—24no—
14C2×#117×C79C2211yescharacteristic
15C22×C4⋊C4×C79C2211yescharacteristic
16#400×C79—21no—
17#398×C79—21no—
18#387×C79—21no—
19#387×C79—21no—
20#385×C79C411yescharacteristic
21#385×C79C411yescharacteristic
22#381×C79C411yescharacteristic
23#381×C79C411yescharacteristic
24#381×C79C2211yescharacteristic
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—
34C23×C4×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×C79—22no—
42C2×C4⋊C4×C79—21no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—22no—
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C22⋊C4×C79C2×C411yescharacteristic
48C2×C22⋊C4×C79C2×C411yescharacteristic
49C2×C22⋊C4×C79C2×C411yescharacteristic
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C42×C79D411yescharacteristic
52C2×C42×C79D411yescharacteristic
53#95×C79—21no—
54#95×C79—21no—
55#95×C79—21no—
56#95×C79—21no—
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×C79C22×C411yescharacteristic
68C22×C4×C79—21no—
69C22×C4×C79—22no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—22no—
74C22×C4×C79—42no—
75C22×C4×C79—41no—
76C22×C4×C79—41no—
77C22×C4×C79—41no—
78C22×C4×C79—41no—
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D412yes—
86C22×C4×C79#4112yes—
87C4⋊C4×C79—22no—
88C4⋊C4×C79—22no—
89C4⋊C4×C79—42no—
90C4⋊C4×C79—42no—
91C4⋊C4×C79—41no—
92C4⋊C4×C79—42no—
93C4⋊C4×C79—41no—
94C22⋊C4×C79—22no—
95C22⋊C4×C79—22no—
96C22⋊C4×C79—41no—
97C22⋊C4×C79—41no—
98C42×C79—21no—
99C42×C79—21no—
100C42×C79—21no—
101C42×C79—21no—
102C23×C79—21no—
103C23×C79—21no—
104C23×C79—41no—
105C23×C79—41no—
106C23×C79C2×C22⋊C411yescharacteristic
107C23×C79C4×D412yes—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—22no—
113C2×C4×C79—24no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79—41no—
121C2×C4×C79—42no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79—41no—
135C2×C4×C79—41no—
136C2×C4×C79—82no—
137C2×C4×C79#12011yescharacteristic
138C2×C4×C79#12112yes—
139C2×C4×C79#12311yescharacteristic
140C79⋊C8—44no—
141C79⋊C8—82no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—22no—
145C22×C79—22no—
146C22×C79—22no—
147C22×C79—42no—
148C22×C79—42no—
149C22×C79#38611yescharacteristic
150C22×C79#41011yescharacteristic
151C22×C79#41011yescharacteristic
152C4×C79—21no—
153C4×C79—22no—
154C4×C79—21no—
155C4×C79—42no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79—41no—
159C4×C79—41no—
160C4×C79—81no—
161C4×C79—81no—
162C4×C79—81no—
163#9514—791nomaximal, Sylow
164C2×C79—21no—
165C2×C79—21no—
166C2×C79—42no—
167C2×C79#189111yescharacteristic
168C2×C79#189911yescharacteristic
169C2×C79#191511yescharacteristic
170#2921—791no—
171#2484D7911yescharacteristic
172#1822—791no—
173#1463—792no—
174#1306—792no—
175C79#951411yescharacteristic, complemented, Sylow
176C2×#117C2×D7911yescharacteristic
177C22×C4⋊C4C2×D7911yescharacteristic
178#433—1584no—
179#423—1581no—
180#407—792no—
181#407—792no—
182C4×M16—1581no—
183#400—21no—
184#398—21no—
185#387—21no—
186#387—21no—
187#385Dic7911yescharacteristic
188#385Dic7911yescharacteristic
189#381Dic7911yescharacteristic
190#381Dic7911yescharacteristic
191#381C2×D7911yescharacteristic
192#343—1581no—
193#343—1581no—
194#324—1584no—
195C23×C4C2×Dic7911yescharacteristic
196C23×C4C2×Dic7911yescharacteristic
197C23×C4C22×D7911yescharacteristic
198C2×M16—1582no—
199C2×M16—1582no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C22⋊D7911yescharacteristic
209C2×C4⋊C4C22⋊D7911yescharacteristic
210C2×C22⋊C4C2×Dic7911yescharacteristic
211C2×C22⋊C4C2×Dic7911yescharacteristic
212C2×C22⋊C4C2×Dic7911yescharacteristic
213C2×C22⋊C4C2×Dic7911yescharacteristic
214C2×C42C22⋊D7911yescharacteristic
215C2×C42C22⋊D7911yescharacteristic
216C4⋊C8—1584no—
217C22⋊C8—794no—
218C22⋊C8—1582no—
219C22⋊C8—1582no—
220C22⋊C8—1582no—
221#97—1582no—
222C4×C8—1582no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C24C22×Dic7911yescharacteristic
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4C22⋊Dic7911yescharacteristic
244C22×C4C22⋊Dic7911yescharacteristic
245C22×C4C22⋊Dic7911yescharacteristic
246C22×C4C22⋊Dic7911yescharacteristic
247C22×C4D4×D7912yes—
248C22×C4C2×C22⋊D7911yescharacteristic
249C22×C4C2×C22⋊D7911yescharacteristic
250C22×C4#1192962312yes—
251M16—3162no—
252M16—3162no—
253M16—3162no—
254M16—3162no—
255C2×C8—3162no—
256C2×C8—1584no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×C22⋊Dic7911yescharacteristic
277C23D4×Dic7912yes—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4#1457912911yescharacteristic
308C2×C4#1457913312yes—
309C2×C4#1457914211yescharacteristic
310C8—3164no—
311C8—6322no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22#1909844311yescharacteristic
320C22#1909853711yescharacteristic
321C22#1909853711yescharacteristic
322C4—21no—
323C4—22no—
324C4—21no—
325C4—42no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2376381911yescharacteristic
337C2#2376385911yescharacteristic
338C2#2376395311yescharacteristic
subgroupG/H
centerC22#19098443
commutator, prime radicalC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2484C2
socleC22×C79#386
cosocleC23×C4×C79C23
hypercenter#2484D79
hypocenterC79#9514
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
175. C79#9514semidirect

Maximal subgroups

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

subgroupindex
#237720382
#237634832
#237617302
#237613082
C79×#24842
#951479