Finite Group Databasebeta

← #34206962

Subgroups — #34206962

Order 20224.

396 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 107 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#23763396C211yesmaximal, characteristic
2C4×#14579080C211yesmaximal, characteristic
3#23761527C212yesmaximal
4#23761527C212yesmaximal
5C79×#2565C211yesmaximal, characteristic
6#19098587—21no—
7C2×#14579080C2212yes—
8C2×#14579080C2212yes—
9C4×#11929608—21no—
10#19098332—21no—
11#19098332—21no—
12C2×#125×C79C411yescharacteristic
13C2×C4×Q8×C79C2211yescharacteristic
14#398×C79—21no—
15#398×C79—21no—
16#391×C79C2211yescharacteristic
17#391×C79C2211yescharacteristic
18#387×C79—21no—
19#387×C79—21no—
20#384×C79C411yescharacteristic
21#382×C79C411yescharacteristic
22C4×C4⋊C4×C79C411yescharacteristic
23C2×#11929608—22no—
24C2×#11929608—22no—
25C4⋊(C79⋊C8)—41no—
26C4⋊(C79⋊C8)—41no—
27#14579080—22no—
28#14579080—24no—
29#14579080Q814yes—
30#14579072—41no—
31C4×C79⋊C8—41no—
32C22×Q8×C79C2311yescharacteristic
33#125×C79—41no—
34#125×C79—41no—
35C4×Q8×C79—41no—
36C4×Q8×C79—41no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—21no—
41C2×C4⋊C4×C79—21no—
42C2×C4⋊C4×C79C2×C411yescharacteristic
43C2×C4⋊C4×C79C2×C411yescharacteristic
44C2×C4⋊C4×C79C2×C411yescharacteristic
45C2×C4⋊C4×C79D411yescharacteristic
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C42×C79C2×C411yescharacteristic
48C2×C42×C79C2×C411yescharacteristic
49C2×C42×C79C2×C411yescharacteristic
50C2×C42×C79D411yescharacteristic
51C2×C42×C79D411yescharacteristic
52#95×C79—21no—
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#11929608—24no—
60#11929608—42no—
61#11929608—42no—
62#11929608—42no—
63C2×C79⋊C8—42no—
64C2×C79⋊C8—42no—
65C2×Q8×C79—41no—
66C2×Q8×C79—41no—
67C2×Q8×C79C2×Q812yes—
68C2×Q8×C79#4112yes—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79C22⋊C411yescharacteristic
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22×C411yescharacteristic
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C4⋊C4×C79—41no—
81C4⋊C4×C79—41no—
82C4⋊C4×C79—41no—
83C4⋊C4×C79—42no—
84C4⋊C4×C79—42no—
85C4⋊C4×C79—42no—
86C4⋊C4×C79—42no—
87C4⋊C4×C79—41no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—41no—
90C4⋊C4×C79—41no—
91C42×C79—21no—
92C42×C79—21no—
93C42×C79—21no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—41no—
98C42×C79—41no—
99C23×C79C2×C22⋊C411yescharacteristic
100Q8×C79—42no—
101Q8×C79—42no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—22no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—42no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79#9911yescharacteristic
127C2×C4×C79#9911yescharacteristic
128C2×C4×C79C4×Q812yes—
129C2×C4×C79C22⋊Q812yes—
130C2×C4×C79#12411yescharacteristic
131C2×C4×C79C4⋊Q811yescharacteristic
132C79⋊C8—44no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79#39111yescharacteristic
137C22×C79C2×#9911yescharacteristic
138C22×C79#41011yescharacteristic
139C4×C79—21no—
140C4×C79—21no—
141C4×C79—22no—
142C4×C79—22no—
143C4×C79—21no—
144C4×C79—21no—
145C4×C79—42no—
146C4×C79—41no—
147C4×C79—41no—
148C4×C79—41no—
149C4×C79—81no—
150C4×C79—81no—
151C4×C79—81no—
152#13303—791nomaximal, Sylow
153C2×C79—21no—
154C2×C79—21no—
155C2×C79#199611yescharacteristic
156C2×C79#213211yescharacteristic
157C2×C79#213311yescharacteristic
158#2565D7911yescharacteristic
159#1801—791no—
160C4×#101—791no—
161#1394—792no—
162#1394—792no—
163C79#1330311yescharacteristic, complemented, Sylow
164C2×#125Dic7911yescharacteristic
165C2×C4×Q8C2×D7911yescharacteristic
166#423—1581no—
167C2×#101—792no—
168C2×#101—792no—
169C4×M16—1581no—
170#398—21no—
171#398—21no—
172#391C2×D7911yescharacteristic
173#391C2×D7911yescharacteristic
174#387—21no—
175#387—21no—
176#384Dic7911yescharacteristic
177#382Dic7911yescharacteristic
178C4×C4⋊C4Dic7911yescharacteristic
179#343—1581no—
180#343—1581no—
181C22×Q8C22×D7911yescharacteristic
182C2×M16—1582no—
183C2×M16—1582no—
184#125—41no—
185#125—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic7911yescharacteristic
194C2×C4⋊C4C2×Dic7911yescharacteristic
195C2×C4⋊C4C2×Dic7911yescharacteristic
196C2×C4⋊C4C22⋊D7911yescharacteristic
197C2×C4⋊C4C22⋊D7911yescharacteristic
198C2×C42C2×Dic7911yescharacteristic
199C2×C42C2×Dic7911yescharacteristic
200C2×C42C2×Dic7911yescharacteristic
201C2×C42C22⋊D7911yescharacteristic
202C2×C42C22⋊D7911yescharacteristic
203C4⋊C8—3161no—
204C4⋊C8—3161no—
205#101—794no—
206#101—1584no—
207#101—1582no—
208#97—3161no—
209C4×C8—3161no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8Q8×D7912yes—
220C2×Q8#1192962512yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4C22⋊Dic7911yescharacteristic
226C22×C4C22⋊Dic7911yescharacteristic
227C22×C4C22⋊Dic7911yescharacteristic
228C22×C4C22⋊Dic7911yescharacteristic
229C22×C4C22×Dic7911yescharacteristic
230C22×C4C2×C22⋊D7911yescharacteristic
231C22×C4C2×C22⋊D7911yescharacteristic
232M16—3162no—
233M16—3162no—
234M16—3162no—
235M16—1584no—
236C2×C8—3162no—
237C2×C8—3162no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—41no—
256C42—41no—
257C23C2×C22⋊Dic7911yescharacteristic
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4#1457907611yescharacteristic
285C2×C4#1457907611yescharacteristic
286C2×C4Q8×Dic7912yes—
287C2×C4#1457913712yes—
288C2×C4#1457915011yescharacteristic
289C2×C4#1457915911yescharacteristic
290C8—3164no—
291C8—6322no—
292C22—21no—
293C22—21no—
294C22#1909847011yescharacteristic
295C22C2×#1457907611yescharacteristic
296C22#1909853711yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2#2376438611yescharacteristic
313C2#2376508511yescharacteristic
314C2#2376508911yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC22×Q8C22×D79
FittingC79×#2565C2
socleC22×C79C2×#99
prime radicalC23×C79C2×C22⋊C4
cosocleC22×Q8×C79C23
hypercenter#2565D79
hypocenterC79#13303
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2565

Complements

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

NH ≅ G/Ntype
163. C79#13303semidirect

Maximal subgroups

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

subgroupindex
#237633962
C4×#145790802
#237615272
C79×#25652
#1330379