Finite Group Databasebeta

← #34186229

Subgroups — #34186229

Order 20224.

544 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 111 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#23772429C211yesmaximal, characteristic
2#23763480C212yesmaximal, complemented
3#23761738C211yesmaximal, characteristic
4#23761306C212yesmaximal, complemented
5C79×#2486C211yesmaximal, characteristic
6#19098635—24no—
7#19098620—21no—
8#19098527C2212yes—
9#19098527C2212yes—
10C2×C22⋊(C79⋊C8)—21no—
11#19098332—24no—
12#19098289—21no—
13#19098287—21no—
14C2×#123×C79C412yes—
15C2×C4×D4×C79C2211yescharacteristic
16C22×C4⋊C4×C79C2211yescharacteristic
17#398×C79—21no—
18#397×C79—21no—
19#389×C79—21no—
20#381×C79C412yes—
21#380×C79—21no—
22#380×C79C2211yescharacteristic
23C2×#11929608—22no—
24C2×#11929608D414yes—
25C22×C79⋊C8—22no—
26C4⋊(C79⋊C8)—22no—
27C22⋊(C79⋊C8)—22no—
28C22⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C22⋊(C79⋊C8)—22no—
31C22⋊(C79⋊C8)—22no—
32#14579072—22no—
33C22×D4×C79C2×C411yescharacteristic
34C23×C4×C79C2×C412yes—
35C23×C4×C79D411yescharacteristic
36C23×C4×C79C2311yescharacteristic
37#123×C79—44no—
38C4×D4×C79—22no—
39C4×D4×C79—22no—
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79—21no—
42C2×C4⋊C4×C79—22no—
43C2×C4⋊C4×C79—21no—
44C2×C4⋊C4×C79—21no—
45C2×C4⋊C4×C79—22no—
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C22⋊C4×C79—22no—
48C2×C22⋊C4×C79C2×C411yescharacteristic
49C2×C22⋊C4×C79C2×C412yes—
50C2×C22⋊C4×C79D411yescharacteristic
51C2×C42×C79D411yescharacteristic
52#95×C79—22no—
53#95×C79—22no—
54#95×C79—21no—
55#11929608—44no—
56#11929608—44no—
57C2×C79⋊C8—24no—
58C2×C79⋊C8—42no—
59C2×C79⋊C8—82no—
60C24×C79C22⋊C411yescharacteristic
61C24×C79C22×C411yescharacteristic
62C2×D4×C79—22no—
63C2×D4×C79—22no—
64C2×D4×C79—41no—
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—22no—
68C22×C4×C79—22no—
69C22×C4×C79—24no—
70C22×C4×C79—42no—
71C22×C4×C79—42no—
72C22×C4×C79—42no—
73C22×C4×C79—42no—
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C2×D412yes—
78C22×C4×C79C2×D411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C22×C4×C79#4112yes—
81C4⋊C4×C79—22no—
82C4⋊C4×C79—42no—
83C4⋊C4×C79—44no—
84C4⋊C4×C79—41no—
85C4⋊C4×C79—42no—
86C4⋊C4×C79—41no—
87C22⋊C4×C79—22no—
88C22⋊C4×C79—22no—
89C22⋊C4×C79—44no—
90C22⋊C4×C79—42no—
91C42×C79—21no—
92C42×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79#10012yes—
102C23×C79C2×C22⋊C411yescharacteristic
103C23×C79C4×D412yes—
104D4×C79—44no—
105C2×C4×C79—24no—
106C2×C4×C79—22no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—41no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—41no—
116C2×C4×C79—41no—
117C2×C4×C79—42no—
118C2×C4×C79—42no—
119C2×C4×C79—42no—
120C2×C4×C79—42no—
121C2×C4×C79—42no—
122C2×C4×C79—42no—
123C2×C4×C79—81no—
124C2×C4×C79—84no—
125C2×C4×C79#12011yescharacteristic
126C2×C4×C79#12112yes—
127C2×C4×C79#12311yescharacteristic
128C79⋊C8—44no—
129C79⋊C8—82no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—22no—
133C22×C79—22no—
134C22×C79—22no—
135C22×C79—22no—
136C22×C79—42no—
137C22×C79—42no—
138C22×C79—42no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—81no—
142C22×C79#38611yescharacteristic
143C22×C79C2×#10011yescharacteristic
144C22×C79#41311yescharacteristic
145C4×C79—21no—
146C4×C79—22no—
147C4×C79—41no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—82no—
151C4×C79—82no—
152C4×C79#45812yes—
153#9507—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79—42no—
157C2×C79—42no—
158C2×C79#189011yescharacteristic
159C2×C79#190011yescharacteristic
160C2×C79#191511yescharacteristic
161#2973—791no—
162#2486D7911yescharacteristic
163#1821—792no—
164#1465—791no—
165#1305—792no—
166C79#950711yescharacteristic, complemented, Sylow
167C2×#123Dic7912yes—
168C2×C4×D4C2×D7911yescharacteristic
169C22×C4⋊C4C2×D7911yescharacteristic
170#435—1584no—
171#432—1581no—
172#407—792no—
173#407—792no—
174C2×C22⋊C8—1581no—
175#398—21no—
176#397—21no—
177#389—21no—
178#381Dic7912yes—
179#380—21no—
180#380C2×D7911yescharacteristic
181#343—1584no—
182#324—1581no—
183#323—1581no—
184C22×D4C2×Dic7911yescharacteristic
185C23×C4C2×Dic7912yes—
186C23×C4C22⋊D7911yescharacteristic
187C23×C4C22×D7911yescharacteristic
188C2×M16—794no—
189C2×M16—1582no—
190C22×C8—1582no—
191#123—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C22⋊D7911yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4C2×Dic7911yescharacteristic
203C2×C22⋊C4C2×Dic7912yes—
204C2×C22⋊C4C22⋊D7911yescharacteristic
205C2×C42C22⋊D7911yescharacteristic
206C4⋊C8—1582no—
207C22⋊C8—1582no—
208C22⋊C8—1582no—
209C22⋊C8—1582no—
210C22⋊C8—1582no—
211C22⋊C8—1582no—
212#97—1582no—
213#95—22no—
214#95—22no—
215#95—21no—
216C24C22⋊Dic7911yescharacteristic
217C24C22×Dic7911yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4C22⋊Dic7911yescharacteristic
231C22×C4C22⋊Dic7911yescharacteristic
232C22×C4C22⋊Dic7911yescharacteristic
233C22×C4D4×D7912yes—
234C22×C4C2×C22⋊D7911yescharacteristic
235C22×C4C2×C22⋊D7911yescharacteristic
236C22×C4#1192962312yes—
237M16—3164no—
238M16—3164no—
239C2×C8—3162no—
240C2×C8—6322no—
241C2×C8—1584no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—44no—
251C22⋊C4—42no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23#1457907812yes—
263C23C2×C22⋊Dic7911yescharacteristic
264C23D4×Dic7912yes—
265D4—44no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—84no—
286C2×C4#1457912911yescharacteristic
287C2×C4#1457913312yes—
288C2×C4#1457914211yescharacteristic
289C8—3164no—
290C8—6322no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1909844311yescharacteristic
304C22C2×#1457907811yescharacteristic
305C22#1909854711yescharacteristic
306C4—22no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#1909875212yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2376381411yescharacteristic
319C2#2376386411yescharacteristic
320C2#2376395311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC24×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2486C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC23×C4×C79C23
hypercenter#2486D79
hypocenterC79#9507
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. #23763480C2semidirect
4. #23761306C2semidirect
166. C79#9507semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237634802
#237617382
#237613062
C79×#24862
#950779