Finite Group Databasebeta

← #34185720

Subgroups — #34185720

Order 20224.

644 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 115 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#23765757C211yesmaximal, characteristic, complemented
2#23761916C212yesmaximal
3#23761730C211yesmaximal, characteristic, complemented
4#23761286C212yesmaximal, complemented
5C79×#2591C211yesmaximal, characteristic
6C2×#14579120—21no—
7#19098527C2212yes—
8C4⋊C4×Dic79—24no—
9C4×C22⋊Dic79—21no—
10C4×C22⋊Dic79C2212yes—
11#19098297—24no—
12#19098289—22no—
13C2×#121×C79C2211yescharacteristic
14C22×C4⋊C4×C79C2211yescharacteristic
15#400×C79C412yescomplemented
16#394×C79C412yescomplemented
17#388×C79—22no—
18#386×C79—21no—
19#386×C79—21no—
20C4×C22⋊C4×C79C2211yescharacteristic
21C2×#11929608—22no—
22#14579120—28no—
23C2×C4⋊Dic79—22no—
24C2×C22⋊Dic79—22no—
25C2×C4×Dic79—22no—
26C2×C4×Dic79—22no—
27C2×C4×Dic79—22no—
28C22⋊(C79⋊C8)—22no—
29C22⋊(C79⋊C8)—22no—
30C79⋊#95—22no—
31C79⋊#95D414yes—
32C22×D4×C79—21no—
33C22×D4×C79C2×C411yescharacteristic
34C23×C4×C79D411yescharacteristic
35C23×C4×C79D411yescharacteristic
36C23×C4×C79C2311yescharacteristic
37#121×C79—22no—
38#121×C79—22no—
39C2×C4⋊C4×C79—22no—
40C2×C4⋊C4×C79—22no—
41C2×C4⋊C4×C79C2×C411yescharacteristic
42C2×C4⋊C4×C79C2×C412yes—
43C2×C4⋊C4×C79D412yes—
44C2×C22⋊C4×C79—21no—
45C2×C22⋊C4×C79—22no—
46C2×C22⋊C4×C79—21no—
47C2×C22⋊C4×C79—22no—
48C2×C42×C79—21no—
49#95×C79—22no—
50#95×C79—22no—
51#95×C79C2×C412yes—
52C22×Dic79—24no—
53#11929608—82no—
54C2×C79⋊C8—42no—
55C4⋊Dic79—28no—
56C22⋊Dic79—28no—
57C4×Dic79—28no—
58C4×Dic79—82no—
59C4×Dic79—82no—
60C24×C79—21no—
61C24×C79C2×D411yescharacteristic
62C2×D4×C79—22no—
63C2×D4×C79—44no—
64C2×D4×C79—41no—
65C2×D4×C79—41no—
66C2×D4×C79—81no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—24no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—22no—
74C22×C4×C79—81no—
75C22×C4×C79—81no—
76C22×C4×C79C22⋊C412yes—
77C22×C4×C79C22⋊C412yes—
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D412yes—
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79#4112yes—
82C4⋊C4×C79—24no—
83C4⋊C4×C79—24no—
84C4⋊C4×C79—22no—
85C22⋊C4×C79—42no—
86C22⋊C4×C79—44no—
87C22⋊C4×C79—42no—
88C22⋊C4×C79—82no—
89C42×C79—81no—
90C23×C79—22no—
91C23×C79—22no—
92C23×C79—41no—
93C23×C79—42no—
94C23×C79—42no—
95C23×C79—41no—
96C23×C79—42no—
97C23×C79C2×C22⋊C411yescharacteristic
98C23×C79#12011yescharacteristic
99C23×C79#12311yescharacteristic
100D4×C79—82no—
101D4×C79—82no—
102D4×C79—82no—
103C2×C4×C79—22no—
104C2×C4×C79—22no—
105C2×C4×C79—24no—
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—42no—
110C2×C4×C79—42no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—42no—
114C2×C4×C79—42no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—42no—
118C2×C4×C79—41no—
119C2×C4×C79—42no—
120C2×C4×C79—81no—
121C2×C4×C79#10414yes—
122C2×C4×C79C4×D412yes—
123C2×C4×C79#12112yes—
124C2×Dic79—28no—
125C2×Dic79—44no—
126C2×Dic79—44no—
127C79⋊C8—82no—
128C22×C79—21no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—21no—
135C22×C79—44no—
136C22×C79—84no—
137C22×C79—81no—
138C22×C79—81no—
139C22×C79—81no—
140C22×C79#38611yescharacteristic
141C22×C79C2×#10412yes—
142C22×C79#45412yes—
143C4×C79—24no—
144C4×C79—22no—
145C4×C79—22no—
146C4×C79—42no—
147C4×C79—82no—
148C4×C79—81no—
149Dic79—48no—
150#9410—791nomaximal, Sylow
151C2×C79—22no—
152C2×C79—22no—
153C2×C79—41no—
154C2×C79—161no—
155C2×C79#188811yescharacteristic
156C2×C79#189611yescharacteristic
157C2×C79#191311yescharacteristic
158#2591D7911yescharacteristic
159#2274—791no—
160#1507—792no—
161#1463—791no—
162#1297—792no—
163C79#941011yescharacteristic, complemented, Sylow
164C2×#121C2×D7911yescharacteristic
165C2×#117—1581no—
166C22×C4⋊C4C2×D7911yescharacteristic
167#407—792no—
168#400Dic7912yescomplemented
169#394Dic7912yescomplemented
170#388—22no—
171#386—21no—
172#386—21no—
173C4×C4⋊C4—1584no—
174C4×C22⋊C4—792no—
175C4×C22⋊C4—1581no—
176C4×C22⋊C4C2×D7911yescharacteristic
177#327—1584no—
178#324—1582no—
179C22×D4—21no—
180C22×D4C2×Dic7911yescharacteristic
181C23×C4C22⋊D7911yescharacteristic
182C23×C4C22⋊D7911yescharacteristic
183C23×C4C22×D7911yescharacteristic
184C2×M16—1582no—
185#121—22no—
186#121—22no—
187#117—1588no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—1582no—
191C2×C4⋊C4C2×Dic7911yescharacteristic
192C2×C4⋊C4C2×Dic7912yes—
193C2×C4⋊C4C22⋊D7912yes—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1582no—
199C2×C42—21no—
200C2×C42—1582no—
201C2×C42—1582no—
202C2×C42—1582no—
203C22⋊C8—1582no—
204C22⋊C8—1582no—
205#95—22no—
206#95—22no—
207#95—794no—
208#95—1582no—
209#95C2×Dic7912yes—
210C24—21no—
211C24C2×C22⋊D7911yescharacteristic
212C2×D4—22no—
213C2×D4—44no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—81no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—1584no—
227C22×C4C22⋊Dic7912yes—
228C22×C4C22⋊Dic7912yes—
229C22×C4C22×Dic7911yescharacteristic
230C22×C4D4×D7912yes—
231C22×C4C2×C22⋊D7911yescharacteristic
232C22×C4#1192962312yes—
233M16—6322no—
234C2×C8—3162no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—1588no—
239C22⋊C4—42no—
240C22⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—82no—
243C22⋊C4—1588no—
244C42—81no—
245C42—6322no—
246C42—6322no—
247C42—1588no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—42no—
255C23C2×C22⋊Dic7911yescharacteristic
256C23#1457912911yescharacteristic
257C23#1457914211yescharacteristic
258D4—82no—
259D4—82no—
260D4—82no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—3164no—
280C2×C4—3164no—
281C2×C4—1588no—
282C2×C4#1457908814yes—
283C2×C4D4×Dic7912yes—
284C2×C4#1457913312yes—
285C8—6322no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—44no—
294C22—84no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1909844311yescharacteristic
299C22C2×#1457908812yes—
300C22#1909873512yes—
301C4—24no—
302C4—22no—
303C4—22no—
304C4—42no—
305C4—82no—
306C4—81no—
307C4—3168no—
308C2—22no—
309C2—22no—
310C2—41no—
311C2—161no—
312C2#2376380411yescharacteristic
313C2#2376384411yescharacteristic
314C2#2376394311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC79×#2591C2
socleC22×C79#386
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenter#2591D79
hypocenterC79#9410
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. #23765757C2semidirect
3. #23761730C2semidirect
4. #23761286C2semidirect
15. #400×C79C4semidirect
16. #394×C79C4semidirect
163. C79#9410semidirect
168. #400Dic79semidirect
169. #394Dic79semidirect

Maximal subgroups

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

subgroupindex
#237657572
#237619162
#237617302
#237612862
C79×#25912
#941079