Finite Group Databasebeta

← #34185956

Subgroups — #34185956

Order 20224.

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

normal subgroups only

This group has 498 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#23772429C211yesmaximal, characteristic
2#23768573C211yesmaximal, characteristic
3#23762118C212yesmaximal, complemented
4#23761255C212yesmaximal, complemented
5C79×#1466C211yesmaximal, characteristic
6C2×#14579161C2211yescharacteristic
7#19098635—24no—
8#19098620C2212yes—
9#19098527—21no—
10C2×C22⋊(C79⋊C8)—21no—
11#19098475—21no—
12#19098475—21no—
13#19098466C412yes—
14#19098428C412yes—
15#19098423—21no—
16#19098423—21no—
17C4×C4⋊Dic79C2211yescharacteristic
18#19098310—24no—
19C2×C4×D4×C79C2211yescharacteristic
20#432×C79C2212yes—
21#323×C79—21no—
22#323×C79—21no—
23C22×Dic158—21no—
24C2×#11929608—22no—
25C22×C79⋊C8—22no—
26#14579161—22no—
27#14579161—22no—
28C2×C4⋊Dic79—21no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79C2×C412yes—
31C2×C22⋊Dic79—22no—
32C2×C22⋊Dic79—22no—
33C2×C4×Dic79C2×C412yes—
34C4⋊(C79⋊C8)D414yes—
35C22⋊(C79⋊C8)—22no—
36C22⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—22no—
38#14579072—22no—
39C79⋊#95—22no—
40C79⋊#95—22no—
41C79⋊#95—21no—
42C22×D4×C79C2×C411yescharacteristic
43C23×C4×C79D411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C4×D4×C79—22no—
46C4×D4×C79—22no—
47C2×C4⋊C4×C79C2×C411yescharacteristic
48C2×C22⋊C4×C79D411yescharacteristic
49C2×C22⋊C4×C79D411yescharacteristic
50C2×C42×C79C2311yescharacteristic
51C4⋊C8×C79—22no—
52C22⋊C8×C79—22no—
53#97×C79—22no—
54C2×Dic158—42no—
55C2×Dic158—82no—
56C22×Dic79—22no—
57C22×Dic79—22no—
58#11929608—82no—
59C2×C79⋊C8—24no—
60C2×C79⋊C8—82no—
61C4⋊Dic79—24no—
62C4⋊Dic79—42no—
63C4⋊Dic79—41no—
64C4⋊Dic79—41no—
65C22⋊Dic79—82no—
66C22⋊Dic79—82no—
67C4×Dic79—42no—
68C24×C79C22⋊C411yescharacteristic
69C24×C79C22⋊C411yescharacteristic
70C2×D4×C79—22no—
71C2×D4×C79—22no—
72C2×D4×C79—41no—
73C22×C4×C79—21no—
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—42no—
78C22×C4×C79—42no—
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22×C411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C2×C8×C79—42no—
85C4⋊C4×C79—22no—
86C22⋊C4×C79—22no—
87C22⋊C4×C79—22no—
88C42×C79C2×D412yes—
89C42×C79#4112yes—
90C23×C79—21no—
91C23×C79—21no—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79#10112yes—
100C23×C79C2×C22⋊C411yescharacteristic
101D4×C79—44no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—22no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—41no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—81no—
113C2×C4×C79—81no—
114C2×C4×C79C4×D412yes—
115C2×C4×C79#12011yescharacteristic
116C2×C4×C79#12112yes—
117C2×C4×C79#12311yescharacteristic
118C2×Dic79—42no—
119C2×Dic79—42no—
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124Dic158—84no—
125C8×C79—82no—
126C79⋊C8—44no—
127C22×C79—21no—
128C22×C79—21no—
129C22×C79—22no—
130C22×C79—42no—
131C22×C79—42no—
132C22×C79—41no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—41no—
136C22×C79—41no—
137C22×C79—81no—
138C22×C79—81no—
139C22×C79#38611yescharacteristic
140C22×C79C2×#10111yescharacteristic
141C22×C79#41311yescharacteristic
142C4×C79—21no—
143C4×C79—22no—
144C4×C79—22no—
145C4×C79—41no—
146C4×C79—41no—
147C4×C79#45512yes—
148Dic79—82no—
149Dic79—82no—
150#9453—791nomaximal, Sylow
151C2×C79—21no—
152C2×C79—21no—
153C2×C79—42no—
154C2×C79—81no—
155C2×C79#188511yescharacteristic
156C2×C79#190111yescharacteristic
157C2×C79#191011yescharacteristic
158#2973—791no—
159#2563—791no—
160#1556—792no—
161#1466D7911yescharacteristic
162#1285—792no—
163C79#945311yescharacteristic, complemented, Sylow
164C2×C4⋊Q8—791no—
165C2×C4×D4C2×D7911yescharacteristic
166#435—1584no—
167#432—792no—
168#432C2×D7912yes—
169#407—1581no—
170C2×C22⋊C8—1581no—
171#393—1581no—
172#393—1581no—
173#390—792no—
174#382—792no—
175#380—1581no—
176#380—1581no—
177C4×C4⋊C4—791no—
178#332—1584no—
179#323—21no—
180#323—21no—
181C22×Q8—1581no—
182C22×D4C4×D7911yescharacteristic
183C23×C4D31611yescharacteristic
184C23×C4D31611yescharacteristic
185C2×M16—1582no—
186C22×C8—1582no—
187C4⋊Q8—1582no—
188C4⋊Q8—1582no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—792no—
192C2×C4⋊C4—1581no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4C4×D7911yescharacteristic
195C2×C22⋊C4—1582no—
196C2×C22⋊C4—1582no—
197C2×C22⋊C4C22⋊D7911yescharacteristic
198C2×C22⋊C4C22⋊D7911yescharacteristic
199C2×C42—792no—
200C2×C42C22×D7911yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—794no—
203C22⋊C8—22no—
204C22⋊C8—1582no—
205C22⋊C8—1582no—
206C22⋊C8—1582no—
207#97—22no—
208#97—1582no—
209#95—1582no—
210#95—1582no—
211#95—1581no—
212C24(C2×D79)⋊C411yescharacteristic
213C24(C2×D79)⋊C411yescharacteristic
214C2×Q8—3162no—
215C2×Q8—6322no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1582no—
226C22×C4—1582no—
227C22×C4(C2×D79)⋊C411yescharacteristic
228C22×C4(C2×D79)⋊C411yescharacteristic
229C22×C4C2×C4×D7911yescharacteristic
230C22×C4C2×C22⋊D7911yescharacteristic
231C22×C4C2×D31611yescharacteristic
232M16—6322no—
233C2×C8—42no—
234C2×C8—6322no—
235C2×C8—1584no—
236C4⋊C4—22no—
237C4⋊C4—3162no—
238C4⋊C4—3161no—
239C4⋊C4—3161no—
240C4⋊C4—1584no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—6322no—
244C22⋊C4—6322no—
245C42—3162no—
246C42D4×D7912yes—
247C42#1192962312yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#1457907912yes—
258C23C2×(C2×D79)⋊C411yescharacteristic
259Q8—6324no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—3162no—
274C2×C4—3162no—
275C2×C4—3162no—
276C2×C4—3162no—
277C2×C4—3162no—
278C2×C4—3162no—
279C2×C4(C22⋊D79)⋊C412yes—
280C2×C4#1457913111yescharacteristic
281C2×C4#1457913312yes—
282C2×C4#1457914511yescharacteristic
283C8—82no—
284C8—3164no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1909844011yescharacteristic
298C22C2×#1457907911yescharacteristic
299C22#1909854511yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—6322no—
306C4—6322no—
307C4#1909874312yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#2376378611yescharacteristic
313C2#2376386611yescharacteristic
314C2#2376392511yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1466C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC2×C42×C79C23
hypercenter#1466D79
hypocenterC79#9453
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #23762118C2semidirect
4. #23761255C2semidirect
163. C79#9453semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237685732
#237621182
#237612552
C79×#14662
#945379